Question

39. Calculate the value of and interest rate on a loan using the option model and the following information.

            Face value of loan (B) = $500,000

            Length of time remaining to loan maturity (t) = 4years

            Risk-free rate (i) = 4%

            Borrower’s leverage ratio (d) = 60%

            Standard deviation of rate of change in the value of the underlying assets (?)= 15%

show me the steps with the correct formula please and thank you

Answer 1

The loan valuation using the Black Scholes option valuation model would first involve determining the market value of the borrower's equity contribution which is equivalent to a call option. This is so because in case of liquidation the lenders have priority over liquidation proceeds as compared to the borrower (who is also the equity contributor) which would imply that if asset value is V and debt value is D, the borrowers would have a payoff of V-D only when V > D and zero if V < D. This payoff is similar to that of a call option and hence the firm's equity value can be determined using the call option valuation model.

Leverage Ratio = 60 % and Borrowings Face Value = $ 500000

Asset Value = 500000 / 0.6 = $ 833333.33

Strike Price = Face Value of Debt = $ 500000, Volatility = 15 %, Maturity of Debt = Maturity of Option = 4 years, Risk-Free Rate = 4 %

Using an online Black Scholes calculator, option value is as given below:

Call Option Value = Equity Market Value = $ 408042.21

Debt Market Value = Asset Value - Equity Value = 833333.33 - 408042.21 = $ 425291.12

Let the interest rate be R

Therefore, 425291.12 = 500000 / (1+R)^(4)

R = 0.04129 or 4.13 % approximately

NOTE: Debt is modelled as a zero coupon bond in this solution.