19.1 Two Methods

Problem. The Value of the Firm (V) is $340 million, the Face Value of the Debt (B) is $160 million, the

time to maturity of the debt (t) is 2.00 years, the riskfree rate ( RF k ) is 5.0%, and the standard deviation of

the return on the firm’s assets () is 50.0%. There are two different methods for valuing the firm’s equity

and risky debt based in an option pricing framework. Using both methods, what the firm’s Equity Value

(E) and Risky Debt Value (D)? Do both methods produce the same result?

Solution Strategy. In the first method, equity is considered to be a call option. Thus, E = Call Price. For

this call option, the underlying asset is the Value of the Firm (V) and the exercise price is the face value

of the debt (B). Hence, the call price is calculated from the Black-Scholes call formula by substituting V

for P and B for X. The rational is that if V > B, then the equityholders gain the net profit V-B. However,

if V < B, then the equityholders avoid the loss by declaring bankruptcy, turning V over to the debtholders,

and walking away with zero rather than owing money. Thus, the payoff to equityholders is Max (V - B,

0), which has the same payoff form as a call option. Further, we can use the fact that Debt plus Equity

equals Total Value of Firm (D + E = V) and obtain the value of debt D = V - E = V - Call.

In the second method, Risky Debt is considered to be Riskfree Debt minus a Put option. Thus, D =

Riskfree Debt - Put. For this put option, the underlying asset is also the Value of the Firm (V) and the

exercise price is also the face value of the debt (B). Hence, the put price is calculated from the Black-

Scholes put formula by substituting V for P and B for X. The rational is that the put option is a Guarantee

against default in repaying the face value of the debt (B). Specifically, if V > B, then the equityholders

repay the face value B in full and the value of the guarantee is zero. However, if V < B, then the

equityholders only pay V and default on the rest, so the guarantee must pay the balance B - V. Thus, the

payoff on the guarantee is Max (B - V, 0), which has the same payoff form as a put option. Further, we

can use the fact that Debt plus Equity equals Total Value of Firm (D + E = V) and obtain E = V - Risky

Debt = V - (Riskfree Debt - Put).

FIGURE 19.1 Spreadsheet for Stocks and Risky Bonds.

How To Build Your Own Spreadsheet Model.

1. Start with the Black Scholes Option Pricing - Basics Spreadsheet and Change the Inputs.

Open the spreadsheet that you created for Black Scholes Option Pricing – Basics and immediately

save the spreadsheet under a new name using the File | Save As command. Relabel the inputs in

the range A4:A8 and enter the new inputs values into the range B4:B8.

2. Riskfree Debt Value. The present value of riskfree debt paying B at maturity is Be−kRFt . Enter

=B7*EXP(-B6*B8) in cell B22.

3. Method One. Based on the first method:

o Equity = Call Price. Enter =B15 in cell C27.

o Risky Debt = V – Call Price. Enter =B4-B15 in cell C29.

o Total Value = Equity + Risky Debt. Enter =C27+C29 in cell C31.

4. Method Two. Based on the second method:

o Equity = V – Riskfree Debt Value + Put Price. Enter =B4-B22+B21 in cell F27.

o Risky Debt = Riskfree Debt Value – Put Price. Enter =B22-B21 in cell F29.

o Total Value = Equity + Risky Debt. Enter =F27+F29 in cell F31.

Both methods of doing the calculation find that the Equity Value (E) = $203.54 and the Risky Debt Value

(D) = $136.46. We can verify that both methods should always generate the same results. Consider what

we get if we equate the Method One and Method Two expressions for the Equity Value (E): Call Price =

V – Riskfree Bond Value + Put Price. You may recognize this as a alternative version of Put-Call

Parity. The standard version of Put-Call Parity is: Call Price = Stock Price - Bond Price + Put Price. To

get the alternative version, just substitute V for the Stock Price and substitute the Riskfree Bond Value for

the Bond Price. Consider what we get if we equate the Method One and Method Two expressions for the

Risky Debt Value (D): V - Call Price = Riskfree Bond - Put Price. This is simply a rearrangement of

the alternative version of Put-Call Parity. Since Put-Call Parity is always true, then both methods of

valuing debt and equity will always yield the same results!

