M .Sc.

UNIVERSITE DE SHERBROOKE
F a c u l t e d ’a d m i n i s t r a t i o n
ST R A T E G IE D ’IN VESTISSEM EN T GUIDE PAR LES PASSIFS ET IMMUNISATION D E PO R TEFEUILLE :
U n e a p p r o c h e DYNAM IQ UE
Par
M ig u e l M o is a n - P o is s o n
Memoire presente a la Faculte d ’adm inistration
en vue de l’obtention du grade de
M AITRE ES SCIENCES
(M .S c.)
A out 2013
© M iguel M oisan-P oisson, 2013
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UNIVERSITE DE SHERBROOKE
F a c u l t e d 'a d m i n i s t r a t i o n
S t r a t e g i e d ' i n v e s t i s s e m e n t g u i d e p a r l e s p a s s i f s e t i m m u n i s a t i o n DE PO RTEFEUILLE :
U ne
appro ch e
d y n a m iq u e
M ig u e l M o is a n - P o is s o n
a ete evalue par un jury compose des personnes suivantes :
___________________________D irecteur de recherche
Alain Belanger
___________________________Lecteur
Guy Bellemare
__________ Lecteur
Anastassios Gentzoglanis
Memoire accepte le
A bstract
In a previous MITACS project in collaboration with Addenda C apital, two basic liability matching
strategies have been investigated: cash flow matching and moment matching. These strategies per­
formed well under a wide variety of tests including historical backtesting. A potential shortcoming
for both of these methods is th a t the optimization process is done only once at the beginning of the
investment horizon and uses determ inistic moment matching constraints to immunize the portfolio
against interest rate movements. Though the portfolio subsequently need to be frequently rebal­
anced, this static optim ization does not take into account the relatively high rebalancing costs it
involves.
The main objective of this present project is to further enhance th e moment matching m ethod
by implementing and testing a stochastic dynamic optim ization and by comparing its efficiency
with the static one. O ur dynam ic optim ization problem is to minimize the portfolio cost and its
expected rebalancing costs one m onth ahead over a set of interest rate scenarios by the use of
stochastic moment matching constraints.
Our backtesting results show some improvements with th e 6 moments m atching strategy as
the dynamic optim ization slightly shrinks the difference in asset-liability gap between scenarios
compared with the static optim ization. However, after analyzing the realized periodic rebalancing
costs each m onth (a constant bid-ask spread has been assigned to each asset’s position change
in the optimal portfolio), the immunization improvements are m itigated by substantialy higher
costs. We also noticed, in the case of the duration/convexity m atching strategy, th a t th e dynamic
optim ization is not th a t much more efficient than the static method.
Thus, these results confirm th a t the 6 moments matching technique is still more efficient with
both the static and stochastic dynamic optimization. Our extensive dynamic analysis of transaction
costs through backtesting showed th a t from an efficiency to cost ratio and an efficiency to simplicity
ratio, the static 6 moments m atching m ethod seems so far to be a more practical solution for liability
matching.
R esum e
Dans le contexte des marches financiers turbulents, les strategies de gestion de portefeuille dont les
actifs doivent etre apparies a des passifs (ex : caisses de retraite, compagnies d ’assurance, etc.) sont
devenues un enjeu im portant. P ar exemple, les caisses de retraite dont les actifs ont litteralem ent
fondus lors de la crise financiere et dont les passifs eventuels augm entent de plus en plus a cause de
la retraite des baby-boomers presentent actuellement des deficits actuariels et doivent reflechir a de
nouvelles strategies pour pallier a ce probleme. Une partie de la solution est dans la gestion accrue
des risques de variations de valeur dans les portefeuilles. C ette gestion du risque provient en partie
de la recherche de strategies optimales d ’immunisation de portefeuilles, nouveau domaine appele
’investissement guide par le passif’ (Liability Driven Investment) . Ceci a pour objectif d ’optim iser
et surpasser l’appariem ent des flux monetaires des actifs et des passifs d ’un portefeuille en utilisant
de nouvelles techniques d ’optim isation dynamique basees sur la duree, la convexite et d ’autres
moments d ’immunisation d ’un portefeuille.
Dans la litterature, on retrouve plusieurs etudes sur l’immunisation de portefeuille. On peut
classer ces techniques d ’im munisation en deux grandes categories : le moment matching et le cash
flow matching. La premiere technique est inspiree de differents travaux classiques comme ceux
de Redington (1952). Fong and Vasicek (1984) et Nawalkha and Chambers (1997). La seconde
implique differentes m ethodes de program m ation lineaire que Ton peut retrouver. par exemple.
dans Kocherlakota et al. (1990).
Dans le cadre d ’un precedent projet MITACS en collaboration avec Addenda C apital. Augustin
et al. (2010 ) etudient les deux strategies d ’appariem ent precedentes dans un contexte de passifs
multiples. Leur strategie de moment matching est inspiree des resultats de Theobald and Yallup
(2010 ) qui m ontrent que l’utilisation de 6 moments offre une efficacite d ’im munisation optimale.
Augustin et al. (2010) m ontrent que ces deux strategies performent bien sous une large variete
de tests, y compris en backtesting. Un inconvenient potentiel de ces deux m ethodes est que le
processus d ’optim isation est seulement effectue une fois au debut de l’horizon de placement et
utilise des contraintes de moment matching determ inistes pour immuniser le portefeuille contre
les fluctuations des taux d ’interet. Alors que le portefeuille necessite ensuite d ’etre frequemment
reequilibre. cette optim isation statique ne tient pas en com pte les couts relativement eleves de ce
reequilibrage.
L’objectif de ce projet est d ’ameliorer la m ethode de moment matching par l’im plantation et
la validation d ’un modele d ’optim isation dynam ique stochastique et en com parant son efficacite
avec l’optim isation statique. Le probleme d ’optim isation dynam ique est de minimiser le cout du
portefeuille ainsi que les couts de reequilibrage esperes sur un horizon d’un mois pour un ensemble
de scenarios de taux d ’interet. Cela est possible via l’utilisation de contraintes stochastiques de
moment matching. D ’autres modeles interessants d ’optm isation stochastique tels que ceux etudies
pas Schwaiger et al. (2010) ont ete envisage, mais n ’ont pu etre utilises faute de performance
computationnelle.
Nos resultats de backtesting m ontrent quelques ameliorations avec la m ethode des 6 moments
car Ton observe que l’optim isation dynamique perm et de reduire la difference de l’ecart actif-passif
entre les differents scenarios com parativem ent a l’optim isation statique. Cependant, apres analyse
des couts de reequilibrage periodiques realises chaque mois, il s ’avere que les ameliorations en
termes d ’efficacite d ’immunisation de portefeuille soient attenuees p ar une hausse substantielles des
couts. Ces couts de transaction ex-post ont ete approxime par un ecart bid-ask constant attribue
au changement de positions de chaque actif du portefeuille optimal. Dans le cas de la strategie
duree/convexite, on rem arque egalement que l’optim isation dynam ique n’apporte pas d ’efficacite
supplementaire par rapport a la methode statique.
Ces resultats confirment done que la technique des 6 moments est. encore une fois, la plus
efficace, a la fois avec l’optim isation stochastique et l’optim isation statique. N otre analyse etendue
des couts de transaction via le backtesting m ontre toutefois que le rapport couts-benefices ainsi que
le rapport parcimonie-couts rend mitige l’efficacite de la methode des 6 moments dans le cadre de
l’optimisation stochastique. Ainsi, dans le cadre de cette etude, il semble que l’optim isation statique
soit une solution plus praticable pour l’appariem ent du passif en comparaison avec l’optim isation
dynamique.
A cknow ledgem ents
I would like to express my gratitude to my supervisor Alain Belanger who has willingly shared his
precious tim e through the learning process of this m aster thesis. I appreciate the useful comments
and remarks which always helped me to develop my thoughts. Furthermore. I would like to thank
MITACS Accelerate C an ad as research internship program (12-13-5629). FQ R N T Acceleration
Quebec (171527) and Addenda Capital for giving me the opportunity to implement and test this
LDI strategy. In particular. I would like to thank Bernard Augustin and the quantitative research
team for their technical support and helpful comments. Finally, I would like to thank the Faculte
d ’adm inistration for their financial support.
C ontents
L ist o f F igu res
vii
L ist o f T ables
viii
In tro d u ctio n
1
1
B ack grou n d in form ation
1
2
T h eo retica l fram ew ork
2.1 Yield curve modeling and shock s c e n a r io s ................................................... .......................
2.2 Bonds and liabilities v alu atio n ...................................................................................................
2.3 Moments c a lc u la tio n ....................................................................................................................
2.4 Optim ization model ....................................................................................................................
3
3
4
4
5
3
B a c k te stin g m eth o d o lo g y
3.1 D ata and lim ita tio n s ....................................................................................................................
3.1.1
Bond universe and liabilities .....................................................................................
3.1.2
Transaction c o s t s ............................................................................................................
3.1.3
Yield curves and shock s c e n a r io s ..............................................................................
3.1.4
Optim ization settings ..................................................................................................
3.2 Backtesting a l g o r i th m ................................................................................................................
3.2.1
Shortfall liquidation algorithm ..................................................................................
3.2.2
Rebalancing adjustm ents alg o rith m ...........................................................................
3.2.3
Asset-Liability gap measures .....................................................................................
7
7
7
8
9
9
10
11
12
14
4
R e su lts o u tco m es and fu tu re research
14
5
C on clu sion
19
6
R eferen ces
20
A
P o rtfo lio p o sitio n s track in g
21
A .l D u ratio n /co n v ex ity -m atch in g ............................................................................................
21
A .2 6 Moments m a tc h in g .................................................................................................................... 24
B
G raph s o f p o rtfo lio im m u n iza tio n resu lts
27
B .l W ithout additional liquidity injection ................................................................................... 27
B.1.1 D uratio n /co n v ex ity -m atch in g ..................................................................................... 28
B .l.2 6 Moments m a tc h in g ...................................................................................................... 33
B.2 W ith additional liquidity injection ......................................................................................... 38
B.2.1 D uratio n /co n v ex ity -m atch in g..................................................................................... 39
B.2.2 6 Moments m a tc h in g ...................................................................................................... 40
vi
C D a ta and se ttle m e n t d a te s
41
C .l Bond universe d e ta ils ................................................................................................................... 41
C.2 L ia b ilitie s ...................................................................................................................................... 43
C.3 Settlem ent d a t e s ..............................................................................
44
D O th er sto ch a stic p rogram m in g m o d els
45
45
D .l Moment matching m e th o d ...................................................................................................
D . 1.1 Stochastic programming (SP) m o d e l ........................................................................ 45
D .l .2 Chance-constrained programming (CCP) m o d e l...................................................... 46
D.2 Cash flow matching m e th o d ...................................................................................................... 46
D.2.1 SP m o d e l ......................................................................................................................... 47
D.2.2 CCP m o d e l ...................................................................................................................... 47
D.2.3 Integrated chance-constrained programming (ICCP) m o d e l .............................. 48
List o f Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Time scale setting for b ac k te stin g ...........................................................................................
D uration/convexity-m atching: asset and liability PV tracking (without additional
liquidity injection a t rebalancing dates) ...............................................................................
6 moments matching: asset and liability PV tracking (w ithout additional liquidity
injection at rebalancing dates) ...............................................................................................
DC matching: portfolio tracking with static optim ization ..............................................
DC matching: portfolio tracking with stochastic o p tim iz a tio n .......................................
6 M matching: portfolio tracking with static o p tim izatio n .................................................
6 M matching: portfolio tracking with stochastic o p tim iz a tio n .......................................
DC matching: asset-liability gap (w ithout additional liquidity injection a t rebalanc­
ing d a t e s ) .......................................................................................................................................
DC matching: asset cash-flows and liability stream (w ithout additional liquidity
injection a t rebalancing dates) ...............................................................................................
DC matching: net investments costs and bid-ask costs (without additional liquidity
injection a t rebalancing dates) ...............................................................................................
DC matching: additional liquidity injection needs at rebalancing d a t e s .......................
DC matching: difference between A-L gap under each scenarios (without additional
liquidity injection a t rebalancing dates) ...............................................................................
6 M matching: asset-liability gap (without additional liquidity injection at rebalanc­
ing d a t e s ) .......................................................................................................................................
6 M matching: asset cash-flows and liability stream (w ithout additional liquidity
injection a t rebalancing dates) ................................................................................................
6 M matching: net investments costs and bid-ask costs (w ithout additional liquidity
injection a t rebalancing dates) ...............................................................................................
6 M matching: additional liquidity injection needs at rebalancing d a t e s .......................
6 M matching: difference between A-L gap under each scenarios (without additional
liquidity injection a t rebalancing dates) ...............................................................................
DC matching: asset-liability gap with additional liquidity injection at rebalancing
d a t e s .................................................................................................................................................
vii
10
15
16
22
23
25
26
28
29
30
31
32
33
34
35
36
37
39
19
6 M matching: asset-liability gap with additional liquidity injection at rebalancing
20
21
22
d a t e s .................................................................................................................................................
Bond u n iv e rse.........................................................................................
L ia b ilitie s ........................................................................................................................................
Optim ization and backtesting settlem ent d a t e s ....................................................................
40
42
43
44
List o f Tables
1
Yield curve PCA scen ario s...........................................................................................................
viii
9
Introduction
Since the recent financial meltdown and because of the current economic instability, many fund
managers tend to shift their total-return-oriented investment approach toward a liability-driven
investment (’LD I’) strategy. Its main objective is to find investment strategies th a t will m atch or
outperform a liability stream (pensions, insurance claims, etc.). The popularity of this strategy
among pension fund managers, in particular, is not surprising because of m ajor changes in the
demographics of developed and emerging countries; among others, life expectancies are increasing.
In N orth America, baby-boomers have also started to retire massively. These effects heighten the
need for pension funds to properly fund their rising liabilities, especially because of th e recent fall
in the equity markets and bond yields reaching their lowest historical levels. Moreover, recent
accounting standards and regulatory changes force pension fund managers to adopt a new view on
their asset allocation to reduce the volatility of their funding statu s and financial results.
This docum ent presents the results of a LDI strategy for portfolio moment m atching immuniza­
tion techniques. This study follows results upon a previous MITACS project. The objective is to
minimize the portfolio cost and its expected rebalancing costs by allowing it to be regularly rebal­
anced over time. In order to take into account the uncertainty of possible movements of the term
structure of interest rates, th a t is the interest rates risk, we use dynamic stochastic optimization.
It is the second MITACS project on this topic in collaboration with Addenda C ap ital1.
The Section 1 gives an overview of the cash flow matching and moment m atching literature
and explains how the present study could improve the previous project results. In Section 2 we
explain, in a generic manner, the theoretical framework of our optimization model. In Section 3 we
give details on d ata used in our analysis and explains the assumptions we had to make. We also
describe how we performed the backtesting of our model and how we analysed its efficiency. We
explain in Section 4 our main results and gives some ideas of possible future research.
1
Background inform ation
There are two large classes of LDI methods: portfolio immunization and cash flow matching. The
purpose of immunization is to construct a portfolio for which the change in value will m atch the
change in liability value over a given horizon. There exist many immunization techniques th a t are
in the lineage of classical papers like Redington (1952) and Fong and Vasicek (1984). The classical
m ethods like duration/convexity matching are used to immunize the portfolio change in value
against parallel movements in the term structure of interest rates (’T S IR ’). More recently, Nawalkha
and Chambers (1997) and Theobald and Yallup (2010) improved immunization techniques by using
non-centered moment matching for multiple liabilities. These more advanced techniques give the
possibility to immunize the portfolio against different movements of th e TSIR. Theobald and Yallup
(2010) conducted a very comprehensive empirical analysis in the UK bond m arket and showed th a t
using the first 6 moments m atching is optim al for an immunized portfolio. The cash flow matching
is a technique which consists in selecting securities w ith a m aturity th at m atch the timing and
am ount of the liabilities. Linear programm ing can be employed to construct a least-cost cash flow
m atching portfolio (see Kocherlakota et al., 1990).
