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## Model of Credit Risk Premia Assignment

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InstructionsComputational Finance Using Excel and Mathematica

© Problem Set 5: Black-Scholes-Merton Model of Credit Risk Premia Using Asset Value and Its Volatility Estimation

Merton Model of Asset, Debt and Equity Valuation

In this problem, worth 25 points, the Black-Scholes-Merton model of Asset, Debt and Equity Valuation will be employed to estimate the risk premiums of a firms debt using the values below computed in Problem Set 4:

1. market asset values (as opposed to book asset values),

2. the market implied volatility of asset returns,

3. risk premium on its debt, senior and junior (subordinated debt),

4. solvency ratio,

5. distance to default (as defined by KMV), and

6. probability of insolvency computed assuming V is a lognormal variable.

Data that is needed will come from the firms balance sheet and includes:

1. The book value of assets as of the last annual report to start the estimation,

2. The sum of debt of the company separated into secured debt and subordinated debt,

3. The risk free interest rates for 1, 2 and 3 years

4. The market capitalization of the firm near the lasts annual report, and

5. An estimate of the volatility over 1 year of the firms stock returns (implied volatility can be used for a 1 year call option).

Use the firms from Problem Set 4 and compute premiums for 1, 2 and 3 years forward.

Computational Requirements

Use Mathematica to perform your calculations. Two approaches will be used:

1. The Mathematica function of FindRoot or Nsolve using the two equations in the Black-Scholes-Merton Model as shown below, and

2. Newtonss iterative method as developed in Mathematica coding added to BlackBoard.Write 3-4 pages analysis/summary based on the results and findings

Equations

The underlying assets are assumed to be stochastic and generated by an Ito process in continuous time. Consistent with this model, equity value of a firm can be considered as a call option on its assets with a strike price being its total promised debt, B, is:

(1)

whereand

and

E = the market value of equity (stock price times number of shares outstanding,

V = the market value of assets,

B = the promised value of firm liabilities discounted at the risk-free rate to time T,

Rf = the risk-free rate with a maturity consistent with the time to asset valuation (bank examination),

T = the time to expiration of the option and time to maturity of the debt,sV = the standard deviation (volatility) of the rate of return on assets,

ln(x) = the natural logarithm of x,

exp(x) = the value e raised to the power of x, and

N(x) = the cumulative standard normal distribution.

Our objective is to estimate two parameters of the contingent claims model of pricing: the market value of assets, V, and the volatility of asset returns, sV. To solve for two variables a second equation is necessary. Ronn and Verma (1986) and Hull (2000, p.630-631) show that by applying Itos lemma to the generating process for the value of assets, the following relationship with observable market value of equity and its volatility can be used as the second equation in our system:

sEE = N(d1)sVV, and by rearranging, . (2)

where sE is the volatility of the return on equity as computed from the market value of equity and all other variables are defined as above. Equation (2) shows that asset volatility is derived from leverage weighted observable volatility of equity. However, it needs to be emphasized that d1 has the asset value and asset volatility as arguments (see equation (1)).

Nonlinear equations (1) and (2) are used to solve for the implied values of V and sV via an iterative process such as that of Newton. An algorithm to solve nonlinear equations based on Newton’s method with numeric derivatives is presented in Blackboard as a Mathematica Notebook. Simply add the necessary equations for the risk premiums to this notebook.

This time, the programming of the risk premiums will be up to you. See the Powerpoiunt side labeled RiskPremia and the word document Hanweck_SpellmanJFSRAug15a.doc on Blackboard and Appendix A below.

Appendix A: The Yield Spread-Solvency Model

A. The Contingent Claims Model for Bank Subordinated Debt and Equity

In order for subordinated debt yields to be a signal of bank insolvency, the market yields should monotonically increase as insolvency is approached. This would appear to be the presumed yield-solvency relationship behind the mandate for subordinated debt. To relate the investors subordinated debt yield relative to solvency we rely on Black and Cox (1976), Smith (1979) and Cox and Rubinstein (1985) (for references see the Hanweck and Spellman paper on the Blackboard course site). who show that the market value of subordinated debt, DSub, equals the value of the difference between two European call options on the value of assets with strike prices of senior debt and total debt and is given by:

DSub = c(V, BDep) c(V, BDep + BSub). (1)

where DSub is the market value of subordinated debt, V is the unobserved market value of assets, BDep is the present value of the promised value of senior debt (deposits for most banks) discounted at the risk-free rate to period T and BSub is the promised value of subordinated debt discounted at the risk free rate.

Applying continuous time approximations to these relationships and the assumptions of the Black-Scholes-Merton options-pricing model gives the market value of the subordinated debt as:

(2)

This can be simplified to the relationship as presented in Gorton and Santomero (1990):where,

and ,

andV = the market value of assets,

B = the promised value of bank liabilities discounted at the risk-free rate to time T,

Rf = the risk-free rate with a maturity consistent with the time to asset valuation (bank examination),

t = the market-perceived time until receivership,sV = the standard deviation (volatility) of the rate of return on assets,

ln(x) = the natural logarithm of x,

exp(x) = the value e raised to the power of x, and

N(x) = the cumulative standard normal distribution.

Consistent with the above model of subordinated debt, the equity value of a firm can be considered as a call option on its assets with a strike price being its total promised debt, B, is:

(3)

where E is the market value of equity and g1, g2 and all other variables are defined as above.

Our objective is to estimate three parameters of the contingent claims model of pricing: the market value of assets, V, the volatility of asset returns, sV, and the investors expected time to resolution, t. To solve for three variables a third equation is necessary. Ronn and Verma (1996) and Hull (2000, p.630-631) show that by applying Itos lemma to the generating process for the value of assets, the following relationship with observable market value of equity and its volatility can be used as the third equation in our system:

sEE = N(g1)sVV (4)

where sE is the volatility of equity of the bank and all other variables are as defined above. Equation (4) shows that the volatility of assets, sV, can be considered a leverage adjusted value of the volatility of equity:

sV = sE(E/(VN(g1)) (5)

The market value of assets can be approximated by the market value of equity and the market value of debt. The value of N(g1) is the likelihood of the normalized value of asset return being less than g1 or the value of the recovery of assets upon default (see Hull, 2000, p. 631).

The relationship in equation (1) can be stated in terms of an interest rate risk premium, or spread, defined as the difference between the yield to maturity on the risky debt, RSub, and a default risk-free security such as a U.S. Treasury security of the same remaining maturity, Rf. This is done by recognizing that the market value of the subordinated debt is the continuous discounted value of the promised amount at the market rate of interest on the debt and treating it as a zero coupon debt instrument (all values are evaluated at time t before expiration):

(6)

Substituting from equation (2) above, the default risk premium, RSub Rf, is:

(7)

Equation (7) is used to simulate the theoretical risk premium, assuming values for all the parameters. In addition, equation (7) is used as one of the three equations to estimate the unobserved parameters of interest: V, sV, and t.

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