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 haveLoss = 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

-60

-40

-20

0

20

40

60

80

0 20 40 60 80 100 120 140 160 180 200

Debito

Risk Free

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 of collateral = 84

Risk-free discount factor = 1

Face value of debt = 72

Value collateral Equity Debt

100

20

28

0

72

20

84

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. “If a bank has a pool residential mortgages with 100 face value, it would approach the FDIC.

2. The FDIC would determine that they would be willing to leverage the pool at a 6-to 1 debt-to-equity ratio

3. 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.

4. Of this $84 purchase price, the FDIC would provide guarantees for $72 of financing, leaving $12 of equity

5. 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

Bid = 96.6

Risk-free discount factor = 1

Face value of debt = 83

Value collateral Equity Debt

100

20

17

0

83

20

84

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

tT

tTrBtVd

tT

tTrBtVd

dBNedNtVf

V

V

V

V

tTr

)2/(/ln

)2/(/ln

)(

2

2

2

1

21

Fair value of debt

• The fair value of debt is recovered as

• The expected loss is a put option

21

21)(

,

dBNedNtV

dBNedNtVtV

ftVTtF

tTr

tTr

21 dBNedNEL tTr tV

DP and LGD in Merton model

• The fair value of debt can be written in a way to isolate default probability and recovery rate

11

tV11

tV,

2

12

21

RRDpBe

dN

dN

BedNBe

dBNedNTtF

tTr

tTrtTr

tTr

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

tVdN

dBNedNtVtf

f

tTr

1

21)(

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)

tT

tTBtVDD

V

V

)2/(/ln 2

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

B

L

dN

dN

1

1 1)(

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.