Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt

Binnenlandse Francqui Leerstoel VUB 2004-2005Options and risky debt

Professor André FarberSolvay Business SchoolUniversité Libre de Bruxelles

VUB 04 Options and risky debt |2April 22, 2023

Today in the Financial Times

• GM bond fall knocks wider markets• GM’s debt downloaded to BBB- (just above junk status)• Stock price: $29 (MarketCap $16.4b)• Debt-per-share: $320 (Total debt $300b)• Cumulative Default Probability 48% (CreditGrade calculation)


Fixed income markets

Companies

Assets Equity

Debt

Investors Banks

Loans EquityDeposits

Corporate bond market

Credit derivatives

Equity, bond and bank markets in EU, US and Japan( 2001 Total assets in €bi and as % of GDP)

75%87%

239%

140%154%

78%122%

58%126%

0

5,000

10,000

15,000

20,000

25,000

Bonds market Equity market Bank Assets

EU15 US Japan

Lannoo, K.and Levin, M. Toward a European Single Market for Financial Services, Presentation, CEPR 2004


Credit risk

• Credit risk exist derives from the possibility for a borrower to default on its obligations to pay interest or to repay the principal amount.

• Two determinants of credit risk:• Probability of default• Loss given default / Recovery rate

• Consequence:• Cost of borrowing > Risk-free rate• Spread = Cost of borrowing – Risk-free rate

(usually expressed in basis points)• Function of a rating

– Internal (for loans)– External: rating agencies (for bonds)


Rating Agencies

• Moody’s (www.moodys.com)• Standard and Poors (www.standardandpoors.com)• Fitch/IBCA (www.fitchibca.com)

• Letter grades to reflect safety of bond issue

S&P AAA AA A BBB BB B CCC D

Moody’s Aaa Aa A Baa Ba B Caa C

Very High Quality

High Quality

Speculative Very Poor

Investment-grades Speculative-grades

http://www.moodys.com/

http://www.standardandpoors.com/

http://www.fitchibca.com/


Spread over Treasury for Industrial Bonds

Reuters Corporate Spreads for IndustrialJanuary 2004

http://bondchannel.bridge.com/publicspreads.cgi?Industrial

AAA AAA AAA AAA AAA AAAAAA

AA AA AA AA AA AA AAA A A A A A ABBB

BBBBBB BBB BBB BBB BBB

BB

BBBB

BB BB BB

BBB

B

BB

BB

B

0

100

200

300

400

500

600

0 5 10 15 20 25 30

Maturity

Spre

ad


Determinants of Bonds Safety

• Key financial ratio used:– Coverage ratio: EBIT/(Interest + lease & sinking fund payments)– Leverage ratio– Liquidity ratios– Profitability ratios– Cash flow-to-debt ratio

• Rating Classes and Median Financial Ratios, 1997-1999Rating Category

Coverage Ratio

Cash Flow to Debt %

Return on Capital %

LT Debt to Capital %

AAA 17.5 55.4 28.2 15.2

AA 10.8 24.6 22.9 26.4

A 6.8 15.6 19.9 32.5

BBB 3.9 6.6 14.0 41.0

BB 2.3 1.9 11.7 55.8

B 1.0 (4.6) 7.2 70.7Source: Bodies, Kane, Marcus 2002 Table 14.3


Standard&Poor’s European Rating Distribution

1985 1990 1995 2000 2002

AAA 20 37 52 49 42

AA 14 59 117 171 185

A 6 14 159 315 350

BBB 0 0 42 141 244

Investment-grade 40 110 370 676 821

BBB 3 2 7 71 103

B 0 1 8 75 81

CCC 0 0 1 12 23

Speculative-grade 3 3 16 158 207

Total 43 113 386 834 1028


Default Rate Calculation

• Incorrect method:– Number defaults/Total number of bonds

• Ignores growth/reduction of bond market over time• Ignores aging effect: takes time to get into trouble

