Get documents about "Market Returns and Risk
Description
Market Returns and Risk document sample
Document Sample
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
Ch 03 Tool Kit 1/28/2001
Chapter 3. Tool Kit for Risk and Return: Portfolio Theory and Asset Pricing Models
PROBABILITY DISTRIBUTIONS
The probability distribution is a listing of all possible outcomes and the corresponding probability.
Probability of
Occurrence Rate of Return Distribution
E F G H
0.10 10% 6% 14% 4%
0.20 10% 8% 12% 6%
0.40 10% 10% 10% 8%
0.20 10% 12% 8% 15%
0.10 10% 14% 6% 22%
1.00
EXPECTED RATE OF RETURN AND STANDARD DEVIATION
The expected rate of return is the rate of return that is expected to be realized from an investment. It is
determined as the weighted average of the probability distribution of returns.
To calculate the standard deviation, there are a few steps. First find the differences of all the possible returns
from the expected return. Second, square that difference. Third, multiply the squared number by the
probability of its occurrence. Fourth, find the sum of all the weighted squares. And lastly, take the square
root of that number.
Calculation of expected return and standard deviation for E
Expected rate of return for E Standard deviation for E
Probability of Deviation from Squared
Occurrence Rate of Return Product k hat deviation Sq Dev * Prob.
10% 10% 1.00% 0% 0.00% 0.00%
20% 10% 2.00% 0% 0.00% 0.00%
40% 10% 4.00% 0% 0.00% 0.00%
20% 10% 2.00% 0% 0.00% 0.00%
10% 10% 1.00% 0% 0.00% 0.00%
100% Sum: 0.00%
Expected Std. Dev. =
Rate of Return, Square root of
k hat = 10% sum = 0.00%
If the probabilities are fairly simple, then a short-cut method is to use the excel functions for AVERAGE and
ay "weigh
STDEVP, but to "trick" them by entering arguments more than once, in a way that "weights" them like
ten probability.
the probabilities. For example, for stock E we would enter 6% once, since it has only a one innten probability
lity.
We would enter 8% twice, since it has a two in ten probability. We would enter 10% 4 times, since it has
a four in ten probability. We can do the same thing with the standard deviation function. Note that we use
STDEVP and not STDEV, since we are measuring the standard deviation for the entire population and not
for a sample. We call this the "indirect" method.
Indirect method k hat = 10% s= 0.00%
Calculation of expected return and standard deviation for F
Michael C. Ehrhardt Page 1 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
Expected rate of return for F Standard deviation for F
Probability of Deviation from Squared
Occurrence Rate of Return Product k hat deviation Sq Dev * Prob.
10% 6% 0.60% -4% 0.16% 0.02%
20% 8% 1.60% -2% 0.04% 0.01%
40% 10% 4.00% 0% 0.00% 0.00%
20% 12% 2.40% 2% 0.04% 0.01%
10% 14% 1.40% 4% 0.16% 0.02%
100% Sum: 0.05%
Expected Std. Dev. =
Rate of Return, Square root of
k hat = 10% sum = 2.19%
Indirect method k hat = 10% s= 2.19%
Calculation of expected return and standard deviation for G
Expected rate of return for G Standard deviation for G
Probability of Deviation from Squared
Occurrence Rate of Return Product k hat deviation Sq Dev * Prob.
10% 14% 1.40% 4% 0.16% 0.02%
20% 12% 2.40% 2% 0.04% 0.01%
40% 10% 4.00% 0% 0.00% 0.00%
20% 8% 1.60% -2% 0.04% 0.01%
10% 6% 0.60% -4% 0.16% 0.02%
100% Sum: 0.05%
Expected Std. Dev. =
Rate of Return, Square root of
k hat = 10% sum = 2.19%
Indirect method k hat = 10.00% s= 2.19%
Calculation of expected return and standard deviation for H
Expected rate of return for H Standard deviation for H
Probability of Deviation from Squared
Occurrence Rate of Return Product k hat deviation Sq Dev * Prob.
