Executive Master in FinanceRisky debt
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
EMF 2006 Risky debt |2April 18, 2023
Recently 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% (CreditGrades calculation)
EMF 2006 Risky debt |3April 18, 2023
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)
EMF 2006 Risky debt |4April 18, 2023
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
EMF 2006 Risky debt |5April 18, 2023
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 AAAA
A AA A A A A
BBBBBB
BBB BBB BBB BBBBBB
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
Sp
read
EMF 2006 Risky debt |6April 18, 2023
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, 1998-2000
Rating Category
Coverage Ratio
Cash Flow to Debt %
Return on Capital %
LT Debt to Capital %
AAA 21.4 84.2 34.9 13.3
AA 10.1 25.2 21.7 28.2
A 6.1 15.0 19.4 33.9
BBB 3.7 8.5 13.6 42.5
BB 2.1 2.6 11.6 57.2
B 0.8 (3.2) 6.6 69.7
Source: Bodies, Kane, Marcus 2005 Table 14.3
EMF 2006 Risky debt |7April 18, 2023
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
EMF 2006 Risky debt |8April 18, 2023
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
EMF 2006 Risky debt |9April 18, 2023
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
EMF 2006 Risky debt |10April 18, 2023
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
EMF 2006 Risky debt |11April 18, 2023
Merton Model: example using binomial option pricing
492.1 teu 670.1
ud
462.670.0492.1
67.05.11
du
drp 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
r
fppff
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
05.1
032,67538.0000,70462.0 D
05.1
0538.0000,80462.0 E
EMF 2006 Risky debt |15April 18, 2023
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 compounding):
• In our example:
Ty
FD
)1(
y
1
000,70146,65
y = 7.45%
Spread = 7.45% - 5% = 2.45% (245 basis points)
EMF 2006 Risky debt |16April 18, 2023
Decomposing the value of the risky debt
f
d
f r
VFp
r
FD
1
))(1(
1
)1(11
pr
Vp
r
FD
f
d
f
146,65
538.827,2667,66
538.05.1
032,67000,70
05.1
000,70
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
146,65
538.840,63462.0667,66
538.05.1
032,67462.0
05.1
000,70
D
EMF 2006 Risky debt |17April 18, 2023
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
V
Dr
V
Err DEA
000,100
146,65
000,100
854,34%11 DE rr
000,100
146,65
000,100
854,341 DE V
D
V
EDEA
EMF 2006 Risky debt |18April 18, 2023
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(E
DDelta
E
VDelta AAE
In example:βA = 1DeltaE = 0.96V/E = 2.87βE= 2.77rE = 5% + 6% × 2.77 = 21.59%
dSuS
ffDelta du
:Reminder
EMF 2006 Risky debt |20April 18, 2023
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/D
In example:βA = 1DeltaD = 0.04V/D = 1.54βD= 0.06rD = 5% + 6% × 0.09 = 5.33%
Putdudu
D DeltadSuS
PutPut
dSuS
PutFPutFDelta
)()(
EMF 2006 Risky debt |21April 18, 2023
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
EMF 2006 Risky debt |22April 18, 2023
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 = 1.327, d = 0.754rf = 2.47% per period =(1.05)1/2-1p = 0.473
# paths Proba/path Proba E D
309,990 1 0.050 0.050 239,990 70,000
233,621
176,065 176,065 4 0.056 0.223 106,065 70,000
132,690 132,690
100,000 100,000 100,000 6 0.062 0.373 30,000 70,000
75,364 75,364
56,797 56,797 4 0.069 0.277 0 56,797
42,804
32,259 1 0.077 0.077 0 32,259
Expected values 46,823 63,427
Present values 42,470 57,530
EMF 2006 Risky debt |23April 18, 2023
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
EMF 2006 Risky debt |24April 18, 2023
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
EMF 2006 Risky debt |25April 18, 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
EMF 2006 Risky debt |26April 18, 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))
TT
XPV
S
d
5.)
