13: Lecture Risk and Return
Reading: RWJ
Chapter 12
Outline:
Rate of returns Defining risk Estimating risk and return Historical risk and return
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Motivation
We have by now developed a fairly complete theory of valuation and capital budgeting under certainty.
In order to extend this theory to situations involving uncertainty (risk), we need to understand
How do we define and measure risk? What is the link between the risk of an investment and its expected rate of return.
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Rate of Return on Financial Assets
The rate of return on a stock at any time t is a random variable:
~ r
~ ~ D1 P P0 1 P0
~ ~ D1 P 1 1 P0
D1 is the dividend at the end of period (random variable)
P0 is the price at the beginning of period ~ is the price at the end of period (random variable) P1 ~
~ is the rate of return earned for investing in the r
stock over the period.
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Statistics Reminder: Random Variables
A random variable is a variable that can take several possible values. We will denote random ~ variables by the superscript tilde (e.g., X).
For simplicity, we consider only discrete random variables that may take on finite number of values.
A probability distribution is a list of all possible outcomes and the probability that each will occur.
~ Pr( X
xj )
p j , j 1,..., n
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Example: Rate of Returns
The rate of returns over future periods is generally uncertain. For example, the S&P 500 index and Dell stock may have the following returns in different states of the world over the next period:
State of the world Probability Return on S&P500 Return on Dell stock 1 0.2 -5% -10% 2 0.6 10% 10% 3 0.2 20% 40%
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Statistics Reminder: Mean
Mean: the expected or forecasted value of a random variable. We will denote the mean of a random variable by the superscript bar (e.g., X ).
~ E( X )
n
X
j 1
pjxj
Example 1: Calculate the expected rate of returns on S&P 500 index and Dell stock in the previous example.
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Example 1: Expected Returns (Sol'n)
E (~ & P ) rS E (~ ) rDell
3
p j rj
j 1 3
(.2)( .05) (.6)(.1) (.2)(.2) .09 (.2)( .1) (.6)(.1) (.2)(.4) .12
p j rj
j 1
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The Required Rate of Return
The required rate of return for a financial investment (i.e., buying a stock), or the appropriate discount rate for its cash flows, is equivalent to the expected rate of return.
r ~ ~ ~ ~ D1 P E ( D1 ) E ( P ) 1 1 E (~ ) E r 1 1 P0 P0 ~ ~ E ( D1 ) E ( P ) 1 P0 1 r
The required rate of returns for non-financial investments equal to the expected returns on financial investments of similar risk.
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Components of Expected Return
An investment's expected return has two components:
Expected return = Risk-free return + Risk premium
Total expected return
Reward for waiting (time value of money) Reward for bearing risk
The risk of an investment (to be defined) determines the risk premium it earns and hence its expected return. The higher the risk, the greater the return.
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Realized vs. Expected Returns
In general realized returns will not equal expected returns, because future returns are random. We can write:
Realized return = Expected return + Unexpected return (or surprise) - or rt = E[rt] + unanticipated return
Example: What are the realized and expected returns for S&P 500 index and Dell stock? Intuitively, which one is "riskier?"
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What Do We Mean by "Risk?"
The fact that realized returns in general differ from expected returns is what makes assets risky.
Investment risk pertains to the probability of earning less than the expected return. Variance of returns is a good measure of risks
A good measure of risk should capture how much the actual returns differ from the expected return. Variance of a random variable measures how much the realized outcome differs from the expected outcome.
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Statistics Reminder: Variance
Variance: the average value of squared deviations from mean.
~ Var ( X )
2 x j 1 n
pj 2 xj
~ E( X )
Standard Deviation: the square root of variance. Also called volatility.
~ StD( X )
x
~ Var ( X )
Example: Calculate the variances and standard deviations of S&P 500 index and Dell stock.