In a previous MITACS project in collaboration with Addenda Capital. B. Augustin. A. Belanger,
’w w w .addenda-capital.com.
1
K. Haniidya and Y. Wagner (hereafter referred to as Augustin et al.. 2010) investigated the previous
two basic liability matching strategies: cash flow matching and moment matching. Their results
first show th at the cash flow matching algorithm works well to reduce deviations from liability cash
flow needs. They also show th a t this strategy is relatively expensive in terms of portfolio cost. For
the second matching strategy, the results of Augustin et al. (2010) show that the moment matching
technique substantially reduces the deviation risk between the portfolio value and the liabilities,
w hat we refer to as the asset-liability gap.
The two basic static liability matching m ethods tested in Augustin et al. (2010) performed well
under a wide variety of tests including historical backtesting. However, in order to m aintain an
optim al portfolio over time, the portfolio has to be frequently rebalanced to meet the matching
constraints. Thus, it involves relatively high periodic rebalancing costs, which are not included in
the optimization problems used in Augustin et al. (2010). As their optim ization can be viewed as
’static’, then one might want to dynamically include theses additional rebalancing costs over time
into the optimization process. This dynamic p art of the cash flow matching and moment matching
optim ization problems becomes an optim al control problem (dynamic optim ization) where the
controls are the positions in the bonds which are now allowed to discretely change with time. For
both matching techniques, the state variable is still the portfolio cost which now also includes
a penalty function for the expected rebalancing costs over different yield curve scenarios for a
given horizon. Moreover, with moment matching, the constraints in th e optim ization problem are
translated into a dynamical way such th a t the first k moments of the portfolio and the liabilities
are now allowed to change with tim e and depend on the TSIR scenarios.
T he main objective of this present project is to further enhance the static moments matching
m ethod described previously. As mentioned before, a potential shortcoming with basic techniques
used by Augustin et al. (2010) is th a t the optim ization process is only done one time at the
beginning of the horizon and uses determ inistic constraints to immunize the portfolio. Our goal
is to model the dynamics of the optim ization process by allowing the portfolio to be rebalanced
a t a minimum cost. In order to take into account the uncertainty of possible movements of the
TSIR, th a t is the interest rate risk, we initially wanted to use the three stochastic programm ing
approaches studied in Schwaiger et al. (2010) for both cash flow matching and moment matching
techniques: a stochastic linear programm ing (’SLP’) model, a chance-constrained programming
(’C C P ’) model and an integrated chance-constrained programm ing (TC C P’) model.
We have to mention th a t for com putational issues, a slightly modified version of only the first
stochastic programming m ethod (the SLP) has been studied here. However, because they could be
useful to use for these types of LDI strategies, these stochastic programming models are explained
with more details in Appendix D. Along the lines of w hat was suggested by a MITACS referee
of this project, we used a stochastic dynam ic approach through ’backward in tim e’ to handle the
problem. The stochastic programming methods mentioned above will be analyzed in a future
research project.
In this study, we investigate th e effects of the dynamic stochastic optim ization on th e portfolio
immunization efficiency and the resulting strategy costs. T he cash flow m atching m ethod being
close to the duration/convexity matching m ethod (2 moments), we tested only the two following
liability moments matching strategies: duration/convexity (2 moments) and 6 moments matching.
We then analysed for both of these matching techniques the efficiency of the dynamic optimization
over the static one. Because of com puter performance issues, we also reduced the number of TSIR
scenarios. We should however try it with more scenarios in a future research project. The two2
stage dynamic stochastic optim ization method developed here as well as the static m ethod were
backtested over a twelve months window with monthly portfolio rebalancing and weekly valuation.
As explained in the ’Results outcom es’ section, we obtained some improvements of the 6 mo­
ments matching strategy by the use of stochastic dynamic optimization but it is m itigated by
substantial higher rebalancing costs. Our results thus highlight the 6 moments matching efficiency
in both static and dynam ic optimization.
The next section explains with more details the technical aspects of the model used in our
analysis.
2
T heoretical framework
Let B be a set of bonds which constitutes the bond universe and £ be a set of actuarial liability
stream th at has to be met over multiple periods prior to an investment horizon H . Note th a t these
liabilities are assumed to be determ inistic and have been given by Addenda C apital. Each bond
n G B. n = 1 , 2 can be described with several characteristics, (fci, &2, &3,
As market
conditions change over time, the bond universe B{t) at tim e t > 0 is described by the following set:
B(t) = {n = n ( k i , k 2 , k 3 ,...) : ki € {c/assi}, &2 € {class 2 }, k 3 € {cZasss},...} .
We will explain later in the ’Backtesting methodology’ section which d a ta and class filter we applied
to our bond universe.
2 .1
Y ie ld c u r v e m o d e lin g a n d s h o c k s c e n a r io s
We define the current (spot) yield curve at time t with a tim e-to-m aturity T as a function r(t, T).
The splines technique has been used to interpolate the spot yield curve function for any m aturity.
We also define the discount function 2 D(t, T ) which gives at time t the discounted value of 1$ paid
at tim e T.
We define a finite set of yield curve scenarios with an horizon of h months. To generate TSIR
shock scenarios, we used th e historical principal com ponent analysis (PCA). We did so by giving
a shock on different combinations of the first three PCA components of the actual spot curve (see
Litterm an and Scheinkman, 1991). T h at is, each shock scenario ui € 0 is of three types of TSIR
change: level (PCA1), steepness (PCA2) or curvature (PCA3). We can define the scenario universe
as:
ft = {to = (L, S, C) : L = {shock in P C A l}, S = {shock in PCA2}, C = {shock in P C A 3}}
The magnitudes M - ^ P C A x ). x = 1,2,3. of these shocks is given by a multiple of the historical
standard deviation of each PCA factor. As the set R is finite, the assigned scenario probability for
each u j is P({w}) = p u and it is calculated by its historical frequency.
T he yield curve function at tim e t for each scenario and for any m aturity is com puted using the
spot curve at time t — h and each of the P C A ’s set of shocks ui. Thus, the yield curve function is
defined as:
r ( t , T ; u ) = f ( r ( t - h,T);uj),
2In our analysis, we have used a discrete tim e discount function.
3
Vw € fb
( 1)
We also define the stochastic discount function D {t,T\uj).
The ’Backtesting m ethodology’ section explains in more details which yield curve d ata we used
and the characteristics of the generated PCA shock scenarios.
2 .2
B o n d s a n d lia b ilitie s v a lu a tio n
Recall the bond index n G B. If tln is the i-th cash flow date after time t of the n -th bond. We
write B P V n(t\uj). the present value of th a t bond evaluated at time t. depending on scenario w. as:
B P V n (t; u) = ' ^ c n {ti„ ) D ( t,tin\u>)
(2)
in
where cn{tin) is the n -th bond’s cash flow at period tin . We let a generic sum m ation over all cash
flow i up to the bond’s m aturity. We define
B P V (t;uj) 4 ( B P V 1(t-,u;),BPV2(t;Lo),...,BPVK (t-,u;))
(3)
as a (1 x /Q -vector which contains the present value of each bond in the portfolio. We also define
P ( t ) ^ ( P 1(t),P 2(t),...,P K (t))
(4)
as a (1 x K )-vector which represents the market price for each bond.
Furthermore, if tj is the j- th liability stream date after tim e t. we write LPV(t;uj). th e present
value of the liabilities, as:
L P V f r u , ) = ' £ i l(tj ) D ( t,tj ;U)
j
(5)
where l(tj) is the liability stream a t period tj and the sum is over all liabilities prior or a t the
investment horizon H .
Finally, let the cheapest (optimal) bond portfolio 11(f) C B(t) which covers the liabilities over
time. Note th a t in our optim ization model, we have a portfolio whose composition depends on tim e
t. As explained later in this section, our optim ization model is used to find the optim al position for
each bond n th a t will minimize the initial portfolio cost and its expected rebalancing cost. These
positions are denoted by the vector u (t) 4 (un (t)), w ith un (t) £ R+. Vn £ 11(f). The portfolio
value at time t, under scenario ui. denoted by A P V (t; tu) for ’Asset P V ’, is expressed as:
un (t)B P V n (t;u) = BPV(f;w)-u'(t)
A P V {t-u )=
(6)
nen(t)
where ' is the transpose operator.
2 .3
M o m e n ts c a lc u la tio n
Following Theobald and Yallup (2010), the k-th moment of the portfolio at tim e t. noted Ik{t;u>),
is computed as:
Ik(t',0j) = A p y U - i J }
zL / ^ „ ltn ( 0 cn ( ^ n ) ^ ( ^ t i n t u;)-
^ ’ ' nen(t) in
4
(7)
Note th a t this portfolio’s moment expression is equivalent to the weighted sum of each bond’s
moment:
n €F I(£ )
where
u ) is the A~-t,h moment of the n-th bond and the weight wn = Un^ p y n ■with
Respectively, the Ar-th moment of the liabilities, noted Jk(t; oj) is computed as:
LPV(t;ui)
wn = 1.
( 8)
For both of these moment measures, we can have A; = 1,2,
where p is the desired number of
moments to be considered. Theobald and Yallup (2010) show th a t p — 6 is optimal. This conclusion
is also supported by Augustin et al. (2010).
2 .4
O p tim iz a t io n m o d e l
Our optim ization model consists of two general steps. First, we need to generate yield curve
scenarios. Since we assume th a t the liability stream is known, the major source of randomness
(risk) in our strategy is th e interest rate (we use high credit quality bonds in our analysis). Thus,
as explained in the previous section, we have to sim ulate multiple TSIR shocks to generate different
yield curve scenarios which are used in our optim ization model. Second, we perform a two-stage
optimization process, which depends on TSIR scenarios.
To incorporate th e rebalancing dynamics and the interest rate risk into the optim ization process,
we use a two-stage stochastic dynamic optim ization w ith stochastic-dependent constraints and an
objective function th a t minimizes the portfolio cost and its expected rebalancing cost one m onth
ahead. As mentioned previously, using the baseline yield curve r(to ,T ) at time to, we first generate
a finite set fi of TSIR scenarios of h = 1 m onth horizon to have different yield curves scenarios
r(ti,T;co) at tim e t\ > to- The number of scenarios generated is explained in th e ’Bakctesting
methodology’ section. After, for each scenario, we find an optimal portfolio I I ( ti; w) C B{t\) with
optimal positions {itn (fi,u;)}. This step is done by minimizing th e portfolio cost subject to the
moment matching constraints. In fact, since we generated multiple TSIR scenarios, the moment
matching constraints and the portfolio cost at tim e t\ are acting as random variables which depend
on these scenarios. These constraints are referred to as stochastic constraints. The ’first stage’
optim ization problem is formulated as follows, for each scenario uj € fi:
minimize
cost(ti;u>) = B P V (fi;w ) • u'(fi;u;)
u(q ,u>)
subject to
A P V{t\\tjj) > L P V {t\\uj)
Ik(ti',u) = Jfc(fi;u;), \/k = 2 m — 1, m = 1 ,2 ,3 ,... (odd moments)
(®)
> Jis(ti;u), Vfc = 2m, m — 1 ,2 ,3 ,... (even moments)
A • u(ti,u>) € a
First note th a t at this stage, we find a set {II(fi; cj)}weQ containing optim al portfolio for each
scenario. In this optim ization problem, the first constraint shows th a t we want the portfolio value
to outperform the liability value. Moreover, we want the portfolio to have even moments greater
th a t those of the liability. This is because of the positive convexity phenomenon. For example,
if k = { 1, 2 } (and thus, m — 1 only), the two moments matching constraints are respectively the
duration (odd moment k = 1) and the convexity (even moment k = 2). W ith 6 moments matching,
we have m, = 1,2 such th a t k = {1, 2,3 ,4 , 5 , 6 }. The m atrix A in the last constraint can include
different m andate constraints such as rating limit, individual weight limit, industry limit, etc. We
will discuss about this below. Finally, note th a t the moment matching are non-linear constraints
because of th e denom inator A P V in the 4 expression (see equation (7)).
At this first stage, when we have the optim al portfolio for each scenario, we go backward through
time to perform a second optimization a t time to to find the needed optimal portfolio Il(to) whose
positions are denoted by {un(4)}- As we have an assigned empirical probability measure pw for
each scenario u € fL we perform the optim ization by minimizing the portfolio cost a t tim e to
and its expected rebalancing cost from to to t\. This is done by using the determ inistic portfolio
cost (market price) and the moment m atching constraints (with the use of the baseline yield curve
r(to,T )). To model the rebalancing costs, we assign a constant bid-ask spread a n as a function
of the positions traded. The bid-ask we used is defined in the ’Backtesting m ethodology’ section.
Thus, if we note 4 ( 4 ) and 4 ( 4 ) respectively the determ inistic fc-th moment of the portfolio and
the liabilities a t time to, we can formulate the ’second stage’ problem as follows:
min
u(«o)
s.t.
cost(to) = P ( 4 ) • u '( t0) +
X PtJ \
X
Q"
A P V ( t 0) > L P V (to )
( 10 )
4 4 o) = 4 ( 4 ) , Vfc = 2m - 1, m = 1, 2 ,3 ,...
4 ( 4 ) > 4 (4 ), Vfc = 2m, m = 1,2,3,...
A • u (4 ) € a
Note th a t the first term in the objective function is the market cost of th e portfolio at time toThe second term is the expectation of the rebalancing cost function over each yield curve scenario
at tim e t\. This term can be viewed as a penalty function for the portfolio rebalancing costs one
m onth ahead. Since the bid-ask cost is calculated with the position changes between the portfolio
II(fo) and Il(<i;u;), A un is defined as follows:
( un (ti\oj) - un (to)
= < un (t\]ui)
{ —un(to)
if n € n ( t 0) D II(fi; w)
ifn £ Il(to) but n € II(<i;u;)
if n € II(to) but n £ II(fi;u/) and m at{n) > t\
( 11 )
where m at(n) is the m aturity d ate of the n -th bond. Note th a t if m a t(n ) < t\. it means th a t this
obligation have m atured between to and t\.
As m andate constraints in A , we included a maximum individual asset weight. This maximum
weight is to force the optim ization to chose a larger number of assets in the portfolio and thus
to limit concentration. We discuss further on this constraint in the ’Backtesting m ethodology’
section. Note th a t if this weight constraint is removed, then when one performs a k moment
matching optim ization, there will only be k assets in the optim al portfolio. For example, with
duration/convexity-m atching w ithout a weight constraint, the optimal portfolio contains only two
bonds. In the ’Research outcomes’ section, we discuss the effects of imposing this constraint on our
results.
6
Finally, as mentioned before, th e three stochastic optim ization models described in Schwaiger
et al. (2010) should be tested in a future research project. These models are: the Stochastic
Linear Programm ing model (SLP). the Chance Constrained Programming model (CCP) and the
Integrated Chance Constrained Program m ing model (ICCP). They are interesting because one
could formulate our optimization problem using a two-stage stochastic linear programming with
recourse decision (control) variables. Instead of finding independently the optim al portfolios under
each scenario at time t\ and going backward through time to the optim ization at time to as it
is the case here, the two-stage SLP formulation would find simultaneously the optim al portfolio
at to by taking into account the expected optim al portfolio and rebalancing costs of the second
stage optim ization (the recourse action) at t\ over all TSIR scenarios. At this stage, the scenarios’
dependent control variables would be used to meet the stochastic moment m atching constraints
(adjustm ent variables). Then, for each scenario at ti, th e objective would still be a function of the
rebalancing costs, which depends on optim al control variables choice. W ith the CCP model, the
objective function and the constraints would still be the same, but we would relax the stochastic
constraints at tim e 11 so th a t there is a non-zero probability of not meeting constraints for a
’sm all’ set of scenarios. In other words, we include a user-specified reliability level of reaching the
stochastic constraints for the TSIR scenarios at t\. Finally, the IC C P model would not only limit
the probability of constraints mismatching, b u t would also constraints the am ount of th e portfolio
underfunding. T h at is, we would include an expected shortfall constraint at t\. which is calculated
over all TSIR scenarios. This can be viewed as a portfolio conditional value-at-risk (’CVaR’) type
of constraint. We give more technical details of these models in Appendix D.
