Transcript Document

Structural Models
Advanced Methods of Risk Management
Umberto Cherubini
Learning Objectives
• In this lecture you will learn
1. Credit risk as a short position in a put
option
2. The seniority structure of a firm as an
option spread
3. How to hedge corporate bonds with
common stock
Credit risk models
• Structural Models
– Credit risk is evaluated from a model of the
structure of the financial structure of the firm and
the dynamics of its business activity.
– The credit risk premium is determined using option
theory.
• Reduced form (intensity based) models
– Credit risk is modelled based on statistical
hypotheses of the default probability and the
recovery rate.
– The credit risk premium is determined using term
structure theory.
Structural models: Merton model
• In Merton (1974), the seminal paper of the
structural approach to credit riskl, the value of
assets of the obligor jointly determines
– Default probability
– The recovery rate in the event of default
– The fair value of equity and debt
• The value of corporate liabilities, depending
on their degree of seniority, is determined
using option theory.
Merton model
• In Merton model debt is assumed to be a
zero coupon bond (that is interest and
principal are paid in one unique cash flow at
maturity), and default is revealed at maturity.
• At maturity T, the payoff of defaultable debt is:
min(B,V(T)).V(T) is the value of the firm (or in
general the collateral of the obligor), at
maturity of the debt.
Equity is a call option
• In Merton model, at expiration T the equity holder
collects the value of the firm, or of the collateral, after
having repaid debt.
• The pay-off of equity capital at T is then
Equity(T) = V(T) – min (B,V(T)) =
= max(V(T) - B, 0)
and it is the payoff of a call written on the collateral
with strike price equal to the face value of debt. The
fair value of equity at time t is then
Equity(t) = Call(V,t;B,T)
Expected loss is a put option
• If we compare the payoff of a default-free investment
and a defaultable investment we find that at time T
we will have
Loss = B – min(B,V(T)) = max(B – V(T), 0)
and the loss will be the payoff of a put option written
on the collateral with strike price equal to the face
value of debt.
• The discounted expected loss (EL) due to credit at
time t is then EL(t) = Put(V,t;B,T) and the fair value of
debt is (v(t,T) = risk free discount factor)
Debt(t) = v(t,T)B – EL(t) = v(t,T)B – Put (V,t;B,T)
Min(B,V(T)) = B – max(B – V(T), 0)
80
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40
20
Debito
0
Risk Free
0
-20
-40
-60
-80
20
40
60
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100
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200
Default Put
Modigliani-Miller
• Adding the fair value of equity and the fair value of
debt we obtain the fair value of the firm, or of the
collateral
• V(t) = Equity(t) + Debt(t) =
= Call(V,t;B,T) + v(t,T)B – Put(V,t;B,T)
• Notice. Both equity and debt are long in the value of
the collateral (i.e. they both benefit from and increase
in V). But equity is long in the volatility of the
collateral and debt is short (i.e. collateral with higher
volatilities increase the value of equity at the expense
of the fair value of debt).
Junior and senior debt
• Assume that debt B is made up by senior debt,
whose face value is S, and junior debt whose face
value is J. Then, B = S + J. We know that:
• Debt(t) = v(t,T)B – Put (V,t;B,T)
• Along the same lines, whe have that
• SeniorDebt(t) = v(t,T)S – Put (V,t;S,T)
• It follows that the value of junior debt is
• Junior Debt(t) = v(t,T)(B – S)
– Put (V,t;B,T) + Put (V,t;S,T)
= v(t,T)J – Put (V,t;B,T) + Put (V,t;S,T)
Junior debt is a call spread
• Expected loss of junior debt is the spread of two put
options. Using put call parity
• v(t,T)B – Put(V,t;B,T) = V(t) – Call(V,t;B,T)
• v(t,T)S – Put(V,t;S,T) = V(t) – Call(V,t;S,T)
we also have
Junior Debt(t) = v(t,T)(B – S)
– Put (V,t;B,T) + Put (V,t;S,T) =
= Call(V,t;S,T) – Call (V,t;B,T)
and the fair value of junior debt is a call spread.
The payoff of junior debt
A binomial example: Obama plan
Value collateral Equity
Debt
100
28
72
20
0
20
84
Value of collateral = 84
Risk-free discount factor = 1
Face value of debt = 72
Fair value calculations
• We compute the risk-neutral probability Q,
corresponding to state high (H):
Q = (V(t)/v(t,T) – V(L))/(V(H) – V(L))
= (84 – 20)/(100 – 20) = 0.8
• Fair value of equity: 0.8 * 28 = 22.4
• Fair value of debt: 0.8*72 + 0.2*20 = 61.6
• Expected loss (default put option)
72 – 61.6 = 10.4
Obama plan
1.
2.
3.
4.
5.
“If a bank has a pool residential mortgages with 100 face
value, it would approach the FDIC.
The FDIC would determine that they would be willing to
leverage the pool at a 6-to 1 debt-to-equity ratio
The pool would then be auctioned by the FDIC, with several
private sector bidders submitting bids. The highest bid from
the private sector – in this example 84 – would be the winner
and would form a Public-Private Investment Fund (PPFI) to
purchase the pool of mortgages.
Of this $84 purchase price, the FDIC would provide
guarantees for $72 of financing, leaving $12 of equity
The Treasury would then provide 50% of the equity funding
required on a side-by-side basis with the investor. In this
example, Treasury would invest approximately $6 with private
investors contributing $6”….” See ppip_fact_sheet.pdf
Obama plan: inconsistency I
• If the fair value of the pool of mortgages is $84, and we
assume a the loss on the pool could be $20, then the
risk-neutral probability of default would be 20%.
• Then, we checked that the fair value of equity woud be
$22.4, compared with a value of cash of $12 paid by the
equity holder.
• This implies a net gain of $10.4 = $22.4 – $12 for the
equity holder, at the expense of FDCI, which would face
an expected loss of the same amount $10.4.
Obama plan: inconsistency II
• Notice that in a typical option pricing problem the
value of the underlying asset is observed on the
market (e.g. options on common, currencies, etc.)
• In Merton model, the underlying asset is not
observed in the market. In the specific Obama
example the market is an auction.
• Notice that in this case, $84 could not emerge as the
optimal bid for the pool of mortgages. Someone
bidding $84 would know that he would pay 12 for
something worth more, so that other would bid more.
A $96.6 bid would break even.
Optimal bid in the Obama plan
Value collateral Equity
100
17
Debt
83
84
Bid = 96.6
Risk-free discount factor = 1
Face value of debt = 83
20
0
20
Overbid = Public EL
• Assuming that the fair value of the pool is $84, we
still have that risk-neutral probability Q,
corresponding to state high (H) is 0.8.
• However, the bid leads to a supply of funds from the
FDIC of $83, and the payment equity of $13.6
• We now compute:
• Fair value of equity: 0.8 * 17 = $13.6
• Fair value of debt: 0.8*72 + 0.2*20 = $70.4
• FDIC Expected Loss (default put option)
83 – 70.4 = $12.6 = 96.6 – 84 = Overbid
A continuous time model
• If the value of collateral follows the dynamics
dV = Vdt + VVdw(t) = (r+V)Vdt + VVdw(t)
the value of equity capital is then given by the Black
and Scholes formula for call options
f  V (t ) N d1   e  r T t  BN d 2 
lnV t  / B   (r   V2 / 2)T  t 
d1 
V T  t
lnV t  / B   (r   V2 / 2)T  t 
d2 
V T  t
Fair value of debt
• The fair value of debt is recovered as
F t , T   V t   f 
 V t   V (t ) N d1   e  r T t  BN d 2 
 V t N  d1   e
 r T t 
BN d 2 
• The expected loss is a put option


