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PVIFA,,12 =

$1,891.20

From Appendix D, a PVIFA of 10.5753 for 12 periods is at i = 2%. The annual interest rate is therefore

2% X 12 = 24%.

Bond Value. What amount should an investor be willing to pay for a $1,000,5-year United States

6.17

government bond which pays $50 interest semiannually and is sold to yield 8 percent?

SOLUTION

The semiannual interest is 4 percent. The value of the bond is:

V = I(PVIFA,,) + M(PVIF,,)

= $50(PVIFA4,10) + $1,OOO(PVIF,,,)

= $50(8.1109) + $1,000(0.6756) = $405.55 + $675.60 = $1,081.15

Bond Values. Calculate the value of a bond with a face value of $1,O00, a coupon interest rate

6.18

of 8 percent paid semiannually, and a maturity of 10 years, Assume the following discount rates:

(a) 6 percent; ( b ) 8 percent; and ( c ) 10 percent.

SOLUTION

8%o($LOoO) = $40

Semiannual interest =

2

Number of periods = 10 X 2 = 20 periods

V = I(PVIA,,) + M(PVIF,,,)

= $4O(PVIFA3,m) -t$1,000(PVIF3,20)

= $40(14.8775) + $1,000(0.5537) = $595.10 + $553.7 = $1,148.80

$40(PVIFA,,,o) -t$l,OOO(PVIF4,,,)

V =

= $40(13.5903) + $1,000(0.4564) = $543.61 + $456.40 = $1,000 (rounded)

V = $40(PVIFAs,,o)+ $1,000(PVIF5,,0)

= $40(12.4622) + $1,000(0.3769) = $498.49 + $376.90 = $875.37

Chapter 7

Risk, Return, and Valuation

7.1 RISK DEFINED

Risk (or uncertainty) refers to the variability of expected returns associated with a given investment.

Risk, along with the concept of return, is a key consideration in investment and financial decisions. This

chapter will discuss procedures for measuring risk and investigate the relationship between risk, returns,

and security valuation.

Probability Distributions

Probabilities are used to evaluate the risk involved in a security. The probability of an event taking

place is defined as the chance that the event will occur. It may be thought of as the percentage chance

of a given outcome.

EXAMPLE 7.1 A weather forecaster may state, вЂњThere is a 30 percent chance of rain tomorrow and a 70 percent

chance of no rain.вЂќ Then we could set up the following probability distribution:

Outcome Probability

30% = 0.3

Rain

07

.

No rain 70% =

100%

- = 1.00

-- -

Expected Rate of Return

Expected rate of return (F) is the weighted average of possible returns from a given investment,

weights being probabilities. Mathematically,

n

P= ripi

i=l

where ri = ith possible return

pi = probability of the ith return

n = number of possible returns

EXAMPLE 7.2 Consider the possible rates of return that you might earn next year on a $50,000 investment in

stock A or on a $50,000 investment in stock B, depending upon the states of the economy: recession, normal, and

prosperity.

For stock A:

Return (r,)

State of Economy Probability (p,)

0.2

-5%

Recession

20To 0.6

Normal

0.2

Prosperity 40Yo

For stock B:

State of Economy Return (r,) Probability (p,)

10% 0.2

Recession

15%

Normal 0.6

20% 0.2

Prosperity

171

172 RISK, RETURN, AND VALUATION [CHAP. 7

Then the expected rate of return (F) for stock A is computed as follows:

2r + (20%)(0.6) + (40%)(0.2) = 19%

ip1 = (-5%)(0.2)

P=

111

Stock Bs expected rate of return is:

'

(1O%)(O.2) + (15%)(O.6) + 2O%(O.2) 15%

P= =

Measuring Risk The Standard Deviation

The standard deviation ( U ) , which is a measure of dispersion of the probability distribution, is

commonly used to measure risk. The smaller the standard deviation, the tighter the probability

distribution and, thus, the lower the risk of the investment.

Mathematically,

I n

To calculate take the following steps:

U,

Step 1. Compute the expected rate of return (F).

Step 2. Subtract each possible return from r' to obtain a set of deviations (r,- r').

Step 3. Square each deviation, multiply the squared deviation by the probability of occurrence for

its respective return, and sum these products to obtain the variunce (2):

n

i= 1

Step 4. Finally, take the square root of the variance to obtain the standard deviation ( U ) .

To follow this step-by-step approach, it is convenient to set up a table.

EXAMPLE 7.3 Using the data given in Example 7.2, compute the standard deviation for each stock and set up

the tables as follows в‚¬or stock A:

Step 2

Step 1 Step 3

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