ch07

(1)

CHAPTER 7

Risk and Return

(2)

Risk and Return

 The greater the risk, the larger the return

investors require as compensation for bearing that risk.

Future Value vs. Present Value

(3)

Quantitative Measures of

Return

 _{Total holding period return consists of two}

components: 1) capital appreciation and 2) income.

Holding Period Returns

R_CA = CapitalAppreciation InitialPrice 

P₁-P₀ P₀ 

P P₀

(4)

Quantitative Measures of

Return

 _{Total holding period return is simply}

R_I= Cash Flow InitialPrice 

CF₁ P₀

 Income component of a return R_I:

Holding Period Returns

1 1

T CA I

0 0 0

CF P CF

P

R R R (7.1)

P P P

  

(5)

Return

Holding Period Returns Example

One year ago today, you purchased a share of Dell Inc. stock for $26.50. Today it is worth $29.00. Dell paid no dividend on its stock. What total return did you earn on this stock over the past year?

1 0 1

T CA I

0

P - P + CF

R = R + R =

P

$29.00 - $26.50 + $0.00 =

(6)

Quantitative Measures of

Return

Expected Returns Example

You estimate that there is 30 percent chance that your total return on your Dell stock investment will be -3.45 percent, a 30 percent chance that it will be 5.17 percent , a 30 percent chance that it will be

12.07 percent and a 10 percent chance that it will be 24.14 percent. Calculate your expected return.

Dell

E(R ) = (0.3 × -0.0345) + (0.3 × 0.0517) + (0.3 × 0.1207) + (0.1 × 0.2414)

(7)

Quantitative Measures of

Return

 _{Expected value represents the sum of products of}

possible outcomes, and probabilities that those outcomes will be realized.

Expected Returns

 Expected return, E(R_Asset), is an average of

possible returns from an investment, where each of these returns is weighted by the probability that it will occur:

n

Asset i i 1 1 2 2 n n

i 1

E(R )  (p ×R )=(p ×R )+(p ×R )+...+(p ×R ) (7.2)



(8)

Quantitative Measures of

Return

Expected Returns

E R



_Asset





R_i

 

i1

n



n 

R₁+R₂ +...+R_n

n

 _{If each of the possible outcomes is equally likely}

(9)

Variance and Standard

Deviation

as Measures of Risk

Calculating the Variance and Standard Deviation

 _{The variance (}2) squares the difference between

each possible occurrence and the mean (squaring the differences makes all the numbers positive), and multiplies each difference by its associated probability before summing them up:









n ₂

2

i i

R _i=1

Var(R) =



=  p × R -E(R) (7.3)

 _{Take the square root of the variance to get the}

(10)

Variance and Standard

Deviation

as Measures of Risk

Calculating the Variance and Standard Deviation

_R2 _

R_i E(R)

 2 i1

n



n

 _{If all possible outcomes are equally likely, the}

(11)

 = > σ = 0.0838 or 8.38%

Profit (Ri)

Probability (Pi) (Ri)(Pi)

[Ri - E(R)](Pi)

-0,10

0,05

-0,0050

(-0,10 - 0,09)2(0,05)

-0,02

0,10

-0,0020

(-0,02 - 0,09)2(0,10)

0,04

0,20

0,0080

(0,04 - 0,09)2(0,20)

0,09

0,30

0,0270

(0,09 - 0,09)2(0,30)

0,14

0,20

0,0280

(0,14 - 0,09)2(0,20)

0,20

0,10

0,0200

(0,20 - 0,09)2(0,10)

0,28

0,05

0,0140

(0,28 - 0,09)2(0,05)

(12)

Deviation

as Measures of Risk

 _{Normal distribution is a symmetric frequency}

distribution that is completely described by its mean (average) and standard deviation.

Interpreting the Variance and Standard Deviation

 _{Normal distribution’s left and right sides are mirror}

images of each other. The mean falls directly in center of distribution. Probability that an outcome is a particular distance from the mean is the

(13)

Risk and Diversification

 By investing in two or more assets whose values

do not always move in same direction at same time, investors can reduce risk of investments or portfolio.

(14)

 _{Returns for individual stocks from one day to}

next are largely independent of each other and approximately normally distributed.

Single-Asset Portfolios

 _{A first pass at comparing risk and return for}

individual stocks is coefficient of variation, CV.

i

R i

i

CV (7.4) E(R )

(15)

Stock A Stock B

E(R) 0,08 0,24

σ 0,06 0,08

(16)

Risk and Diversification

 Coefficient of variation has a critical shortcoming not quite evident when only a single asset is

considered.

Portfolios with More than One Asset

E(R_Portfolio)  x₁E(R₁)  x₂E(R₂)

 _{Expected return of portfolio made up of two}

(17)

 _{Expected return of portfolio made up of multiple}

assets:







1₁ ₁

 

₂ ₂





_n



( ) ( ) (7.5)

= x ( ) x ( ) ... x ( )

n

Portfolio i i

i

n

E R x E R

E R E R E R





 

(18)

Risk and Diversification

Expected return of Portfolio Example

You invested $100,000 in Treasury bills that yield 4.5 percent; $150,000 in Proctor and Gamble stock, which has an expected return of 7.5 percent; and $150,000 in Exxon Mobil Corporation stock, which has an expected return of 9.0 percent. What is the expected return of this $400,000 portfolio?

&

$100,000

0.25 $400,000

$150,000

0.375 $400,000

TB

P G EMC

x

x x

 

(19)

Risk and Diversification

Expected return of Portfolio Example -continued

( ) (0.25 0.045) (0.375 0.075)

(0.375 0.090) =0.0731 or 7.31%

Portfolio

E R    

(20)

 Expected return of each asset must be found

before applying either of the two above formulas; fraction of portfolio invested in each asset must also be known.

 _{Prices of two stocks in a portfolio will rarely}

(21)

Risk and Diversification

 _{When stock prices move in opposite directions,}

the change in price of one stock offsets at least some of the change in price of other stock.

 _{Level of risk for portfolio of two stocks is less than}

average of risks associated with individual shares. The risk can be calculated with the variance

equation below:

1,2

2 Asset portfolio 1 2

2 2 2 2 2

1 2 R

1 2

R R R

(22)

Risk and Diversification

Portfolio Variance Example

The variance of the annual returns of CSX and Wal-Mart stocks in Exhibit 7.6 are 0.03949 and 0.02584 respectively. The covariance between the annual returns of these stocks is 0.00782. Calculate the variance of the portfolio that consists of 50 percent CSX stock and 50 percent Wal-Mart stock.

2 Portfolio

2 2 2

R

σ = (0.5) (0.03949) + (0.5) (0.02584)

(23)

Risk and Diversification

 _{In order to ease interpretation of covariance, we}

divide it by the product of the standard

deviations of returns for the two assets. This gives the correlation coefficient between the returns on the two assets:

1,2

1 1

(7.8)

R

R R





(24)

Correlation Coefficient Example

Find the correlation coefficient between the annual returns of CSX and Wal-Mart in Exhibit 7.6.

1 2 CSX 1 2 Wal-Mart 1,2 1 1 = = (0.03949) =0.199 (0.02584) =0.161 0.00782

= = =0.244