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â€œ M=

r= 1 (1+ r)â€˜ (1+ r)â€œ

where V is the market price of the bond.

Finding the bondâ€™s yield, r, involves trial and error. It is best explained by example.

EXAMPLE 7.17 Suppose you were offered a 10-year, 8 percent coupon, $1,000 par value bond at a price of

$877.07. What rate of return could you earn if you bought the bond and held it to maturity? Recall that in Example

7.11 the value of the bond, $877.07, was obtained using the required rate of return of 10 percent. Compute this

bondâ€™s yield to see if it is 10 percent.

181

RISK, RETURN, AND VALUATION

CHAP. 71

First, set up the bond valuation model:

$80

+ $1,000

--

'O

V = $877.07 =

(I + r)' (1 + r)"

= $80(PVIFA,,lo) + $1,000(PVIF,lo)

Since the bond is selling at a discount under the par value ($877.07 versus $l,OOO), the bond's yield is above the

going coupon rate of 8 percent. Therefore, try a rate of 9 percent. Substitutingfactors for 9 percent in the equation,

we obtain:

V = $80(6.4177)+ $1,000(0.4224) = $513.42 + $422.4 = $935.82

The calculated bond value, $935.82, is above the actual market price of $877.07, so the yield is not 9 percent. To

lower the calculated value, the rate must be raised.Trying 10 percent, we obtain:

+ $l,000(0.3855) = $491.57 + $385.50 = $877.07

V = $80(6.1446)

This calculated value is exactly equal to the market price of the bond; thus, ZOpercent is the bond's yield to

maturity.

The formula that can be used to find the approximate yield to maturity on a bond is:

+ (M- V)ln

I

Yield =

(M+ V ) / 2

I = dollars of interest paid per year

where

M = the par value, typically $1,000 per share

V = a bond's value

n = number of years to maturity

This formula can also be used to obtain a starting point for the trial-and-error method discussed in

Example 7.17.

EXAMPLE 7.18 Using the same data as in Example 7.17 and the shortcut method, the rate of return on the

bond is:

+ (M- V)/n - $80 + ($l,OOO - $877.60)/10 - $80 + $12.24 - $92.24

Yield = I - --= 9.8%

(M+ V)/2 ($1,000 f $877.60)/2 $938.80 $938.80

which is very close to the exact rate of 10 percent.

Expected Rate of Return on Common Stock

The formula for computing the expected rate of return on common stock can be derived easily from

the valuation models.

The single-period return formula is derived from:

PO=-+- D1 p1

(1 + r ) (1 + r )

Solving for r gives:

r = D1+ (PI - P )

O

P

O

In words,

dividends + capital gain

Rate of return =

beginning price

= dividend yield + capital gain yield

182 RISK, RETURN, AND VALUATION [CHAP. 7

EXAMPLE 7.19 Consider a stock that sells for $50. The company is expected to pay a $3 cash dividend at the

end of the year, and the stockâ€™s market price at the end of the year is expected to be $55 a share. Thus, the expected

return would be:

or:

$3.00

-= 6%

Dividend yield =

$50

$5.00

Capital gain yield = -= 10%

$50

r = dividend yield + capital gain yield

= 6% + 10% = 16%

Assuming a constant growth in dividend, the formula for the expected rate of return on an

investment in stock can be derived as follows:

EXAMPLE 7.20 Suppose that ABC Companyâ€™s dividend per share was $4.50, expected to grow at a constant rate

of 6 percent. The current market price of the stock is $30. Then the expected rate of return is:

7.4 DETERMINING INTEREST-RATE RISK

Interest-rate risk of a debt instrument such as a bond can be determined in two ways. One way is

to look at the term structure of a debt security by measuring its average term to maturity - a duration.

The other way is to measure the sensitivity of changes in a debt securityâ€™s price associated with changes

in its yield to maturity. We will discuss two measurement approaches: Macaulayâ€™s duration coefficient

and the interest elasticity.

Duration

Duration (D), exactly known as Macaulayâ€™sDurution Coeficient,is an attempt to measure risk

more

in a bond by considering the maturity and the time pattern of cash inflows (i.e., interest payments and

principal). It is defined as the number of years until a bond pays back its principal.

EXAMPLE 7.21 A bond pays a 7 percent coupon rate annually on its $l,OOO face value if it has 3 years until its

maturity and has a YTM of 6 percent. The computation of duration involves the following three steps:

Step 1 Calculate the present value of the bond for each year.

Step 2 Express present values as proportions of the price of the bond.

Step 3 Multiply proportions by yearsâ€™ digits to obtain the weighted average time.

183

RISK, RETURN, AND VALUATION

CHAP. 71

(Step 2) (Step 3)

(Step 1)

(2) (3) (4) (6)

(5)

(1)

PV of PV as proportion (6)= (1) X (5)

PV factor

@p 6% Column (5)

CashfIow of price of bond

Year Cash flow

0.0643 0.0643

$66.04

0.9434

$70

1

0.0607 0.1214

62.30

0.8900

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