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# Homework week 4

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The following are the historic returns for the Chelle Computer Company: Year Chelle Computer General Index
Year chelle computer general index
1 37 15
2 9 13
3 -11 14
4 8 -9
5 11 12
6 4 9

Based on this information, compute the following:
a. The correlation coefficient between Chelle Computer and the General Index. Answer : r= .1305
b. The standard deviation for the company and the index
Answer: sd of company= 14.209, sd of index= 8.266
c. The beta for the Chelle Computer Company
Problem 8
8. As an equity analyst, you have developed the following return forecasts and risk estimates for two different stock mutual funds (Fund T and Fund U): Forecasted Return CAPM Beta
Fund T 9.0% 1.20
Fund U 10.0% 0.80
a. If the risk-free rate is 3.9 percent and the expected market risk premium (i.e. E( RM) – RFR) is 6.1 percent, calculate the expected return for each mutual fund according to the CAPM. b. b. Using the estimated expected returns from Part a along with your own return forecasts, demonstrate whether Fund T and Fund U are currently priced to fall directly on the security market line (SML), above the SML or below the SML.

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c. According to your analysis, are Funds T and U overvlued, undervalued or properly valued? CAPM and the SML

8). As an equity analyst, you have developed the following return forecasts
and risk estimates for two different stock mutual funds (Fund T and Fund U):

Forecasted
Return CAPM
Beta
Fund T 9.0 1.20
Fund U 10.0 .80

a). If the risk-free rate is 3.9% and the expected market risk premium (i.e., E(RM) – RFR) is 6.1%, calculate the expected return for each mutual fund according to the CAPM. b). Using the estimated expected returns for Part a along with your own return forecasts, demonstrate whether Fund T and Fund U are currently priced to fall directly on the security market line (SML), above the SML, or below the SML. c). According to your analysis, the Funds T and U overvalued, undervalued, or properly valued? Solution to Question 8

a.Using the following equation we find the expected returns:

Expected Return on stock = Risk-free rate + Beta x (Market return – Risk-free rate)

Expected Return for Fund T = 3.9% + 1.2(6.1%) = 11.22%

Expected Return for Fund U = 3.9% + 0.8(6.1%)
= 8.78%

*Fund T 10.0% * Fund U

3.9%

1.8 1.2
c) Even though, both funds seem to fall directly on the SML line in the graph above; FUND T could be said to be undervalued at 9.0% of return, whereas the
expected return is 11.22% FUND U is overvalued at 10.0% being estimated by the analyst, whereas the expected return is actually 8.78%. The recommendation here would be to sell this stock, while buying FUND T. Problem 10

10. Draw the security market line for each of the following conditions: a) (1) RFR = 0.08; RM(proxy) – 0.12
(2) Rz = 0.06; RM(true) = 0.15
b) Rader Tire has the following results for the last six periods. Calculate and compare the betas using each index. RATES OF RETURN

Proxy Specific Index

True General Index
Period

(%)

(%)

(%)
1

29

12

15
2

12

10

13
3

-12

-9

-8
4

17

14

18
5

20

25

28
6

-5

-10

0

c) If the current period return for the market is 12 percent and for Rader Tire it is 11percent, are superior results being obtained for either index beta?

Solution

a. R

0.15
0.12

0.08
0.06

1.0 Beta

b. = Cov i,m/(m)2
Cov i,m = 187.4
m2 = 190.4
Using the proxy:
using proxy = 187.4/190.4 = .984

The true index the covariance = 176.4
using true = 176.4/168 = 1.05

c.Proxy
E(RR) = 0.08 + 0.984(0.12 – 0.08)
= 0.08 + 0.0394
= .1194 or 11.94 percent

True market
E(RR) = 0.06 + 1.05(0.12 – 0.06)
= 0.06 + 0.063
= 0.123 or 12.3 percent

Chapter 9
Problem 3
You have been assigned the task of estimating the expected returns for three different stocks: QRS, TUV, and WXY. Your preliminary analysis has
established the historical risk premiums associated with three risk factors that could potentially be included in your calculations: the excess return on a proxy for the market portfolio (MKT), and two variables capturing general macroeconomic exposures (MACRO1 and MACRO2). These values are: ƛMKT=7.5%, ƛMACRO1= -0.3%, and ƛMACRO2= 0.6%. You have also estimated the following factor betas (i.e. loadings) for all three stocks with respect to each of these potential risk factors: FACTOR LOADING

