1.
Explain the relation between risk and return.
2.
Describe the two components of a total holding period return, and calculate this return for an asset.
3.
Explain what an expected return is, and calculate the expected return for an asset.
4.
Explain what the standard deviation of returns is, explain why it is especially useful in finance, and be able to calculate it.
5.
Explain the concept of diversification.
6.
Discuss which type of risk matters to investors and why.
7.
Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to evaluate whether the expected return of an asset is sufficient to compensate an investor for the risks associated with that asset.
7.1 Risk and Return

The greater the risk, the larger the return investors require as compensation for bearing that risk.

Higher risk means you are less certain about the ex post level of compensation.
Which stock would you invest in?
Prepared by Jim Keys 1

7.2 Quantitative Measures Return
A.
Holding Period Returns

The total holding period return consists of two components: (1) capital appreciation and (2) income.

The capital appreciation component of a return, R
CA
:
R =
CA
Capital Appreciation

Initial Price P
0
0


P
P
0

The income component of a return R
I
:
Cash Flow

CF
Initial Price P
0
1

The total holding period return is simply R = R + R =
T CA I

P

CF
P P
0 0
1


P + CF
1 .
P
0
Suppose a stock had an initial price of $78 per share, paid a dividend of $1.25 per share during the year, and had an ending share price of $87. Compute the percentage total return.
Total Percentage Return

$87

$ 78
$78

$1.25
$78

($87 $78

$1.25)
$78

$10.25
$78

.
1314

13 .
14 %
What was the dividend yield? The capital gains yield?
Dividend yield

D t

1
P t

$ 1 .
25
$78

.
01603

1 .
60 %
Capital gains yield

P t

1
P t

P t

($87

$ 78 )
$78

$ 9
$78

.
11538

11 .
54 %
Total holding period return = 13.14% = 1.60% + 11.54%
Prepared by Jim Keys 2

B.
Expected Returns

Expected value represents the sum of the products of the possible outcomes and the probabilities that those outcomes will be realized.

The expected return, E(R
Asset
), is an average of the possible returns from an investment, where each of these returns is weighted by the probability that it will occur:

Asset
  i n 

1
 p i

R i
  p
1

R
1
  p
2

R
2

....
 p n

R n
 where R i
is possible return i and p i
is the probability that you will actually earn return
R i
.

If each of the possible outcomes is equally likely (that is, p
1
= p
2
= p
3
= … = p n
= p =
1/ n ), this formula reduces to:

Asset
  i n 

1 n
R

R + R +...+ R
1 2 n n
.

Expected Return – The return on a risky asset expected in the future. Given all possible outcomes for a particular investment, the average rate of return is called the expected return. The actual return can differ from the expected return.
Risk premium = Expected return – Risk-free rate = E(R) – R f
Based on the following information, calculate the expected return.
E(R) = [.20 x (-.07)] + [.55 x .13] + [.25 x .30] = (-.014) + (.0715) + (.075) = .1325 = 13.25%
7.3 The Variance and Standard Deviation as Measures of Risk
A.
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
Prepared by Jim Keys 3

difference by its associated probability before summing them up:
Var (R)
 
2
R
 i n 

1
 p i

 R i
   2


Variance measures the dispersion of points around the mean of a distribution. In this context, we are attempting to characterize the variability of possible future security returns around the expected return. In other words, we are trying to quantify risk and return. Variance measures the total risk of the possible returns.

If all of the possible outcomes are equally likely, then the formula becomes:
Variance = n 

2
R
 i

1

R i

E(R)

2 n
Some experience confusion in understanding the mathematics of the variance calculation. They may have the feeling that they should divide the variance of an expected return by (n-1). We point out that the probabilities account for this division. We divide by n-1 in the historical variance because we are looking at a sample. If we looked at the entire population (which is what we are doing with expected values), then we would divide by n to get our historical variance.
This is the same as saying that the “probability” of occurrence is the same for all observations and is equal to 1/n.

