advanced finance 2006-2007 risky debt (2)
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Advanced Finance 2006-2007 Risky debt (2). Professor André Farber Solvay Business School Université Libre de Bruxelles. Toward Black Scholes formulas. Value. Increase the number to time steps for a fixed maturity. The probability distribution of the firm value at maturity is lognormal. - PowerPoint PPT PresentationTRANSCRIPT
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
Advanced Finance 2007 Risky debt - Merton |3April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |4April 22, 2023
Black-Scholes using Excel
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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)
Advanced Finance 2007 Risky debt - Merton |5April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |6April 22, 2023
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
+
Advanced Finance 2007 Risky debt - Merton |7April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |8April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |9April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |10April 22, 2023
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%
Advanced Finance 2007 Risky debt - Merton |11April 22, 2023
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)
Advanced Finance 2007 Risky debt - Merton |12April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |13April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |14April 22, 2023
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
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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%
Advanced Finance 2007 Risky debt - Merton |15April 22, 2023
Agency costs
• Stockholders and bondholders have conflicting interests• Stockholders might pursue self-interest at the expense of creditors
– Risk shifting– Underinvestment– Milking the property
Advanced Finance 2007 Risky debt - Merton |16April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |17April 22, 2023
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
Advanced Finance 2007 Risky debt - Merton |18April 22, 2023
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