Question

1. Assume the value of the assets of a firm are $22.503 million and the volatility of the logarithmic returns of the asset is 21.02%. The firm has issued a zero coupon bond with face value 20m dollars due at the end of five years. Assume the riskless five year interest rate is 5% per year continuously compounded. Compute the value of the bond, the credit spread of the bond, and the value of the equity. Show all calculations.

Answer 1

Here, we can consider Value of the assets is the Value of the underlying assets of a call option using which we will value the equity of the company as a call option.

Value of the asset = S = $ 22.503 million

Value of debt = Excercise Price = X = $20 million

Life of the option = T = Life of debt = 5 years

Volatility of the underlying asset = Sigma = 21.02%

Risk-free rate = r = 5% continuously compounded

We can use Black Scholes Option pricing model for valuing the call option.

Value of Call Option Formula = N(d1) * S - N(d2) * X * e^-RT

where, d1 = (Ln(S/X) + (r + Sigma² / 2)*(T-t)) / (Sigma * (T - t)^1/2)

d2 = (Ln(S/X) + (r - Sigma² / 2)*(T-t)) / (Sigma * (T - t)^1/2)

Putting values in d1 and d2.

d1 = (Ln(22.503/20) + (5% + 21.02%² / 2)*(5 - 0)) / (21.02% * (5 - 0)^1/2)

Solving the above expression:

d1= 1.017776

d2 = (Ln(22.503/20) + (5% - 21.02%² / 2)*(5 - 0)) / (21.02% * (5 - 0)^1/2)

Solving the above expression:

d2 = 0.547754

Putting values of d1 and d2 in Value of the option formula.

Value of the call option = N(1.10776) * 22.503 - N(0.547754) * 20 * e^{-5% * 5}

The N(.) is the cumulative probability of d1 and d2 for a Standard Normal distribution

Using Excel's NORM.DIST function and input the value of d1 and d2 as x, Mean as 0, Standard Deviation as 1 and Cumulative as True, we can find the N(d1) and N(d2) values.

Below is the screenshot of how N(d1) and N(d2) were calculated using Excel.

N(d1) = 0.845608 | N(d2) = 0.70807

Value of the call option = 0.845608 * 22.503 - 0.70807 * 20 * 0.778801

Value of the call option = 19.02872 - 11.0289

Value of the call option = $8.00 million

Value of the call option is the Value of equity.

Hence, Value of the equity = $ 8.00 million

As Assets = Debt + Equity

We can find the value of the Debt using the above expression.

Value of outstanding Debt = Assets - Equity = 22.503 - 8.00

Value of outstanding zero-coupon bond = $14.503 million

Now using the Value of the bond and its Face value, we can find the interest rate or YTM on the bond.

PV of Zero bond = Face value / (1+Rate)^T

PV of Bond = $14.503 million | Face value = $20 million | T = 5 years

=> 14.503 = 20 / (1 + R)⁵

=> (1+R) = (20 / 14.503)^1/5

=> R = (20 / 14.503)^1/5 - 1

=> R = 1.06639 - 1

Interest Rate or YTM on Zero coupon bond = 0.06639 or 6.64%

As Risk-free rate is 5%, Credit Spread of the bond = YTM - Risk-free rate = 6.64% - 5%

Credit Spread of the bond = 1.64%

Below are all the answers combined:

Value of the equity = $ 8.00 million

Value of the bond = $ 14.503 million

Credit Spread of the bond = 1.64%