)(ln(
1 TT
XPV
S
d
5.)
)(ln(
2
EMF 2006 Risky debt |27April 18, 2023
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)
EMF 2006 Risky debt |28April 18, 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
EMF 2006 Risky debt |29April 18, 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
dN
dNVdNFeD rT
Value of risk-free
debt
Probability of no default
Probability of default
× ×Discounted
expected recovery
given default
+
EMF 2006 Risky debt |30April 18, 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
dNdN
dNVdNFeD rT
031,466985.01
8612.01000,100
)(1
)(1
2
1
dN
dNV
EMF 2006 Risky debt |31April 18, 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
EMF 2006 Risky debt |32April 18, 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
EMF 2006 Risky debt |33April 18, 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%
EMF 2006 Risky debt |34April 18, 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)²
58,228 = 60,000/(1+y)²
y = 9.64%
Spread = 364 basis points (bp)
EMF 2006 Risky debt |35April 18, 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
Sp
rea
d
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
Sp
read
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<1
d>1
Quasi debt PV(F)/V Volatility
Maturity
EMF 2006 Risky debt |36April 18, 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
Sp
read
))(1
)(ln(1
12 dNd
dNT
s
Proba of no default - Delta of put option
EMF 2006 Risky debt |37April 18, 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(-
d1)
Del
ta o
f p
ut
op
tio
n
d=0.6
d=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
MaturityP
rob
a o
f b
ankr
up
tcy
d=0.6
d=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%
EMF 2006 Risky debt |38April 18, 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
EMF 2006 Risky debt |39April 18, 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 Debt
30% 46,626 53,374
40% 48,506 51,494
+1,880 -1,880
EMF 2006 Risky debt |40April 18, 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
EMF 2006 Risky debt |41April 18, 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
EMF 2006 Risky debt |42April 18, 2023
Where are we?
• 1. Modigliani Miller 1958
• V = E + D = VU
• WACC = rA
• 2. Debt and taxes: PV(Interest tax shield)
• V = E + D = VU +VTS
• WACC < rA
• 3. Risky debt : Merton model – No tax shield
• Agency costs
• The tradeoff model: Leland
EMF 2006 Risky debt |43April 18, 2023
Still a puzzle….
• If VTS >0, why not 100% debt?
• Two counterbalancing forces:
– cost of financial distress
• As debt increases, probability of financial problem increases
• The extreme case is bankruptcy.
• Financial distress might be costly
– agency costs
• Conflicts of interest between shareholders and debtholders (more on this later in the Merton model)
• The trade-off theory suggests that these forces leads to a debt ratio that maximizes firm value (more on this in the Leland model)
EMF 2006 Risky debt |44April 18, 2023
Trade-off theory
Market value
Debt ratio
Value of all-equity firm
PV(Tax Shield)
PV(Costs of financial distress)
EMF 2006 Risky debt |45April 18, 2023
Leland 1994
• Model giving the optimal debt level when taking into account:
– limited liability
– interest tax shield
– cost of bankruptcy
• Main assumptions:
– the value of the unlevered firm (VU) is known;
– this value changes randomly through time according to a diffusion process with constant volatility dVU= µVU dt + VU dW;
– the riskless interest rate r is constant;
– bankruptcy takes place if the asset value reaches a threshold VB;
– debt promises a perpetual coupon C;
– if bankruptcy occurs, a fraction α of value is lost to bankruptcy costs.
