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.
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 - P
1 0
P
0


P
P
0

The income component of a return R
I
:

The total holding period return is simply
R = R + R =
T CA I

P

CF
P
0
P
0
1


P + CF
1 .
P
0
Cash Flow

CF
Initial Price P
0
1
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


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
  n  i

1 n
R

R + R +...+ R
1 2 n n
.
2

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 difference by its associated probability before summing them up: Var (R)
  2
R
 i n 

1
 p i

 R i
   2


If all of the possible outcomes are equally likely, then the formula becomes: n 

2
R
 i

1

R i

E(R)

2 n

Take the square root of the variance to get the standard deviation (

) .
B. 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 Fraction of Total
3

Deviations from the Mean Observations
1.000
1.645
68.26%
90%
1.960 95%
2.575 99%
C. Historical Market Performance

The key point is that, on average, annual returns have been higher for riskier securities. For instance, Exhibit 7.4 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.
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,
4

CV i


R i
E (R ) i
.

A lower value for the CV is what we are looking for.
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
  n   i

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 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.

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.
 
R
2
2 Asset Portfolio
 x
1
2

2
R
1
 x 2
2

2
R
2

2 x x
1 2

R
12
5

 
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:
Cov(R , R )
1 2
 
R
12
 i n 

1
 p i

(R
1, i

E(R )
1
(R
2, i

E(R )
2


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.
6


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.
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 or systematic risk .

Most of the risk-reduction benefits from diversification can be achieved in a portfolio with 15 to 20 assets.
7.5 Systematic Risk
7


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.

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.
8


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 .

We call the slope of the line of best fit beta.

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.
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
9


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

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
).
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.
B.
The Capital Asset Pricing Model and Portfolio Returns
10


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
 i n 

1 x
 i i
 x

1 1
 x

2 2
 x

3 3 x
 n n
.
11