Take the square root of the variance to get the standard deviation (

) .
Standard Deviation = σ
R

(σ
R
2
) 2
1
Based on the following information, calculate the variance and standard deviation.
From our previous calculations, E(R) = .1325 or 13.25%.
 2
R
= Variance(R) = .20(-.07 - .1325) 2 + .55(.13 - .1325) 2 + .25(.30 - .1325) 2

2
= Variance(R) = (.00820125) + (.00000344) + (.00701406) = .01521875
R
Prepared by Jim Keys 4

B.

R
= Standard Deviation(R) = (.01521875)
.5
= .123364 = 12.34%
Interpreting the Variance and Standard Deviation

The normal distribution is a symmetric frequency distribution that is completely described by its mean (average) and standard deviation.

The normal distribution is symmetric in that the left and right sides are mirror images of each other. The mean falls directly in the center of the distribution, and the probability that an outcome is a particular distance from the mean is the same whether the outcome is on the left or the right side of the distribution.

The standard deviation tells us the probability that an outcome will fall a particular distance from the mean or within a particular range:
Number of Standard
Deviations from the Mean
Fraction of Total
Observations
1.000
1.645
1.960
2.575
68.26%
90%
95%
99%
Prepared by Jim Keys 5

C. Historical Market Performance

The key point is that, on average, annual returns have been higher for riskier securities. For instance, Exhibit 7.3 shows that small stocks, which have the largest standard deviation of total returns, also have the largest average return. On the other end of the spectrum,
Treasury bills have the smallest standard deviation and the smallest average annual return.

The following are the basis for the nominal pretax rates of return reported by Ibbotson and
Sinquefield. o Large-company stocks – S&P 500 index, which contains 500 of the largest companies in terms of total market value in the U.S. o Small-company stocks – Smallest 20% of stocks listed on the New York Stock
Exchange based on market value of outstanding stock. o Long-term corporate bonds – High quality corporate bonds with 20 years to maturity. o Long-term government bonds – Portfolio of U.S. government bonds with 20 years to maturity. o U.S. Treasury bills – Portfolio of T-bills with a three-month maturity.
Prepared by Jim Keys 6

Prepared by Jim Keys 7

The average (or mean) rate of return is simply the arithmetic average, total returns divided by the number of observations. The average return is the best guess of what returns will be in any given year in the future.
Prepared by Jim Keys 8

7.4 Risk and Diversification

By investing in two or more assets whose values do not always move in the same direction at the same time, an investor can reduce the risk of his or her investments, or portfolio.
This is the idea behind the concept of diversification.
A.
Single-Asset Portfolios

Returns for individual stocks from one day to the next have been found to be largely independent of each other and approximately normally distributed.

A first pass at comparing risk and return for individual stocks is the coefficient of variation, CV,
CV i

E

R i .

The coefficient of variation is a measure of the risk associated with an investment for each one percent of expected return.

A lower value for the CV is what we are looking for.
Prepared by Jim Keys 9

B.
Portfolios with More Than One Asset

The coefficient of variation has a critical shortcoming that is not quite evident when we are only considering a single asset.

The expected return of a portfolio is made up of two assets:
E (R
Portfolio
)
 

The expected return of a portfolio is made up of multiple assets:
Portfolio
 
 i
 n 

1 x
1

 x i

E(R ) i
E(R )
1
 
 x
2

E(R )
2
    x n

E(R ) .
n


The expected return of each asset must be found before applying either of the two above formulas. The fraction of the portfolio invested in each asset, x n
, must also be known.

The prices of two stocks in a portfolio will rarely, if ever, change by the same amount and in the same direction at the same time.

When the stock prices move in opposite directions, the change in the price of one stock offsets at least some of the change in the price of the other stock.
Prepared by Jim Keys 10


As a result, the level of risk for a portfolio of the two stocks is less than the average of the risks associated with the individual shares.
 
