Returns on Bonds If you invest in a bond, you realize a return from the interest it pays (if

Returns on Bonds If you invest in a bond, you realize a return from the interest it pays (if

it is a coupon bond) and from either the sale, the maturity, or call of the bond. We calculate the return on a bond in the same way we calculate

Valuation of Securities and Options

the return for a stock, except in the case of stock the cash flow is divi- dend income, rather than interest income.

There is another dimension to consider with bonds that we needn’t consider with common stocks: Bonds have a finite life since they either mature or are called. Therefore, we are interested in:

■ the realized return, which is the return over a specific period of time, ■ the yield if the bond is held to maturity, which is the return assuming

the bond is held to maturity, and ■ the yield to call, which is the return on the bond assuming the bond is called.

Realized Return

A bond’s return comprises the return from the appreciation or deprecia- tion in the value of the bond over the period—the capital yield—and the return from the interest received during the period—the coupon yield.

Return = Capital yield + Coupon yield

Let’s look at an investment in 100 Olympic Power bonds that

is, 96.5% of face value, or $965.00 per bond); at the end of 1997, they 8.875% on the par value of $1,000, or $88.75 for the year. If interest

were paid at the end of the year, the return on 100 bonds for 1997 is:

-------------------------------------------------------------------------- $97,500 $96,500 – + $8,875 --------------------- = $9,875 = = 10.2332%

Breaking down this return into its capital yield and coupon yield: Return on Olympic

= $97,500 $96,500 ------------------------------------------------- – + --------------------- $8,875 = 10.2332% maturing in 2007

Capital yield

Coupon yield

THE FUNDAMENTALS OF VALUATION

Because the interest is paid semiannually (each bond pays $44.375 on June 30th and December 31st), what return could you have earned if you bought 100 of these bonds on January 1, 1997 and held them through December 31, 1997? The semiannual interest payments make our computations a bit more complicated. But we can make our job eas- ier if we lay out the cash flows in an orderly fashion:

Beginning of

End of

End of

January 1997 June 1997 December 1997

Bond value $96,500.00 $97,500.00 Interest

4,437.50 Total

$101,937.50 The yield on these bonds is such that an investment of $96,500.00

will produce cash flows of $4,437.50 after six months and $101,947.50 after 12 months. Stated in the form of a present value equation, with r d representing the six-month yield,

$96,500.00 = -------------------------- + ---------------------------------

( 1 + r d ) Where do we start to solve for r d ? We can begin at either of two places.

For one, we know these bonds are selling at a discount from their par value of $1,000. This tells us the yield is greater than the coupon rate because investors are not willing to pay the maturity value, $1,000, to get interest of 8.875% per year. Therefore, the market rate must be something greater than 8.875%. So, we know the effective annual yield must be greater than 8.875%, which means that the six-month yield must be greater than 8.875%/2 = 4.4375%.

What we know, then, is that the semiannual yield is above 4.992% Using a financial calculator, r d is 5.1087%. If the yield over six months is 5.1087%, the effective annual yield for a year is 5.1087% com- pounded for two six-month periods:

Effective annual yield = (1 + 0.051087) 2 − 1 = 10.48% Now let’s look at an example of the return on a zero-coupon bond.

Suppose on January 1, 2001, you bought 10 Dot.com zero-coupon

What is your effective annual return on these bonds during 2000?

Valuation of Securities and Options

Return on Dot.com bonds during 2000 ------------------------------------------ $5,020 $4,775 – ------------------ = $245 = = 5.13%