# Equity Call Option: Risky & Risk-Free Debt

Instructor: David Bartosiak

Dave draws off his years of experience as a Financial Advisor and Analyst to teach others all about finance and the investing world.

By understanding equity as a call option, equity holders own on a company with a strike price equal to the company's total debt, we can break down risk-free and risky debt held by debt holders and bond holders of a company. Using the Black-Scholes formula, all of these values can be calculated.

## Rethinking Equity as a Call Option

One way to conceptualize a company's value is to think of it in terms of debt holders and equity holders. Debt holders have loaned a company money in exchange for interest payments and the eventual return of principal at a specified time. These are also known as bondholders. The company is indebted to debt holders and eventually must pay them back. These debt holders have a claim on the cash flows of the company.

What's left over after the company fulfills their debt obligations belongs to the equity holders. Equity holders are the owners and shareholders of a company. Equity holders own the cash flows in excess of the company's obligations. They also hold all the value of the company above and beyond what is owed to the debt holders.

## Calculating Equity: An Example

Let's say XYZ Corp is a company whose stock trades publicly with both debt holders and equity holders. Today, the total value of the company is \$100 million. XYZ Corp also has \$75 million in face value debt set to mature in 5 years with zero coupon. Knowing there is this looming liability down the line, calculating equity isn't as simple as subtracting total debt from the total value of the company.

Equity can be thought of as a call option on the company's assets with a strike equal to the face value of the debt. This is true because of the concept of limited liability. Limited liability reduces the risk of loss for equity investors if the firm is valued less than the value of the outstanding debt.

## Call Options

On liquidation of XYZ Corp, the most value that equity holders could possibly lose in XYZ Corp is equity they currently have. Limited liability protects the equity holders from having negative equity. Now draw the parallel to call options. Call options give the option holder the right to purchase value of the total debt which, in this case, equals \$75 million. Since we know the debt matures in 5 years, we can also say this equity call option on the company expires in 5 years.

## Using the Black-Scholes formula

The value of this equity call option can be calculated using the same Black-Scholes formula for determining theoretical options pricing. Since the math involved is beyond the scope of this lesson, using an online financial calculator is a perfectly acceptable way of calculating the value of a call option. Using the example, we can calculate what the theoretical value of the equity call option would be as well as account for the risky and the risk-free portion where:

T = 5 years

S = 100 million

K = 75 million

R = 2%

σ = 20%

## Calculating Call and Put Options

To calculate the theoretical value of a call option, first calculate d1 and d2. Then use those values in the cumulative distribution function for the normal distribution curve to get the theoretical value of the call option.

The theoretical value of the call option here, given the inputs we have above, would be \$36 million. Using the same Black-Scholes methodology, we can calculate the theoretical value of a put option at the same strike. Put options give the owner the right to sell the underlying security at an agreed price at any time from now until expiration.

The same values for d1 and d2 can be used in the put option calculation. Based on this calculation using the same parameters, the theoretical value of the put option is \$3.87 million. The risky debt value is equal to the total value of the firm minus the equity value. In this case, it's \$100 million minus the \$36 million or \$64 million.

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