Lecture 3

c 2015 Juliusz Jablecki: Equity and Fixed Income
Equity and Fixed Income
Juliusz Jabłecki
Banking, Finance and Accounting Dept.
Faculty of Economic Sciences
University of Warsaw
jjablecki@wne.uw.edu.pl
and
Head of Monetary Policy Analysis Team
Economic Institute, National Bank of Poland
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c 2015 Juliusz Jablecki: Equity and Fixed Income
Lecture 3:
Option-theoretic perspective stock
valuation
So far, we have discussed two practitioner models for equity valution. From a conceptual point of view, both were rather shallow
and based on the idea that PV of a stock can be calculated by
discounting the stream of future dividends:
T Dividendi
PV =
i=1 (1 + r)i
X
This lecture introduces a deeper theoretical model that looks at
equity valuation from the perspective of option pricing theory.
This approach was originally suggested by Black and Scholes but
refined substantially by Merton (1974) – hence, we call it “Merton
model”.
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Debtholders have priority in securing their claims (=face value of debt), but do not
participate in the upside of the firm.
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Equityholders have nothing guaranteed, but do participate in the upside of the firm.
c 2015 Juliusz Jablecki: Equity and Fixed Income
Take a look at the diagrams below:
Default: firm value
falls below the
default point
No default: the firm
is worth more than
the value of debt
Financial position
Equityholders
leave the whole value
of the firm to the
bondholders and walk
away with no
obligation
Bondholders
only receive the firm
value and lose some
or all of the principal
pocket the residual
firm value
only collect bond
principal
long call on the firm’s
assets with strike
equal level of debt
long risk-free bond
and short put on the
firm’s assets (credit
protection)
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Scenario
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c 2015 Juliusz Jablecki: Equity and Fixed Income
To add structure to our model, consider the following assumptions.
The Merton model is based on three very simple assumptions:
1. The firm is capitalized with common stock and a zerocoupon bond payable at T .
2. At maturity date, the firm defaults if its value is less than
the face value of the bond.
3. In default (which can happen only at T ), bondholders receive
the entire firm and equityholders get nothing.
Then the value of debt at T is
Debt(VT , T ) = B − max(0, B − VT ) = Be−rT − P ut(B, T )
The value of equity at T is:
Equity(VT , T ) = max(VT − B, 0) = Call(B, T )
But how can we value Debt(VT , T ) and Equity(VT , T )?
We need to specify a process for the value of the firm:
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c 2015 Juliusz Jablecki: Equity and Fixed Income
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c 2015 Juliusz Jablecki: Equity and Fixed Income
Merton’s idea was to think about the value process of the firm just
as we would about any other financial asset: as a combination
of something deterministic (drift) and stochastic (diffusion). We
have a way of formalizing this. Letting ∆V /V be the change in
the firm’s value over some period ∆T we write:
√
∆V
µ∆T + σ ∆T
√
= 

µ∆T − σ
V
∆T





with probability 0.5
with probability 0.5
We already know that CLT guarantees that under the risk-neutral
measure, the limiting distribution is:

log VT = log V0 + r −




2
σ
2



√
T + σ T W, with W ∼ N (0, 1)
We can use these insights to price stocks/bonds qua options. By
the fundamental theorem
Call(B, T ) = Z(t, T )E? (VT − B)+|V0
We know that


log VT |Vt ∼ N log Vt + r −



2
σ
2




(T − t), σ 2(T − t)

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c 2015 Juliusz Jablecki: Equity and Fixed Income
Substitute
Yt = log Vt, then Yt ∼ N (ν, σ 2(T − t)), ν = log Vt +


2
r − σ  (T − t), and all we need to do is calculate the integral:
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ˆ ∞
(ey −B) √
Call(B, T ) = Z(t, T )
log B
1
√
2πσ T − t
(y−ν)2
−
e 2σ2(T −t) dy
Note that:
ˆ ∞
B√
log B
1
√
2
− (y−ν)
2
e
2σ (T −t) dy
= P (y ≥ log B) =
2πσ T − t



y
−
ν
−
log
B
+
ν
log
B
−
ν




P  √
≥ √
= Φ  √
σ T −t
σ T −t
σ T −t

where Φ(·) is the standard normal CDF. Substituting for ν, we
obtain
ˆ ∞
√
log B
B
√
e
2πσ T − t
2
− (y−ν)
2σ 2 (T −t)

