CAPM provides that investors are rewarded for bearing systematic risk which is attributed to stock’s response to a change in the market. Systematic risk can not be eliminated and the investor has to bear this risk. Though investors are generally risk averse but they have to assume systematic risk and more they assume it the higher will be the reward. Whereas, security’s individual risk or unsystematic risk can be eliminated by diversification. So the investors are actually rewarded for assuming systematic risk.
The CAPM can be explained like this that expected return on a security is equal to its level of systematic risk. The capital asset pricing model further provides that the beta of a stock remains stable over a period of time.
Systematic risk is market-related risk that cannot be diversified away. Because systematic risk cannot be diversified away, investors are rewarded for assuming this risk.
Each of the individual securities has its own risk (and return) characteristics, described as specific risk. By including a sufficiently large number of holdings, the specific risk of the individual holdings offset each other, diversifying away much of the overall specific risk and leaving mostly nondiversifiable or market-related risk.
Covariance measures the extent to which two securities tend to move, or not move, together. The level of covariance is heavily influenced by the degree of correlation between the securities (the correlation coefficient) as well as by each security’s standard deviation. As long as the correlation coefficient is less than 1, the portfolio standard deviation is less than the weighted average of the individual securities’ standard deviations. The lower the correlation, the lower the covariance and the greater the diversification benefits (negative correlations provide more diversification benefits than positive correlations).
The variance of an individual security is the sum of the probability-weighted average of the squared differences between the security’s expected return and its possible returns. The standard deviation is the square root of the variance. Both variance and standard deviation measure total risk, including both systematic and specific risk. Assuming the rates of return are normally distributed, the likelihood for a range of rates may be expressed using standard deviations. For example, 68 percent of returns may be expressed using standard deviations. Thus, 68 percent of returns can be expected to fall within + or -1 standard deviation of the mean, and 95 percent within 2 standard deviations of the mean.
The capital asset pricing model (CAPM) asserts that investors will hold only fully diversified portfolios. Hence, total risk as measured by the standard deviation is not relevant because it includes specific risk (which can be diversified away).
Under the CAPM, beta measures the systematic risk of an individual security or portfolio. Beta is the slope of the characteristic line that relates a security’s returns to the returns of the market portfolio. By definition, the market itself has a beta of 1.0. The beta of a portfolio is the weighted average of the betas of each security contained in the portfolio. Portfolios with betas greater than 1.0 have systematic risk higher than that of the market; portfolios with betas less than 1.0 have lower systematic risk. By adding securities with betas that are higher (lower), the systematic risk (beta) of the portfolio can be increased (decreased) as desired.
Systematic Risk
CAPM provides that investors are rewarded for bearing systematic risk which is attributed to stock’s response to a change in the market. Systematic risk can not be eliminated and the investor has to bear this risk. Though investors are generally risk averse but they have to assume systematic risk and more they assume it the higher will be the reward. Whereas, security’s individual risk or unsystematic risk can be eliminated by diversification. So the investors are actually rewarded for assuming systematic risk.
The CAPM can be explained like this that expected return on a security is equal to its level of systematic risk. The capital asset pricing model further provides that the beta of a stock remains stable over a period of time.
Systematic Risk
Political
Economic
Economic Factors
Change in
Political Factors
Unsystematic Risk
Each of the individual securities has its own risk (and return) characteristics, described as specific risk. By including a sufficiently large number of holdings, the specific risk of the individual holdings offset each other, diversifying away much of the overall specific risk and leaving mostly nondiversifiable or market-related risk.
Beta
Under the CAPM, beta measures the systematic risk of an individual security or portfolio. Beta is the slope of the characteristic line that relates a security’s returns to the returns of the market portfolio. By definition, the market itself has a beta of 1.0. The beta of a portfolio is the weighted average of the betas of each security contained in the portfolio. Portfolios with betas greater than 1.0 have systematic risk higher than that of the market; portfolios with betas less than 1.0 have lower systematic risk. By adding securities with betas that are higher (lower), the systematic risk (beta) of the portfolio can be increased (decreased) as desired.
Beta
If beta = 1.0, the security is just as risky as the average stock.