2
R
2 Asset Portfolio
 x
2

2
1 R
1
 x
2

2
2 R
2

2 x x

R
12
Prepared by Jim Keys 11

 
R1,2
is the covariance between stocks 1 and 2. The covariance is a measure of how the returns on two assets covary, or move together:
2
 
R
12
 i n 

1
 p i

(R
1, i

E(R ) (R
2, i

E(R )


The covariance calculation is very similar to the variance calculation. The difference is that, instead of squaring the difference between the value from each outcome and the expected value for an individual asset, we calculate the product of this difference for two different assets.

In order to ease the interpretation of the covariance, we divide the covariance by the product of the standard deviations of the returns for the two assets. This gives us the correlation coefficient between the returns on the two assets,
  

R
12
 
R
1
R
2
.

The value of the correlation between the returns on two assets will always have a value between –1 and +1.

A negative correlation means that the returns tend to have opposite signs.

A positive correlation means that when the return on one asset is positive, the return on the other asset also tends to be positive.

A correlation of 0 means that the returns on the assets are not correlated.

If we have imperfect correlation between assets, or a correlation coefficient less than
+1, then we have a benefit from diversification by holding more than one asset with different risk characteristics.

As we add more and more stocks to a portfolio, calculating the variance becomes increasingly complex because we have to account for the covariance between each pair of assets.
Prepared by Jim Keys 12

C.
The Limits of Diversification

If the returns on the individual stocks added to our portfolio do not all change in the same way, then increasing the number of stocks in the portfolio will reduce the standard deviation of the portfolio returns even further.

However, the decrease in the standard deviation for the portfolio gets smaller and smaller as more assets are added.

As the number of assets becomes very large, the portfolio standard deviation does not approach zero. It only decreases up to a point.

That is because investors can diversify away risk that is unique to the individual assets, but they cannot diversify away risk that is common to all assets.

The risk that can be diversified away is called diversifiable, unsystematic, or unique risk, and the risk that cannot be diversified away is called nondiversifiable, systematic risk, or market risk .

Most of the risk-reduction benefits from diversification can be achieved in a portfolio with 15 to 20 assets.
Prepared by Jim Keys 13

7.5 Systematic Risk

With complete diversification, all of the unique risk is eliminated from the portfolio, but the investor still faces systematic risk.
A.
Why Systematic Risk Is All That Matters

Diversified investors face only systematic risk, whereas investors whose portfolios are not well diversified face systematic risk plus unsystematic risk.

Because diversified investors face less risk, they will be willing to pay higher prices for individual assets than other investors.

Therefore, expected returns on individual assets will be lower than the total risk
(systematic plus unsystematic risk) of those assets suggests they should be.
Prepared by Jim Keys 14


The bottom line is that only systematic risk is rewarded in asset markets, and this is why we are only concerned about systematic risk when we think about the relation between risk and return in finance.
B.
Measuring Systematic Risk

If systematic risk is all that matters when we think about expected returns, then we cannot use the standard deviation as a measure of risk since the standard deviation is a measure of total risk.

Since systematic risk is, by definition, risk that cannot be diversified away, the systematic risk (or market risk ) of an individual asset is really just a measure of the relation between the returns on the individual asset and the returns on the market.

We quantify the relation between the returns on a stock and the general market by finding the slope of the line of best fit between the returns of the stock and the general market .
Prepared by Jim Keys 15


We call the slope of the line of best fit beta.
Prepared by Jim Keys 16


If the beta of an asset is:

Equal to one, then the asset has the same systematic risk as the market.

Greater than one, then the asset has more systematic risk than the market.
 Less than one, then the asset has less systematic risk than the market.

Equal to zero, then the asset has no systematic risk.
Prepared by Jim Keys 17

Beta coefficients for selected companies
7.6 Compensation for Bearing Systematic Risk

The difference between required returns on government securities and required returns for risky investments represents the compensation investors require for taking risk:
E(R i
) = R rf
+ Compensation for taking risk i .