!
2
V
σ
t + r −
 (T − t) 
 log


2
B






dy = Φ 


√
σ T −t



Now consider the exponent of the second integral:
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c 2015 Juliusz Jablecki: Equity and Fixed Income
!
(y − ν)2
−1
2
2
2
y− 2
=
y − 2yν + ν − 2yσ (T − t) =
2σ (T − t)
2σ 2(T − t)

!
2
−1
2
 y − (ν + σ (T − t)
×
−
2
2σ (T − t)
2νσ 2(T − t) − σ 4(T − t)2
−
#
Hence,
ˆ ∞
(y−ν)2
−
e 2σ2(T −t) dy
ey
√
√
=
log B 2πσ T − t

2
2
log B − ν − σ (T − t) 
 y − ν − σ (T − t)
2

P


√
σ T −t
√
≥
σ (T −t)
ν+
× e 2


σ T −t
And thus,
ˆ ∞
(y−ν)2
y−
e 2σ2(T −t)
√
log B
√
2πσ T − t


σ2
 log( V t ) + r +

2
B


dy = Φ 

√



(T − t) 
σ T −t




×
× elog Vt+r(T −t)
This leads to the famous Black-Scholes formula for the price of
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c 2015 Juliusz Jablecki: Equity and Fixed Income
a European call
CallB (t, T ) = VtΦ(d1) − BZ(t, T )Φ(d2)
!
Vt
log B + (r + 12 σ 2)(T − t)
√
d1 =
√σ T − t
d2 = d1 − σ T − t
Similarly, we can calculate the price of a put (and hence of debt):
P ut(t, T ) = Z(t, T )BΦ(−d2) − VtΦ(−d1)
To see the relevance of Merton’s perspective consider the following example.
Consider a company on the verge of default characterized by the
following:
• Value of the firm = $ 50 million
• Face value of outstanding debt = $ 80 million
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c 2015 Juliusz Jablecki: Equity and Fixed Income
• Maturity of the debt = 10 years
• Variance in the value of the underlying asset = σ 2 = 0.16
• Riskless rate = r = 10Y Treasury bond rate = 3%
Using the data we obtain d1 = 0.5, d2 = −0.77, Φ(d1) = 0.69,
Φ(d2) = 0.22 and the value of eqiuty= 50 × 0.69 − 80 ×
exp(−0.03 × 10) × 0.22 = $21.4 million. This leads to an important point:
Equity will have value even if the value of the firm falls well below
the face value of the outstanding debt.
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c 2015 Juliusz Jablecki: Equity and Fixed Income
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Source: Damodaran (1999)
Change
Bond
Equity
Explanation
Intuition
Asset value increases
↑
↑
If asset value increases, it
↑ V =⇒ the chance
makes the call more
that the firm will not be
valuable and the put less
able to make its debt
Leverage increases
payments ↓
Increase in
=⇒ V /B ↓ =⇒ d1 , d2 ↓
leverage =⇒ there is
which increases the value
more debt for the firm to
of the short put and
service and make it more
decreases the value of the
likely that it will not be
long call.
able to make these
↑ σV =⇒ the short put
payments.
Higher asset volatility
and long call become
makes it more likely that
more valuable. This
the firm value will move
widens the credit spread
either below the default
and increases the value of
point (lower bond value)
equity.
or far above the debt level
Leverage increases
Asset volatility increases
↓
↓
↓
↑
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(higher equity value).
Let’s see how our theoretical insights square with empirical data.
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The Bloomberg implementation of the Merton model shows nice
alignment with market CDS prices
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Implementation problem: knowing Vt, B, r, T, and σ we can calculate the fair value of a stock, but:
• Vt and σ are unobservable to investors
• B comprises typically different bonds
To estimate Vt and σ we use the fact that a company’s equity
value E and equity volatility σE are observable and can be estimated. Then, Vt and σ are given by the following system of
equations:
E = VtΦ(d1) − DZ(t, T )Φ(d2)
σE E = Φ(d1)σV V
The solution can typically easily be found e.g. using MS Solver.
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The link between σE and σV (finance geeks only!)
To express σV V as a function of observable variables we use a
formal mathematical result called Ito’s lemma. Specifically, the
assumptions of the Merton model imply that the dynamics of V
and E is given by the following SDEs (omitting the drift terms):
= ... + V σV dW
(1)
dE = ... + EσE dU
(2)
dV
where U, W are Wiener processes. Since we also know that E is
a call option on V , i.e. E = Call(V ), we can use Ito’s lemma
to obtain that the martingale part of the dynamics of E should