If beta > 1.0, the security is riskier than average.
If beta < 1.0, the security is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5
.
Beta
If beta = 1.0, the security is just as risky as the average stock.
If beta > 1.0, the security is riskier than average.
If beta < 1.0, the security is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5
.
Beta of a Stock
Market
Stock
Standard Deviation
0.124721913
0.062360956
Correlation
1
Beta
0.5
Year
Market Return
Return of Stock L
2001
10%
10%
2002
20%
15%
2003
-10%
0%
Calculate Beta
Year
Market Return
Return of Stock H
2001
10%
10%
2002
20%
30%
2003
-10%
-30%
Calculate Beta
Year
Market Return
Return of Stock A
2001
10%
10%
2002
20%
20%
2003
-10%
-10%
Calculate Beta
Beta =
(Cov of Stock or Portffolio with Market)
Variance of Market
Beta =
Excel
Slope
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the of your portfolio was 0.20 and M was 0.16, what would be the of the portfolio.
Beta of a Portfolio:
The beta of a portfolio is the weighted average of each of the stock’s betas
Calculate Portfolio Beta
Stock
Beta
Portfolio Weight
A
1.539
0.25
B
0.769
0.15
C
0.985
0.4
D
1.423
0.2
Stock
Beta
Portfolio Weight
Portfolio Beta
A
1.539
0.25
0.38475
B
0.769
0.15
0.11535
C
0.985
0.4
0.394
D
1.423
0.2
0.2846
1.179
Portfolio Beta = 1.179
Sakina has Rs. 35,000 invested in a stock which has a beta of 0.8 and Rs. 40,000 in a stock with a beta of 1.4. If these are only two investment in her portfolio what is her portfolio beta
Investment Beta
Rs. 35,000 0.8
40,000 1.4
Total 75,000
Portfolio Beta = (35,000/75,000)(0.8) + (40,000/75,000)(1.4)
= 1.12.
Briefly explain whether investors should expect a higher return from holding Portfolio A versus Portfolio B under capital asset pricing theory (CAPM). Assume that both portfolios are fully diversified.
` Portfolio A Portfolio B
Systematic risk (beta) 1.0 1.0
Specific risk for each individual security High Low
Under CAPM, the only risk that investors should be compensated for bearing is the risk that cannot be diversified away (systematic risk). Because systematic risk (measured by beta) is equal to one for both portfolios, an investor would expect the same return for Portfolio A and Portfolio B.
Since both portfolios are fully diversified, it doesn’t matter if the specified risk for each individual security is high or low. The specific risk has been diversified away for both portfolios
Required Rate of Return = RFR+ (MR-RFR)(BETA)
= 6 +(8 – 6) *(1.5)
= 9
Stock
Beta
U
0.85
N
1.25
D
–0.20
You expect an RFR of 10 percent and the market return (RM) of 14 percent. Compute the expected (required) return for the following stocks.
E(Ri) = RFR + i(RM - RFR)
= .10 + i(.14 - .10)
= .10 + .04i
Stock Beta (Required Return) E(Ri) = .10 + .04i
U 85 .10 + .04(.85) = .10 + .034 = .134
N 1.25 .10 + .04(1.25) = .10 + .05 = .150
D -.20 .10 + .04(-.20) = .10 - .008 = .092
Stock
Beta
A
1.72
B
1.14
C
0.76
D
0.44
E
0.03
F
-0.79
Calculate the required return for each of the following stocks when the risk-free rate is 0.08 and you expect the market return to be 0.14.
E(Ri) = RFR + bi (RM - RFR)
= .068 + bi (.14 - .08)
= .08 + .06bi
(a). E(RA) = .08 + .06(1.72) = .08 + .1050 = .1850 = 18.50%
(b). E(RB) = .08 + .06(1.14) = .08 + .0684 = .1484 = 14.84%
(c). E(RC) = .08 + .06(0.76) = .08 + .0456 = .1256 = 12.56%
(d). E(RD) = .08 + .06(0.44) = .08 + .0264 = .1064 = 10.64%
(e). E(RE) = .08 + .06(0.03) = .08 + .0018 = .0818 = 8.18%
(f). E(RF) = .08 + .06(-0.79) = .08 - .0474 = .0326 = 3.26%
Suppose you are the money manager of a Rs 4 Million investment fund. The fund consists of the following investments and betas:
Stock Investment Beta
A Rs 400,000 1.50
B 600,000 (0.50)
C 1,000,000 1.25
D 2,000,000 0.75
Total 4,000,000
If the market required rate of return is 14% and the risk free rate is 6%, what is the fund’s required rate of return.