10% 4% 0.40% -6% 0.36% 0.04%
20% 6% 1.20% -4% 0.16% 0.03%
40% 8% 3.20% -2% 0.04% 0.02%
20% 15% 3.00% 5% 0.25% 0.05%
10% 22% 2.20% 12% 1.44% 0.14%
100% Sum: 0.28%
Expected Std. Dev. =
Rate of Return, Square root of
k hat = 10.00% sum = 5.27%
Indirect method k hat = 10.00% s= 5.27%
Michael C. Ehrhardt Page 2 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
COVARIANCE
The covariance is a measure that combines the variance of a stock's return with the tendency of those returns
to move up or down at the same time another stock moves up or down.
To calculate the covariance, there are a few steps. First find the differences of all the possible returns from
the expected return; do this for both stocks. Second, multiply the differences of both stocks. Third, multiply
the previous product by the probability of its occurrence. Fourth, find the some of all the weighted products.
The result is the covariance.
Calculation of covariance between F and G
Probability of Deviation of F Deviation of G Product of Product *
Occurrence from k hat from k hat deviations Prob.
10% -4% 4% -0.1600% -0.02%
20% -2% 2% -0.0400% -0.01%
40% 0% 0% 0.0000% 0.00%
20% 2% -2% -0.0400% -0.01%
10% 4% -4% -0.1600% -0.02%
100%
Covariance =
sum = -0.048%
Calculation of covariance between F and H
Probability of Deviation of F Deviation of H Product of Product *
Occurrence from k hat from k hat deviations Prob.
10% -4% -6% 0.2400% 0.02%
20% -2% -4% 0.0800% 0.02%
40% 0% -2% 0.0000% 0.00%
20% 2% 5% 0.1000% 0.02%
10% 4% 12% 0.4800% 0.05%
100%
Covariance =
sum = 0.108%
Calculation of covariance between F and E
Probability of Deviation of F Deviation of E Product of Product *
Occurrence from k hat from k hat deviations Prob.
10% -4% 0% 0.0000% 0.00%
20% -2% 0% 0.0000% 0.00%
40% 0% 0% 0.0000% 0.00%
20% 2% 0% 0.0000% 0.00%
10% 4% 0% 0.0000% 0.00%
100%
Covariance =
sum = 0.000%
Michael C. Ehrhardt Page 3 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
CORRELATION COEFFICIENT
Like covariance, the correlation coefficient also measures the tendency of two stocks to move together, but it is
standardized and it is always in the range of -1 to +1. The correlation coefficient is equal to the covariance
divided by the product of the standard deviations.
Calculation of the correlation between F and G
rFG = Covariance FG ÷ SigmaF * SigmaG
= -0.048% ÷ 2.19% 2.19%
= -0.048% ÷ 0.048%
rFG = -1.0
Calculation of the correlation between F and H
rFH = Covariance FH ÷ SigmaF * SigmaH
= 0.108% ÷ 2.19% 5.27%
= 0.108% ÷ 0.116%
rFH = 0.935
PORTFOLIO RISK AND RETURN: THE TWO-ASSET CASE
Suppose there are two assets, A and B. X is the percent of the portfolio invested in asset A.
Since the total percents invested in the asset must add up to 1, (1-X) is the percent of the
portfolio invested in asset B.
The expected return on the portfolio is the weighted average of the expected returns on
asset A and asset B.
^ ^ ^
k p = X k A + (1 - X ) k B
The standard deviation of the portfolio, sp, is not a weighted average. It is:
sp = X2 s 2 + (1 - X) 2 sB + 2X(1 - X) rAB s A sB
A
2
ATTAINABLE PORTFOLIOS: THE TWO ASSET-CASE
Asset A Asset B
Expected return, k hat 5% 8%
Standard deviation, s 4% 10%
Using the equations above, we can find the expected return and standard deviation of a
portfolio with different percents invested in each asset.