EL  V t N  d1   er T t BN d2 
DP and LGD in Merton model
• The fair value of debt can be written in a way
to isolate default probability and recovery rate
F t , T   Vt N  d1   e  r T t  BN d 2 


Vt  N  d1   
e
B 1  N  d 2 1  r T t 

B N  d 2  
 e

 e r T t  B1  Dp1  RR
 r T t 
KMV Model
• In case the stock is quoted in an efficient
market, the value of equity can be used to
back out the probability of default of a firm
• This is done in the KMV model using the price
of the stock and its volatility:
f t   V (t ) N d1   e  r T t  BN d 2 
V t 
 f  N d1 
f
Distance to default (DD)
• Default probability in the Merton model is given by
N(-d2). Since the normal distribution delivers default
probability figures which are much lower than those
in the market, even in the pre-crisis period, KMV
propose to focus on the argument, This measure is
called distance to default (DD)
lnV t  / B   (   V2 / 2)T  t 
DD 
V T  t
Hedging credit risk
• Assume you have a position in L euros of
exposure to an obligor. According to the
structural model the sensitivity of the value of
the exposure debt to a change in the value of
the firm is
dDL(t,T) = dF(t,T)L/B= LN(-d1) dV(t)/B
where DL is discounted value of exposure L.
• Notice that the value of the exposure is long
in the value of the firm (it is a short position in
a put option)
Hedging credit risk
• Assume the equity is made up by n stocks,
and the value of each piece of common stock
is S(t). Then, the sensitivity of the price of the
stock to a change in the value of the firm is
dS(t) = df(t,T)/n= N(d1) dV(t)/n
• Notice that also equity is long in the value of
the firm (in fact it is a long position in a call
option)
Hedging credit risk
• Credit risk can be transferred from one party
to the other using credit derivatives, but in the
end can only be hedged by taking an
offsetting position in the value of the firm. This
is not a traded asset. So, either we hedge
with options on assets correlated with the
value of the firm, or we hedge by using
another derivative written on the value of the
value of the firm…
Delta hedging with equity
• Assume to take a position in  stocks in such
a way as to offset the impact of an
infinitesimal change in the value of of the firm
V(t). We want
dDL(t,T) +  dS(t) = 0
from which we solve
N (d1 )  1 L

n
N d1  B
What may go wrong?
• Assume that the value of the firm becomes
more uncertain (for example because the
manager switches to a riskier business). In
this case the value of equity increases at the
expense of the value of debt.
• In other words:
• Equity and debt agree on the first moment of
the return on investment distribution
• Equity and debt disagree on the second
moment of the distribution.