Security Exp. Ret. Beta
HT 17.4% 1.30
Market 15.0 1.00
USR 13.8 0.89
T-Bills 8.0 0.00
Coll. 1.7 -0.87
Risk Free Rate of Return is 8%
Calculate (a) Required Rate of Return
(b) State whether the security is
undervalued or over valued
Security Exp. Ret. Req. Return Valuation
HT 17.4% 17.1 Under-Valued
Market 15.0 15.0 Fairly Valued
USR 13.8 14.2 Over Valued
T-Bills 8.0 8.0 Fairly Valued
Coll. 1.7 1.9 Over valued
Excess Stock Return = Stock Return – Risk Free Rate of Return
Exess Market Return = Market Return – Risk Free Rate of Return
Excess Return = Exess Market Return * Beta
Excess Return = C + Excess Market Return +
Excess Return = 0+ Excess Market Return + 0
Required Rate of Return = RFR + (MR-RFR)(BETA)
= 6 +(8 – 6) *(1.5)
= 9
Suppose you are the money manager of a Rs 4 Million investment fund. The fund consists of the following investments and betas:
Stock Investment Beta
A Rs 400,000 1.50
B 600,000 (0.50)
C 1,000,000 1.25
D 2,000,000 0.75
Total 4,000,000
If the market required rate of return is 14% and the risk free rate is 6%, what is the fund’s required rate of return.
Security Exp. Ret. Beta
HT 17.4% 1.30
Market 15.0 1.00
USR 13.8 0.89
T-Bills 8.0 0.00
Coll. 1.7 -0.87
Risk Free Rate of Return is 8%
Calculate (a) Required Rate of Return
(b) State whether the security is
undervalued or over valued
Security Exp. Ret. Req. Return Valuation
HT 17.4% 17.1 Under-Valued
Market 15.0 15.0 Fairly Valued
USR 13.8 14.2 Over Valued
T-Bills 8.0 8.0 Fairly Valued
Coll. 1.7 1.9 Over valued
Expected Return
Standard Deviation
Beta
Stock X
12.00%
20%
1.3
Stock Y
9
15
0.7
Market Index
10
12
1
Risk-free rate
5
The following information describes the expected return and risk relationship for the stocks of two of WAH’s competitors.
Using only the data shown in the preceding table:
a. Draw and label a graph showing the security market line and position stocks X and Y relative to it.
b. Compute the alphas both for Stock X and for Stock Y. Show your work.
c. Assume that the risk-free rate increases to 7 percent with the other data in the preceding matrix remaining unchanged. Select the stock providing the higher expected risk-adjusted return and justify your selection. Show your calculations.
(a). The security market line (SML) shows the required return for a given level of systematic risk. The SML is described by a line drawn from the risk-free rate: expected return is 5 percent, where beta equals 0 through the market return; expected return is 10 percent, where beta equal 1.0.
(b). The expected risk-return relationship of individual securities may deviate from that suggested by the SML, and that difference is the asset’s alpha. Alpha is the difference between the expected (estimated) rate of return for a stock and its required rate of return based on its systematic risk Alpha is computed as
ALPHA () = E(ri) - [rf + b(E(rM) - rf)]
where
E(ri) = expected return on Security i
rf = risk-free rate
bi = beta for Security i
E(rM) = expected return on the market
Calculation of alphas:
Stock X: = 12% - [5% + 1.3% (10% - 5%)] = 0.5%
Stock Y: = 9% - [5% + 0.7%(10% - 5%)] = 0.5%
In this instance, the alphas are equal and both are positive, so one does not dominate the other.
Another approach is to calculate a required return for each stock and then subtract that required return from a given expected return. The formula for required return (k) is
k = rf + bi (rM - rf ).