be
dE = ... + Φ(d1)V σdW
But since we know from eq. (2) that dE = ... + EσE dU , this
implies that EσE = Φ(d1)V σ, as desired.
Another problem is the maturity adjustment of the company’s
debt structure:
• Most firms have more than one debt issue on their books,
and much of the debt comes with coupons
• These multiple bonds issues and coupon payments have to
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be compressed into one measure (equivalent zerocoupon
bond)
We tackle this by creating a synthetic ZC bond:
• estimate the duration of each debt issue and calculate a
face-value-weighted average of the durations of the different
issues
• this value-weighted duration is then used as a measure of
the time to expiration of the option
• the face value of debt has to include all of the principal
outstanding on the debt plus expected coupons
Consider the following example
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100
120
130
140
+5×
+6×
+ 10 ×
= 6.5
490
490
490
490
KZC = 100 + 120 + 130 + 140 = 490
TZC = 4 ×
Unfortunately, this sort of maturity adjustment causes a pricing
distortion
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• The default probability of the liability portfolio is entirely
concentrated on the maturity date of the synthetic ZC bond
instead of being spread among all the actual servicing dates
• Misspecification of the debt servicing =⇒ reduction of
credit risk & overestimation of equity value
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Case study: Olivetti has a face value of debt of EUR 13.14 bn with a weighted
average maturity of 5Y, book value of assets of EUR 22.69 bn and thus book value of
equity of EUR 9.55 bn. Using the process above and observing Olivetti’s equity market
capitalization as EUR 9.20 bn and equity volatility of 42% implies an asset value of EUR
20 billion (hence implied debt value of EUR 10.80 billion) and asset volatility of 21% in
the Merton model.
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Having estimated Olivetti’s asset value and volatility, we can examine changes in the equity volatility due to changes in the firm’s
capital structure – “leverage effect”. For our modeling, we assume that small changes in the share price leave the asset volatility constant, and use the formula above to calculate the equity
volatility for changes in the share price with constant V and σV .
Since we can treat σE as the price of an option on Olivetti stock,
we have thus produced a simple option-pricing calculator. The
usefulness of being able to estimate skew from capital structure
comes in the case of pricing out-of-the -money (OTM) options
on stocks where the options market is illiquid or nonexistent.
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Exam-like problems
1. Find a formula for the risk-neutral probability of default in
the Merton model. Discuss how relevant it can be to reallife applications, given that it is calculated under risk-neutral
(rather than objective) probability measure.
2. Build a spreadsheet to empirically illustrate:
(a) how asset value paths change with underlying volatility
(b) the dependence of equity price on asset volatility
3. The value of a company’s equity is $3 million, and the
volatility of the equity is 80%. The debt that will have
to be paid in 1Y is $10 million. The risk free rate is 5%.
Find the value of the company’s assets, equity and the 1Y
probability of default.
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Homework: Alcoa valuation exercise
Alcoa Inc. is an American producer of aluminum. Your task
will be to value Alcoa using the dividend discount model and (as
a robustness check) the Merton model. A full valuation sheet
should include:
1. A brief review of the company (what it does, what its balance
sheet looks like, its recent history) and its market prospects;
2. A review of earnings estimates coupled by assumptions on
growth stages, discount rates etc.;
3. An estimation of market value of assets and asset volatilities;
4. A discussion and comparison of model results.
Please submit your solutions via email no later than May 5th
(late submission will not be considered).
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