Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt Professor André Farber Solvay Business School Université Libre de Bruxelles Up to now, we have not explicitly model the risk associated with the debt. Risky debt require higher rates of return. In this lecture, we will use option pricing theory to estimate the cost of debt and the required rate of return on risky debt.

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) VUB 04 Options and risky debt

Fixed income markets Corporate bond market Investors Banks Companies Investors Assets Equity Debt Banks Loans Equity Deposits Credit derivatives 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 Rate, Journal of Finance, 29 (May 1974) Merton, R. Continuous-Time Finance Basil Blackwell 1990 Lannoo, K.and Levin, M. Toward a European Single Market for Financial Services, Presentation, CEPR 2004 VUB 04 Options and risky debt

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) VUB 04 Options and risky debt

Rating Agencies Investment-grades Speculative-grades Moody’s (www.moodys.com) Standard and Poors (www.standardandpoors.com) Fitch/IBCA (www.fitchibca.com) Letter grades to reflect safety of bond issue Very High Quality High Quality Speculative Very Poor S&P AAA AA A BBB BB B CCC D Moody’s Aaa Aa Baa Ba Caa C Reference Bodie, Kane and Marcus, Investments, 4th ed. McGraw-Hill 1999 Moody’s, The Evolving Meaning of Moody’s Bond Rating – August 1999 Standard & Poors, European Default Rates Inch Closer to the Global Average, April 2002 Investment-grades Speculative-grades VUB 04 Options and risky debt

Spread over Treasury for Industrial Bonds This figure illustrate a pattern for the spread with different maturities and different credit ratings. The spread increases as the rating declines. It also increases with maturity for investment-grades bonds. For speculative-grades bonds the relationship with maturity is more complex. VUB 04 Options and risky debt

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, 1997-1999 Rating 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 70.7 Traditionally, rating agencies base their ratings on financial ratios. The key ratios used are: Coverage ratios: EBIT/Interest Cash flow to debt: Free operating cash flow/Total debt Profitability ratio: Return on Capital Leverage ratio: Total Debt / Capital Source: Bodies, Kane, Marcus 2002 Table 14.3 VUB 04 Options and risky debt

Standard&Poor’s European Rating Distribution 1985 1990 1995 2000 2002 AAA 20 37 52 49 42 AA 14 59 117 171 185 A 6 159 315 350 BBB 141 244 Investment-grade 40 110 370 676 821 3 2 7 71 103 B 1 8 75 81 CCC 12 23 Speculative-grade 16 158 207 Total 43 113 386 834 1028 Bond rating is new in Europe. Before the introduction of the euro, the corporate bond market was almost non existent. This was essentially due to the segmentation based on the national currencies. For instance, financial institutions had to match the currencies for assets and liabilities. VUB 04 Options and risky debt

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 VUB 04 Options and risky debt

Moody’s:Average cumulative default rates 1920-1999 % 3 4 5 10 15 20 Aaa 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 0.27 0.60 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 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 VUB 04 Options and risky debt

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 VUB 04 Options and risky debt

Merton (1974) Limited liability: equity viewed as a call option on the company. D Market value of debt E Market value of equity Loss given default F Bankruptcy Why is debt risky? The key reason is limited liability: a company can default on its obligation to pay the interests or to repay the principal. From the stockholders’ perspective, going bankrupt means that they lose the money that they have invested in the company. But their loss is limited. They are not required to pay additional amounts. Black and Scholes, in their seminal paper, discuss the implications of their option pricing model for the bond valuation. This insight was later expanded by Robert Merton in 1974 (hence the name “Merton Model” for risky debt valuation models based on an option pricing model). We present here the problem in a simplified setting: 1. The value of a the firm (V) is given and is independent of the capital structure (the MM 58 proposition is satisfied) 2. The firm issues one single debt: a zero-coupon giving the holder the right to a fixed sum of money at maturity. 3. There are no cost of bankruptcy. Two main conclusions appear is this setting. 1. The market value of equity is equal to the value of a call option 2. The market value of the debt is equal to the value of the riskless debt minus a put option. F Face value of debt V Market value of comany F Face value of debt V Market value of comany VUB 04 Options and risky debt