EMF 2006 Risky debt |47April 18, 2023
Exogeneous level of bankruptcy
• Market value of levered company V = VU + VTS(VU) - BC(VU)
– VU: market value of unlevered company
– VTS(VU): present value of tax benefits
– BC(VU): present value of bankruptcy costs
• Closed form solution:
• Define pB : present value of $1 contingent on future bankruptcy
²
2
r
U
BB V
Vp
EMF 2006 Risky debt |48April 18, 2023
Example
Value of unlevered firm VU = 100
Volatility σ = 34.64%
Coupon C = 5
Tax rate TC = 40%
Bankruptcy level VB = 25
Risk-free rate r = 6%Simulation: ΔVU = (.06) VU Δt + (.3464) VU ΔW
1 path simulated for 100 years with Δt = 1/12
1,000 simulations
Result: Probability of bankruptcy = 0.677 (within the next 100 years)
Year of bankruptcy is a random variable
Expected year of bankruptcy = 25.89 (see next slide)
),0(~ tNW
EMF 2006 Risky debt |49April 18, 2023
Year of bankruptcy – Frequency distribution
Number default each year
0
5
10
15
20
25
30
35
40
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99
Year
# d
efau
lts
EMF 2006 Risky debt |50April 18, 2023
Understanding pB
25.0100
25 ²3464.
06.2²
2
r
U
BB V
VpExact value
Simulation 248.1
ˆ1
N
n
rYB
neN
p
N =number of simulations
Yn = Year of bankruptcy in simulation n
EMF 2006 Risky debt |51April 18, 2023
Value of tax benefit
)1()( BC
U pr
CTVTB
Tax shield if no default
PV of $1 if no default
Example: 2575.033.33)25.01(06.
540.)(
UVTB
EMF 2006 Risky debt |52April 18, 2023
Present value of bankruptcy cost
BBU pVVBC )(Recovery if default
PV of $1 if default
Example: BC(VU) = 0.50 ×25×0.25 = 3.13
EMF 2006 Risky debt |53April 18, 2023
Value of debt
BBU V
r
Cp
r
CVD )1()(
BBBU Vpr
CpVD )1()1()(
Risk-free debt
Loss given default
PV of $1 if default
63.6583.7025.033.83)255.06.
5)(25.0(
06.
5)( UVD
EMF 2006 Risky debt |54April 18, 2023
Endogeneous bankruptcy level
• If bankrupcy takes place when market value of equity equals 0:
²)5.0(
)1(
r
CTV C
B
25²)3464.5.06(.
5)0401(
BV
EMF 2006 Risky debt |55April 18, 2023
Leland 1994 - Summary
• Notation
• VU value of unlevered company
• VB level of bankruptcy
• C perpetual coupon
• r riskless interest rate (const.)
• σ volatility (unlevered)
• α bankruptcy cost (fraction)
• TC corporate tax rate
• Present value of $1 contingent on bankruptcy
• Value of levered company:
Unlevered: VU
Tax benefit: + (TCC/r)(1-pB)
Bankrupcy costs: - α VB pB
• Value of debt
• Endogeneous level of bankruptcy
BBB pVpr
CD )1()1(
²2
r
U
BB V
Vp
²5.0
)1(
r
CTV C
B
EMF 2006 Risky debt |56April 18, 2023
Inside the model
• Value of claim on the firm: F(VU,t)
• Black-Scholes-Merton: solution of partial differential equation
•
• When non time dependence ( ), ordinary differential equation with general solution:
F = A0 + A1V + A2 V-X with X = 2r/σ²
• Constants A0, A1 and A2 determined by boundary conditions:
• At V = VB : D = (1 – α) VB
• At V→∞ : D→ C/r
CrFFVFrVF VVUVUt "'' ²²5.0
0' tF
EMF 2006 Risky debt |57April 18, 2023
Unprotected and protected debt
• Unprotected debt:
• Constant coupon
• Bankruptcy if V = VB
• Endogeneous bankruptcy level: when equity falls to zero
• Protected debt:
• Bankruptcy if V = principal value of debt D0
• Interpretation: continuously renewed line of credit (short-term financing)
EMF 2006 Risky debt |58April 18, 2023
The Pecking Order Theory
• Developed by S. Myers (1984)
• Starts with asymmetric information:
• Managers know more than outside investors
– Use equity if stock overvalued
– Use debt if stock undervalued
• Issuing equity is a signal of overvaluation =>stock price drops
• Main implication: stock issues costly
• Order of preference for financing:
• 1.Internal funds
• 2. Debt
• 3. Stock issue
Consider the following story:
The announcement of a stock issue drives down the stock price because investors believe managers are more likely to issue when shares are overpriced.