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Example: Variance (Sol'n)
Since E (~ & P ) .09 (calculated before) rS Var (~ & P ) rS
3
p j rj
j 1
E (~ & P ) rS
2
(.2)( .05 .09) 2 (.6)(.1 .09) 2 (.2)(.2 .09) 2 .0064 StD(~ r ) Var (~ & P ) rS .0064 .08
S &P
S &P
Similarly for Dell Stock Var (~ ) .0256, rDell
Dell
.16
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Risk in Asset Returns is Substantial
Annual Asset Returns (1926-1996)
60
Percentage Return
40 20 0 -20 -40 -60 26 30 35 40 45 50
Common Stocks Long T-Bonds T-Bills
55
60
65
70
75
80
85
90
95
Year
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Estimating Risk: Some Key Assumptions
Investor Preferences
Higher mean in return is preferred Lower standard deviation (StD) in return preferred Investors care only about mean and StD (or variance)
Asset returns
Asset returns are IID, that is, successive returns are independently and identically distributed.
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Estimating Risk and Return
It suffices to estimate the mean and standard deviation of asset returns IID return implies that future return distribution can be estimated from past returns.
Return distribution over a particular horizon provides sufficient information on returns for all horizons. IID assumption is consistent with information-efficient market.
Estimation can be done using sample moments
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Statistics Reminder: Sample Moments
Let r1, r2, ..., rT be the historical returns on an asset from time t=1 to T. 1 T Sample Mean: ^ r rt T t1
Sample Variance:
^r
2
1 T 1t
T
rt
1
^ r
2
Sample Standard Deviation:
^r
^r
2
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Cumulative Historical Returns
1000 S&P Corp Bonds Long Bond T Bill 10
0.1 1925 1933 1941 1949 1957 1965 1973 1981 1989 1997
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Average Returns and StD 1926 - 1998
Average Average Average Standard Nominal Real Risk Deviation Return Return Premium of Returns 3.8% 0.7% 0.0% 3.2% 5.7 2.6 1.9 9.2 6.1 3.0 2.3 8.6 13.2 10.1 9.4 20.3 17.4 14.3 13.6 33.8
Portfolio Treasury Bills Government Bonds Corporate Bonds Common Stocks Small-Firm Stocks
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Historical Risk and Return
Historically, riskier securities offer higher expected returns.
This is because investors demand a risk premium for holding risky securities. In equilibrium, prices adjust to offer higher expected returns for riskier assets. More on the equilibrium pricing theory later
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Relation Between Different Asset Returns
If the return on asset A exactly offsets the return on asset B, i.e., rA = - rB, then if we buy A and B together, we can cancel all the risks and need not concern about risk premium on either asset. In general, we would need to know how the risks of different assets related to each other, in order to understand the risk of total investments. We need a measure of how much two random outcomes "vary together":
Covariance does exactly that!
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Statistics Reminder: Covariance
~ ~ Covariance: Let X , Y be two random variables
with a joint distribution of
~ Pr( X ~ xi , Y yj)
n xy i 1 j 1 m
pij , i 1,..., n, j 1,..., m.
pij xi ~ E( X ) y j ~ E (Y )
Then
~ ~ Cov( X , Y )
Correlation: A standardized measure of covariation.
~ ~ Corr ( X , Y )
xy
~ ~ Cov( X , Y ) ~ ~ StD( X ) StD(Y )
xy x y
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Example: Covariance
Calculate the covariance and correlation between S&P 500 index and Dell stock.
State of the world 1 2 3
Probability Return on S&P500
Return on Dell stock
0.2 -5%
-10%
0.6 10%
10%
0.2 20%
40%
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Example: Covariance (Sol'n)
We have calculated E (~ & P ) .09, E (~ ) .12, S & P .08, Dell .16, rS rDell Cov(~ & P , ~ ) (.2)( .05 .09)( .1 .12) rS rDell S & P , Dell (.6)(.1 .09)(.1 .12) (.2)(.2 .09)(.4 .12) .0122
S & P , Dell S & P , Dell S &P Dell
.0122 (.08)(.16)
.95
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