advanced finance 2006-2007 risky debt (2)

Advanced Finance2006-2007Risky debt (2)

Professor André FarberSolvay Business SchoolUniversité Libre de Bruxelles

Advanced Finance 2007 Risky debt - Merton |2April 22, 2023

Toward Black Scholes formulas

Increase the number to time steps for a fixed maturity

The probability distribution of the firm value at maturity is lognormal

Time

Value

Today

Bankruptcy

Maturity


Black-Scholes: Review

• European call option: C = S N(d1) – PV(X) N(d2)

• Put-Call Parity: P = C – S + PV(X)• European put option: P = + S [N(d1)-1] + PV(X)[1-N(d2)]

• P = - S N(-d1) +PV(X) N(-d2)

Delta of call option Risk-neutral probability of exercising the option = Proba(ST>X)

Delta of put option Risk-neutral probability of exercising the option = Proba(ST<X)

(Remember: 1-N(x) = N(-x))

TTXPV

S

d

5.)

)(ln(

1 TTXPV

S

d

5.)

)(ln(

2


Black-Scholes using Excel

23456789

10111213141516171819202122232425

A B C D EData Variable Comments and formulas

Stock price S 100.00Strike price Strike 70.00Maturity T 2Interest rate rf 4.88% with continuous compoundingVolatility Sigma 40.00%

Intermediate resultsPV(Strike price) PVStrike 63.49 D10. =Strike*EXP(-rf*T)ln(S/PV(Strike)) 45.43% D11. =LN(S/PVStrike)Sigma*t0.5 AdjSigma 56.57% D12. =Sigma*SQRT(T)Distance to exercice DTE 0.803 D13. =LN(S/PVStrike)/AdjSigmad1 1.0859 D14. =DTE+0.5*AdjSigmad2 0.5202 D15. =DTE-0.5*AdjSigma

CallCall 41.77 D18. =S*NORMSDIST(D14)-PVStrike*NORMSDIST(D15)Delta 0.86 D19. =NORMSDIST(D14)Proba in-the-money 0.30 D20. =1-NORMSDIST(D15)

PutPut 5.26 D23. =-S*NORMSDIST(-D14)+D10*NORMSDIST(-D15)Delta 0.14 D24. =NORMSDIST(-D14)Proba in-the-money 0.70 D25. =1-NORMSDIST(-D15)


Merton Model: example

DataMarket value unlevered firm €100,000Risk-free interest rate (an.comp): 5%Beta asset 1Market risk premium 6%Volatility unlevered 40%

Company issues 2-year zero-couponFace value = €70,000Proceed used to buy back shares

Using Black-Scholes formulaPrice of underling asset 100,000Exercise price 70,000Volatility 0.40Years to maturity 2Interest rate 5%

Value of call option 41,772Value of put option (using put-call parity) C+PV(ExPrice)-Sprice 5,264

Details of calculation:PV(ExPrice) = 70,000/(1.05)²= 63,492log[Price/PV(ExPrice)] = log(100,000/63,492) = 0.4543√t = 0.40 √ 2 = 0.5657

d1 = log[Price/PV(ExPrice)]/ √ + 0.5 √ t = 1.086

d2 = d1 - √ t = 1.086 - 0.5657 = 0.520

N(d1) = 0.861

N(d2) = 0.699

C = N(d1) Price - N(d2) PV(ExPrice)= 0.861 × 100,000 - 0.699 × 63,492= 41,772


Valuing the risky debt

• Market value of risky debt = Risk-free debt – Put Option

D = e-rT F – {– V[1 – N(d1)] + e-rTF [1 – N(d2)]}

• Rearrange:D = e-rT F N(d2) + V [1 – N(d1)]

)](1[)(1)(1 )( 2

2

12 dN

dNdNVdNFeD rT

Value of risk-free

debt

Probability of no default

Probability of default× ×

Discounted expected recovery

given default

+


Example (continued)

D = V – E = 100,000 – 41,772 = 58,228

D = e-rT F – Put = 63,492 – 5,264 = 58,228

228,583015.0031,466985.0492,63

)](1[)(1)(1 )( 2

2

12

dNdNdNVdNFeD rT

031,466985.018612.01000,100

)(1)(1

2

1

dNdNV


Expected amount of recovery

• We want to prove: E[VT|VT < F] = V erT[1 – N(d1)]/[1 – N(d2)]• Recovery if default = VT

• Expected recovery given default = E[VT|VT < F] (mean of truncated lognormal distribution)

• The value of the put option:• P = -V N(-d1) + e-rT F N(-d2)

• can be written as• P = e-rT N(-d2)[- V erT N(-d1)/N(-d2) + F]