Stock

MKT

MACRO1

MACRO2

QRS

1.24

-0.42

0.00

TUV

0.91

0.54

0.23

WXY

1.03

-0.09

0.00

a) Calculated expected returns for the three stocks using just the MKT risk factor. Assume a risk-free rate of 4.5%. b) Calculate the expected returns for the three stocks using all three risk factors and the same 4.5% risk-free rate. c) Discuss the differences between the expected return estimates from the single-factor model and those from the multifactor model. Which estimates are most likely to be more useful in practice? d) What sort of exposure might MACRO2 represent? Given the estimated factor betas, is it really reasonable to consider it a common (i.e. systematic) risk factor? Problem 5

Suppose that three stocks (A, B, and C) and two common risk factors (1 and 2) have the following relationship: E( RA)= (1.1)ƛ1 + (0.8)ƛ2
E( RB)= (0.7)ƛ1 + (0.6)ƛ2
E( RC)= (0.3)ƛ1 + (0.4)ƛ2
a) If ƛ1 = 4% and ƛ2 = 2%, what are the prices expected next year for each of the stocks? Assume that all three stocks currently sell for \$30 and will not pay a dividend in the next year. Answer:

E (RA) = 1.1×0.04 + 0.8×0.02 = 0.044 + 0.016 = 0.06 or 6% E (Price A) = \$ 30 (1.06) = \$ 31.80 E(RB) = 0.7×0.04 + 0.6×0.02= 0.28 + .012= 0.04 or 4% E(Price B) = \$30(1.04) = \$31.20 E (RB) = 0.7×0.04 + 0.6×0.02 = 0.28 + .012 = 0.04 or 4% E (Price B) = \$ 30 (1.04) = \$ 31.20 E(RC) = 0.3×0.04 + 0.4×0.02= 0.12 + 0.008= 0.02 or 2% E(Price C) = \$30(1.02) = \$30.60 E (RC) = 0.3×0.04 + 0.4×0.02 = 0.12 + 0.008 = 0.02 or 2% E (Price C) = \$ 30 (1.02) = \$ 30.60

b) Suppose that you know that next year the prices for Stocks A, B, and C will actually be \$31.50, \$35.00, and \$30.50. Create and demonstrate a riskless, arbitrage investment to take advantage of these mispriced securities. What is the profit from your investment? You may assume that you can use the proceeds from any necessary short sale. Answer:

by the answer seen from the theoretical price, A, C is overvalued, B is undervalued, should buy B, both A and buy space C, construct an arbitrage portfolio is as follows: Buy 2 Unit B, respectively, while short one shares of A and a share C, stocks share portfolio weights are: WA = -0.5, WB = 1 and WC = -0.5. initial investment is: 30 +30-2 * 30 = 0 因素λ1的风险暴露为(-0.5)x1.1+ (+1.0)x0.7+(-0.5)x0.3=0 Risk factors λ1 exposed as (-0.5) x1.1 + (+1.0) x0.7 + (-0.5) x0.3 = 0 因素λ2的风险暴露为(-0.5)x0.8+ (+1.0)x0.6+ (-0.5)x0.4=0 The exposure factor λ2 is (-0.5) x0.8 + (+1.0) x0.6 + (-0.5) x0.4 = 0 收益= (\$30 – \$31.50) + 2x(\$35 – 30) + (\$30 – \$30.50)= -\$1.50 + \$10 – \$0.50= \$8Revenue = (\$ 30 – \$ 31.50) + 2x (\$ 35 – 30) + (\$ 30 – \$ 30.50) = – \$ 1.50 + \$ 10 – \$ 0.50 = \$ 8

Problem 7
a. Using regression analysis, calculate the factor betas of each stock associated with each of the common risk factors. Which of these deviations of the estimated factor correlation coefficients are statically significant? b. How well does the factor model explain the variation in portfolio returns? On what basis can you make an evaluation of this nature? c. Suppose you are now told that the three factors in Exhibit 9.12 represent the risk exposures in the Fama-French characteristic-based model (i.e., excess market, SMB, and HML). Based on your regression results which one of these factors is the most likely to be the market factor? Explain why. d. Suppose it is further revealed that Factor 3 is the HML factor. Which of the two portfolios is most likely to be a growth-oriented fund and which is the value-oriented fund? Explain why.

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