3
B ack testin g m ethod ology
In our analysis, we compared the backtesting results of the stochastic dynamic optim ization with
the static optim ization used in A ugustin et al. (2010). We performed this by comparing the
immunization efficiency (asset-liability gap) and rebalancing costs a t each m onth w ith the two
following liability matching techniques: duration/convexity and 6 moments matching. For each
immunization strategy, we first optimized the portfolio a t the beginning of the first m onth of
our backtesting window. We then evaluated the portfolio each week (and under different TSIR
scenarios) until the next month. At this time, we performed a new portfolio optim ization and
compared the change of each asset’s position to calculate the realized bid-ask costs. After, we
re-evaluated the new portfolio each week up to the next rebalancing m onth and so on for a total
of twelve months.
3 .1
D a t a a n d lim it a t io n s
There are several assum ptions/lim itations th a t must be made on inputs d a ta of our optimization
model and backtesting algorithm.
3.1.1
B o n d u n iverse an d lia b ilities
A ddenda has a large universe of over 300 liquid bonds th a t can be used to construct an optimal
portfolio th a t would best m atch the liabilities (by cash flow matching an d /o r by moment matching).
However, we needed to apply different filters for credit quality, m andate policy and other technical
7
reasons. First, for credit quality, we limited the universe Group to ’G overnm ent’. For m andate
policy, we added an additional filter on Sector to include only ’Provincial’ issuers. We limited
Industry to issuers with highest credit quality. We included ’Agency’ and ’Non-agency’ issuers.
For technical reasons, we also excluded bonds with m aturity greater than 2020-01-01. The set
of deterministic liabilities C is detailed in Appendix C.2. One can see th a t the last liability is on
2015-12-31. which can be seen as the liabilities horizon H (or the investment horizon). However,
our 12 months backtesting window range only in [2010-07-01. 2011-07-01]. which is smaller than
the investment horizon H (see Appendix C.3). Note also th a t the m aturities of the initial large
universe spread up to year 2050. which is far beyond the last liability date. We thus applied the
m aturity filter for m aturity dates beyond 2020-01-01 to avoid some difficulties within the M atlab
optim ization algorithm.
We furtherm ore included additional money market assets, i.e. bonds with m aturity less th an 1
year and 1 month C anadian Government Index as ’Tbills’. These adjustm ents were made since we
have a large liability stream very close to some settlem ent dates of optimization.
In fact, the m aturity filter and the inclusion of money market assets are made so to have a
m aturity distribution in the bond universe th a t spreads over the 22 liability dates. It thus allows
the moment matching to be more efficient. W ith those filters, the bond universe B(t) for each date
t becomes:
n
n{k\ = Group, k? = Sector, k 3 = Industry, k4 = Maturity)
ki =
&2 =
k3 e
k4 <
Government
Provincial
{BC,AB,QC,ON,MA,NB}
2020 - 01-01
and it includes a new 1 m onth TBill a t each date t. The details of the tim e zero filtered universe
are in Appendix C .l.
3.1.2
T ran saction c o sts
For simplicity, we assumed a constant bid-ask spread measure a n = a for all bonds to calculate the
transaction costs. However, for Tbills, we assigned a zero bid-ask spread because these are very
liquid assets. O ur bid-ask spread is defined as ’basis points’ per bond unit, or ’dollars’ per bond
per 100$ notional. Thus, we have the following definition for a:
0.05$ per 100$ notional
0
if the asset is a bond
if the asset is a TBill
Note th a t, instead of a constant bid-ask, we could also use a bid-ask spread measure which would
be defined as a fraction of the market mid-price. In such case however, the transaction costs would
have been overstated if bonds were priced at premium and understated if bonds were priced at
discount. Overall, the results would have been relatively similar.
Finally, a m ajor assum ption is th a t our bid-ask spread measure does not depend on bonds char­
acteristics. We should use a bid-ask spread a function of several param eters such as bond-specific
characteristics or market liquidity/credit conditions as explained later in the ’Results outcom es’
section.
3.1.3
Y ield cu rves and sh ock scen arios
Our baseline yield curve r(t, T) is the Canadian Government yield curve at each d ate t. We used
it for all bonds and we did not add any bond-specific spread.
According to Addenda C apital’s PCA of the Canadian Government yield curve historical move­
ments. the different shocks are classified by three types: level, steepness and curvature. As explained
in the previous section, these shocks have different m agnitude and frequency. For com putational
purpose, we limited the number of scenarios by generating shocks only 011 PCA1 (level) and PCA2
(steepness), but not on PC A3 (curvature) because of its small contribution to the TSIR movements.
We generated a total of 9 yield curve scenarios given by the function r(t, T ; uj) defined in equation
( 1). As the portfolio is rebalanced each month, we used a TSIR shock scenario horizon of h = 1
m onth for each optim ization date, for a total of 12 settlem ent dates (see the tim e scale setting on
Fig. 1). For the weekly backtesting evaluation process, we took a scenario horizon of h = 0.25
month, or one week.
The m agnitude M ^ { P C A X) of each PCA shock is defined as a multiple of their respective
historical standard deviation. Each scenario has its assigned probability (historical frequency).
The following Tab. 1 contains a sum m ary of the scenarios used in our analysis. Note th a t the 5th
scenario has no meaning because it is equivalent to th e baseline curve. In our analysis (e.g. on
graphs in Appendix B). we voluntarily om itted this scenario and labeled the scenarios as 1 to 8 .
T a b . 1: P C A ’s scenario sets
Scenario ui
Pu> (%)
PCA1
1
2
3
4
5
6
7
8
9
3.1.4
6.25
12.50
6.25
12.50
25.00
12.50
6.25
12.50
6.35
100.00
-1.4
0
1.4
-1.4
0
1.4
-1.4
0
1.4
M U( P C A X)
PCA2 PCA3
-1.4
0
-1.4
0
-1.4
0
0
0
0
0
0
0
1.4
0
1.4
0
1.4
0
Shock type
negative steepness
negative level
positive level
positive steepness
O p tim iza tio n se ttin g s
Since our optim ization model involves a lot of d a ta (large scale optimization), we used the ’Global
Search’ option in M atlab to avoid sub-optimal solutions. We also let a relatively high tim e limit
to give enough tim e for the algorithm to come up with a solution. Note th a t with the dura­
tion/convexity case, we almost reached this maximum time limit, but th at w ithout any unfeasible
solution message.
To limit large variations in portfolio’s positions between scenarios, we forced the first stage
optimization process a t time t.\ to begin with a pre-computed optim al portfolio. This portfolio was
calculated using a determ inistic moment matching optim ization w ith the baseline TSIR, th a t is,
9
by the simple static optim ization described previously in the ’Background inform ation’ section.
As m andate constraint in A. as explained in the previous section, we imposed a maximum
individual weight of 8 % for each asset. This was to increase the number of assets in the optimal
portfolio and limit concentration. Note th a t without these constraints, we found respectively an
optimal portfolio composed of only two assets with duration/convexity-m atching and only G assets
with G moments matching.
Finally, to avoid nonlinear constraints in optim ization problems (9) and (10). we forced the
portfolio value to be equal to liability value, th a t is, we assumed th a t A P V = L P V . We then
optimized to find optim al weights {u>n } instead of unit positions {un }. Thus, we com puted the
portfolio fc-th moment constraint as the weighted sum of each bond’s fc-th moment. However,
because the cost in the objective function depends of each bond’s unit position {un }, we used the
assumption above to calculate these positions as: un = wn g p y . We thus added the additional
constraint
wn = 1 in the m atrix A.
3 .2
B a c k t e s t in g a lg o r ith m
Let
a set of m onth indices with M being the number of rebalancing months w ithin the back­
testing window. For this analysis, we took a one year backtesting- window and initially optimized
the portfolio at the beginning of the first m onth and then re-optimized to rebalance adequately
the portfolio a t th e beginning of the remaining eleven months, for a to tal of M = 12 optim izations
(months). Define also { A j } ^ the number of weeks in each m onth i (note th a t they can be different
because of working holidays, etc.) As we evaluate the portfolio each week within each month, we
have a set of evaluation dates index {?C( } ^ 0 for each m onth i.
We thus have the following time scaling:
are the optimization dates, where i = 1
is the initial portfolio creation index and i > 1 are the rebalancing period indexes. For each index
i. the set {tWltirit
contains the weekly evaluation dates within each month. In our settings,
the settle date of the first optimization is for example t w0,mi = 2010-07-01 (the complete lists of
optimization and evaluation settlem ent dates are in Appendix C.3). Note th a t tWo<mi are the dates
of the beginning of the first week of each m onth m* (where the optimization is done) and tWhTni are
the dates of the end of each week in each m onth mi. Note also th a t the last week d ate of a given
month is approxim ately the same date as the next m onth’s first date, th a t is tWN,_ mi_j « tWQjnt.
T he Fig. 1 illustrates this time scale setting.
M o n th # 1
i
1
M onth # A I
M o n th # 2
11
*
i
i
^U'v, jn2
i
^wy
C0,m
^h-0,wv/
i
o p tim . 1
*
o p tim . M
o p tim . 2
F ig . 1 : T im e scale settin g . For th is analysis, we have M = 12.
The value of the portfolio over time is equal to the asset value plus the available cash, net of any
shortfall. Here we will describe how we have tracked these measures over the backtesting window.
10
The end of week’s cash equation is defined as:
.ha
\ - I max { cash(tu!,-.},mi )(l + r 0) + A C F (tll.(,mi) , 0 }
CaS 1 Wl'mi) ~ {
- netRebalAdj(t.Wo,mi)
if 1 <1 < Ni, Vi
if I = 0
'
'
where 3 A C F ( tWhl1li) = C ( tWh71lj) — L (tWh7Tli) and with cash(tWOtlJll) A 0. In fact, at time zero, we
inject enough cash to construct the starting portfolio. Thus, immediately after, there is no available
cash in the portfolio. So The netRebalAdj term comes from the net rebalancing costs (investment
cost and bid-ask cost) and is computed a t the beginning of each m onth (at th e rebalancing date).
Note th a t this net rebalancing cost can be either positive or negative, the negative case occurs
when we are net seller (th at is, we sell more assets th an we buy to rebalance the portfolio). In our
analysis, we assumed a money market rate of ro = 0 .
Because there are some weeks in which a liability stream can be greater than the available cash
in the portfolio, we define the following shortfall equation:
shfl{tWumi) = min {cash{tWl_umi) (l + r0) + A C F ( t Whirii),0} - shflAdj(tWhini)
(14)
for 1 < I < Ni, Vi. In presence of a shortfall (shfl > 0). after using all available cash, we need to
liquidate the corresponding am ount of the portfolio. This is w hat is called here shflAdj. Note th a t
the value of (14) should equal zero since shflAdj is a cash inflow which comes from the portfolio
liquidation to meet the shortfall.
In the following subsections, we explain how we computed the shflAdj and netRebalAdj.
3.2.1
S h ortfall liq u id ation a lg o rith m
Let n(tW
Jim4) be th e optim al portfolio at m onth rril which contains optimal positions { u n } ^ ^ . Note
th a t K* can be different each m onth (the optim al portfolio has not necessarily the same number
of positions each month). The net shortfall value (if different from zero), is assumed to be equally
distributed for each asset for liquidation purpose. Thus, the shortfall value for each bond n is:
n
\ _ shfl{twi,mi)
sh fln itw i.m i) —
( 15)
■
However, to take into account the bid-ask spread, we need to liquidate a few more positions to fund
these transaction costs. The quantity of th e n -th asset to be liquidated must be4:
A un = shfln (tWhTn() ————
vn\J'Wi,rrn)
—
(16)
01
3Since we can have m any coupons w ithin a w eek, n o te th a t C ( t w,,m i ) — X)t€u'| S)n6n(t„.| _ 1,m J Cr,(t)un , for
1 < I < Ni , where n ( t UJi_ ] ,m i) is th e optim al portfolio a t th e end of th e previous week (or a t th e beginning o f the
actual week) and Cn and u n are the n -th bond cou p on and position (c(t) = 0 if there is no cou p on at t for a given
bond). T h e tota l week liability L ( t wliT,l t ) is calcu lated in a sim ilar manner.
4W e w ant to liquidate som e part o f the portfolio to fund th e bid-ask, B A n and have a net value o f shfln from the
liquidation. T h en , w e m ust have
shfln = liq u id a tio n n — B A n <=> sh,flrl = A u n B P V n — A u na
Aun
11
—
shfln
BPVn - a
where B P V n is the present value of the bond. In our analysis, as explained previously, we used the
constant bid-ask spreads defined in equation (12). Hence, for each asset, the liquidation value is
A unB P V n(tWhmi). with A un defined in equation (16). and the transaction cost is B A n = A una.
Thus, the available cash from selling (net liquidation value) is
UquidatioTin(tWl,ni) — A unB P V n(tu./ nii)
BAn
(17)
and it must be equal to shfln (tWhmi).
Note th a t in fact, we must liquidate the quantity min{Aun , u n}. since short sells are not allowed
(we cannot liquidate more units than the actual position in the portfolio). So we have to calculate
the liquidated quantity in a iterative m anner, starting with th e smallest position in the portfolio.
For example, let the smallest position u\ < A iti. In this case, we can only sell the entire position
u\ and will receive a net liquidation value of liquidation = u \B P V \ — B A \. Then, the initial
total shortfall is reduced to shfl’ = s h fl— liquidation. To compute again if a new position has to
be liquidated, we iteratively use equation (15) and (16) using shfl’ and the new number of assets
becomes K* —¥ K* —1 for the remaining n > 1 . Hereafter, we compare one more tim e m in{A un , un }
and so on.
3 .2 .2
R eb a la n cin g a d ju stm en ts alg o rith m
At rebalancing dates, we optimize to find an optim al portfolio, called th e target portfolio B i9i(tWo<mi)
w ith a value approxim ately equal to the present value of the liabilities. Since the real portfolio a t this
date, n ^ ^ o ^ ) is different from the target portfolio (because market conditions have changed and
some bonds may have m atured), we need to rebalance it. If we define the positions in B t9t(twomi)
as
we have, similarly to (11) in the ’Theoretical framework’ section, the following definition
for the change in each position:
( u T - u n if n €
(two,mi) n n(tu,0,TOi)
A un = <
if n €
but n £ n(ttlJo,mi)
[ -u n
if n ^ n i9t{tW0^mi) but n £ n ( t jj;0imi) and m a t(n ) > t WOtTni
(18)
where m a t( n ) is the m aturity date of the n -th asset.
Thus, for each asset, the net rebalancing cost (the net investment cost) is calculated by
netRebalCostnftwQ^) — Au^iB PVji(tmQ^rn^.
Note th a t if A u n < 0. we have a negative rebalancing cost. This could be possible if we are net
seller when rebalancing. The corresponding bid-ask cost is calculated by
B An — [Au/ilcr,
where the bid-ask spread a is defined in equation (12). Finally, we have the following definition of
the total rebalancing costs adjustm ent a t rebalancing dates:
netRebalAdj(tWOtmi) =
[netRebalCostn(tWOtmi) + B A n}.
nen<9<((u,0,mi)un(tu,0,mj)
12
(19)
Now. let
A P V t9t(tW0,mi)
£
u ^ B P V n ( t u,0,m i ),
nen' 9f(tu.0,mi)
A P V (iu.-o.mJ
=
y '
i(tiB P v n (fifo,mj),
n ( / u 'Q . rt l j )
be respectively the present value of the target portfolio and the present value of the actual (real)
portfolio before rebalancing. At rebalancing dates, there are two possible cases:
1. A P V t9t(tWo<mi) < A
P V : The actual portfolio value is greater than the target portfolio
value. In this case, we can easily rebalance the portfolio by using cash an d /o r liquidating
part of the actual portfolio to meet the net rebalancing costs (net investment plus bid-ask).