Stock = Call + PV(Strike) – Put 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 The value of risky debt is equal to the value of a default-free debt minus a put option. This put option appears because of limited liability. Stockholders have the insurance that the market value of equity will never become negative. If the value of the company is below the face value of the debt, they do not have to contribute to fund the difference. This insurance is provided by the bondholders. There is no free lunch in finance. Stockholders have to pay for this guarantee that the value of their equity will not become negative. The price to pay is the value of the put option. VUB 04 Options and risky debt

Merton Model: example using binomial option pricing Data: Market Value of Unlevered Firm: 100,000 Risk-free rate per period: 5% Volatility: 40% Company issues 1-year zero-coupon Face value = 70,000 Proceeds used to pay dividend or to buy back shares Binomial option pricing: review Up and down factors: V = 149,182 E = 79,182 D = 70,000 Risk neutral probability : V = 100,000 E = 34,854 D = 65,146 V = 67,032 E = 0 D = 67,032 1-period valuation formula We first use the binomial model to illustrate the Merton model. The calculation is a straightforward application of binomial option pricing. All securities are valued by discounting the risk-neutral expected value with the risk-free interest rate. The analysis proceeds as follow: 1. Calculate the up and down factors 2. Calculate the risk neutral probabilities p and 1 – p 3. Calculate the value of the securities at maturity 4.Calculate the present value using the 1-period valuation formula. Note that, in this example, the value of the risk-free debt would be: PV(F) = 70,000 e-5% = 66,586 The difference between the value of the risk-free debt (66,586) and the value of the (risky) debt (65,146) is the value of the put option. ∆t = 1 VUB 04 Options and risky debt

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: y = 7.45% Spread = 7.45% - 5% = 2.45% (245 basis points) Once we know how to value risky debt, we can calculate the interest rate on the risky debt. This rate is defined the yield to maturity on the bond issue (or the discount rate required to set the value of the debt equal to the present value of the future cash flows). In our simplified setting, the debt is a zero coupon. VUB 04 Options and risky debt

Decomposing the value of the risky debt 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 Bank and financial institutions devote considerable resources to credit risk. They quantify this risk by estimating: 1. The probability that the firm will default 2. The loss given that a default occurs. These two notions determine the crediworthiness of corporate debts. Assume that Vd <F (otherwise, there would be no bankruptcy). First, the value of the debt can be expressed is the risk neutral expected value discounted with the risk-free interest rate. Rearranging the term, it is also be stated as: Value of risk-free debt – (Probability of default) × (Loss given default) The loss given default is equal to the difference between the face value of the debt and the recovery. In our simple 1-period binomial model, the recovery is equal to the value of the firm if default occurs. The important point to note is the pricing of the risky debt is based on the risk-neutral probability of default and not the real-world probability. VUB 04 Options and risky debt

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 We now wish to understand the effect of leverage on the cost of equity and the cost of debt when debt is risky. Our starting point in the Modigliani Miller (1958) propositions. The value of the firm is unaffected by the capital structure. Hence, the weighted average cost of capital of the levered firm is equal to the cost of capital of the unlevered firm. Similarly, the weighted average beta of equity and debt is equal to the asset beta. Combining the option pricing model with the CAPM, we can work out either the values of rE and βE. The cost of debt and the beta of the debt can than be calculated using the WACC formula. VUB 04 Options and risky debt

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) In example: βA = 1 DeltaE = 0.96 V/E = 2.87 βE= 2.77 rE = 5% + 6% × 2.77 = 21.59% To calculate the beta of equity, we proceed as follow. 1. The equity of the levered company is assimilated to a call option on the firm’value. 2. A call option is a portfolio composed of on a long position on delta shares combined with borrowing. The fraction of this portfolio invested in the shares is (delta V)/E. 3. The beta of a portfolio is equal to the weighted average of the beta of the underlying securities. In this setting, the betas of the underlying securities are the asset beta and the beta of borrowing in the replicating portfolio which is equal to zero. 4. Therefore βE = βA × Delta × (V/E) This expression generalizes the formula that had been established when the debt is risk-free: βE = βA × (V/E) Remember that 1≥Delta≥0. This implies that, for a given level of leverage, the equity is less risky with risky debt than when the debt is riskless. Remember also that delta varies whenever V varies. As a consequence, the beta of equity will not be constant. VUB 04 Options and risky debt