Therefore firms prefer internal finance since funds can be raised without sending adverse signals.
If external finance is required, firms issue debt first and equity as a last resort.
The most profitable firms borrow less not because they have lower target debt ratios but because they don't need external finance.
EMF 2006 Risky debt |59April 18, 2023
Implications of the pecking order theory
• Firms do not have target debt ratios
• Debt absorbs difference between retained earnings and investments
• Debt increases when investments > retained earnings
• Debt decreases when investments < retained earnings
EMF 2006 Risky debt |60April 18, 2023
The message from CFO’s: debt
What factors affect how you choose the appropriate amount of debt for your firm?Source: US Graham and Harvey J FE December 2001 n = 392
Europe Bancel and Mittoo The Determinants of Capital Structure Choice: A Survey of European Firms, WP 2002
0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00%
Potential costs of financial distress
Debt level of other firms in industry
Transactions costs for issuing debt
Tax advantage of interest deductibility
Volatility of earnings and cash flows
Credit rating
Financial flexibility
% important or very important
Europe
US
EMF 2006 Risky debt |61April 18, 2023
Survey evidence and capital structure theories
• Trade-off theory Corporate interest deduction
moderately important Cash flow volatility important 44% have strict or somewhat
strict target/range
But: Expected distressed costs not
important Personal taxes not important
• Pecking order theory Firm value flexibility Issue debt when internal funds
are insufficient Equity issuance affected by
equity undervaluation
But: Equity issuance decision
unaffected by ability to obtain funds from debt,…
Debt issuance unaffected by equity valuation
EMF 2006 Risky debt |62April 18, 2023
Event studies
Security Issued
Security Retired
Two-Day Announcement Period
Retun
Leverage Increased
Stock Repurchase Debt Common 21.9%
Exchange offer Debt Common 14.0%
Exchange offer Preferred Common 8.3%
Leverage reduced
Exchange offer Common Debt -9.9%
Security Sales Common Debt -4.2%
Conversion-forcing call Common Convertible -0.4%
Conversion-forcing call Common Preferred -2.1%Source: Smith, C. Raising Capital: Theory and Evidence
Against tradeoff story
EMF 2006 Risky debt |63April 18, 2023
Problems with empirical studies
• Require data basis + computing capacities
• Accounting convention obscure relevant variables
• Problem for isolating capital structure decisions from other decisions
• Which econometric techniques to use?
• What are the testable hypothesis?
• How to measure the relevant variables?
• Contradictory results
• Harris & Ravis (1990) “The second major trend in financial structure has been the secular increase in leverage.” (p.331)
• Barclay, Smith, Watts (1995) “When viewed over the entire 30-year period, however, both market leverage ratios and dividend yields appear to be remarkably stable.” (p. 5)
EMF 2006 Risky debt |64April 18, 2023
Rajan Zingales 1995
• International data – 1987-1991
• Large listed companies
• Difference in accounting rules: pensions, leases
• Do leverage ratios vary across countries?
• Are determinants of leverage identical across countries?