Problem. The Value of the Firm (V) is $340 million, the Face Value of the Debt (B) is $160 million, the

time to maturity of the debt (t) is 2.00 years, the riskfree rate ( RF k ) is 5.0%, and the standard deviation of

the return on the firm’s assets () is 50.0%. There are two different methods for valuing the firm’s equity

and risky debt based in an option pricing framework. Using both methods, what the firm’s Equity Value

(E) and Risky Debt Value (D)? Do both methods produce the same result?

Solution Strategy. In the first method, equity is considered to be a call option. Thus, E = Call Price. For

this call option, the underlying asset is the Value of the Firm (V) and the exercise price is the face value

of the debt (B). Hence, the call price is calculated from the Black-Scholes call formula by substituting V

for P and B for X. The rational is that if V > B, then the equityholders gain the net profit V-B. However,

if V < B, then the equityholders avoid the loss by declaring bankruptcy, turning V over to the debtholders,

and walking away with zero rather than owing money. Thus, the payoff to equityholders is Max (V - B,

0), which has the same payoff form as a call option. Further, we can use the fact that Debt plus Equity

equals Total Value of Firm (D + E = V) and obtain the value of debt D = V - E = V - Call.

In the second method, Risky Debt is considered to be Riskfree Debt minus a Put option. Thus, D =

Riskfree Debt - Put. For this put option, the underlying asset is also the Value of the Firm (V) and the

exercise price is also the face value of the debt (B). Hence, the put price is calculated from the Black-

Scholes put formula by substituting V for P and B for X. The rational is that the put option is a Guarantee

against default in repaying the face value of the debt (B). Specifically, if V > B, then the equityholders

repay the face value B in full and the value of the guarantee is zero. However, if V < B, then the

equityholders only pay V and default on the rest, so the guarantee must pay the balance B - V. Thus, the

payoff on the guarantee is Max (B - V, 0), which has the same payoff form as a put option. Further, we

can use the fact that Debt plus Equity equals Total Value of Firm (D + E = V) and obtain E = V - Risky

Debt = V - (Riskfree Debt - Put).

FIGURE 19.1 Spreadsheet for Stocks and Risky Bonds.

How To Build Your Own Spreadsheet Model.

1. Start with the Black Scholes Option Pricing - Basics Spreadsheet and Change the Inputs.

Open the spreadsheet that you created for Black Scholes Option Pricing – Basics and immediately

save the spreadsheet under a new name using the File | Save As command. Relabel the inputs in

the range A4:A8 and enter the new inputs values into the range B4:B8.

2. Riskfree Debt Value. The present value of riskfree debt paying B at maturity is Be−kRFt . Enter

=B7*EXP(-B6*B8) in cell B22.

3. Method One. Based on the first method:

o Equity = Call Price. Enter =B15 in cell C27.

o Risky Debt = V – Call Price. Enter =B4-B15 in cell C29.

o Total Value = Equity + Risky Debt. Enter =C27+C29 in cell C31.

4. Method Two. Based on the second method:

o Equity = V – Riskfree Debt Value + Put Price. Enter =B4-B22+B21 in cell F27.

o Risky Debt = Riskfree Debt Value – Put Price. Enter =B22-B21 in cell F29.

o Total Value = Equity + Risky Debt. Enter =F27+F29 in cell F31.

Both methods of doing the calculation find that the Equity Value (E) = $203.54 and the Risky Debt Value

(D) = $136.46. We can verify that both methods should always generate the same results. Consider what

we get if we equate the Method One and Method Two expressions for the Equity Value (E): Call Price =

V – Riskfree Bond Value + Put Price. You may recognize this as a alternative version of Put-Call

Parity. The standard version of Put-Call Parity is: Call Price = Stock Price - Bond Price + Put Price. To

get the alternative version, just substitute V for the Stock Price and substitute the Riskfree Bond Value for

the Bond Price. Consider what we get if we equate the Method One and Method Two expressions for the

Risky Debt Value (D): V - Call Price = Riskfree Bond - Put Price. This is simply a rearrangement of

the alternative version of Put-Call Parity. Since Put-Call Parity is always true, then both methods of

valuing debt and equity will always yield the same results!