If we recognize that the compensation for taking risk varies with asset risk, and that systematic risk is what matters, we find:
E(R i
) = R rf
+ (Units of systematic risk i

Compensation per unit of systemic risk)
 If beta, β, is the appropriate measure for the number of units of systematic risk, we find:
Compensation for taking risk = β 
Compensation per unit of systemic risk

The required rate of return on the market, over and above that of the risk-free return, represents compensation required by investors for bearing a market (systematic) risk:
Compensation per unit of systemic risk = E(R m
) – R rf
→ this is referred to as the
“market risk premium.

Which brings us to the equation for expected return:
E(R i
) = R rf
+ β i
(E(R m
) – R rf
)
7.7 The Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is a model that describes the relation between risk and expected return: E(R i
) = R rf
+ β i
(E(R m
) – R rf
).
Prepared by Jim Keys 18

The CAPM demonstrates that the expected return for a given asset is a function of the following:
 the pure time value of money, R f
 the reward for bearing systematic risk, [E(R
M
) – R f
]
 the amount of systematic risk, β i
A stock has a beta of 0.9, the expected return on the market is 13 percent, and the risk-free rate is 6 percent. What must the expected return on this stock be?
E(R i
) = R f
+ [E(R
M
) – R f
] x β i
E(R i
) = .06 + (.13 - .06)(0.9) = .1230 = 12.30%
A stock has an expected return of 17 percent, the risk-free rate is 5.5 percent, and the market risk premium is 8 percent. What must the beta of this stock be?
.17 = .055 + (.08)(β i
)
.17 - .055 = (.08)(β i
)
β i
= .1150 / .08 = 1.4375
A stock has an expected return of 11.90 percent and a beta of .85, and the expected return on the market is 13 percent. What must the risk-free rate be?
.1190 = R f
+ (.13 - R f
)(.85)
.1190 = R f
+ .1105 - .85(R f
)
.1190 - .1105 = .15(R f
)
R f
= .0085 / .15 = .056667 = 5.67%
Prepared by Jim Keys 19

A.
The Security Market Line

Security Market Line (SML) is the line described by: E(R i
) = R rf
+ β i
(E(R m
) – R rf
)

The SML illustrates what the CAPM predicts the expected total return should be for various values of beta. The actual expected total return depends on the price of the asset: R =
T

P + CF
1
P
0
. If an asset’s price implies that the expected return is greater than that predicted by the CAPM, that asset will plot above the SML.
Prepared by Jim Keys 20

B.
The Capital Asset Pricing Model and Portfolio Returns

The expected return for a portfolio: E(R n Asset portfolio
) = R rf
+ β n Asset portfolio
(E[R m
] – R rf
)

The above can be found by applying the expected return and the beta of a portfolio:

The expected return of a portfolio:
Portfolio
  n   i


1
 x
1
 x i

E(R ) i

  x
2

E(R )
    x n
 
  n Asset Portfolio
 n  i

1 x
 i i
 x

1 1
 x

2 2
 x

3 3 x
 n n
.
Example:
Stock Amount Invested E(R i
)
IBM
GM
$6,000
$4,000
14.0%
9.0%
Portfolio Weight
$6,000 / $12,000 = 50%
Beta Product
1.47
$4,000 / $12,000 = 33.33% 1.19
.735
.397
Wal-Mart
Portfolio
$2,000
$12,000
8.0% $2,000 / $12,000 = 16.67% 0.91
100%
.152
1.284
E(R p
) = (.50)(14%) + (.3333)(9%) + (.1667)(8%) = 11.33%
Β p
= (.50)(1.47) + (.3333)(1.19) + (.1667)(.91) = 1.284
Prepared by Jim Keys 21