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 = 1 DeltaD = 0.04 V/D = 1.54 βD= 0.06 rD = 5% + 6% × 0.09 = 5.33% We now turn to the debt. The beta of the debt is equal to minus the beta of the put option (remember that the beta of the put option is negative, therefore, the beta of the debt is positive). An interesting point to note here is the difference between the cost of borrowing (y) calculated previously and the cost of debt (rD). Remember that the cost of borrowing is defined as the yield to maturity on the debt whereas the cost of debt is equal to the expected return on the debt. In our example, the cost of borrowing is 7.45% whereas the cost of debt is 5.33%. To understand the difference, compare the formulas for y and rD. where Π is the true probability of an up movement. This true probability can be calculated by solving the following equation: rA = Π (u – 1) + (1 – Π) (d – 1) In the example: Π = 0.535 The cost of borrowing calculation is based on the face value of the debt whereas the cost of debt calculation is based on the (true) expected future value. VUB 04 Options and risky debt

Multiperiod binomial valuation Risk neutral proba u4V p4 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 u3V u²V u3dV 4p3(1 – p) uV u2dV V udV 6p²(1 – p)² u2d²V dV ud²V Δt ud3V 4p (1 – p)3 d²V d3V The price change in one period in independent of the price change in previous periods. The probability of a path with k ups and n-k downs (where n is the total number of steps) is pk (1 – p)n-k The number of path is given by the binomial coefficient: The Excel function COMBIN(N,n) provides the desired result. For small binomial tree, Pascal’s triangle is useful: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Each subsequent row is obtained by adding the two entries diagonally above. This is known in China as the Yanghui triangle as it had been studied earlier by the Chinese mathematician Yanghui about 500 years earlier. Reference mathworld.wolfram.com d4V (1 – p)4 VUB 04 Options and risky debt

Multiperiod binomial valuation: example Firm issues a 2-year zero-coupon Face value = 70,000 V = 100,000 Int.Rate = 5% (annually compounded) Volatility = 40% Beta Asset = 1 4-step binomial tree Δt = 0.50 u = 1.332, d = 0.751 rf = 2.47% per period =(1.05)1/2-1 p = 0.471 We calculate the values using a binomial tree with 4 steps. We first calculate the parameters of the binomial tree. As the final maturity is 2 years, the length of one period is Δt =2/4=0.50 The up and dow factors are: and d = 1/1.332=0.751 The risk-free interest rate is 5% per annum with annual compounding. The risk-free interest per period in the binomial tree is thus rf = (1.05)1/2-1=2.47% The risk-neutral probability of an up movement is: p = (1.0247 – 0.751)/(1.332 – 0.751) = 0.471 The next step is to calculate the possible values of the firm at maturity and the associated probabilities. We then calculate the values of equity and debt at maturity and their risk-neutral expected values We get the present value by discounting the expected values at the risk-free interest rate. VUB 04 Options and risky debt

Multiperiod valuation: details Binomial models are easier to implement in Excel using diagonal matrices instead of branching trees. V uV u²V u3V … dV udV u²dV … d²V ud²V … … The upper part of the slide shows the binomial evolution of the value of the company. The lower part shows the evolution of the value, the delta and the beta of the equity and of the debt. For the equity, the delta increases (decreases) and the beta decreases (increases) when the value of the company goes up (down). The opposite applies for the debt. Note that, as the value of the firm goes down, the debt becomes more and more similar to equity. VUB 04 Options and risky debt

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 The value of the risky debt can also be expressed as follow: D = {F – (RN probability of default) ×(Loss|Default)}×Discount factor Default take place if the value of the firm at maturity is less than 70,000. Based on previous calculations, we have: VT RN Proba Loss|Default 56,797 0.277 13,203 32,259 0.077 37,741 Based on these number, we get: Probability of default = 0.277 + 0.077 = 0.354 Expected loss given default = (0.277 ×13,203 + 0.077 37,741)/0.354 = 18,552 VUB 04 Options and risky debt