Calculations of required returns:
Stock X: k = 5% + 1.3(10% - 5%) = 11.5%
= 12% - 11.5% = 0.5%
Stock Y: k = 5% + 0.7(10% - 5%) = 8.5%
= 9% - 8.5% = 0.5%
(c). Calculations of revised alphas:
Stock X = 12% - [7% + 1.3 (10% - 7%]
= 12% - 10.95% = 1.1%
Stock Y = 9% - [7% + 0.7(10% - 7%)]
= 9% - 9.1% = -00.1%
By increasing the risk-free rate from 5 percent to 7 percent and leaving all other factors unchanged, the slope of the SML flattens and the expected return per unit of incremental risk becomes less. Using the formula for alpha, the alpha of Stock X increases to 1.1 percent and the alpha of Stock Y falls to -0.1 percent. In this situation, the expected return (12.0 percent) of Stock X exceeds its required return (10.9 percent) based on the CAPM. Therefore, Stock X’s alpha (1.1 percent) is positive. For Stock Y, its expected return (9.0 percent) is below its required return (9.1 percent) based on the CAPM. Therefore, Stock Y’s alpha (-0.1 percent) is negative. Stock X is preferable to Stock Y under these circumstances.
An analyst expects a risk-free return of 4.5 percent, a market return of 14.5 percent, and the returns for Stocks A and B that are shown in the following table.
Stock Beta Analyst’s Estimated Return
A 1.2 16%
B 0.8 14%
a. Show on the graph
(1) Where Stock A and B would plot on the security market line (SML) if they were fairly valued using the capital asset pricing model (CAPM)
(2) Where Stock A and B actually plot on the same graph according to the returns estimated by the analyst and shown in the table
b. State whether Stock A and B are undervalued or overvalued if the analyst uses the SML for strategic investment decisions.
(a). Security Market Line
Fair-value plot. The following template shows, using the CAPM, the expected return, ER, of Stock A and Stock B on the SML. The points are consistent with the following equations:
ER on stock = Risk-free rate + Beta x (Market return – Risk-free rate)
ER on stock = Risk-free rate + Beta x (Market return – Risk-free rate)
ER for A = 4.5% + 1.2(14.5% - 4.5%)
= 16.5%
ER for B = 4.5% + 0.8(14.5% - 4.5%)
= 12.5%
Over vs. Undervalue
Stock A is overvalued because it should provide a 16.5% return according to the CAPM whereas the analyst has estimated only a 16.0% return.
Stock B is undervalued because it should provide a 12.5% return according to the CAPM whereas the analyst has estimated a 14% return.
Company
ai(Intercept)
i
riM
Intell
0.22
12.10%
0.72
Ford
0.1
14.6
0.33
Busch
0.17
7.6
0.55
Merck
0.05
10.2
0.6
S&P 500
0
5.5
1
Based on five years of monthly data, you derive the following information for the companies listed:
a. Compute the beta coefficient for each stock.
b. Assuming a risk-free rate of 8 percent and an expected return for the market portfolio of 15 percent, compute the expected (required) return for all the stocks and plot them on the SML.
c. Plot the following estimated returns for the next year on the SML and indicate which stocks are
undervalued or overvalued.
• Intel—20 percent
• Ford—15 perent
• Anheuser Busch—19 percent
• Merck—10 percent
then COVi,m = (ri,m)(i)( sm)
For Intel:
COV i,m = (.72)(.1210)(.0550) = .00479
For Ford:
COV i,m = (.33)(.1460)(.0550) = .00265
For Anheuser Busch:
COV i,m = (.55)(.0760)(.0550) = .00230
For Merck:
COV i,m = (.60)(.1020)(.0550) = .00337
E(Ri) = RFR + Bi(RM - RFR)
= .08 + Bi(.15 - .08)
= .08 + .07Bi
Stock Beta E(Ri) = .08 + .07Bi
Intel 1.597 .08 + .1118 = .1918
Ford .883 .08 + .0618 = .1418
Anheuser Busch .767 .08 + .0537 = .1337
Merck 1.123 .08 + .0786 = .1586