Note th a t this situation did not appear in our backtesting analysis;
2. A P V t9t(tW0tini) > A P V (fmo.mj : The value of the target portfolio is greater than the actual
portfolio value. In this case, the rebalancing algorithm will depend on the am ount of cash
available in the portfolio:
(a) cash(tWOjmi) > netRebalA dj(tWo^mi) : There is enough cash to meet the target portfolio,
i.e. paying the net investment costs and bid-ask costs. In this case, the new positions
in the portfolio becomes un (tWOiTni) -> un(tw0,mi) + A un . We also have A P V n (tW()jrii) -»
AP V n9t(tWo<mi) and the available cash is given by the second case of (13). Note th a t in
this case, this term (cash in the portfolio) is greater or equal to zero after rebalancing.
(b) cash(tWo<mi) < netRebalA dj(tWo<rni) : In this case, we can only partially rebalance the
portfolio because we do not allow for additional liquidity injection. In fact, we can only
invest an am ount corresponding to cash{tWQ^mi) minus bid-ask costs from this investment.
As we want to reach th e target portfolio, we need to find what fraction of each optim al
new positions we are able to invest. Define this fraction /? such th a t we have our limited
am ount of net investment netRebalAdj*(tWo<TTli) equal to cash(tWOtTni):
cash(tWo<mi)
=►cash(tWo,mi)
=
netRebalAdj* (two<mi)
= y^{Pui9t -Un)BPVn + Y , W 9t - U n\a
n
=
n
J 2 ( P u t9t- u n)B P V n +
n
_
/3 (< s t - « n ) a n(buy)
£
P (ut9t- u n)a
n(sell)
c a s h ( t u i o , mi ) + Y i n u n B P V n + Y n ( b u y ) u n a — Y n ( s e l l ) u n a
Y n Un B P V n + Y n ( b u y ) u n a ~~ Y n ( s e l l ) Un a
where n(buy) are the n -th asset such th a t u 9t — un > 0 (we need to buy additional
positions) and n(sell) are such th a t u 91 —un < 0 (we need to sell positions). Note th a t
we always have 0 < (3 < 1, the case of (} = 1 returns to (a).
In the ’Results outcom es’ section, we will see th a t (3 is frequently slightly less than one, so it
creates system atic patterns of negative gaps at rebalancing dates. Thus, we also analyzed what
happens if we allow additional liquidity injections to meet th e entire target portfolio.
13
3.2.3
A sset-L ia b ility gap m easu res
Our goal is to create and m aintain (rebalancing) an optimal portfolio at a minimum cost that
will best match the liability value when there are variations in interest rates (th at is. portfolio
immunization). Thus, the first measure of immunization efficiency we analyzed is the asset-liabilitv
absolute gap (both under the baseline yield curve scenario and the TSIR shocks scenarios):
gap(t;u>) =
portfolio value — liability value
= APV(t]ui) — L PV{t\uj),
w ithout adding available cash
(21)
or
=
[APV(t-,u) + cash(t-,uj)] — LPV(t-,Lj),
including available cash
(22)
Note th a t because of the large value of the portfolio (~ 65 M$), the difference in the gaps between
each scenario is very small. In fact, we were not able to see the difference on graphs. We thus only
analyzed the gap efficiency using the baseline yield curve r ( t,T ) . However, the difference becomes
more evident when we look at the relative gaps, i.e. the difference between the gap under each
scenario and the baseline gap:
Agap(t-, u ) = gap(t; u ) - gap(t),
(23)
where gap(t) is th e asset-liability gap computed with the baseline yield curve r ( t,T ) .
4
R esu lts ou tcom es and future research
In this section, we give
a big picture of th e results we obtained by analysing graphs and other
information given in the Appendices. All these figures contain more detailed explanation in them ­
selves. Appendix A illustrates the optimal portfolio positions tracking over time for both strategies
(static optim ization and stochastic dynamic optim ization) and for both matching techniques (du­
ration/convexity and 6 moments). Moreover, graphs in Appendix B illustrate the portfolio immu­
nization efficiency and costs for both strategies and matching techniques. Note th a t th e first graph
section B .l shows results for which we do not perm it additional liquidity injection at rebalancing
dates. Appendix B.1.1 presents the duration/convexity-m atching and Appendix B .l.2 presents
the 6 moments matching. The second graph section B.2 illustrates what happens if we allow for
additional liquidity injection.
At first glance, we observed th a t our results are different of what we initially expected. T h at is,
the stochastic dynamic optim ization technique is not th a t much more efficient in comparison with
the static version. However, in accordance with results of Augustin et al. (2010), the immunization
is better as we increase the number of moments to be matched but with an associated higher
rebalancing cost (with both optimization techniques). T h at is, by comparing the PV tracking
results of the duration/convexity-m atching on Fig. 2 with the 6 moments m atching on Fig. 3
on the next pages, one can remark th a t the immunization is better as we increase th e num ber of
moments to be m atched (for both static and stochastic optimization). We can see th a t the time-zero
optimal portfolio has a value ~ 65M $. In ’absolute’ dollars terms, there is no big difference between
graphs. Thus, in Appendices, we also illustrate other ’relative’ measures as gap (equation(22)) and
A gap (equation (23)).
14
DurCvx - P rese n t v a lu e s T im e se r ie s
\ f ' VA.
2 01 0-07-01
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
-A P V
I
LPV
I
R eb a la n c in g d a te s
AT
A
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
x tg ’
2 011 -0 1 -0 4
2 0 1 1 -0 2 -0 2
-\J
\J
V
\
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2011-07-01
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2011-0 7-0 1
DurCvx • A vailable c a sh Tim e se r ie s
_ y \
20 10-0 7 -0 1
2 0 1 0 -1 2 -0 2
j
A
2 0 1 0 -0 0 -0 3
/
“ V
2 0 1 0 -0 9 -0 2
i
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
20 11-0 1 -0 4
2 0 1 1 -0 2 -0 2
DurCvx - P re se n t v a lu e s Tim e s e r ie s (Including available c a sh in th e portfolio)
- A PV * C a sh
LPV
R eb a la n c in g d a te s j
20 1^-0 7 -0 1
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
(a )
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2011-0 7-0 1
D u ra tio n /C o n v ex ity S tatic case.
DurCvx (STO CH ASTIC ) - P resen t v a lu e s T im e se r ie s
" — \T "
-A P V
i
LPV
R eb a la n c in g d a te s |
A/
NT..........
-I
2 0 1 6-07-01
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
* , o’
2010 07-01
.
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
20 11-0 1 -0 4
I 2 0 1 1 -0 2 -0 2
I _
2 0 1 1 -0 3 -0 2
l
I
2 011 -0 4 -0 2
2 0 1 1 -0 S -0 3
_
2010 08 03
2011-0 7-0 1
Z 7 Vr ~
2010-09 02
j
l
2010-10 05
2010-11-02
2010 12 02
2011-01-04
2011-02 02
1/
2011-03-02
1____ / i _____ A_
2011 04-02
2011-05-03
2011-06-02
2011-07-01
DurCvx (STO C H A STIC ) - P r e se n t v a lu e s Tim e se r ie s (including availab le c a s h in th e portfolio)
«io’
- t *—
*----^ r-
|— —-' i
— ( •— —[— - i
r
•
4 k
2 0 1 0-07-01
i
2 0 1 1 -0 6 -0 2
DurCvx (STO C H A STIC) - A vailable c a s h T im e se r ie s
. I.
2 0 1 0 -0 8 -0 3
L
2 0 1 0 -0 9 -0 2
I
. .. . I .
2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2
...
i................................... 1 _
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
I _ ..........
i
2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2
L.
2 0 1 1 -0 4 -0 2
I
2 0 1 1 -0 5 -0 3
A PV ♦ C a sh
LPV
j
R eb alan cin g d a te s
I
2 0 1 1 -0 6 -0 2
2011 -0 7-0 1
( b ) D u ra tio n /C o n v ex ity S toch astic case.
F ig . 2: In both graphs, there is a decrease o f th e liability P V at regular d a tes caused by a liability stream (cash
outflow ). We have a total o f four liability stream s in our backtesting w indow. T h e negative gaps betw een th e asset
P V and liability P V are because som e bonds are m atured and becom e cash (see th e portfolio cash-flow s on Fig. 9
in A p pendix B .1 .1 ). B u t th e available cash from assets (given by equation (1 3 )) alm ost fills th ese n egative gaps.
T h e large cash peak before th e 6-th rebalancing d a te is due to three large m atured position s and drop to zero after
because it entirely funds th e rebalancing costs. A s we can see on (b ), the use o f stoch astic op tim ization d oes not
im prove the d u ra tio n /co n v ex ity m atching efficiency. It is also confirmed by observing th e relative m easures graphs
in A ppendix B .1.1. See the a sset-liab ility gap graph on Fig. 8 and the A in A-L gap on F ig. 12.
As we can see on the portfolio’s positions tracking graphs in Appendix A, the higher rebalancing
costs associated with higher moments (6 moments in th a t case) are because it needs a larger number
of assets to reach an optim al portfolio and it consequently increases the frequency of rebalancing.
By comparing portfolio positions for duration/convexity (static case) on Fig. 4 w ith portfolio
positions for 6 moments (static case) on Fig. 6 , one can see th a t the number of assets and the
15
6MM - P r esen t v a lu e s T im e s e n e s
-A P V
LPV
R eb alan cin g c
" V
45
1
2010-07-01
„
2 0 1 0 -0 8 -0 3
I
I
2 0 1 0 -0 9 -0 2
I
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 2 -0 2
I
2 0 1 1 -0 1 -0 4
"\
N_
i
2 01 1-0 2-0 2 2 0 1 1 -0 3 -0 2
:
i
201 1-04-02
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
I
I.
2011-07-01
6MM- A vailable c a s h T im e s e n e s
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
x I q7
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 6 -0 2
2011-07-01
6MM - P r e se n t v a lu e s T im e s e r ie s (including availab le c a s h in t h e portfolio)
.............................................
' • ? ....................
1
"1
I
V ....... .
2010-07 -01
v -
I
2 0 1 0 -1 1 -0 2
in*
2010-07 -01
i
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
(a )
I
I
!
......................................................................................................................................................................... |
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
1
A PV * C ash
^
2 0 1 1 -0 5 -0 3
R eb alan cin g d a te s ]
2 0 1 1 -0 6 -0 2
2011-0 7-0 1
6 M om ents S tatic case.
6MM (STO C H A STIC) • P r esen t v a lu e s T im e s e n e s
-A P V
LPV
R eb alan cin g a
" V J
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
A_
2 011 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 011 -0 4 -0 2
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2011 -07 -0 1
2011-04-02
2011-05-03
2011-08-02
2011 07 01
6MM (STO CH ASTIC ) • Available c a s h T im e s e r ie s
2010-07-01
2010-08-03
2010-09-02
2010- 10-06
2010-11-02
2010-12-02
2011-01-04
2011-02-02
2011-03-02
6MM (STO C H A STIC ) • P r e se n t v a lu e s T im e s e r ie s (including a v a ilab le c a s h in th e portfolio)
.T
I---
1 ------I'
!
I
*
•
20 10-07-01
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
APV + C a sh
I
LPV
R e b a la n cin g d a te s]
2 0 1 1 -0 6 -0 2
2011-07-01
(b ) 6 M om ents S toch astic case.
F ig . 3: We can see th a t th e im m unization is b etter w hen using higher m om ents. The negatives gaps are sm aller
and th e available cash is more lim ited (see the portfolio cash flow tracking on Fig. 14 in A ppendix B . l . 2). N ote th at
the difference betw een the sta tic case and the sto ch a stic case is b etter illustrated in A ppendix B . l . 2 with the relative
m easures graphs on the A-L gap graph Fig. 13 and the A in A-L gap Fig. 17.
rebalancing frequency is larger with higher moments. This is also in accordance with the results
of Augustin et al. (2010). The marginal beneficial effect on immunization with 6 moments by
using dynamic stochastic optim ization seems less prominent as it is a substantially more expensive
strategy (we discuss its costs below). The reason is th a t stochastic optimization involves a higher
number of positions and higher asset turnover in the portfolio. One can observe this phenomenon
by comparing portfolio positions for 6 moments (static case) 011 Fig. 6 with the stochastic case
on Fig. 7. Note th a t for duration/convexity, both static and stochastic cases are the same for the
portfolio positions. The explanation follows below.
16
I
LPV
We first thought th a t the use of dynamic stochastic optim ization would be less efficient as
we increase the number of moments to be matched because immunization with higher moments
itself is already an efficient matching strategy against TSIR shocks. Thus, the use of stochastic
optimization would b etter improve the immunization results with duration/convexity matching. In
fact, as explained before. This is not the case here. As we impose a maximum asset weight of &%..
these constraints are always tight with duration/convexity matching and this impedes all possible
improvements by the use of dynamic stochastic optimization. Thus, it explains why there is no
difference between the immunization results obtained by the static and stochastic optimizations.
In fact, we noticed th a t the optimal portfolio positions were practically the same with both static
and stochastic optim ization with this matching technique. We can observe this by comparing
portfolio positions for duration/convexity (static case) on Fig. 4 with the stochastic case on Fig.
5. Note th a t we performed the same optim ization with unconstrained asset weight to see if the
optim al portfolios would be different with the use of dynamic stochastic optimization but we found
th a t there is still little difference between them. The optimal portfolio positions chosen over time
are still the same with both optimization methods. The optim ization algorithms do alm ost only
roll short term assets (e.g. 1 m onth Tbills) to meet the duration/convexity constraints. Thus,
the cumulative rebalancing bid-ask costs, which is approxim ately 40k $, is less than w ith higher
moments, which is approxim ately 60k $ in the static case (see Fig. 10 in Appendix B.1.1). As
explained in ’Backtesting m ethodology’, this is because th e assigned Tbills’ bid-ask spread is defined
as zero basis point because they are very liquid assets. Moreover, we can see th a t this Tbills rolling
p attern involves large periodic cash-fiows (at each month) on Fig. 9 and no shortfall. This can
explain why duration/convexity-m atching is close to the cash flow matching technique.
Now turning our attention to the higher moment matching methods, we noticed th a t as the
number of moments increases (6 moments in this case), the use of dynamic stochastic optim ization
does a better im m unization against interest rates shocks. It seems to shrink positive and negative
differences in asset-liability gap between scenarios (th at is, the A gap in equation (23)) in comparison
with the static optim ization (compare Fig. 17b with Fig. 17a in Appendix B.1.2). Note th a t the
larger peaks on these graphs show th a t a p art of the portfolio has been liquidated because of a
shortfall and thus it created an imbalance in optimal positions. See these three shortfall occurrences
on Fig. 14. We will discuss further about it below. Note also th a t for the duration/convexitymatching, the differences in asset-liability gap are far more larger than with 6 moments matching
(see Fig. 12 in Appendix B.1.1). These findings are in accordance with results of Augustin et al.
( 2010 ).
We also noticed th a t as we do not allow for additional liquidity injection a t rebalancing dates
(self-financing strategy), there are slight shortfalls in cash at rebalancing dates to meet the target
(optimal) portfolio. T h at is, we do not always have enough cash to fund bid-ask costs and buy all
new optimal positions (net of cash from selling some other positions). At rebalancing dates, we
can observe system atic negative asset-liability gap. This is because the rebalancing factor /? from
equation (20) is always slightly less than one, especially with the 6 moments matching. Thus, it
means th a t the m atching constraints cannot be perfectly met and it leads to system atic mismatch
patterns a t rebalancing dates. See the asset-liability gap graphs (including available cash) on Fig.
13 in Appendix B.1.2. These mismatch are also in greater m agnitude with the use of stochastic
optimization. This is because this strategy involves less available cash in the portfolio and is more
costly. We discuss further on these costs below. In dollar terms. Fig. 16 illustrates how much cash
should be added in the portfolio to perfectly meet the target portfolio at rebalancing dates. Note
17
that these observations are less significant with duration/convexity-m atching. In the next graph
section. Appendix B.2. we allowed the optim ization algorithm for additional liquidity injections at
rebalancing dates. T h at is. we forced the fraction j3 to be equal to one and we com puted how much
additional cash it needs. We can see on Fig. 19 th a t it removes the systematic negative gap patterns
and the remaining (small) gaps are almost noise. It does not change the other immunization results
(for both duration/convexity and moment matching).