Michael C. Ehrhardt Page 4 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
Correlation = 1
Proportion of
Proportion of Portfolio in
Portfolio in Security A Security A
(Value of X) (Value of 1-X) kp sp
1.00 0.00 5.00% 4.0%
0.90 0.10 5.30% 4.6%
0.80 0.20 5.60% 5.2%
0.70 0.30 5.90% 5.8%
0.60 0.40 6.20% 6.4%
0.50 0.50 6.50% 7.0%
0.40 0.60 6.80% 7.6%
0.30 0.70 7.10% 8.2%
0.20 0.80 7.40% 8.8%
0.10 0.90 7.70% 9.4%
0.00 1.00 8.00% 10.0%
rAB = +1: Attainable Set of Risk/Return
Combinations
10%
Expected return
5%
0%
0% 5% 10% 15%
Risk, sp
Correlation = 0
Proportion of
Proportion of Portfolio in
Portfolio in Security A Security A
(Value of X) (Value of 1-X) kp sp
1.00 0.00 5.00% 4.0%
0.90 0.10 5.30% 3.7%
0.80 0.20 5.60% 3.8%
0.70 0.30 5.90% 4.1%
0.60 0.40 6.20% 4.7%
0.50 0.50 6.50% 5.4%
0.40 0.60 6.80% 6.2%
0.30 0.70 7.10% 7.1%
0.20 0.80 7.40% 8.0%
0.10 0.90 7.70% 9.0%
0.00 1.00 8.00% 10.0%
Michael C. Ehrhardt Page 5 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
rAB = 0: Attainable Set of Risk/Return
Combinations
10%
Expected return 5%
0%
0% 5% 10% 15%
Risk, sp
Correlation = -1
Proportion of
Proportion of Portfolio in
Portfolio in Security A Security A
(Value of X) (Value of 1-X) kp sp
1.00 0.00 5.00% 4.0%
0.90 0.10 5.30% 2.6%
0.80 0.20 5.60% 1.2%
0.70 0.30 5.90% 0.2%
0.60 0.40 6.20% 1.6%
0.50 0.50 6.50% 3.0%
0.40 0.60 6.80% 4.4%
0.30 0.70 7.10% 5.8%
0.20 0.80 7.40% 7.2%
0.10 0.90 7.70% 8.6%
0.00 1.00 8.00% 10.0%
rAB = -1: Attainable Set of Risk/Return
Combinations
10%
Expected return
5%
0%
0% 5% 10% 15%
Risk, sp
Expected
Portfolio Efficient Set
Michael C. Ehrhardt Page 6 4/19/2011
Return, kp
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
ATTAINABLE AND EFFICIENT PORTFOLIOS: MANY ASSETS
Expected
Portfolio Efficient Set
Return, kp
Feasible Set
Risk, sp
Feasible and Efficient Portfolios
.
OPTIMAL PORTFOLIOS
Expected
IB2 I
Return, kp B1
Optimal Portfolio
IA2 Investor B
IA1
Optimal Portfolio
Investor A
Risk sp
Optimal Portfolios
.
Efficient Set with a Risk-Free Asset
Michael C. Ehrhardt Page 7 4/19/2011
Expected Z
Return, kp
.B
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
EFFICIENT SET WITH A RISK-FREE ASSET
Efficient Set with a Risk-Free Asset
Expected Z
Return, kp
.B
^
kM .
M
The Capital Market
kRF
A . Line (CML):
New Efficient Set
sM Risk, sp
.
OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET
Expected
Return, kp
CML
I2
I1
. .
^ M
kM
^ R
k R
R = Optimal
kRF Portfolio
sR sM Risk, sp
.
Michael C. Ehrhardt Page 8 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
CALCULATING BETAS
We downloaded stock prices and dividends from http://finance.yahoo.com for Wal-Mart,
using its ticker symbol, WMT. We also downloaded data for the Wilshire 5000 Index
(^WIL5), which contains most actively traded stocks, and the Fidelity Magellan mutual
fund (FMAGX). We computed returns, as shown in Chapter 2. We also obtained the
monthly rates on Treasury bills from the Federal Reserve, http://www.federalreserv.gov/.