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 Bankruptcy What happens if we increase the number of steps of a given maturity? The size of the binomial tree increases but the underlying logic remains unchanged. In a binomial tree with n steps (the length of each step is Δt=T/n), the firm can take n+1 different values at maturity depending on the number of ups (k) and downs (n – k): Vk = ukdn – k k = 0, 1, …,n The risk-neutral probability of Vk is: Let fk be the value at maturity of a derivative if VT = Vk. Risk-neutral pricing implies that the value at time 0 is: f = e-rT (p0 f0 + p1 f1 + … + pk fk + … + pn+1 fn+1) [1] As n→∞, the probability distribution of VT tends converges to a lognormal distribution. ln(VT) ~ N[ln(V)+(r-0.5σ²)T, σ√T] The Black-Scholes formulas is the continuous equivalent of [1] when the probability distribution of the underlying value at maturity is lognormal. Today Maturity Time VUB 04 Options and risky debt

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) The next step is to use the Black-Scholes formula to value the securities. This slide is a reminder of the formulas. Delta of put option Risk-neutral probability of exercising the option = Proba(ST<X) (Remember: 1-N(x) = N(-x)) VUB 04 Options and risky debt

Black-Scholes using Excel Comments: Stock price: for dividend paying stocks, the stock price should be reduced by the present value of the dividends paid before the option’s maturity. Interest rate: the easiest way is to use the interest rate with continuous compounding. In that case, the t-year discount factor is exp(-rFt). ln(S/PV(Strike)): this number is equal to minus the total excess return (with continuous compounding) in order for the stock price to reach the exercise price at maturity. To see this, note that: PV(Strike) = S e-x → x=ln(S/PV(Strike)) and Strike = S e(rT – x) Adjusted sigma: σ √T is the standard deviation of the total return until the option’s maturity. Distance to exercice: VUB 04 Options and risky debt

Merton Model: example Data Market value unlevered firm €100,000 Risk-free interest rate (an.comp): 5% Beta asset 1 Market risk premium 6% Volatility unlevered 40% Company issues 2-year zero-coupon Face value = €70,000 Proceed used to buy back shares Details of calculation: PV(ExPrice) = 70,000/(1.05)²= 63,492 log[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 Using Black-Scholes formula Price of underling asset 100,000 Exercise price 70,000 Volatility s 0.40 Years to maturity 2 Interest rate 5% Value of call option 41,772 Value of put option (using put-call parity) C+PV(ExPrice)-Sprice 5,264 Using the Black-Scholes-Merton with a handheld calculator is tedious. Moreover, you need a table of the normal probability distribution to get N(d1) and N(d2). The two important results in the calculation are: - The delta of the call option (the equity) N(d1) = 0.86 - The risk neutral probability of no default N(d2) = 0.70 Keep in mind that this is a probability in a risk neutral world. As the stock doesn’t pay any dividend, the expected growth rate (equal to the expected return) of the value of the unlevered company is set equal to the risk-free interest rate (5% per annum with annual compounding). In the real world, the expected return would normally be higher if the beta of the unlevered firm is positive. As a consequence, the real probability of no default would be higher. VUB 04 Options and risky debt

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)] The value of the risky debt is equal to the value of the risk-free debt minus the value of the put option. It is easier to express the risk-free interest rate r with continuous compounding.The T-year discount factor is e-rT . The value of the put option is equal to the value of the replicating portfolio. The delta of the put option is N(d1) – 1 = –(1 – N(d1)) As N(d2) is the risk neutral (RN) probability that the option will be exercised. Therefore, 1 – N(d2) is the RN probability that the put option will be exercised. In the Merton model, this is the probability of default. Rearranging, we show that risky debt is a mixture of risk-free debt and of the asset of the firm. When distinction between debt and equity blurs. If the probability of default is high, the risky debt looks more like equity than like debt. Value of risk-free debt Probability of no default Discounted expected recovery given default Probability of default × + × VUB 04 Options and risky debt

Example (continued) D = V – E = 100,000 – 41,772 = 58,228 D = e-rT F – Put = 63,492 – 5,264 = 58,228 VUB 04 Options and risky debt