• Correct method: cohort style analysis– Pick up a cohort– Follow it through time


Moody’s:Average cumulative default rates 1920-1999 %

1 2 3 4 5 10 15 20

Aaa 0.00 0.00 0.02 0.09 0.20 1.09 1.89 2.38

Aa 0.08 0.25 0.41 0.61 0.97 3.10 5.61 6.75

A 0.08 0.27 0.60 0.97 1.37 3.61 6.13 7.47

Baa 0.30 0.94 1.73 2.62 3.51 7.92 11.46 13.95

Inv. Grade 0.16 0.49 0.93 1.43 1.97 4.85 7.59 9.24

Ba 1.43 3.45 5.57 7.80 10.04 19.05 25.95 30.82

B 4.48 9.16 13.73 17.56 20.89 31.90 39.17 43.70

Spec. Grade 3.35 6.76 9.98 12.89 15.57 25.31 32.61 37.74

All Corp. 1.33 2.76 4.14 5.44 6.65 11.49 15.35 17.79


Modeling credit risk

• 2 approaches:• Structural models (Black Scholes, Merton, Black & Cox, Leland..)

– Utilize option theory– Diffusion process for the evolution of the firm value– Better at explaining than forecasting

• Reduced form models (Jarrow, Lando & Turnbull, Duffie Singleton)– Assume Poisson process for probability default– Use observe credit spreads to calibrate the parameters– Better for forecasting than explaining


Merton (1974)

• Limited liability: equity viewed as a call option on the company.

E Market value of equity

FFace value

of debt

VMarket value of comany

Bankruptcy

D Market value of debt

FFace value

of debt

VMarket value of comany

F

Loss given default


Using put-call parity

• Market value of firm: V = E + D

• Put-call parity (European options)Stock = Call + PV(Strike) – Put

• In our setting:• V ↔Stock The company is the underlying asset• E↔Call Equity is a call option on the company• F↔Strike The strike price is the face value of the debt

• → D = PV(Strike) – Put• D = Risk-free debt - Put


Merton Model: example using binomial option pricing

492.1 teu 670.1

ud

462.670.0492.1

67.05.11

dudr

p f

Data:Market Value of Unlevered Firm: 100,000Risk-free rate per period: 5%Volatility: 40%

Company issues 1-year zero-couponFace value = 70,000Proceeds used to pay dividend or to buy back shares

f

du

rfppf

f

1)1(

V = 100,000E = 34,854D = 65,146

V = 67,032E = 0D = 67,032

V = 149,182E = 79,182D = 70,000

∆t = 1

Binomial option pricing: reviewUp and down factors:

Risk neutral probability :

1-period valuation formula

05.1032,67538.0000,70462.0

D

05.10538.0000,80462.0

E


Calculating the cost of borrowing

• Spread = Borrowing rate – Risk-free rate• Borrowing rate = Yield to maturity on risky debt• For a zero coupon (using annual compouding):

• In our example:

TyFD

)1(

y

1000,70146,65

y = 7.45%

Spread = 7.45% - 5% = 2.45% (245 basis points)


Decomposing the value of the risky debt

f

d

f rVFp

rFD

1

))(1(1

)1(11

pr

Vp

rFD

f

d

f

146,65538.827,2667,66

538.05.1

032,67000,7005.1000,70

D

In our simplified model:

F: loss given default if no recovery

Vd : recovery if default

F – Vd : loss given default

(1 – p) : risk-neutral probability of default

146,65538.840,63462.0667,66

538.05.1032,67462.0

05.1000,70

D


Weighted Average Cost of Capital

• (1) Start from WACC for unlevered company– As V does not change, WACC is unchanged– Assume that the CAPM holds

WACC = rA = rf + (rM - rf)βA

– Suppose: βA = 1 rM – rf = 6%

WACC = 5%+6%× 1 = 11%

• (2) Use WACC formula for levered company to find rE

VDr

VErr DEA 000,100

146,65000,100854,34%11 DE rr

000,100146,65

000,100854,341 DE V

DVE

DEA


Cost (beta) of equity

• Remember : C = Deltacall × S - B– A call can is as portfolio of the underlying asset combined with borrowing B.