kRF, Risk-free
kM, Market kp, Fidelity rate (Monthly Excess stock
Return (Wilshire kS, Wal-Mart Magellan Fund return on T- Excess market return
Date 5000 Index) Return Return bill) return (kM-kRF) (ks-kRF)
Apr-00 -5.3% -2.0% -4.3% 0.47% -5.7% -2.5%
Mar-00 7.5% 16.2% 9.1% 0.47% 7.1% 15.7%
Feb-00 0.5% -11.0% 1.0% 0.46% 0.0% -11.4%
Jan-00 -4.2% -20.8% -4.8% 0.44% -4.7% -21.2%
Dec-99 5.6% 20.1% 6.9% 0.43% 5.2% 19.7%
Nov-99 5.0% 2.3% 2.0% 0.42% 4.6% 1.9%
Oct-99 6.3% 18.4% 6.0% 0.41% 5.9% 18.0%
Sep-99 -2.7% 7.6% -1.5% 0.39% -3.1% 7.2%
Aug-99 -3.4% 4.9% -1.2% 0.39% -3.8% 4.5%
Jul-99 -0.9% -12.4% -3.4% 0.38% -1.3% -12.8%
Jun-99 5.1% 13.4% 6.5% 0.38% 4.7% 13.1%
May-99 -3.1% -7.3% -3.0% 0.38% -3.5% -7.7%
Apr-99 5.6% -0.2% 2.4% 0.36% 5.2% -0.6%
Mar-99 3.7% 7.3% 5.4% 0.37% 3.4% 6.9%
Feb-99 -2.9% 0.1% -3.2% 0.37% -3.3% -0.2%
Jan-99 2.7% 5.6% 5.6% 0.36% 2.3% 5.2%
Dec-98 6.3% 8.4% 6.4% 0.37% 5.9% 8.0%
Nov-98 5.7% 9.0% 7.8% 0.37% 5.4% 8.7%
Oct-98 7.8% 26.4% 7.7% 0.33% 7.4% 26.1%
Sep-98 6.4% -7.2% 6.0% 0.38% 6.0% -7.5%
Aug-98 -15.7% -6.5% -15.5% 0.41% -16.1% -6.9%
Jul-98 -2.3% 3.9% -0.7% 0.41% -2.7% 3.5%
Jun-98 3.4% 10.5% 4.3% 0.42% 3.0% 10.1%
May-98 -2.8% 9.0% -2.1% 0.42% -3.2% 8.6%
Apr-98 1.1% -0.5% 1.2% 0.41% 0.7% -0.9%
Mar-98 4.9% 10.1% 5.0% 0.42% 4.5% 9.6%
Feb-98 7.1% 16.3% 7.6% 0.42% 6.7% 15.9%
Jan-98 0.5% 1.0% 1.1% 0.42% 0.0% 0.5%
Dec-97 2.1% -1.2% 1.2% 0.43% 1.7% -1.6%
Nov-97 2.7% 14.5% 1.9% 0.43% 2.3% 14.0%
Oct-97 -3.4% -4.4% -3.4% 0.41% -3.8% -4.9%
Sep-97 5.8% 3.6% 5.9% 0.41% 5.4% 3.1%
Aug-97 -3.9% -5.3% -4.4% 0.43% -4.3% -5.8%
Jul-97 7.6% 10.9% 8.4% 0.42% 7.1% 10.5%
Jun-97 4.5% 13.6% 4.1% 0.41% 4.0% 13.2%
May-97 6.9% 6.2% 7.2% 0.42% 6.5% 5.8%
Apr-97 4.2% 0.9% 4.5% 0.43% 3.8% 0.5%
Mar-97 -4.5% 6.2% -3.4% 0.43% -5.0% 5.8%
Feb-97 -0.3% 11.1% -1.3% 0.42% -0.7% 10.6%
Jan-97 5.2% 4.4% 4.4% 0.42% 4.8% 4.0%
Dec-96 -1.3% -10.4% -2.0% 0.41% -1.7% -10.8%
Nov-96 6.4% -3.8% 6.5% 0.42% 6.0% -4.2%
Oct-96 1.3% 0.5% 2.4% 0.42% 0.9% 0.1%
Sep-96 5.2% 0.0% 4.0% 0.42% 4.7% -0.4%
Aug-96 3.0% 10.3% 2.5% 0.42% 2.6% 9.9%
Jul-96 -5.5% -5.4% -4.7% 0.43% -6.0% -5.8%
Michael C. Ehrhardt Page 9 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
Jun-96 -1.0% -1.5% -0.1% 0.42% -1.4% -2.0%
May-96 2.5% 8.4% 0.4% 0.42% 2.1% 8.0%
Average (Annual) 19.8% 45.3% 21.5% 4.9% 14.9% 40.3%
Standard deviation
(Annual) 16.6% 32.3% 16.8% 0.1% 16.7% 32.3%
Correlation with market. 0.54 0.97 -0.15 1.00 0.54
Slope 1.05 0.98 0.00 1.00 1.05
R-square 0.29 0.94 0.02 1.00 0.29
Using the AVERAGE function and the STDEV function, we found the average historical
returns and standard deviations. (We converted these from monthly figures to annual
figures. Notice that you must multiply the monthly standard deviation by the square root
of 12, and not 12, to convert it to an annual basis.) These are shown in the rows above.