Another point to be noted is that the average available cash with moment m atching is larger
as we decrease the number of moments. As explained before, this is logical since it is like we are
approaching cash flow matching. This is also explained by the fact th a t as we decrease the number
of moments, each asset position is larger and the num ber of assets in th e portfolio decreases.
Consequently, these larger positions increase th e am ount of periodic cash flow in the portfolio.
This cash flow behavior is even more amplified when we remove the asset weight constraints.
The use of stochastic optim ization seems to reduce th e available cash in the portfolio since it
increases the number of assets to find an optim al portfolio (as we explained previously with portfolio
position tracking in Appendix A). Moreover, the use of stochastic optimization seems to decrease the
shortfall m agnitude (with the 6 moments matching). This can be verified by comparing shortfalls
in the static case on Fig. 14a with shortfalls in th e stochastic case on Fig. 14b. As explained in the
’Backtesting methodology’ section, when there is a shortfall, we need to liquidate the corresponding
am ount of assets. Small shortfalls translate into small additional transaction costs and the portfolio
liquidation causes an imbalance in the optim ality of th e positions (see Fig. 15). It thus impedes
moment matching efficiency as discussed before.
The small gains in immunization efficiency when increasing moments comes at a cost, as de­
scribed above. We can notice th a t the cumulative periodic rebalancing bid-ask costs are increasing
with the use of stochastic optimization, especially with higher moments to be m atched (6 moments
here). As we can see on Fig. 15, the cumulative bid-ask cost in the static case is approxim ately 60k
$ while it is approxim ately 120k $ in the stochastic case. As stated before, it is greater than with
the duration/convexity matching. This is because the optim al portfolio with 6 moments matching
contains more assets and have a higher asset turnover at almost each rebalancing date. Although
this behavior is also observed with the static optim ization, it is even more amplified by the use of
stochastic optim ization.
Finally, as mentioned in the ’Theoretical framework’ section, it could be interesting to investi­
gate the three other stochastic programming techniques described in Schwaiger et al. (2010): the
Stochastic Linear Programm ing model (SLP). the Chance Constrained Programm ing model (CCP)
and the Integrated Chance Constrained Program m ing model (ICCP). They are interesting because
one could formulate our optim ization problem using a two-stage stochastic linear programming
with recourse decision (control) variables. W ith the C C P model, we adjust the optim ization prob­
lem in order th a t there can be a non-zero probability of not meeting a matching constraint for a
’small’ set of scenarios. W ith the ICCP model, we do not only allow for a non-zero probability of
constraints mismatching, but we also constraint the am ount of the portfolio underfunding. We do
so by including an expected shortfall constraint which is calculated over all TSIR scenarios. This
can be viewed as a portfolio conditional value-at-risk (’CVaR’) type of constraint. We give more
technical details of these models in Appendix D. Since these models are based on a two-stage op­
tim ization framework, we could ultim ately extend these models to a multistage framework to have
a more realistic model of periodic rebalancing costs and therefore produce more efficient portfolios.
18
This would allow the portfolio facing TSIR shock scenarios to be rebalanced and optimized over
multiple periods prior to our investment horizon. An issue which comes from the different stochas­
tic optim ization methods described above is th a t they require a lot of computing capacity. Thus,
we could use parallel computing or com puter clustering to improve com putation performances and
optim ization efficiency. On the software side, we could also use more specialized tools for large-scale
stochastic programming like AMPL language or FortSP solver5.
A nother improvement could be done by refining the rebalancing cost function. In the present
study, we only use a constant bid-ask spread for each asset, independently of their characteristics
and their volume of trades. We could add the possibility to assign a user-defined bid-ask spread
for each asset or develop a more sophisticated measure of bid-ask spread. As it is closely linked
to the liquidity modeling literature, this could be even a new parallel project to model a bid-ask
spread function a = /( a s s e t’s param eters and trades’ volume) and use it in the objective function
of our optim ization problem. This would be another way of making a more realistic modeling of
the rebalancing costs and could eventually lead to more efficient portfolios as well.
In addition, we could also validate the robustness of our results by doing the same study on a
wider backtesting window (e.g. to include more liability stream s in our backtesting) which could be
divided into periods where results could be compared. This could reduce outlier results and help to
distinguish between noise variations and system atic variations among immunization techniques and
optim ization methods. We could also use a more complex liability universe to test our technique’s
robustness. Furtherm ore, we could perform daily valuation instead of weekly valuation to have a
larger d ata sample and consequently improve our statistics.
A final way to improve the immunization strategy would be to include other types of assets in
the universe (e.g. corporate bonds, swaps, other derivatives, etc.) This larger universe of assets
would provide a much greater flexibility in producing the desired portfolios.
5
C onclusion
In summary, our main objective of this present project was to further enhance the static moment
m atching m ethods used by Augustin et al. (2010) by allowing the portfolio to be rebalanced at a
minimum cost. We did so by using a two-stage stochastic optimization model to incorporate the
uncertainty of th e interest rate. We thus included the expected rebalancing costs in th e objective
function and used stochastic moment m atching constraints.
As the static 6 moments m atching immunization technique is already very efficient, we first
thought th a t the benefits of stochastic optim ization should be better reflected in the duration/convexity
m atching technique. Instead, we found the duration/convexity m ethod producing unsatisfactory
portfolios with both static and stochastic optim izations (with or w ithout asset concentration con­
straints). The stochastic optim ization does improve the 6 moments m atching m ethod but the
improvement is marginal, especially when we factor in the higher transaction costs.
There are many other im plem entations of the stochastic optimization b u t our im plem entation
in the context of this project and th e extensive dynam ic analysis we did of transaction costs through
backtesting did provide evidence of the great robustness of the static 6 moment m atching method.
From both an efficiency to cost ratio and an efficiency to simplicity ratio, the static 6 moments
m atching m ethod appears to be th e most practical solution so far.
5See h t t p ://w w w . o p t i r i s k - s y s t e m s . c o m /p r o d u c ts _ f o r ts p . a sp .
19
6
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20
A
P ortfolio positions tracking
In the following sets of graphs, we illustrate the optim al portfolio compositions at the beginning
of each m onth and the position changes from the preceding month. There are twelve rebalancing
dates (so twelve optimization runs).
Graphs interpretation:
The label axis contains the bond issuer along with their maturity, noted ' issuer.maturity' . This
axis can be viewed as the bond labels in th e set |Jt=i n*(f). where t is the month index. T h at is. we
are keeping all bond labels over the 12 dates even if some bonds are m atured after a given month
date (so they no longer exist in the universe). The red bars show the optimal portfolio composition
(the new rebalanced portfolio) a t the beginning of each month. The blue bars show th e changes in
positions from the previous month. The M onth # 1 is the portfolio creation date, so the changes
are identical to the actual positions. After a given month, a bond n can either m ature, be bought
or be sold. In the first case, the position falls to zero (there is no longer a red bar for this bond).
In the two other cases, the new red bar equals th e previous m onth red bar to which we applied the
actual position changes (the blue bar). If red bars remain the same over time, it is because there
is no change in those positions.
A .l
D u r a t io n /c o n v e x it y - m a t c h in g
(On next page)
21
Month #2: 2010-08-02
Month 41: 2010-07-01
Month 43: 2010-09-01
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F ig . 4: F irst, remark th a t the number o f bonds in each portfolio (th e number o f red bars at each m onth) and the bond turnover (blue bars) is less than
w ith the 6 m om ents m atching for b oth sta tic and stoch astic optim ization (see A ppendix A .2). If we look at the bond labels, we can see that the changes
in positions are alm ost alw ays the 1 m onth T B ills ( ’G C A N in d ex’). For exam ple, we can see th at the very first T bill at M onth # 1 matured before Month
# 2 because at th a t tim e, there is no longer a red bar (and no blue bar). However, we can see th at a new 1 m onth Tbill has been bought. N ote that each
position corresponds approxim ately alw ays to a weight of 8% o f th e portfolio, i.e. the im posed m axim um weight constraint. B ecause Tbills have a zero
assigned bid-ask spread, th is T B ills rolling behavior explains why the transaction costs are lower with d uration/convexity.
, *CtiM»Poi-torn
Month 01: 2010-07-01
Month 02: 2010-08-02
Month 03:2010-09-01
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F ig . 5: As explained in th e ’Research ou tcom es’ section , because the m axim um weight constraints are alm ost always tight in the duration/convexitym atching, the use stoch astic op tim ization cannot enhance the im m unization efficiency. W e can observe on these graphs th at the portfolio com position
over tim e is alm ost the sam e as the sta tic case (see previous graph, Fig. 4).
A .2
6 M o m e n t s m a tc h in g
(On next page)
24
Month #2: 2010-08-02
Month HI: 2010-07-01
Month #3: 2010-09-01
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in each portfolio is larger (m ore red bars) than w ith d u ration/convexity. W ith the stochastic case on n ext graph (see Fig. 7), we see th at the asset
turnover is even higher and also increases in m agnitude (i.e. the position changes are larger).
Month *1: 2010-07-01
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F ig . 7: In com parison w ith th e previous sta tic case, we can see on these graphs th at the turnover is higher (there are more blue bars at each m onth)
and th e p osition s changes are larger. It explains why th is strategy is more expensive in term s of transaction costs.
• Op'ii Povment
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B .l
Graphs o f portfolio im m unization results
W it h o u t a d d itio n a l liq u id ity in je c tio n
This subsection presents backtesting results with both duration/convexity-m atching and 6 moments
matching but w ithout allowing for additional liquidity injection at rebalancing dates.
27
B .1 .1
D u ra tio n / co n v ex ity -m a tch in g
DurCvx • Asset-Liab<iity g a p Tima s a n e s
g a p « APV - LPV
R eb alan cin g d a te s
-15
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 -07 -0 1
2 0 1 1 -0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 -0 7 -0 1
DurCvx • A vailable c a s h T im e se r ie s
2
1.5
1
0 .5
,8x 105
2 0 1 0 - 0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 12 -0 2
2 0 1 1 - 0 1 -0 4
2 0 1 1 - 0 2 -0 2
2 0 1 1 - 0 3 -0 2
DurCvx - Asset-Liability g a p T im e s e r ie s (including availab le c a s h in th e portfolio}
•
2 0 1 0 - 10 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 12 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 - 0 2 -0 2 2 0 1 1 -0 3 -0 2
2 0 1 1 - 04 -0 2
g a p * APV ♦ C a sh - LPV
R eb alan cin g d a te s ______
2 0 1 1 - 0 5 -0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 -0 7 -0 1
(a ) D u ra tio n /co n v ex ity Static case.
DurCvx (STO C H A STIC ) - Asset-Liability g a p T im e se r ie s
•
2 0 1 1 - 0 1 -0 4
2 0 1 1 -0 2 -0 2 2 0 1 1 - 0 3 -0 2
g a p -A P V -L P V
R eb alan cin g d a le s
2 0 1 1 - 0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 0 7 -0 1
2 0 1 1 - 0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 -0 7 -0 1
DurCvx (STO C H A STIC ) - A vailable c a s h T im e se r ie s
2
1 .5
1
0 .5
,8
2 0 1 1 - 0 1 -0 4
2 0 1 1 - 0 2 -0 2
2 0 1 1 -0 3 -0 2
DurCvx (STO CH ASTIC ) • A sset-U ab ility g a p Tim e s e r ie s (including a v a ila b le c a sh in th e portfolio)
•
2 0 1 0 - 10 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 - 0 2
2 0 1 1 - 0 3 -0 2
2 0 1 1 - 0 4 -0 2
g a p » APV * C a sh - LPV
R eb alan cin g d a t e s ______
2 0 1 1 -0 5 - 0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 -0 7 -0 1
( b ) D u ra tio n /co n v ex ity S toch astic case.
F ig . 8 : T h e first graph on b oth (a) and (b ) show s th e a sset-liab ility gap w ithout adding available cash in the portfolio.
W e can see th a t the gaps are larger than th e case w ith higher m om ents (see for instance F ig 13). W hen we add the
available cash, it fills in part the negatives gaps, but there still have large variations in th e gap. It is also m ore like
noise than a sy stem a tic m ism atch pattern as w ith 6 m om ents-m atching. W e can see on Fig. 11 th a t there still som e
m ism atch (additional liquidity needs), b u t less frequently (but larger in m agnitude) than w ith 6 m om ents. In the
n ext graph section in A ppendix B .2, th e Fig. 18 illu strates w hat happen to th e gaps when allow ing for additional
liquidity injection. As explained in th e ’R esu lts o u tco m es’ section , the use o f stoch astic program m ing d oes not bring
any additional efficiency in the d u ra tio n /co n v ex ity m atching case.
28
DurCvx - Asset total cash-flows and Liability stream Series
I Asset total cash-flow
LiabHiy stream
2
0
Mi l n
2
i i ij i i I
•4
j ______________l_
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04
2011-02-02
2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
DurCvx - Available cash and Shortfall tracking Time Series
x 10'
■
•
Available cash in the portfolio
Net shortfall amount
Rebalancing dates_________
0.5
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
( a ) D u ra tio n /co n v ex ity S ta tic case.
DurCvx (STOCHASTIC) -
10
8
6
total cash-flows and liability stream Series
I Asset total cash-flow
%Liabiliy stream
2
0
•2
-4
1
• 6 -------2010-07-01
20
2010-10-05
2010-11-02
2010- 12-02
2011 -04-02
2011-05-03
2011-06-02
2011-07-
DurCvx (STOCHASTIC) - Available cash and Shortfall tracking Time Series
x 10
— Available cash in the portfolio
Net shortfall amount
• Rebalancing dates_________
10
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
( b ) D u ra tio n /co n v ex ity S toch astic case.
F ig . 9: On b oth second graph o f (a) and (b ), we can see th at there is no shortfall w ith th e d u ratio n /co n vexitym atching case (it is close to th e cash flow m atching case). T h e drop after each peak is th e use o f available cash to
invest in th e target portfolio at each rebalancing d ates and to pay the bid-ask co sts (these co sts are illu strated on
Fig. 10).
29
2010-07-01
T ..........................
i ......................
1
1
DurCvx - Net investments and Shortfall adjustments Times series {excluding time zero data)
!
:
!
i
Shortfall adjustment amount (portfolio liquidation)!
■ H i Net real investment amount
Net target investment amount
[
I
i
i
2010-08-03
2010-09-02
l
i
2010-10-05 2010-11-02
2010-12-02
l
l
i
1
1
1
l
1
2011-01-04
2011-02-02
2011-03-02
2011-04-02
2011-05-03
'
I
2011-06-02 2011-07-01
DurCvx • 8id-Ask costs Times series (excluding time zero
10000
Bid-Ask cost at rebalancing dates
Bid-Ask cost for shorttaN adjustements
8000 6000
4000
2000
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04
2011-02-02
2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
DurCvx - Cumulative Bid-Ask costs Time series (excluding time zero data)
- Cumulative rebalancing Bid-Ask cost
- Cumulative shortfall adjustment Bid-Ask cost
2010-07-01
2010-08-03
2010-09-02
~r
2010-10-05 2010-11-02
~r
2010-12-02
~r
_ r
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
(a ) D u ra tio n /co n v ex ity S tatic case.
DurCvx (STOCHASTiC) - NNet investments and Shortfall adjustments Times series (excluding time zero data)
20f
T"
T
~r
i Shortfall adjustment amount (portfolio liquidation)
I Net real investment amount
Net target investment amount________________
15 10►
5-
J
0-
I
1
I
_ l_
2010-07-01
2010-08-03
~T~
15000
2010-09-02
2010-10-05 2010-11-02
1
2010-12-02
2011-01-04
2011-02-02
2011-03-02
_L_
2011-04-02
2010-08-03
~T~
2010-09-02
x jo 4
I Bid-Ask cost at rebalancing dates
BBid-Ask cost for shortfall adjustements
J
L
2010-10-05 2010-11-02
2010-12-02
2011-01-04
J
2011-02-02
I
2011-03-02
2011-04-02
DurCvx (STOCHASTIC) - Cumulative Bid-Ask costs Time series (excluding time zero data)
_T_
~r
~r
i
- Cumulative rebalancing Bid-Ask cost
- Cumulative shortfall adjustment Bid-Ask cost
2010-07-01
2010-08-03
2010-09-02
2011-06-02 2011-07-01
DurCvx (STOCHASTIC) • Bid-Ask costs Times series (excluding time zero data)
5000
201&07-01
2011-05-03
L
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
I
2011-05-03
.