• The fraction invested in the underlying asset is X = (Deltacall × S) / C

• The beta of this portfolio is X βasset

• When analyzing a levered company:– call option = equity– underlying asset = value of company– X = V/E = (1+D/E)

)1(EDDelta

EVDelta AAE

In example:βA = 1DeltaE = 0.96V/E = 2.87βE= 2.77rE = 5% + 6% × 2.77 = 21.59%

dSuSff

Delta du

:Reminder


Cost (beta) of debt

• Remember : D = PV(FaceValue) – Put

• Put = Deltaput × V + B (!! Deltaput is negative: Deltaput=Deltacall – 1)

• So : D = PV(FaceValue) - Deltaput × V - B

• Fraction invested in underlying asset is X = - Deltaput × V/D

• βD = - βA Deltaput V/DIn example:βA = 1DeltaD = 0.04V/D = 1.54βD= 0.06rD = 5% + 6% × 0.09 = 5.33%

Putdudu

D DeltadSuSPutPut

dSuSPutFPutF

Delta

)()(


Multiperiod binomial valuation

V

uV

u²V

u3V

u4V

dV

d²V

udV

u2dV

u3dV

u2d²V

ud3V

d4V

ud²V

d3V

p4

4p3(1 – p)

6p²(1 – p)²

4p (1 – p)3

(1 – p)4

Δt

Risk neutral proba

For European option, (1) At maturity, calculate

- firm values;- equity and debt values- risk neutral probabilities

(2) Calculate the expected values in a neutral world(3) Discount at the risk free rate


Multiperiod binomial valuation: example

Firm issues a 2-year zero-couponFace value = 70,000V = 100,000Int.Rate = 5% (annually compounded)Volatility = 40%Beta Asset = 1

4-step binomial tree Δt = 0.50u = 1.332, d = 0.751rf = 2.47% per period =(1.05)1/2-1p = 0.471

# paths Proba/path Proba E D309,990 1 0.050 0.050 239,990 70,000

233,621176,065 176,065 4 0.056 0.223 106,065 70,000

132,690 132,690100,000 100,000 100,000 6 0.062 0.373 30,000 70,000

75,364 75,36456,797 56,797 4 0.069 0.277 0 56,797

42,80432,259 1 0.077 0.077 0 32,259

Expected values 46,823 63,427Present values 42,470 57,530


Multiperiod valuation: details

Down Firm value0 100,000 132,690 176,065 233,621 309,9901 75,364 100,000 132,690 176,0652 56,797 75,364 100,0003 42,804 56,7974 32,259

Equity value42,470 69,427 109,399 165,308 239,990

20,280 36,828 64,377 106,0656,388 13,843 30,000

0 00

Delta0.86 0.95 1.00 1.00

0.70 0.88 1.000.43 0.69

0.00Beta

2.02 1.82 1.61 1.412.62 2.39 2.06

3.78 3.78#DIV/0!

Debt value57,530 63,262 66,667 68,313 70,000

55,084 63,172 68,313 70,00050,409 61,521 70,000

42,804 56,79732,259

Delta0.14 0.05 0.00 0.00

0.30 0.12 0.000.57 0.31

1.00Beta

0.25 0.10 0.00 0.000.40 0.19 0.00

0.65 0.371.00


Multiperiod binomial valuation: additional details

• From the previous calculation, we can decompose D into:• Risk-free debt• Risk-neutral probability of default• Expected loss given default

• Expected value at maturity:• Risk-free debt = 70,000• Default probability = 0.354• Expected loss given default = 18,552• Risky debt = 70,000 – 0.354 × 18,552 = 63,427

• Present value:• D = 63,427 / (1.05)² = 57,530


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

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

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

Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt

Documents

Transcript of Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt