Callable Bonds Some bonds have a feature, referred to as a call feature, that allows the

bond issuer to buy back the bonds from the investor at a speciﬁed price—the call price—during a speciﬁed period prior the bond’s maturity date. A bond with this feature is referred to as a callable bond. If a bond is callable, investors are concerned with not just its yield-to-maturity, but also its return if the bond is called away. Yield-to-call is a concept similar to the yield-to-maturity. It is the yield to the date when the bond is expected to be called, instead of a yield to a bond’s maturity. The yield- to-call is calculated like the yield-to-maturity, except:

■ instead of the number of periods to the maturity date, N is the number of periods to some date when the bonds are expected to be called, and

THE FUNDAMENTALS OF VALUATION

■ the call price of the bond is used as the maturity value, M. The call price is speciﬁed in the bond indenture. When the bond may be

called is also specified in the indenture, but it is usually a range of dates, so the precise date the firm will actually call the bond is not specified. Therefore, some assumption has to be made regarding when the bond will be called away.

Let’s look at a callable bond to see how this works. Illinois Bell maturity until the year 2011. But the call price depends on the year

called. For example, if Illinois Bell calls in the bonds in 1991, the company must pay 104.95, or $1,049.50 per bond; if called in 2011, it’s 100.24 or $1,002.40 per bond.

On January 1, 1991, the price was 88, or $880 per bond. This bond pays $82.50 interest each year, or $41.25 every six months. As of Janu- ary 1, 1991, there were 52 interest payments remaining to maturity. Therefore, the yield to maturity as of January 1, 1991 will be the annual yield equivalent to the six-month rate that solves:

--------------------- $41.25 + ------------------------ ∑ $1,000

t = 1 ( 1 + r d ) ( 1 + r d ) We know that since this is a discount bond (the value is less than its par

a ﬁnancial calculator, the six-month rate is 0.0475 or 4.75%. The yield- to-maturity is therefore 4.75% × 2 = 9.5%. The yield-to-call is calculated is a similar manner. For example, if the bonds are called at the end of 1991, Illinois Bell must pay $1,049.50 per bond at the call date and prior to the call has paid two interest payments since January 1, 1991: June and December of 1991.

Using this information ( M = $1,049.50 and N = 2), the yield-to-call is calculated by determining the six-month yield and translating it into an annual yield:

--------------------- + ∑ --------------------------

( 1 + r 2 d ) The six-month rate is 13.7% and the yield-to-call is 27.4% per year.

Though the yield-to-call is usually calculated using the ﬁrst available call date, we can calculate the yield-to-call for any possible call date. For

Valuation of Securities and Options

bonds called at the end of the year 2000, the yield-to-call is the annual yield equivalent to the six-month yield solving using the call price for 2000 of 102.83:

--------------------- $41.25 + -------------------------- ∑ $1,028.30

t = 1 ( 1 + r d ) ( 1 + r d ) The six-month yield is 5.19% and the yield-to-call is 10.38% per year.

VALUATION OF OPTIONS In Chapter 4, we discussed options and how the price of an option can

be decomposed into intrinsic value and time premium. The factors that affect the time value of an option are:

1. the value of the underlying asset;

2. the exercise price;

3. the time value of money;

4. the expected volatility in the value of the underlying asset; and

5. the time to maturity. To see how these factors inﬂuence the value of an option, let’s look

at a simple option—a stock option. A stock option is the right to buy or sell a particular common stock at a speciﬁed price within a speciﬁed period. These options are not created by the company that issued the underlying stock; rather, they are created by the exchange on which the option is to be traded.

To illustrate the inﬂuence of these factors an option’s value, con- sider the following stock option:

The right to buy a share of ABC stock at $40 a share before December 15th.

Since this is a right to buy an asset, we refer to this as a call option. This option gives the investor the right to buy a share of ABC stock at $40 per share—the exercise price, also called the strike price—before December 15th—the expiration date.

If ABC stock is currently trading for $35 a share, this option is referred to as out-of-the-money; that is, the current stock price is less that the exercise price of $40. Is this option worthless? The answer is no. The option to buy ABC stock at $40 a share is valuable (that is, the

THE FUNDAMENTALS OF VALUATION

option is worth more than $0) since there is some chance that the price of ABC stock will rise above $40 a share prior to December 15th. 5

If ABC stock is currently trading for $40 a share, this option is referred to as at-the-money; that is, the current stock price is equal to the strike price. Again, the option would be worth something since ABC stock may rise above the exercise price prior to December 15th.