2011-06-02 2011-07-01
i
2011-05-03
2011-06-02 2011-07-01
( b ) D u ra tio n /co n v ex ity S toch astic case.
F ig . 10: T h e net investm ent c o sts and bid-ask co sts at each rebalancing d ates are less than th e 6 m om ents m atching
case (see Fig. 15). T h is is because th e algorithm alm ost only d o roll T b ills and because th ey have a zero assigned
bid-ask cost.
30
DurCvx - Asset-Liabrtty gap Time series (including available cash in the portfolio)
•
gap - APV + Cash • LPV
Rebalancing dates
0.5
-0.5
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
DurCvx - Additional liquidity injection needs to meet the target portfolio at rebalancing dates
[-------- 1----------1----------1----------1----------j---------- j—
I
J
J ________________ L________________ I..
2010-07-01
2010-08-03
2010-09-02
J _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . l . . . .
2010-10-05 2010-11-02
(a )
2010-12-02
2011-01-04
J ________________ L _
2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
D u ratio n /co n v ex ity S tatic case.
DurCvx • (STOCHASTIC) Asset-Liability gap Time
x 10'
2011-02-02
(including available cash in the portfolio)
»
gap - APV + Cash ■LPV
Rebalancing dates
0.5
-0.5
2010-07-01
x iq 4
2010-08-03
2010-09-02
!
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
DurCvx • (STOCHASTIC) Additional liquidity injection needs to meet the target portfolio at rebalancing dates
I
2010-07-01
2010-10-05 2010-11-02
J ____________ L
2010-08-03 2010-09-02
j--------j--------- !--------- 1-----
i
i--------- 1--------- r
1
l
l
2010-10-05 2010-11-02
2010-12-02
_±_
2011-01-04
2011-02-02
2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
( b ) D u ra tio n /co n v ex ity S toch astic case.
F ig . 11: W e can see th a t there is som e shortfall in cash (liquidity needs) when rebalancing th e portfolio to m eet the
target optim al portfolio, but it is less than th e 6 m om ents case (see Fig. 16).
31
DurCvx • Asset-liability gap differences between yield curve scenanons and baseline yield curve
0.8
o> - ( 1 . 4 , 0 . 0 )
o»7 - ( 0 , 1 .4 .0 )
ti>6 « (1 .4 , 1 .4 ,0 )
0.2
*
J
Rebalancing dates
- 0 .2
-0 .4
- 0.6
2 0 1 0 -0 7 -0 1
2 0 1 0 - 0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 -0 6 - 0 2
2 0 1 1 -0 7 -0 1
(a ) D u ra tio n /co n v ex ity S tatic case.
DurCvx (STOCHASTIC) - Asset-liability gap differences between yield curve scenarions and baseline yield curve
x 10'
w2 - ( 0 , -1 .4 , 0)
0.8
(1 .4 ,- 1 .4 , 0)
« * - ( - 1 . 4 , 0 , 0)
0.6
o»7 - (0, 1 .4 , 0)
0 .4
< 1 4 . 1 . 4 ,0 )
Rebalancing dales
0.2
-0.2
rtj»
-0 .4
- 0.6
- 0.8
2 0 1 0 -0 7 -0 1
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 -0 6 - 0 2
2 0 1 1 -0 7 -0 1
( b ) D u ra tio n /co n v ex ity S toch astic case.
F ig . 12: F irst, the shocks on th ese graphs are num ber o f standard d eviation s o f respectively P C A 1, P C A 2 and
P C A 3. We only have height scenarios and we did not shocked th e P C A 3 because its m inim al im pact on the curve.
T he variations o f gap betw een th e scenarios in this case are larger in th is com pared to the 6 m om ents m atching (see
Fig. 17). It is indeed because we reduce th e portfolio im m unization again st yield curve m om ents w hen we decrease
th e num ber o f m om ents.
32
B .1 .2
6 M o m en ts m atch in g
6MM - Asset-Liabtfity g a p T im e se r ie s
•
2010 -
6
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 - 0 1 -0 4
2 0 1 1 - 0 2 -0 2
g a p -A P V -L P V j
R eb a la n c in g d a te s \.
2 0 1 1 -0 3 -0 2
2 0 1 1 - 0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 - 0 7-01
2 0 1 1 - 0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 -07-0 1
6MM • A vailable c a s h T im e se r ie s
x 10*
4
2
A
2 0 1 1 - 0 1 -0 4
2 0 1 1 - 0 2 -0 2
6MM - A sset-Liability g a p Tim e s e r ie s (including availab le c a s h m th e portfolio)
5000
•
0
g a p - APV + C a sh - LPV
R eb a la n cin g d a te s _______
-5 0 0 0
W
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 - 0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 0 2
2 0 1 1 - 07-01
( a ) 6 M om ents S tatic case.
6M M (STO CH A STIC) • Asset-Liability g a p T im e se r ie s
1
1
:
2 0 1 0 - 0 7 -0 1
1
^
i
2 0 1 0 - 0 8 -0 3
r
T
2 0 1 0 - 1 2 -0 2
1
2 0 1 1 -0 1 -0 4
7
1
!
1
1,------ ---------------1-------- — ------ :
--------- g a p - A P V - L P V
\J
i
2 0 1 0 - 0 9 -0 2
i
i
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
x jo *
1
2 0 1 1 -0 2 -0 2
I
2 0 1 1 - 0 3 -0 2
I
2 0 1 1 -0 4 -0 2
1
2 0 1 1 -0 5 - 0 3
1
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 0 7-0 1
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 -0 6 * 0 2
2 0 1 1 - 0 7-0 1
6MM (STO CH ASTIC) • A vailable c a sh Tim e s e r ie s
2 0 1 0 - 07*01
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 - 03*02
6 MM (STO C H A STIC ) - Assel-Liability g a p T im e s e r ie s (including av a ila b le c a s h in th e portfolio)
■
i
— t
u
^
1
2 0 1 0 - 0 7 -0 1
2 0 1 0 -0 8 -0 3
....... r
—
1
]
[
1
2 0 1 0 - 0 9 -0 2
2 0 1 0 - 1 0 -0 5
1
i
i
i
*
y
V
1
2 0 1 0 - 1 1 -0 2
I
2 0 1 0 - 1 2 -0 2
i
i
2 0 1 1 -0 1 -0 4
vy
—
I
2 0 1 1 -0 2 -0 2
I
i
' V
V
1
!
2 0 1 1 - 0 3 -0 2
---------g a p - APV + C a s h - LPV
2 0 1 1 -0 4 -0 2
I
2 0 1 1 - 0 5 -0 3
1
i
2 0 1 1 -0 6 -0 2
2 0 1 1 - 0 7 -0 1
( b ) 6 M om ents S toch astic case.
F ig . 13: W hen we include the available cash in the portfolio, it fills (in part) the n egative asset-liab ility gaps.
However, we can see a sy stem a tic pattern o f negative gaps. It is because at rebalancing d ates, w e have not enough
cash to m eet th e target optim al portfolio (i.e. th e fraction f3 discussed in the ’B acktesting m eth o d o lo g y ’ section is
slightly less than 1). We can see on Fig. 16 hon much cash in $ th ese gaps m ism atch would need a t rebalancing
d ates to m eet the target portfolio. In th e n ext graphs section in A ppendix B .2, we can see w h at happen if we allow
for th ese additional cash injection at rebalancing d ates (see Fig. 19). W hen we use stoch astic program m ing, w e can
observe on (b) th at the sy stem a tic n egative gaps at rebalancing d ates increase in m agnitude. It is explained by two
things: first, th e available cash is less in th e sto ch a stic case (see Fig. 14b), and second, th e n et rebalancing costs
are greater (see Fig. 15b). W e can see n ex t on Fig. 16b how m uch cash in $ th ese gaps m ism atch would need at
rebalancing d a tes to m eet the target portfolio. Finally, on e can observe th at allow ing for th is ad d ition al liquidity
rem oves th e system atic negative gaps (see Fig. 19 in A ppendix B .2).
33
6MM - Asset total cash-flows and Liability stream Series
~r
I Asset total cash flow
Liabtty stream
2
±
«A 0
_
J
■2
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04
2011-02-02
2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
6MM - Available cash and Shortfall tracking Time Series
*
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
Available cash in the portfolio
Net shortfall amount
Rebalancing dates_________
2011-05-03
2011-06-02 2011-07-01
(a ) 6 M om ents S ta tic case.
6MM (STOCHASTIC) - Asset total cash-flows and Liability stream Series
I Asset total cash flow
S Liabiliy stream
6MM (STOCHASTIC) - Available cash and Shortfall tracking Time Series
x 10
•
Available cash in the portfolio
Net shortfall amount
Rebalancing d a te s ________
w
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
( b ) 6 M om ents S to ch a stic case.
F ig . 14: F irst, we can see the four liability stream in our b ack testin g w indow (th ere is a to ta l o f 22 liab ility stream ).
On b oth th e second graph o f (a) and (b ), th e available cash is th e cum ulative a sse ts’ cash flow, n et o f th e liability
stream and rebalancing co sts (equation 13). N ote th a t in th e cases o f shortfalls, we forced th e portfolio to b e liquidated
to m eet th e liability stream . C onsequently, it im pacted th e im m unization efficiency as the p o sitio n s in th e portfolio
becom e no longer optim al. T here are th ree shortfall dates: 2010-10-05, 2011-04-02 and 2011-07-03. In th e stoch astic
case, as stated previously, th e available cash is sligh tly less than th e sta tic case and shortfalls are in sligh tly less in
m agnitude.
34
6MM - Net investm ents and Shortfall adjustm ents Tim es series {excluding tim e zero data)
Shortfall adjustment amount (portfolio liquidation)
I Net real investment amount
Net target investment amount_____________
_L
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
_ l_
2011-01-04
2011 02-02
2011-03-02
_1_
2011-04-02
2011-05-03
2011-06-02 2011-07-01
6MM - Bid-Ask costs Times series (excluding time zero data)
Bid-Ask cost at rebalancing dates
Bid-Ask cost for shortfall adjustements
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
x jg 4
2010-12-02
2011-01-04
2011-02-02
2011-03-02 2011-04-02
2011-05-03
2011-06-02 2011-07-01
6MM • Cumulative Bid-Ask costs Time series (excluding time zero data)
Cumulative rebalancing Bid-Ask cost
- Cumulative shortfall adjustment Bid-Ask cost
2010-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02 2011-07-01
(a ) 6 M om ents S ta tic case.
1
1
-
6MM (STOCHASTIC) - NNet investments and Shortfall adjustments Times series (excluding time zero data)
1
'— 1...................... T...........
t
1
Shortfall adjustment amount (fjortfolio liquidation)
Net real investment amount
I
Net target investment amount
______________________
2010-07-01
1
2010-08-03
...
1...
2010-09-02
, iq 4
2018 -07-01
)
1
i
l
i
2010-12-02
2011-01-04
2011-02-02
i
i _______
2011-03-02 2011-04-02
i
2011-05-03
i
2011-06-02 2011-07-01
_r_
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04
2011-02-02
2011-03-02
I Bid-Ask cost at rebalancing dates
I Bid-Ask cost for shortfall adjustements [•
2011-04-02
6MM (STOCHASTIC) • Cumulative Bid-Ask costs Time series (excluding time zero data)
_r
T
~r
T
- Cumulative rebalancing Bid-Ask cost
- Cumulative shortfall adjustment Bid-Ask c
2010-08-03
2010-09-02
2011-05-03
~r
2011-06-02 2011-07-01
~r
r « » -----------------
__
---------------------- A
2010-07-01
*
r
6MM (STOCHASTIC) - Bid-Ask costs Times series (excluding time zero data)
*r
*r
T
T
, -jg4
15r
10
.
2010-10-05 2010-11-02
1
i
_____ I______L--------- 1------2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
_ l -----------------------1,.
2011-04-02
2011-05-03
2011-06-02 2011-07-01
( b ) 6 M om ents S toch astic case.
F ig . 15: B oth the first graph on (a) and (b) show s th e n et target cost to rebalance th e portfolio and th e net
realized cost, th at is, the cost o f buying new p osition s less cash from selling other positions. W e excluded th e first
tim e-zero d a ta because it is too large com pared to th e other d a ta (th e first d a te correspond in fact to th e initial
portfolio investm ent value o f ~ 65M $ ). M oreover, because additional liquidity injection is n ot allowed, th e target
investm ent is frequently greater than th e real investm ent (which caused the fraction f3 slig h tly less than on e as
discussed previously). T h e difference betw een the n et real investm ent and the target investm ent corresponds to the
liquidity needs illustrated on Fig. 16. W e can also observe the value o f th e portfolio liquidation value a t th e three
shortfall dates. On b oth th e second graph o f (a) and (b), there are th e bid-ask co sts o f rebalancing th e portfolio and
liquidating for shortfalls.
35
6MM - Asset-Liability g a p Tima s e rie s (including available c a sh in the portfolio)
— gap - APV ♦ Cash - LPV
• Rebalancing dates_____
•2000
-4000
-6000
-8000
1-07-01
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04 2011-02-02 2011-03-02
2011-04-02
2011-05-03
2011-06-02
2011-07-01
6MM - Additional liquidity injection n eeds to meet the target portfolio at rebalancing dales
n-------------r
T
_r~
8000
6000
4000
2000
2010-07-01
■ill
2010-08-03
2010-09-02
2010-10-05 2010-11-02
(a )
0.5
2010-12-02
2011-01-04
2011-02-02
2011-03-02
2011-04-02
I I
2011-05-03
2011-06-02 2011-07-01
6 M om ents S tatic case.
6MM - (STOCHASTIC) Asset-Liability gap Time series (including available cash in the portfolio)
x 10*
•
gap - APV + Cash • LPV
Rebalancing dates
,
-0.5
-2
-
I______I______
I_____ I______I______
I______I_____ I______ I______ I______ I______I
.95!______
2010-08-03 2010-09-02
2010-10-05 2010-11-02 2010-12-02
2011-01-04 2011-02-02 2011-03-02 2011-04-02 2011-05-03 2011-06-02 2011-07-01
.2010-07-01
6MM • (STOCHASTIC) Additional liquidity injection needs to meet the target portfolio at rebalancing dates
2010-07.01
u
2010-08-03
2010-09-02
2010-10-05 2010-11-02
2010-12-02
2011-01-04
ll
2011-02-02
2011-03-02
2011-04-02
I I
2011-05-03
2011-06-02 2011-07-01
(b ) 6 M om ents S toch astic case.
F ig . 16: T h e sy stem a tic n egative gaps due to th e rebalancing fraction less than o n e (shortfall in cash) can be view ed
as an add itional cash needs at rebalancing d ates. N o te th a t th is difference in $ is not so large com pared to the
portfolio value ( ~ 65M S). T h e n ext graph section ’W ith additional liquidity in jection ’ illustrates w h at happen when
we allow injection o f these additional cash needs to m eet th e optim al target portfolio. N o te th a t th e available cash
(first graph) plus th e additional cash injection (second graph) should fill (n ot ex a c tly because o f noise) the negative
gaps. W ith th e use o f sto ch a stic program m ing, th e additional liquidity needs are greater. It is explained by th e fact
th at there is less available cash in the portfolio w ith th is m ethod and it is more c o stly (in term s o f rebalancing costs
and bid-ask costs).