Beta, Beta, Who's Got the Beta?
Based on what we've studied so far, you can see that beta is a pretty important topic. You might wonder then, are all published betas created equal? Read on for a partial answer to this question.
We did some checking on betas and found some interesting results. The Value Line Investment Survey is one of the best-known sources for information on publicly traded companies. However, with the explosion of online investing, there has been a corresponding increase in the amount of investment information available online. We decided to compare the betas presented by Value Line to those reported by Yahoo! Finance ( finance.yahoo.com
) and CNN Money ( money.cnn.com
). What we found leads to an important note of caution.
Consider Amazon.com
, the big online retailer. Its beta reported on the Internet was 3.75, which is much larger than Value Line's beta of 1.25. Amazon.com
wasn't the only stock that showed a divergence in betas from different sources. In fact, for most of the technology companies we looked at, Value Line reported betas that were significantly lower than their online cousins. For example, the online beta for
Dell was 1.36, but Value Line reported 0.95. The online beta for computer antivirus company McAfee was 2.88 versus a Value Line beta of 1.60. Value Line's betas are not always lower. For example, the online beta for Yahoo! was 0.68, compared to Value Line's 1.55.
We also found some unusual, and even hard to believe, estimates for beta. Starwood Hotels had a very low online beta of 0.00, while Value Line reported 1.35. The online estimate for Hormel Foods, the famous maker of Spam (the lunch meat, not junk e-mail), was 0.06, compared to Value Line's 0.75.
Perhaps the most outrageous reported betas were the online betas for the International Fight League and
Nano Jet Corp., with betas of 77.4 and –64.06 (notice the minus sign!), respectively. Value Line did not report a beta for these companies. How do you suppose we should interpret a beta of –64.06?
There are a few lessons to be learned from all of this. First, not all betas are created equal. Some are computed using weekly returns and some using daily returns. Some are computed using 60 months of stock returns; some consider more or less. Some betas are computed by comparing the stock to the S&P
500 index, while others use alternative indices. Finally, some reporting firms (including Value Line) make adjustments to raw betas to reflect information other than just the fluctuation in stock prices.
The second lesson is perhaps more subtle. We are interested in knowing what the betas of the stocks will be in the future, but betas have to be estimated using historical data. Anytime we use the past to predict the future, there is the danger of a poor estimate. The moral of the story is that, as with any financial tool, beta is not a black box that should be taken without question.
Prepared by Jim Keys 22

Note on Arithmetic vs. Geometric Average

Arithmetic Versus Geometric Average – The arithmetic average return answers the question
“What was your return in an average year over a particular time period?” The geometric average return answers the question “What was your average compound return per year over a particular time period?”

Calculating Geometric Average Returns
The geometric average return over T periods is calculated as shown:
Geometric average return

[(1

R
1
) x (1

R
2
) x ...
x (1

R
T
)]
1/T 
1
A stock has had returns of 36 percent, 19 percent, 27 percent, −7 percent, 6 percent, and 13 percent over the last six years. What are the arithmetic and geometric returns for the stock?
Arithmetic return

(36

19

27

7

6

13)
6

94
6

15.6666

15.67%
Geometric return

[(1.36)(1.
19)(1.27)( 0.93)(1.06
)(1.13)]
1/6 
1
Geometric return

(2.2895854
04)
1/6 
1

.1480

14.80%

Arithmetic Average Return or Geometric Average Return?
If you are using averages calculated over a long period to forecast returns over a shorter period, the arithmetic average should be used. If you are forecasting for very long periods, you should use the geometric average.
Prepared by Jim Keys 23