If the ABC stock is currently trading for $45 a share, this option is referred to as in-the-money; that is, the current stock price is greater than the strike price. The option will be worth more than $5. Why? Because an investor today can buy the stock at $40 (exercising the option) and then sell it for $45 in the market, making a $5 proﬁt. There- fore, the option is at least worth $5. Again, since the stock price has a chance of rising further prior to December 15th, the option will be worth more than $5.

From this analysis, we can see that the greater the price of the underlying asset (the stock, in this case), the greater the value of the call option. That is, there is a direct relation between the price of the underlying asset and the value of the call option.

The option value is also affected by the exercise price. For a given price of ABC stock, the lower the exercise price, the greater the value of the option. For example, assume that the price of ABC stock is $45. The option with an exercise price of $40 will have a value greater than $5. Compare this with an option on ABC with an exercise price of $35. In this latter case, the ABC option will trade for some value greater than $10. Therefore, there is an inverse relation between the exercise price and the value of the call option.

The value of the option is also affected by the time value of money. The call option is the right to buy an asset sometime in the future. Since the option represents buying in the future, the greater the opportunity cost of funds, the greater the value of the option. By delaying the purchase of the asset, you can invest your funds in other assets—the greater the return available, the greater is the value of deferring the purchase of the asset. In other words, the greater the opportunity cost, the more valuable it is to have the option, which allows you to purchase the asset in the future (instead of today).

The value of the option is also inﬂuenced by the volatility of the value of the underlying asset. If ABC stock is currently trading for $35 a

5 Of course, the price of the stock may fall below $40. If the price of the stock is below $40, the option owner will not choose to exercise the option (that is, the option

owner will not purchase the stock). Since a call option is an option to buy and not an option to sell the stock, the option’s value depends on the probability that the stock’s price will be above the strike price prior to expiration.

Valuation of Securities and Options

share, the value of the option to buy ABC stock at $40 a share is inﬂu- enced by the probability that ABC stock will rise above $40 prior to the expiration date. What affects this probability? The more volatile the value of the underlying asset is expected to be, the more likely that it may increase in value prior to the expiration date. Therefore, there is a direct relation between the expected volatility of the underlying asset’s value and the value of the call option.

The time remaining to expiration also affects the value of the option. For example, if today is October 15th, and the ABC stock is trading for $35 a share, there would be two months prior to the option’s expiration. If, instead, today is November 15th and the ABC stock is trading for $35 a share, there is one month prior to expiration. In which case is it more likely that the option will become in-the-money before expiration? October 15th, because there is more time for the stock price to move upward. Therefore, there is a direct relation between the time to maturity of an option and the option’s value.

If we alter our example to make the option a put option in ABC stock—that is, an option to sell ABC stock—we would have a different set of relations between these factors and the value of the option. Con- sider a put option on ABC stock:

The right to sell a share of ABC stock at $40 a share before December 15th.

The put becomes more valuable in the following circumstances: ■ the lower the value of the asset, since the investor in the put option

gains when the exercise price is more than the asset’s actual value, ■ the higher the exercise price, since this means the investor can sell the asset for a higher price, ■ the lower the time value of money, since the investor is delaying selling the asset and getting the proceeds from that sale, ■ the more the expected volatility of the underlying asset’s value, since there is no proﬁt if the price of the underlying asset does not move, and ■ the longer the time to maturity, since there is more time for the underlying asset’s price to move below the exercise price.

The factors that affect the value of call and put options and their relation to the value of an option are summarized in Exhibit 9.6. Incor- porating these factors mathematically into the valuation of an option or option-like security is quite complex. That’s part of your advanced stud- ies in ﬁnance.