36
6MM - Assel-LiaCility g ap differences betw een yield curve scenarions a n d baseline yield curve
( 0 .- 1 . 4 .0 )
(1 .4 . -1 .4 , 0)
{ • 1 .4 . 0 ,0 )
<o? * ( 0 ,1 . 4 , 0)
i
(1 .4 , 1 .4 ,0 )
•
0 .5
Rebalancing dates |
-0 .5
2 0 1 0 -0 7 -0 1
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 - 0 5
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 -0 7 -0 1
( a ) 6 M om ents S tatic case.
6MM (STOCHASTIC) - Asset-LiabiBty gap differences between yield curve scenarions and baseline yield curve
x 10
t»t • (-1 .4 , -1 .4 ,0)
o» - ( 0 . - 1 . 4 , 0)
o) - (1 .4 , .1 .4 , 0)
0 .5
-0 .5
2 0 1 0 0 7 -0 1
2 0 1 0 -0 8 0 3
2 0 1 0 -0 9 -0 2
2 0 1 0 -1 0 -0 5
2 0 1 0 -1 1 -0 2
2 0 1 0 -1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2011-03-1
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 -0 7 -0 1
( b ) 6 M om ents S toch astic case.
F ig . 17: T h e large peaks are explained by th e shortfall portfolio liquidation. T h ese portfolio liquidation betw een
to op tim ization d ate im balance th e optim al portfolio positions. W e can see on (b ) that th e stoch astic op tim ization
shrinks these peaks. T h is is because shortfalls are in less m agnitude in th is case and consequently, w e need to liquidate
a sm aller part o f the portfolio.
37
B .2
W it h a d d itio n a l liq u id ity in je c tio n
This subsection presents the asset-liability gap when allowing for additional liquidity injection at
rebalancing dates.
38
B . 2.1
D u r a tio n /c o n v e x ity -m a tc h in g
By allowing additional cash at rebalancing dates, we observe only a very small effect of removing
the the system atic negative gaps on the following Fig. 18. In fact, with the duration/convexitymatching. there are not th a t much additional liquidity needs than with 6 moments. On can compare
these liquidity needs on Fig. 11 with those with 6 moments m atching on Fig. 16.
DurCvx - Asset-Liability
T im e s e r ie s
g a p * APV - LPV
R eb a la n c in g d a te s
2 0 1 0 - 0 8 -0 3
2
2 0 1 0 - 0 9 -0 2
2 0 1 1 - 0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 - 0 3 -0 2
2 0 1 1 - 04 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 -07-0 1
2 0 1 1 - 0 3 -0 2
2 0 1 1 - 0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 -0 2
2 0 1 1 - 07-0 1
DurCvx - A vailable c a s h T im e se r ie s
107
1 .5
1
0 .5
8.
2 0 1 1 -0 1 -0 4
2 0 1 1 - 0 2 -0 2
DurCvx • Asset-Liability g a p Tim e s e r ie s {including a v a ilab le c a s h in th e portfolio)
g a p » A PV ♦ C a sh - LPV
R eb alan cin g d a te s ______
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 11 -0 2
2 0 1 0 - 1 2 -02
2 0 1 1 -0 1 -0 4
2 0 1 1 - 0 2 -0 2
2 0 1 1 - 0 3 -0 2
2 0 1 1 - 0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 07-01
( a ) D u ra tio n /co n v ex ity S ta tic case.
DurCvx (STO CH ASTIC) • Asset-Liability g a p T im e
g a p . A PV - LPV
R eb alan cin g d a te s
2 0 1 0 -0 8 -0 3
2 0 1 0 -0 9 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 - 0 2 -0 2
2 0 1 1 - 0 3 -0 2
2 0 1 1 -04 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 - 0 2
2 0 1 1 - 0 7 -0 1
2 0 1 1 -0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 -0 6 - 0 2
2 0 1 1 - 07-0 1
DurCvx (STO CH ASTIC) - A vailable c a s h T im e s e r ie s
0 .5
2 0 1 0 - 0 8 -0 3
2 0 1 0 - 0 9 -0 2
2 0 1 0 - 1 0 -0 5 2 0 1 0 - 11 -0 2
2 0 1 0 - 12 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 - 0 2 -0 2
2 0 1 1 - 0 3 -0 2
DurCvx (STO C H A STIC ) • A sset-Liability g a p T im e s e r ie s (including availab le c a s h in th e portfolio)
g a p - A PV ♦ C a sh • LPV
R eb alan cin g d a te s
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
(b)
2 0 1 0 - 12 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2
2 0 1 1 -0 3 -0 2
D u ra tio n /co n v ex ity S toch astic case.
F ig . 18
39
2 0 1 1 - 04 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 -0 6 - 0 2
2 0 1 1 - 07-0 1
B .2 .2
6 M o m en ts m atch in g
Allowing for additional liquidity injection at rebalancing dates removes the system atic negative
gaps observed on Fig. 13 with 6 moments matching.
6MM - Asset-LiabMity g a p T im e se r ie s
•
2 0 1 1 *0 1 -0 4
6
2 0 1 1 -0 2 -0 2
g a p « APV • LPV
R eb alan cin g d a t e s
2 0 1 1 - 0 3 -0 2
2 0 1 1 - 0 4 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 07-01
2 0 1 1 - 0 2 -0 2 2 0 1 1 - 0 3 -0 2
2 0 1 1 - 04 -0 2
2 0 1 1 -0 5 - 0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 0 7-01
6MM - A vailable c a s h T im e se r ie s
x 10*
4
2
8
2 0 1 0 - 07-0 1
2 0 1 0 - 0 8 -0 3
2 0 1 0 - 0 9 -0 2
2 0 1 0 - 10 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 -0 1 -0 4
6MM - A sset-U ability g a p T im e s e r ie s (including availab le c a s h in th e portfolio)
1000
I
I
I
I
I
1
1
1
---------g a p « APV + C a sh - LPV
0*
-1000
r
-2000
0 1 0 - 07 -0 1
1
2 0 1 0 - 0 8 -0 3
•
1
2 0 1 0 -0 9 -0 2
I
I
2 0 1 0 - 1 0 -0 5 2 0 1 0 - 1 1 -0 2
1
2 0 1 0 - 1 2 -0 2
(a )
1
2 0 1 1 - 0 1 -0 4
I
I
2 0 1 1 -0 2 -0 2 2 0 1 1 - 0 3 -0 2
^
2 0 1 1 - 04 -0 2
~
i
2 0 1 1 -0 5 -0 3
"
i
2 0 1 1 -0 6 -0 2
i
2 0 1 1 - 07-01
6 M om ents S ta tic case.
6MM (STO CH A STIC) - Asset-LiabiHty g a p Tim e se r ie s
g a p - A PV • LPV
R eb a la n c in g d a te
2 0 1 0 - 0 7 -0 1
2 0 1 0 - 0 8 -0 3
2 0 1 0 - 0 9 -0 2
2 0 1 0 - 1 0 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 - 0 2 -0 2 2 0 1 1 - 0 3 -0 2
2 0 1 1 -0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 07-01
2 0 1 1 -0 4 -0 2
2 0 1 1 - 0 5 -0 3
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 0 7 -0 1
6MM (STO CH A STIC) - A vailable c a s h T im e s e r ie s
5
4
3
2
1
18-
2 0 1 0 - 0 8 -0 3
2 0 1 0 - 0 9 -0 2
2 0 1 0 - 10 -0 5
2 0 1 0 - 1 1 -0 2
2 0 1 0 - 1 2 -0 2
2 0 1 1 -0 1 -0 4
2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2
6 MM (STO C H A STIC) - A sset-U ab ility g a p T im e s e r ie s (including availab le c a sh in th e portfolio)
1000
O
i
i
'
_____'
1
„ 1
1
1
^
---------------- g a p » APV ♦ C a sh • LPV
•
H e o aian ctn g g a te s
\
-1000
-2000
-3 0 0 0
5- 0 7 -0 1
i
2 0 1 0 -0 8 -0 3
i
2 0 1 0 - 0 9 -0 2
i
2 0 1 0 - 10 -0 5
2 0 1 0 - 1 1 -0 2
(b )
i
2 0 1 0 - 1 2 -0 2
i
2 0 1 1 -0 1 -0 4
l
l
2 0 1 1 - 0 2 -0 2 2 0 1 1 - 0 3 -0 2
6 M om ents S toch astic case.
F ig . 19
40
2 0 1 1 -0 4 -0 2
l
2 0 1 1 -0 5 -0 3
I
2 0 1 1 - 0 6 -0 2
2 0 1 1 - 0 7 -0 1
C
C .l
D ata and settlem en t dates
B o n d u n iv e r s e d e t a ils
This is the bond universe used as of beginning of the backtesting window (at time zero), after
applying filter on Maturity. Group. Sector and Industry and including money market assets (bonds
with m aturity < 1Y and 1M Canadian Government Index as ’TBills'). Recall however th a t this
universe change as tim e goes by.
20100801GCAN1M
110709EY4
683234NX2
748148QU0
642866EY9
748140KC6
448814002
6428660M6
642866E26
683234P05
683234YM4
1107098SO
110709FD9
748148KE2
448814CV3
7481488G7
683234ZK7
683234RX8
642866FB8
6428660U8
642866GC5
683234TF5
642866DY0
110709DG4
748148RK1
683234UF3
1107090F6
110709FT4
110709FK3
11070ZAH7
6832348C5
683078DJ5
748148Rkfi
642866FT9
683234WM6
642866EF0
683234C30
748148RP0
642866FV4
683234YC6
642866FX0
74814ZDH3
683234YX0
S42866EP8
110709FR8
74814ZDR1
642866EQ6
683234Z04
642866GB7
11070ZDE1
683234TQ1
74814Z0U4
110709FX5
683234B80
6428668Z3
11070ZAG9
11070ZDK7
74814ZEE9
110709FZ0
683078DK2
683234WT1
6832348J0
642869A87
448814DG5
110709887
683078DS5
683078DQ9
74814ZEG4
GCAN1M Index
British Columbia
Ontario
Quebec
New Brunswick
Quebec
Hydro-Quebec
New Brunswick
New Brunswick
Ontario
Ontario
British Columbia
British Columbia
Quebec
Hydro-Quebec
Quebec
Ontario
Ontario
New Brunswick
New Brunswick
New Brunswick
Ontario
New Brunswick
British Columbia
Quebec
Ontario
British Columbia
British Columbia
British Columbia
British Columbia
Ontario
Ontario Hydro
Quebec
New Brunswick
Ontario
New Brunswick
Ontario
Quebec
New Brunswick
Ontario
New Brunswick
Quebec
Ontario
New Brunswick
British Columbia
Quebec
New Brunswick
Ontario
New Brunswick
British Columbia
Ontario
Quebec
British Columbia
Ontario
New Brunswick
British Columbia
British Columbia
Quebec
British Columbia
Ontario Hydro
Ontario
Ontario
New Brunswick
Hydro-Quebec
British Columbia
Ontario Hydro
Ontario Hydro
Quebec
6,38
6.10
6.25
5.80
9,50
10,00
10.13
5.85
6,10
4.40
9,50
5.75
9,00
10,25
6.00
4.50
5,38
5.88
9,25
3.35
4,75
8.50
8.50
5.25
5.00
7.50
4,25
5.30
8,50
3.25
10,00
5.50
4.50
4.50
8.75
3.15
5.00
4,30
4,40
4.70
4.50
4.30
6,75
4.70
4,50
6.00
4.20
4,45
5.60
5.50
4,50
4,65
4,40
4.40
9,00
5.30
4,50
4.10
10.00
4.85
4,20
4,50
11,00
10.60
11,00
11.50
4.50
2010-08-01
2010-08-23
2010-11-19
2010-12-01
2011-07-12
2011-09-02
2011-09-26
2011-10-31
2011-12-01
2011-12-02
2011-12-02
2012-01-09
2012-01-09
2012-02-10
201207-16
2012-1001
2012-12-02
2012-1202
2012-1206
201301-18
20130601
2013-0602
201306-28
2013-08-23
2013-1001
20140308
2014-06-09
2014-06-18
2014-06-18
2014-06-20
2014-0908
2014-10-17
2014-1201
201502-04
20150308
2015-05-12
20150908
2015-1201
2015-1203
20160308
201607-21
2016-1201
20170306
201706-27
2017-1201
2017-1201
2017-12-27
20180308
20180326
2018-0601
20180602
2018-1201
201312-16
2019-0602
201906-03
2019-06-17
20190317
2019-1201
201312-18
202002-06
2020-0602
2020-0602
20200302
202008-15
20230305
20231001
202311-27
20231201
99.98
100.77
101,93
102.17
104.59
109,41
110,42
111.30
106,12
106,48
104,13
111,95
106,40
111.66
116,74
108,69
105.83
107,87
109,09
117,59
103,07
106,97
117,72
118,65
108.86
108,49
118,23
106,30
110.20
122,13
102.03
128,93
111.10
107,15
107.10
126,10
100.65
109,38
106,07
106,34
107.75
106,39
105.15
119,80
107.66
105.61
115j54
103,67
105,32
113,22
112.09
104.76
106.88
103,99
103.95
138,53
111,12
103.98
102.04
14724
106.96
101,72
104,02
156,83
154.88
157,79
162.42
103,41
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
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Government
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Government
Government
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(continued on next page)
41
Provincial
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Provincial
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New Brunswick
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New Brunswick
New Brunsvwck
Ontario
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British Columbia
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Quebec
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Ontario
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New Brunswick
New Brunswick
New Brunswick
Ontario
New Brunswick
British Columbia
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Ontario
British Columbia
British Columbia
British Columbia
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Ontario
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Quebec
New Brunswick
Ontario
New Brunswick
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New Brunswick
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New Brunswick
Quebec
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New Brunswick
British Columbia
Quebec
New Brunswick
Ontario
New Brunswick
British Columbia
Ontario
Quebec
British Columbia
Ontario
New Brunsvwck
British Columbia
British Columbia
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British Columbia
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New Bnjnsvwck
Quebec
British Columbia
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Quebec
AAA
AA
A
AA
A
A
AA
AA
AA
AA
AAA
AAA
A
A
A
AA
AA
AA
AA
AA
AA
AA
AAA
A
AA
AAA
AAA
AAA
AAA
AA
AA
A
AA
AA
AA
AA
A
AA
AA
AA
A
AA
AA
AAA
A
AA
AA
AA
AAA
AA
A
AAA
AA
AA
AAA
AAA
A
AAA
AA
AA
AA
AA
A
AAA
AA
AA
A
1107090X7
110709FM9
683078DW6
448814DW0
683078FQ7
110709BK7
683234HC5
448814DZ3
683078FV6
110709BL5
748148NX7
748148PA5
1107090KS
683234HM3
683234HL5
110709DP4
683078GB9
683078GD5
74814ZDE0
683234JA7
683234JQ2
748148PZ0
683234JT6
683078GG8
683234KN7
683234KG2
110709EJ7
11070ZCC6
683234LN6
642866ET0
683234U 5
110709EK4
746148QJ5
683234NM6
110709EX6
448814GY3
748148QT3
683234SL3
642866FR3
44889ZBF2
44889ZCM6
683234VR6
110709FJ6
642866PW2
748148RL9
642866FZ5
683234YD4
110709FL1
74814ZDK6
683234ZP6
683234MM7
642866GA9
44889ZCN4
110709FY3
683234B98
642869AA9
74814ZEF6
683234PS1
448814HZ9
448814JA2
British Columbia
British Columbia
Ontario Hydro
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Ontario Hydro
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Ontario
Hydro-Quebec
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Quebec
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British Columbia
Ontario Hydro
Ontario Hydro
Quebec
Ontario
Ontario
Quebec
Ontario
Ontario Hydro
Ontario
Ontario
British Columbia
British Columbia
Ontario
New Brunswick
Ontario
British Columbia
Quebec
Ontario
British Columbia
Hydro-Quebec
Quebec
Ontario
New Brunswick
Hydro-Quebec
Hydro-Quebec
Ontario
British Columbia
New Brunswick
Quebec
New Brunswick
Ontario
British Columbia
Quebec
Ontario
Ontario
New Brunswick
Hydro-Quebec
British Columbia
Ontario
New Brunswick
Quebec
Ontario
HvdroOuabec
Hydro-Quebec
9.95
4,80
10.75
10,50
10,13
9,50
9.50
9,63
8,90
8,75
9.38
9.50
8.00
8,10
7,50
9,00
8,50
9,00
5.35
9,50
8.50
8.50
8,00
8,25
8,00
7,60
6.15
5.62
6.25
5.65
6.50
5,70
6.00
6.20
6.35
6,00
6.25
5,85
5,50
6,50
6,50
5,60
5.40
4,65
5,75
4,55
4,70
4.70
5.00
4,60
5.65
4,80
6.00
4,95
4.65
4.80
5.00
6,20
5.00
5,00
2021-05-15
2021-06-15
2021-08-06
2021-10-15
2021-10-15
2022-06-09
2022-07-13
2022-07-15
2022-08-18
2022-08-19
2023-01-16
2023-03-30
2023-09-08
2023-09-08
2024-02-07
2024-08-23
2025-05-26
2025-05-26
2025-06-01
2025-06-02
2025-12-02
2026-04-01
2026-06-02
2026-06-22
2026-12-02
2027-06-02
2027-11-19
2028-08-17
2028-08-25
2028-12-27
2029-03-08
2029-06-18
2029-10-01
2031-06-02
2031-06-18
2031-08-15
2032-06-01
2033-03-08
2034-01-27
2035-01-16
203502-15
2035-0602
203506-18
203509-26
2036-1201
203703-26
2037-0602
2037-06-18
2038-1201
2039-0602
203907-13
203909-26
204002-15
2040-06-18
2041-0602
20410603
2041-1201
2041-1202
2045-02-15
205002-15
151.79
106.82
158.47
156,24
153.66
150,26
150.27
150,20
144.28
143,97
148.86
150,27
138.81
139,09
132.93
150,37
144.11
149,28
106.77
155,19
145.35
144,09
140,12
142,52
140,61
136,80
121,00
113.88
120,81
113.29
125,01
116,09
117,59
122,25
125.34
117_,84
121,52
117,44
111,44
125,12
125.50
114,69
112.79
99,61
115.42
98,28
101.74
102,75
104,75
1X .57
116.11
102,47
120.92
107.33
101.72
102,53
105,37
125,80
105.62
106.11
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Govern mem
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Government
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provinciai
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
Provincial
British Columbia
British Columbia
Ontario
Quebec
Ontario
British Columbia
Ontario
Quebec
Ontario
British Columbia
Quebec
Quebec
British Columbia
Ontario
Ontario
British Columbia
Ontario
Ontario
Quebec
Ontario
Ontario
Quebec
Ontario
Ontario
Ontario
Ontario
British Columbia
British Columbia
Ontario
New Brunswick
Ontario
British Columbia
Quebec
Ontario
British Columbia
Quebec
Quebec
Ontario
New Brunswick
Quebec
Quebec
Ontario
British Columbia
New Brunswick
Quebec
New Brunswick
Ontario
British Columbia
Quebec
Ontario
Ontario
New Brunswick
Quebec
British Columbia
Ontario
New Brunswick
Quebec
Ontario
Quebec
Quebec
B i f e l l li M
AAA
AAA
AA
A
AA
AAA
AA
A
AA
AAA
A
A
AAA
AA
AA
AAA
AA
AA
A
AA
AA
A
AA
AA
AA
AA
AAA
AAA
AA
AA
AA
AAA
A
AA
AAA
A
A
AA
AA
A
A
AA
AAA
AA
A
AA
AA
AAA
A
AA
AA
AA
A
AAA
AA
AA
A
AA
A
A
F ig . 2 0 : Filtered bond universe a t tim e zero o f the backtesting w indow.