1.
2.
In a game of chance, the probability of winning a $50 prize is 40 percent, and the probability of winning a $100 prize is 60 percent. What is the expected value of a prize in the game? a. b. c. d.
$50
$75
$80
$100
Use the following table to calculate the expected return for the asset.
3.
4.
5.
Return Probability
0.1 0.25
0.2
0.25
0.5
0.25 a. b. c. d.
15.00%
17.50%
18.75%
20.00%
The expected return for the asset below is 18.75 percent. If the return distribution for the asset is described as in the following table, what is the variance for the asset's returns?
Return Probability
0.1
0.2
0.25
0.25
0.5
0.25 a. b. c. d.
0.002969
0.000613
0.015195
0.054486
Ahmet purchased a stock for $45 one year ago. The stock is now worth $65. During the year, the stock paid a dividend of $2.50. What is the total return to Ahmet from owning the stock? (Round your answer to the nearest whole percent.) a. 5% b. c. d.
44%
35%
50%
Babs purchased a piece of real estate last year for $85,000. The real estate is now worth $102,000. If
Babs needs to have a total return of 25 percent during the year, then what is the dollar amount of income that she needed to have to reach her objective? a. b. c.
$3,750
$4,250
$4,750 d. $5,250
Prepared by Jim Keys 24

6.
7.
8.
9.
10.
Tommie has made an investment that will generate returns that are subject to the state of the economy during the year. Use the following information to calculate the standard deviation of the return distribution for Tommie's investment.
State Return Probability
Weak
OK
Great
0.13
0.2
0.25
0.3
0.4
0.3 a. b. c. d.
0.0453
0.0467
0.0481
0.0495
You invested $3,000 in a portfolio with an expected return of 10 percent and $2,000 in a portfolio with an expected return of 16 percent. What is the expected return of the combined portfolio? a. 6.2% b. c. d.
12.4%
13.0%
13.6%
The beta of Elsenore, Inc., stock is 1.6, whereas the risk-free rate of return is 8 percent. If the expected return on the market is 15 percent, then what is the expected return on Elsenore? a. 11.20% b. c. d.
19.20%
24.00%
32.00%
The expected return on Kiwi Computers stock is 16.6 percent. If the risk-free rate is 4 percent and the expected return on the market is 10 percent, then what is Kiwi's beta? a. 1.26
b. c. d.
2.10
2.80
3.15
The expected return on KarolCo. stock is 16.5 percent. If the risk-free rate is 5 percent and the beta of
KarolCo is 2.3, then what is the risk premium on the market? a. 2.5% b. c. d.
5.0%
7.5%
10.0%
Prepared by Jim Keys 25

MULTIPLE CHOICE
1. ANS: C
Learning Objective: LO 3
Level of Difficulty: Easy
Feedback: $50(0.4) + $100 (0.6) = $80
2. ANS: C
Learning Objective: LO 3
Level of Difficulty: Easy
Feedback: (0.1)(0.25) + (0.2)(0.5) + (0.25)(0.25) = 0.1875
3. ANS: D
Learning Objective: LO 4
Level of Difficulty: Easy
Feedback:
Return Probability
0.1
0.2
0.25
0.25
0.5
0.25
E(R) = .1875 (given)
Variance = .25(.10 - .1875)
2
+ .50(.20 - .1875)
2
+ .25(.25 - .1875)
2
Variance = .0019140625 + .000078125 + .000976562 = .0029687495
Standard Deviation = Variance
1/2
= .0029687495
1/2
= .054486 = 5.45%
4. ANS: D
Learning Objective: LO 3
Level of Difficulty: Easy
Feedback:
5. ANS: B
Learning Objective: LO 2
Level of Difficulty: Medium
Feedback:
6. ANS: B
Learning Objective: LO 4
Level of Difficulty: Medium
Prepared by Jim Keys 26

Feedback:
7. ANS: B
Learning Objective: LO 5
Level of Difficulty: Medium
Feedback:
8. ANS: B
Learning Objective: LO 6
Level of Difficulty: Hard
Feedback:
9. ANS: B
Learning Objective: LO 6
Level of Difficulty: Hard
Feedback:
10. ANS: B
Learning Objective: LO 6
Level of Difficulty: Hard
Feedback:
Prepared by Jim Keys 27