We also use the CORREL function to find the correlation of the market with the other assets
Using the function Wizard for SLOPE, we found the slope of the regression line, which is
the beta coefficient. We also use the function Wizard and the RSQ function to find the
R-Squared of the regression.
Using the Chart Wizard, we plotted the Wal-Mart returns on the y-axis and the market
returns on the x-axis. We also used the menu Chart > Options to add a trend line, and to
display the regression equation and R2 on the chart. The chart is shown below. We also
used the regression feature to get more detailed data. These results are also shown below.
Wal-Mart Analysis
The beta coefficient is about 1.05, as shown by the slope coefficient in the regression
equation on the chart. The beta coefficient has a t statistic of 4.39, and there is virtually a
zero chance of getting this if the true beta coefficient is equal to zero. Therefore, this is a
statistically significant coefficient. However, the confidence interval ranges from 0.57 to
1.54, which is very wide. The R2 of about .29 indicates that 29% of the variance of the stock
return can be explained by the market. The rest of the stock's variance is due to factors
other than the market. This is consistent with the wide scatter of points in the graph.
Historic Wal-Mart Regression Results (See columns J-N)
Realized Beta
y = 1.054x + 0.0203 Returns
Coefficient 1.05
R² = 0.2948 on Wal-Mart, kS
t statistic 4.39
30%
Probability of t stat. 0.0%
Lower 95% confidence interval 0.57
20% Upper 95% confidence interval 1.54
Intercept
10%
Coefficient 0.02
t statistic 1.68
0% Probability of t stat. 10.0%
-30% -20% -10% 0% 10% 20% 30% Lower 95% confidence interval 0.00
Historic Realized Returns Upper 95% confidence interval 0.04
-10% on the Market, kM
-20%
-30%
Michael C. Ehrhardt Page 10 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
Magellan Analysis
The beta coefficient is about 0.98, as shown by the slope coefficient in the regression
equation on the chart. The beta coefficient has a t statistic of 25.67, and there is virtually a
zero chance of getting this if the true beta coefficient is equal to zero. Therefore, this is a
statistically significant coefficient. The confidence interval ranges from 0.90 to 1.05, which
is very small compared to the confidence interval for a single stock. The R2 of about 0.98
indicates that 98% of the variance of the portfolio return can be explained by the market.
This is consistent with the very narrow scatter of points in the graph. The estimate of the
intercept is equal to 0.00, and has a t statistic with a probability of 34.3%. Since this is
greater than 5%, we would say that the coefficient is not statistically significant-- in other
words, the true intercept might well be equal to zero.
Wal-Mart Regression Results (See columns J-N)
Historic
Realized Beta
y = 0.9757x + 0.0018 Returns Coefficient 0.98
R² = 0.9347 on Magellan, t statistic 25.67
kP
20% Probability of t stat. 0.0%
Lower 95% confidence interval 0.90
Upper 95% confidence interval 1.05
10%
Intercept
Coefficient 0.00
t statistic 0.96
Probability of t stat. 34.3%
Lower 95% confidence interval 0.00
0%
Upper 95% confidence interval 0.01
-30% -20% -10% 0% 10% 20% 30%
Historic Realized Returns
on the Market, kM
-10%
-20%
Michael C. Ehrhardt Page 11 4/19/2011
7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model
The Market Model vs. CAPM
We have been regressing the stock (or portfolio) returns against the market returns.
However, CAPM actually states that we should regress the excess stock returns (the stock
return minus the short-term risk free rate) against the excess market returns (the market
return minus the short-term risk free rate). We show the graph for such a regression
below. Notice that it is virtually identical to the market model regression we used earlier
for Wal-Mart. Since it usually doesn't change the results whether we use the market model
to estimate beta instead of the CAPM model, we usually use the market model.
Excess Returns
y = 1.0549x + 0.0205 on Wal-Mart,
R² = 0.2956 kS-kRF
30%
20%
10%
0%
-30% -20% -10% 0% 10% 20% 30%
Excess Returns
-10% on the Market, kM-kRF
-20%
-30%
Michael C. Ehrhardt Page 12 4/19/2011