• But, given default: VT = F – Put

• So: E[VT|VT < F]=F - [- V erT N(-d1)/N(-d2) + F] = V erT N(-d1)/N(-d2)

Discount factor

Probability of default

Expected value of put given

F

F

Default

Put

Recovery

VT


Another presentation

Discount factor

Face Value

Probability of default

Expected loss given default

Loss if no recovery

Expected Amount of recovery given default

])(1)(1[)](1[

2

12 dN

dNVeFdNFeD rTrT

]749,50000,70[3015.0000,1009070.0 D


Example using Black-Scholes

DataMarket value unlevered company € 100,000Debt = 2-year zero coupon Face value € 60,000

Risk-free interest rate 5%Volatility unlevered company 30%

Using Black-Scholes formula

Market value unlevered company € 100,000Market value of equity € 46,626Market value of debt € 53,374

Discount factor 0.9070N(d1) 0.9501N(d2) 0.8891

Using Black-Scholes formula

Value of risk-free debt € 60,000 x 0.9070 = 54,422

Probability of defaultN(-d2) = 1-N(d2) = 0.1109

Expected recovery given defaultV erT N(-d1)/N(-d2) = (100,000 / 0.9070) (0.05/0.11)= 49,585

Expected recovery rate | default= 49,585 / 60,000 = 82.64%


Calculating borrowing cost

Initial situation

Balance sheet (market value)Assets 100,000 Equity 100,000

Note: in this model, market value of company doesn’t change (Modigliani Miller 1958)

Final situation after: issue of zero-coupon & shares buy back

Balance sheet (market value)Assets 100,000 Equity 41,772

Debt 58,228

Yield to maturity on debt y:D = FaceValue/(1+y)²58,228 = 60,000/(1+y)²

y = 9.64%Spread = 364 basis points (bp)


Determinant of the spreads

0

200

400

600

800

1000

1200

1400

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Quasi debt

Spre

ad

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Volatility of the firm

Spre

ad

0

500

1000

1500

2000

2500

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Maturity

d<1d>1

Quasi debt PV(F)/V Volatility

Maturity


Maturity and spread

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Maturity

Spre

ad

))(1)(ln(112 dN

ddN

Ts

Proba of no default - Delta of put option


Inside the relationship between spread and maturity

Delta of put option

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

19.0

20.0

21.0

22.0

23.0

24.0

25.0

26.0

27.0

28.0

29.0

30.0

Maturity

N(-d

1) D

elta

of p

ut o

ptio

n

d=0.6d=1.4

Probability of bankruptcy

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

11.0

12.0

13.0

14.0

15.0

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20.0

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23.0

24.0

25.0

26.0

27.0

28.0

29.0

30.0

MaturityPr

oba

of b

ankr

uptc

y

d=0.6d=1.4

Probability of bankruptcy

d = 0.6 d = 1.4

T = 1 0.14 0.85

T = 10 0.59 0.82

Delta of put option

d = 0.6 d = 1.4

T = 1 -0.07 -0.74

T = 10 -0.15 -0.37

Spread (σ = 40%)

d = 0.6 d = 1.4

T = 1 2.46% 39.01%

T = 10 4.16% 8.22%


Agency costs

• Stockholders and bondholders have conflicting interests• Stockholders might pursue self-interest at the expense of creditors

– Risk shifting– Underinvestment– Milking the property


Risk shifting

• The value of a call option is an increasing function of the value of the underlying asset

• By increasing the risk, the stockholders might fool the existing bondholders by increasing the value of their stocks at the expense of the value of the bonds

• Example (V = 100,000 – F = 60,000 – T = 2 years – r = 5%)Volatility Equity Debt30% 46,626 53,37440% 48,506 51,494+1,880 -1,880


Underinvestment

• Levered company might decide not to undertake projects with positive NPV if financed with equity.

• Example: F = 100,000, T = 5 years, r = 5%, σ = 30%V = 100,000 E = 35,958 D = 64,042

• Investment project: Investment 8,000 & NPV = 2,000∆V = I + NPV

V = 110,000 E = 43,780 D = 66,220∆ V = 10,000 ∆E = 7,822 ∆D = 2,178

• Shareholders loose if project all-equity financed:• Invest 8,000• ∆E 7,822

Loss = 178


Milking the property

• Suppose now that the shareholders decide to pay themselves a special dividend.

• Example: F = 100,000, T = 5 years, r = 5%, σ = 30%V = 100,000 E = 35,958 D = 64,042

• Dividend = 10,000∆V = - Dividend

V = 90,000 E = 28,600 D = 61,400∆ V = -10,000 ∆E = -7,357 ∆D =- 2,642

• Shareholders gain: • Dividend 10,000• ∆E -7,357

advanced finance 2006-2007 risky debt (2)

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