Coupon Bonds The present value of a bond is its current market price, which is the dis-

Coupon Bonds The present value of a bond is its current market price, which is the dis-

counted value of all future cash flows of the bond—the interest and principal. The yield to maturity on a coupon bond is the discount rate, put on an annual basis, that equates the present value of the interest and principal payments to the present value of the bond. So, in the case of a bond that pays interest semiannually, we first solve for the six-month yield, and then translate it to its equivalent annual yield-to-maturity.

Now let’s look at a the yield-to-maturity on a coupon bond. Going with interest paid semiannually, what is the yield to maturity on these

bonds if you bought them on January 1, 1997 for $96,500.00? Or, put another way, what annual yield equates the investment of $96,500.000 with the present value of the 22 interest cash flows and maturity value?

In this example, we know the following:

V = $1,000 × 96.5% × 100 = $96,500.00

C = (0.08875/2) × $1,000 × 100 bonds = $4,437.50

Valuation of Securities and Options

M = $1,000 × 100 bonds = $100,000.00 N = 11 years ×2

= 22 six-month periods and t identifies the six-month period we’re evaluating. Therefore,

$4,437.50 -------------------------- + --------------------------------- ∑ $100,000.00

( 1 + r d ) Where do we start looking for a solution to r d ? Before we revert to our

financial calculators, let’s think about the value of r d . If the bonds bonds). This would be equivalent to a six-month value of r d = 4.4375%

for six months. But these bonds are priced below par. That is, investors are not will- ing to pay the maturity value for these bonds because they can get a better return on similar bonds elsewhere. As a result, the price of the bonds is driven downward until these bonds provide a return or yield-to-maturity equal to that of bonds with similar risk.

Given this reasoning, the yield on these bonds must be greater than the coupon rate, so the six-month yield must be greater than 4.4375%. Using the trial and error approach, we would start with 5% as the six-month yield and look at the relation between the present value of the cash inflows (interest and principal) discounted at 5% and the price of the bonds (the $96,500.00):

Present value of bonds using a 5% discount rate

---------------------------- + --------------------------------- = ∑ $92,595.81

or, Present value of bonds using a 5% discount rate ≠ Present value of bonds

In fact, using 5%, we have discounted too much, since the present value of the bonds using 5% is less than the present value of the bonds. Therefore, we know that r d should be less than 5%. We now have an idea of where the yield lies: between 4.4375% and 5%. Using a financial calculator, we find the value of r d = 4.70%, a six-month yield:

THE FUNDAMENTALS OF VALUATION

Hewlett-Packard Hewlett-Packard Hewlett-Packard Texas Instruments 10B

12C

17B

BA-II Plus

4437.5 PMT 4437.5 PMT

4437.5 PMT 22 N

4437.5 PMT

22 N 100000 FV

22 n

22 N

100000 FV 96500 ± PV

100000 FV

100000 FV

96500 ± PV I/YR

96500 CHS PV

96500 ± PV

CPT I/Y Translating the six-month yield into an annual yield, we find that these

I%YR

bonds are valued such that the yield-to-maturity is 9.4%:

Yield to maturity = 4.7% × 2 = 9.4%

Another way of saying this is that the bonds are priced to yield 9.4% per year.

Why is the yield to maturity different from the annual yield of 10.48% that we calculated earlier? The annual yield was calculated using the begin- ning and end-of-year values of the bonds ($96,500.00 and $97,500.00), as well as the two interest payments. But the yield-to-maturity assumes that we buy the bonds for $96,500 and hold them until 2007, getting 22 interest payments and the $100,000 principal. So, we know that if we buy and hold these bonds for one year, we would have gotten a 10.48% annual return on our investment. But if we held onto these bonds, we would have gotten a 9.4% annual return. Remember: When we bought the bonds at the begin- ning of 1997, we didn’t know if the price of the bonds was going to go up, down, or stay the same since we didn’t know what was going to happen to interest rates during the year. But when we buy the bonds at the beginning of 1997 we do know what we will get at maturity—assuming the bond issuer is able to pay the principal at that time.

The bond’s price changes from January 1, 1997 to December 31, 1997 for two reasons:

■ As time progresses, the value of a bond tends toward its maturity value (we’ll show why and how next). ■ The value of the bonds change as yields change.

We now take a brief look at both of these considerations. The Value of Bonds as They Approach Maturity

Let’s focus on maturity, holding the yield constant at the January 1, 1997 yield. What is value of the bond if the yield-to-maturity is 9.4% per year and there are now 20 interest payments left, instead of 22? This is the

Valuation of Securities and Options

same as asking: What is the value of the bonds as of December 1997— two six-month periods later—if the yield-to-maturity does not change?

Present value of bonds on December 31, 1997

------------------------------- $4,437.50 + ---------------------------------- $100,000.00 = ∑ $96,643.83

t = 1 ( 1 + 0.047 ) ( 1 + 0.047 )

Moving ahead one more year, to December of 1992: Present value of bonds on December 31, 1998

18 ------------------------------- $4,437.50 ---------------------------------- = $100,000.00 + = $96,858.27 ∑

t = 1 1 + 0.047 t ) ( 1 + 0.047 ( 22 ) In Exhibit 9.4, we continue this calculation for each year to maturity. We

see that the value of the bond increases until it approaches the maturity value. The interest payments contribute less to the bond’s present value as time goes on since there are fewer interest payments through time, yet the maturity value contributes more as the bond approaches maturity—and hence more valuable—as we get closer to maturity. The change in the value of the bond as it approaches maturity is referred to as the time path of the bond.