42
C .2
L ia b ilitie s
These are all the liability stream over the horizon. However, note th a t in our twelve months
backtesting window, we only observe the first four liability stream . Note however the two large
bullets in 2014. which im pact the liability present value in our analysis.
LiabOl
Liab02
Liab03
Uab04
Liab05
Uab06
Liab07
Liab08
Liab09
LiablO
Uab11
Uab12
Liab13
Liab14
Liabl5
Liab16
Liab17
Uab18
Liab19
Uab20
Liab21
Liab22
2010-09-30
2010-12-31
2011-03-31
2011-06-30
2011-09-30
2011-12-31
2012-03-31
2012-0630
2012-09-30
2012-1231
2013-03-31
201306-30
201309-30.
2013-1231
2014-03-31
2014-06-30
2014-09-30
2014-12-31
2015-03-31
2015-06-30
2015-09-30
2015-1231
F ig . 21
43
5675 000
4 675 000
3605 000
3 605 000
3605 000
3 605 000
2 756 000
2 756 000
2756 000
2 756 000
2616 000
2616 000
2616 000
2 616 000
7112 000
2 072 000
2072 000
7112 000
625 000
625 000
625 000
625 000
C .3
S e t t le m e n t d a t e s
In the following figure, the first optim ization date consist of the portfolio construction. The optim al
portfolio is then evaluated each week after until the beginning of the next month, where we perform
a new optimization to rebalance the portfolio.
Optimization #1
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #2)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #3)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #4)
Evaluation
Evaluation
Evaluation
Evaluation
R ebalancing (Optim. #S)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #S)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #6)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #7)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #8)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #9)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. *10)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
Rebalancing (Optim. #11)
Evaluation
Evaluation
Evaluation
Evaluation
Evaluation
2010-07-01
2010-07-05
20 1 0 0 7 -1 2
2 0 1007-19
2 0 1007-26
20100802
20100802
2 0 1 0 -0 8 0 9
2 0 1008-16
2010-08-23
2010-08-30
2010-09-01
20100801
2010-09-06
2010-09-13
2010-09-20
2010-09-27
2 0 1 0 -1 0 0 4
2 0 1 0 -1 0 0 4
2010-10-11
2010-10-18
2010-10-25
2010-11-01
2018-1101
2010-11-08
2010-11-15
2010-11-22
2010-11-29
2010-1201
2010-1201
2010-12-06
2010-12-13
2010-12-20
2010-12-28
2 0 1 1-0103
20110103
2 0 1 1 0 1 -1 0
2 0 1 1 0 1 -1 7
2 0 1 1 0 1 -2 4
201101-31
20110201
20110201
20110207
20 1 1 0 2 -1 4
201102-21
20 1 1 0 2 -2 8
2011-0301
20110301
20 1 1 -0 3 0 7
20 1 1 0 3 -1 4
2011-03-21
2011-03-28
20110401
2011-0401
20 1 1-0404
2011-04-11
2011-04-18
2011-04-25
20 1 1 0 5 -0 2
20110502
2011-05-09
2011-05-16
2011-05-23
2011-05-30
2011-0601
2011-0601
2011-06-06
2011-06-13
2011-06-20
2011-06-27
20110701
F ig . 22
44
PF1
PF1
PF1
PF1
PF1
PF1
PF2
PF2
PF2
PF2
PF2
PF2
PF3
PF3
PF3
PF3
PF3
PF3
PF4
PF4
PF4
PF4
PF4
PF5
PF5
PF5
PF5
PF5
PF5
PF6
PF6
PF6
PF6
PF6
PF6
PF7
PF7
PF7
PF7
PF7
PF7
PF8
PF8
PF 8
PF8
PF8
PF8
PF9
PF9
PF9
PF9
PF9
PF9
PF10
PF10
PF10
PF10
PF10
PF10
PF11
PF11
PF11
PF11
PF11
PF11
PF12
PF12
PF12
PF12
PF12
PF12
D
O ther stoch astic program m ing m odels
We want to formulate a two stage optim ization problem th a t minimizes the cost of the portfolio
and its expected rebalancing costs. Note th a t we are going forward through time in these models
compared to stochastic dynam ic optimization. At tim e to (’today’), the first stage finds optim al
positions u(to) = (un (to)). with un G R+. for each bond n G fl(fo) th a t minimizes the actual cost
of immunization in addition to the expected cost of adjusting the solution at second stage at time
ti- At to- we know the current yield curve. r(to,T), and we have a set of determ inistic constraints.
At second stage, we have the simulated yield curve r (ti,T ;iv ) for each scenario. In th e following,
the second stage problem will be denoted Qtx- It involves the rebalancing cost a t time t,\ > to,
which includes control variables y{t\\uj) = (j/n (h ;<*>)), with yn G R, for each n G II(ii;u ;) to adjust
optim al positions from time to to meet the stochastic constraints a t time t\. T he second stage
optim ization is performed for each oj G fh Suppose here we have fI = {w*, : k = 1,2, ...,5 } . Note
th a t {yn } can be either positive or negative, respectively in the case of buying new positions or
selling positions. We however avoid short selling.
D .l
D .1 .1
M o m e n t m a tc h in g m e t h o d
S to ch a stic p rogram m in g (S P ) m od el
We can formulate the two stage moment matching optim ization problem as follows:
min
{ u ( t o ) ,y ( t i ; u n ) ,...,y ( t i ; u ; s )}
s.t.
{ P ( io ) - u ,(io) + E [Q t l (u (t 0) , y ( t i ; wi ) , . . . , y ( t 1,ws))|w]}
A P V u{t° \ t 0) > L P V ( t 0)
= Jk(to), k = 2m, m = 1 ,2 ,3 ,...
4u(to)(io) > Jfe(to), k = 2 m + l , m = 1,2, 3,...
where the second stage, for each w G fi, is expressed as
Qh
= min
y ( t i; w )
s.t.
y ^ \ y n (ti\uj)\an
^
A P P y(ti;^ ( f i ) + A P F u(to)(fi) > LP V (tr,uj)
+ I ^ to\ t i ; u ) = J k (ti;u ), k = 2m, m = 1 ,2 ,3 ,...
I p tl,ui\ t i ; u ) + I ^ to\ t i ; u ) > J k (ti;u ), k = 2m + l, m = 1 ,2 ,3 ,...
u (*o) + y(<i;w) > 0
where the term s A P V U and A P V y mean th a t we are com puting the asset PV respectively with the
positions {un } or the control positions { y n }- Equation (6 ) show how these quantities are calculated.
Similarly, the term s
and / y mean th a t we are com puting the fc-th moment respectively with
positions {u„} or the controls positions { y n }- See Equation (7) show how these quantities are
calculated. P (to) is the m arket price of each position as defined in the ’Theoretical framework’
section. a n is the bid-ask spread cost for each position. The fourth constraint is to avoid short
selling at second stage.
The algorithm for this problem is in fact two embedded optim ization processes. T he first stage
and the second stage (for each scenario). Thus, it explains why this algorithm is dem anding lot of
45
com putation performances. One should use stochastic programm ing an algorithm like ’determ inistic
equivalent program s’ with a specialized stochastic optim ization solver.
D .1 .2
C h a n c e - c o n s tr a in e d p ro g r a m m in g ( C C P ) m o d e l
The param eters for this model are the same as th e SP model b u t we need to introduce a reliability
level, v. with 0 < v < 1. which is the probability of meting constraints. We also define 7 , a desired
fraction of the total liabilities value with respect to the portfolio value. Thus, if 7 = 1. we are
trying to meet or overperform all the liabilities value; if 0 < 7 < 1 . we force the portfolio value to
be equal or greater than a fraction 7 of the liabilities value. For convenience, we will fix 7 = 1 in
the following.
In the CC P model, the two stage chance-constrained problem is formulated as follows:
min
{u(«o),y(ti;wi),...,y(ti;us )}
s.t.
{P(<o) •u'(io) + E [Qtl (u(t0), y ( i x; u q ),..., y ( tu w s ) ) M }
P { A P V u{to)(t0) > L P V { t 0)\uj} > 1/
P { / " (to)(t0) = J fc(to)|w} > *7 k — 2m, m = 1, 2, 3,...
p {jfc(t° V o ) > Jk(to)\u} > V, k = 2m. + 1 , m = 1 ,2 ,3 ,...
where the second stage, for each ui € fh is expressed as
Qti = min
y(ti;w)
s.t.
n
p { A P T /y(‘i;w)(fi) + A P I / u(io)(ti) > L P V (ti;w )|w }
>V
W ^ l {tl'u3){tl -,u) + l " it0\ t l -uj) = J fc(ti ; u;)|u>} > v , k = 2m, m = 1 ,2 ,3 ,...
P
> Jfc(ti;w )|cj| > v , k = 2 m + l , m = 1 , 2 ,3 ,...
u(to) + y(<i;w) > 0
The notations in this optim ization problem are th e same as in the previous model.
D .2
Cash flow m atching m ethod
For this algorithm , to trace the portfolio’s cash flow shortage or surplus in the optim ization process,
we define two variables, 0 ( t) € M+ and U (t) € M+, respectively a surplus variable (overperforming)
and a shortage variable (underperforming) between the portfolio’s cash flow and the liabilities’
stream at time t. These variables are respectively defined as:
0 ( t) = m a x { s m of net portfolio cash flow over liabilities fo r periods beyond t, 0 }
U(t) = —min{suTO of net portfolio cash flow over liabilities fo r periods beyond t, 0}.
Thus, for b etter immunization, we will add a fraction (penalty) r] > 1 of the sum of these variables
in the optim ization objective function, since we want to include in the objective minimization
function the net deviation of the portfolio’s cash flow against the liability stream .
46
D .2 .1
S P m odel
In this model, the first stage problem of the cash flow matching algorithm is w ritten as follows:
mm
( u ( t 0) . y ( t i ; u ; i )
y ( t i ; w s )}
+ E[ Qt , ( u( t 0) , y ( < i ; w i ) ,
s.t.
where the second stage, for each u> €
Q t , = min
y (tuu)
ws ))M
j
A P V u{lo){t0) > L P V ( t 0)
is expressed as
1 r7[0y{tl)(fi;u;) + t/ y(tl)(ti;w )] +
^
^
\yn(h;<jj)\an 1
J
A P V y ^ ' ^ i h ) + A P V u(to\ t \ ) > LPV{t\\u})
s.t.
u (to)
+ y(*i;w) > 0
where the term s 0 U and 0 y mean th a t we are computing the to tal cash surplus respectively with
the positions u or the control positions y. Similarly, the term s Uu and Uy mean th a t we are
com puting the total shortage respectively with the positions u or the control positions y.
D .2 .2
C C P m od el
T he first stage problem of the cash flow matching algorithm is w ritten as follows:
min
{u(f0),y(ti;u>i),...,y(ti;u;s )}
L [ O ^ H t 0) + U ^ H t Q)] + P ( t o ) - u \ t 0)
^
+ E [Q tl(u(* 0), y(t i ; wi ) , ...,y ( tu ujs))\u}
s.t.
j
P { A P V u{to)(t0) > L P V (io)|w} > v
where the second stage, for each ui G fl, is expressed as
Qti = min
y(h-,u>)
s.t.
i
{
77[Oy(tl)( t 1;aj) + Uy(ti)(ti;u>)} + V ] |r/„(*i; w )|an 1
^
J
P jA P V y(ti;w)(fi) + A P V ult0\ t i ) > L P V ( ti\u j) \u } > v
u (*o) + y ( t \ ; u ) > 0
47
D .2 .3
In teg ra ted ch a n ce-co n stra in ed p ro g ra m m in g (IC C P ) m odel
F irst this model, we add a constraint involving a maximum expected cash flow shortage like a
’CVaR’. For this, we define a new param eter A. 0 < A < 1. which is an upper limit of the al­
lowed maximum expected cash flow shortage. Thus, the two stage problem formulated as follows,
beginning with the first stage:
111111
{ u ( t o ) , y ( t i ; w i y ( l i ;w s)}
+ E [Qt,( u ( t0),y(<i;wi), ...,y(*i,cjs ))M
s.t.
A P V u^ { t 0) > L P V ( t 0)
where the second stage, for each u € fi. is expressed as
Q tx = min
y(ti;w)
s.t.
\r][Oy{tl)(ti-,uj) + [ /y(tl)(£i;u;)] + V
[
\yn
^
A P V ^ ' ^ i h ) + L P V u{t0\ t i ) > J 0(ti;w)
u (t0) + y(<i;w) > o
with the additional expected shortfall constraint
48