THE FUNDAMENTALS OF VALUATION

EXHIBIT 9.6 Relation between Call and Put Option Features and the

Value of an Option

Relation to a Feature

Relation to a

Call Option Value Put Option Value

Value of the Direct relation Inverse relation underlying

The greater the value of the The greater the value of the asset

underlying asset, the greater underlying asset, the lower the value of the option.

the value of the option. Exercise price Inverse relation

Direct relation The lower the exercise price,

The greater the exercise the greater the value of the

price, the greater the value option.

of the option. Time value of

Direct relation Inverse relation money

The greater the time value of The greater the time value of money, the greater the value

money, the lower the value of the option.

of the option. Volatility of

Direct relation Direct relation the underly- The greater the volatility of

The greater the volatility of ing asset’s

the value of the underlying the value of the underlying value

asset, the greater the value of asset, the greater the value the option.

of the option. Time to

Direct relation Direct relation maturity

The greater the time remaining The greater the time remain- to maturity, the greater the

ing to maturity, the greater value of the option.

the value of the option. In addition to options on stocks, as we discussed in our ABC exam-

ple, there are other types of option securities. ■

A warrant is the right to buy a specified stock at a specified price in a specified time period, generally attached to a corporate bond as a “sweetener” to make the bond more attractive. A warrant is therefore

a call option. ■

A detachable warrant is a warrant that can be sold separately from the bond and traded as a security. ■

A right is a call option given to shareholders to buy additional stock in the issuing corporation (usually at a discount from the current market price) for a limited period of time. Rights can be sold by shareholders or exercised. If they are sold to another investor, they are traded as securities.

Valuation of Securities and Options

In addition to these option securities, there are also securities with option-like features. A convertible bond is a bond that can be converted into common stock at the option of the investor. This bond is therefore

a combination of a straight bond (a bond without such a conversion feature) and an option to convert the bond to shares of stock. Another example is the putable bond. A putable bond is a bond that gives the investor the right to put or sell the bonds back to the issuer at a speci- ﬁed price, under certain speciﬁed conditions.

There are many option-like features that may affect the value of the security. These features include callability and convertibility. A bond with a call feature gives the bond issuer the right to buy back the bond from the investor for a specified price during a specified period. This feature provides the issuer with flexibility—for example, if interest rates decline, the issuer can call, or buy back, the bonds and then sell new bonds with a lower interest rate. Since the issuer is likely to call the bond when interest rates have declined below the bond’s coupon rate, the investor must reinvest the proceeds received when the bond is called at a lower interest rate. Conse- quently, a call feature increases the risk to the investor because the investor is exposed to the risk that the proceeds received will have to be reinvested at a lower rate. As a result of this risk, investors demand a higher yield to invest in a bond that has a call feature relative to an otherwise comparable bond that does not have this feature. Looked at from the issuer’s perspective, the issuer must pay a higher cost (in the form of a higher coupon rate) by issuing a bond with a call feature than one without a call feature.

A bond with a convertible feature gives the investor the right to exchange the bond for common stock of the issuer at a specified rate of exchange. This feature gives the investor flexibility. For example, if the common stock’s price increases sufficiently, the investor could exchange the bond for common stock. A convertible feature therefore increases the potential return on the bond since it could be turned into stock when it is attractive to do so.

We already know the value of a debt security is affected by its return (in the form of interest and principal payments) and the uncertainty associated with these interest and principal payments. Now we know features such as callability and convertibility also affect the value of debt securities.

In addition to the options found in securities, the ﬁnancial manager faces investment decisions that have options. In deciding whether or not to invest in a new product, the ﬁnancial manager has the option to post- pone or defer investment. This is a call option—the option to invest in the product at some future point in time.

Another example is the abandonment option. In evaluating an investment that was made in the past, the ﬁnancial manager has the

THE FUNDAMENTALS OF VALUATION

option to abandon the investment—stop production and sell off the assets. The option to abandon is a put option, since it is an option to sell the investment.

Looking at options in a broader perspective, we see that the owners of

a ﬁrm have the option to not pay the creditors, halting operations, selling off assets, and distributing the proceeds. This is a put option held by the owners since they control whether or not to pay off creditors or to default.

Whether we are talking about securities that are options, securities with option-like features, or financial decisions that contain options, the same five factors listed in Exhibit 9.6 apply in valuing them. Though the precise calculation of the value of options is beyond the scope of this text, you should be able to recognize the factors affecting the value of an option and how they could influence the financial decisions you will have to make.

Callable Bonds Some bonds have a feature, referred to as a call feature, that allows the