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) Put F Recovery Discount factor Probability of default Expected value of put given F This slide shows that V[1-N(d1)]/[1-N(d2)] is equal to the discounted expected recovery given default. To understand this result, first note that if default take place (if VT<F), recovery is equal to F – Put (see figure). Next, express the value of the put option as: P = E(Put|default) ×(Probability of default)×(Disc.Factor) After some manipulation, you get: E(Put|default) = –V erT [1-N(d1)]/[1-N(d2)] + F Therefore, the expected recovery given default is: E(Recovery|Default) = F – E(Put|Default) = V erT [1-N(d1)]/[1-N(d2)] Hence, the discounted expected recovery given default is: E(Recovery|Default) ×(Disc. Factor) = V [1-N(d1)]/[1-N(d2)] Default VT VUB 04 Options and risky debt

Another presentation Discount factor Face Value Probability of default Loss if no recovery Expected Amount of recovery given default Expected loss given default VUB 04 Options and risky debt

Example using Black-Scholes Data Market value unlevered company € 100,000 Debt = 2-year zero coupon Face value € 60,000 Risk-free interest rate 5% Volatility unlevered company 30% Using Black-Scholes formula Value of risk-free debt € 60,000 x 0.9070 = 54,422 Probability of default N(-d2) = 1-N(d2) = 0.1109 Expected recovery given default V 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% Using Black-Scholes formula Market value unlevered company € 100,000 Market value of equity € 46,626 Market value of debt € 53,374 Discount factor 0.9070 N(d1) 0.9501 N(d2) 0.8891 VUB 04 Options and risky debt

Calculating borrowing cost 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) 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) VUB 04 Options and risky debt

Determinant of the spreads Quasi debt PV(F)/V Volatility Maturity The spread is determined by 3 factors: The quasi debt ratio defined as the ratio between the present value of the face value calculated with the risk-free interest rate; The volatility of the firm’s value The maturity VUB 04 Options and risky debt

Maturity and spread Proba of no default - Delta of put option The most surprising result of the Merton model is the relationship between the spread and maturity. Let d be the quasi debt ratio: d = F e-rT / V The value of the debt can be written as: D = F e-rT N(d2) + V N(-d1) = F e-rT [N(d2) + N(-d1)/d] The cost of borrowing (with continuous compounding) is: y = ln (F/D) / T = - ln(D/F) / T = r – ln[N(d2) + N(-d1)/d]/T The spread s is the difference between the cost of borrowing and the risk-free rate: s = y – r = – ln[N(d2) + N(-d1)/d]/T For low levels of quasi debt ratio the spread is an increasing function of maturity. For high levels of quasi debt ratio (highly levered firms), the spread is a decreasing of maturity. The explanation is the following: N(d2), the probability that the company will not default at maturity, is a decreasing function of maturity. However, for N(-d1), minus the delta of the put option, the relationship with maturity is more complex. For low values of d, N(-d1) is an increasing function of T. Therefore, the spread increases. But, for higher values of d, it decreases and the spread goes down as the maturity increases. VUB 04 Options and risky debt

Inside the relationship between spread and maturity d = 0.6 d = 1.4 T = 1 2.46% 39.01% T = 10 4.16% 8.22% 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 The relationship between the spread and maturity is complex. Two factors determine this relationship. 1. The probability of bankruptcy. This probability increases sharply with maturity for low levels of leverage. However, this probability is much higher for high levels of leverage. Moreover, it doesn’t change a lot when the maturity changes. Therefore, the probability of bankruptcy is an important determinant of the relationship between the spread and maturity for low levels of leverage, but not for high levels. As it goes up, the spread goes up. 2. The delta of the put option. Remember that delta is a measure of the sensitivity of the option to changes in the value of the underlying asset. If the level of leverage is low, the delta of the put option is slightly negative and close to 0: the value of risky debt is not very sensitive to changes in the value of the company. The delta does not play an important role in the relationship between the spread and maturity. If, on the other hand, the level of leverage is high, the delta of the put is highly negative and close to -1. The sensitivity of the risky debt to the value of the firm is high (and negative). The delta plays an important role. As it goes up, the spread goes down. VUB 04 Options and risky debt

Agency costs Stockholders and bondholders have conflicting interests Stockholders might pursue self-interest at the expense of creditors Risk shifting Underinvestment Milking the property VUB 04 Options and risky debt

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 VUB 04 Options and risky debt

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 VUB 04 Options and risky debt

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 VUB 04 Options and risky debt