SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins

### Valuation of Convertible Bonds

A convertible bond is one that gives the owner the option to convert it into shares of common stock. For example, in 1987 the Atari Corporation issued \$5 million of 15 year bonds that could be converted into common stock in a ratio 30.66 shares per bond. This ratio is called the conversion ratio.

The payoff to the bondholder as a function of the value of the company's assets is shown in Figure 1. The profile is similar to someone holding a share of stock for which they have sold a call at one exercise price and bought a fraction of a call at a higher exercise price.

```

|                         .

|       _____________._______

|      /

|    /

|  /

|/________________________________

X            X(1+1/q)

Market Value of

Assets of Corp.

Figure 1

```

Suppose the number of shares outstanding is n and let V be the value of the compay assets. Let X be the face value of all the m convertible bonds and let c be the conversion ratio. The market price S of the stock on the day of maturity of the bonds would be S = (V-X)/n, if the bonds are not converted. If they are converted then S = V/(n+cm). The bonds will be converted only if

```                       X/m < cS

X/m < cV/(n+cm)

V > X(n+cm)/cm

V  > X(1+q)/q,

```
where q = cm/n is the dilution ratio.

The value of the portfolio consisting of all the convertible bonds at the date of maturity is equal to the maximum of:

• 1. the value of firm's assets - call with exercise price equal to the face value of the convertible bonds),
• 2. q/(1+q)(value of firm's assets).

As an equation this condition is:

#### Portf = max[(V-C(V,X),Vq/(1+q)],

This formula is not convenient for making computations of value. The value of the convertible bonds is also equal to

#### Portf = V - C(V,X) + (q/(1+q))C(V,X(1+q)/q)

At a time before the expiration day the value of the portfolio is equal to its expected value on the day of expiration (the maturity date of the bonds).

Illustration:

Suppose a company's debt consists of \$20 million of convertible bonds with a maturity date two years from now. The value of the assets of the company is \$30 million. The risk-free interest rate is five percent per year and the volatility of the value of the company's assets is 0.4 per year. There are 200,000 shares of stock outstanding and 20,000 convertible bonds with a conversion ratio of 20. The dilution ratio q is then (20)(20,000)/200,000=2. Assuming that the Black-Scholes formula for call value applies, then the value of all the convertible bonds is:

#### 30,000,000 - 13,178,469 + (2/3)(7,857,024) = 16,821,531 + 5,238,016 = 22,059,547

```

Value of    Value of     Value of  Value of

Company     Stockholder  Debt      Convertible

Assets      Equity                 Bonds

(\$mill)      (\$mill)     (\$mill)  (\$mill)

5.000        0.020       4.980     4.982

10.000        0.559       9.441     9.534

15.000        2.320      12.680    13.239

20.000        5.238      14.762    16.352

25.000        8.958      16.042    21.223

30.000       13.178      16.812    22.060

35.000       17.701      17.299    24.949

40.000       22.405      17.595    27.916

```

The optimal strategy for the bondholders is to postpone conversion if the current dividend is low in comparison with interest received on the bonds. This would enable the bondholders to obtain the benefits of the capital gains without giving up the interest and relative security of bond ownership. For this reason, convertible bond issuers often include a call provision that can be used to force early conversion. A call provision gives the company the right to buy back the bonds. If convertible bonds are called the holders have a short time to convert to common stock rather than sell their bonds back to the company.