EMF 2006 Risky debt |65April 18, 2023
Table II - Balance Sheets for Non-Financial Firms - 1991
US J ap Germ F I UK Can AverageCash 11.2 18.4 8.8 10.3 10.5 11.4 8.2 11.3Ac.Rec. 17.8 22.5 26.9 28.9 29.0 22.1 13.0 22.9Inv. 16.1 13.9 23.6 17.4 15.6 17.7 11.0 16.5Cur.As.Other 2.9 3.0 0.1 1.7 1.6 3.7 1.9 2.1Fixed As 52.0 42.2 40.6 41.7 43.3 45.1 65.9 47.3
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0Debt in cur.liab. 7.4 16.4 9.9 11.6 16.2 9.6 7.3 11.2Acc.pay. 15.0 15.4 11.5 17.0 14.7 13.7 13.3 14.4Cur.Liab. Other 11.0 10.4 8.7 14.8 12.2 16.7 2.8 10.9
33.4 42.2 30.1 43.4 43.1 40.0 23.4 36.5Def. Taxes 3.2 0.1 0.8 1.3 1.5 0.9 4.4 1.7LT Debt 23.3 18.9 9.8 15.7 12.1 12.4 28.1 17.2Minority Int 0.6 0.9 1.6 3.9 3.4 1.1 2.0 1.9Reserve Untaxed 0.0 0.0 1.7 0.0 0.0 0.0 0.0 0.2Liab.other 5.8 4.8 28.7 6.3 7.8 3.4 2.6 8.5Liab. total 66.3 66.9 72.7 70.6 67.9 57.8 60.5 66.1Equity 34.1 33.2 28.0 31.2 32.6 42.2 39.7 34.4
100.4 100.1 100.7 101.8 100.5 100.0 100.2 100.5
EMF 2006 Risky debt |66April 18, 2023
Table III Leverage in different countries
Book Book adjusted
Market Market adjusted
EBITDA/Interest
United States 37% 33% 28% 23% 4.05x
Japan 53% 37% 29% 17% 4.66x
Germany 38% 18% 23% 15% 6.81x
France 48% 34% 41% 28% 4.35x
Italy 47% 39% 46% 36% 3.24x
United Kingdom 28% 16% 19% 11% 6.44x
Canada 39% 37% 35% 32% 3.05x
Median debt to total capital in 1991
Adjusted debt = Net Debt = Debt – Cash
Book: using book equity, Market: using market value of equity
EMF 2006 Risky debt |67April 18, 2023
Determinants of leverage
• Tangibility of assets: Fixed Assets/Total Assets Debt
• Collateral => lower agency cost of debt
• More value in liquidation
• Market to book Debt
• Growth opportunities - underinvestment
• Costs of financial distress
• Size Debt
• Lower probability of bankruptcy
• Less asymmetry of information
• Profitability
• Myers Majluf: profitable companies prefer internal funds
EMF 2006 Risky debt |68April 18, 2023
Table IX Factors Correlated with Debt to Market Capital
US J ap Germ F I UK CanTangibility 0.33*** 0.58*** 0.28* 0.18 0.48** 0.27*** 0.11
(0.03) (0.09) (0.17) (0.19) (0.22) (0.06) (0.07)Market-to-book -0.08*** -0.07*** -0.21*** -0.15** -0.18* -.06** -0.13***
(0.01) (0.02) (0.06) (0.06) (0.11) (0.03) (0.03)Logsale 0.03*** 0.07*** -.06*** -0.00 0.04 0.01 0.05***
(0.00) (0.01) (0.02) (0.02) (0.03) (0.01) (0.01)Profitability -0.6*** -2.25*** 0.17 -0.22 -0.95 -0.47** -0.48***
(0.07) (0.32) (0.47) (0.53) (0.77) (0.24) (0.17)
Nb observations 2207 313 176 126 98 544 275
Pseudo R² 0.19 0.14 0.28 0.12 0.19 0.30
Standard errors are in parentheses.*,** and ***, significant at the 10, 5, 1 percent respectively.
EMF 2006 Risky debt |69April 18, 2023
References
• Altman, E., Resti, A. and Sironi, A., Analyzing and Explaining Default Recovery Rates, A Report Submitted to ISDA, December 2001
• Bohn, J.R., A Survey of Contingent-Claims Approaches to Risky Debt Valuation, Journal of Risk Finance (Spring 2000) pp. 53-70
• Merton, R. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates Journal of Finance, 29 (May 1974)
• Merton, R. Continuous-Time Finance Basil Blackwell 1990
• Leland, H. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure Journal of Finance 44, 4 (September 1994) pp. 1213-
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