**Chapter 8**

**Problem 6**

The formula is:

- The correlation coefficient between Chelle Computer and the General Index.

=SQRT (2.48/6)

- Standard deviation of C computers
- = SQRT [{(Chelle return – expected return) ^2}/n]

**=1.57**

Standard deviation of General Index

= SQRT [{(General Index return – expected return) ^2}/n]=SQRT (1.1714/6)

**= 0.108**

COV (Chelle, G.Index)

= Sum of [(Chelle return – expected return) x (General return – expected general return)]/n

= 6.40%/6

**=0.011**

Coefficient of correlation = 0.11/(0.157 x 0.108)

**= ****0.6260**

Beta = [Cov (Ri, Rm) / var. (Rm)

- The beta for the Chelle Computer Company

= 8.83%/ 1.1714%

**=0.911**

**Problem 8**

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

- 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.

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

- If the risk-free rate is 3.9 percent and the expected market risk premium (i.e. E (RM) – RFR) 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.

10.0% * Fund U

3.9%

1.2

Assuming that the market is equilibrium, all securities should plot on the SML. Any security with an estimated rate of return that plot above the SML is considered undervalued because the rate of return that would be received will be above the required rate of return based on its systematic risk. In contrast, a security with estimated rate of return that plot below the SML is considered overvalued. This position relative to the SML means that the estimated rate of return is below the desired return based on the asset’s systematic risk.

- According to your analysis, are Funds T and U overvalued, undervalued or properly valued?

Given the pegged points we can conclude that Find U is undervalued because the investor will earn better rate that are above the required rate of return based on its systematic risk. Fund T is overvalued as it’s below the SML.

**Problem 10**

a. Draw the security market line for each of the following conditions:

R

- (1) RFR = 0.08; RM (proxy) – 0.12
- (2) Rz= 0.06; RM (true) = 0.15

0.15

0.12

0.08

0.06

1.0 Beta

b.Calculate and comparethe betas using each index.

= Covi,m/(m)2

Covi,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.If the current period return for the market is 12 percent and for Rader Tire it is 11percent, aresuperior results being obtained for either index beta?

Proxy

E (RR) = 0.08 + 0.984(0.12 – 0.08)

= 0.08 + 0.0394

**= .1194 or 11.94%**

True market

E (RR) = 0.06 + 1.05(0.12 – 0.06)

= 0.06 + 0.063

**= 0.123 or 12.3%**

**Chapter 9**

**Problem 3**

RQRS = 4.5 +7.5×1.24

- Calculated expected returns for the three stocks using just the MKT risk factor. Assume a risk-free rate of 4.5%.

= 4.5 + 6.825

**= 13.8%**

RTUV= 4.5 + 7.5×0.91

= 4.5 + 6.825

**= 11.325%**

RWXY = 4.5 + 7.5×1.03

= 4.5 +7.725

**= 12.225%**

RQRS = 4.5 + 7.5×1.24 + (-0.3) x (-0.42) + 0.6×0.00

- Calculate the expected returns for the three stocks using all three risk factors and the same 4.5% risk-free rate.

= 4.5 + 9.30 + 0.126 +0.00

**= 13.926%**

RTUV= 4.5 + 7.5×0.91 + (0.3) x (0.54) + 0.6×0.23

= 4.5 + 6.825 – 0.162 + 0.138

**= 11.301%**

RWXY = 4.5 + 7.5×1.03 + (-0.3) x (-0.09) +).6×0.00

= 4.5 + 7.725 + 0.027 + 0.00

**= 12.252%**

Assuming that the factor loadings are significant the three factor model should be more useful to the extent that the non-market factors pick up movements in returns not captured by the market return.

- 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?

Because the factor loadings on MACRO2 are zero for two of the stocks, it appears that MACRO2 is not a systematic factor, i.e., one that generally affects all stocks. It may represent industry- or firm-specific factors.

- 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 **

Because no stock pays a dividend, all return is due to price appreciation.

- Ifƛ1 = 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.

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**

In order to create a riskless arbitrage investment, an investor would short 1 share of A and one share of C, and buy 2 shares of B. The weights of this portfolio are WA = -0.5, WB = +1.0, and WC = -0.5.

- 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

The net investment is:

Short 1 share A = +$30

Buy 2 shares B= – $60

Short 1 share C= +$30

Net investment= $ 0

At the end of the period the profit is given by:

Profit = ($30 – $31.50) + 2x ($35 – 30) + ($30 – $30.50)

= -$1.50 + $10 – $0.50

**= $8**

**Problem 7**

Using a basic regression package, we get the following results (t-statistics given in parentheses):

- Using regression analysis, calculate the factor betas of each stock associated with each of the common risk factors. Which of these deviation of the estimated factor correlation coefficients are statistically significant?

RA = .0057 + 0.99x (Factor 1) – 0.201x (Factor 2) – 0.133x (Factor 3)

= (2.92*) (22.27*) (-4.61*) (-1.90)

adj. R2 **= .967**

RB = .0080 + 0.969x (Factor 1) + 0.059x (Factor 2) + 0.344x (Factor 3)

(2.85*) (14.67*) (0.91) (3.29*)

adj. R2 **= .900**

Significant at 5% level of significance.

The adjusted R2’s in both regressions are very high (.967 and .900, respectively.) This means the regressions account for over 90% of the variation of each portfolio’s returns around its respective mean.

- How well does the factor model explain the variation in portfolio returns? On what basis can you make an evaluation of this nature?

Factor 1 is the most likely candidate for the market factor, because it has a large, significant, and positive effect on both portfolios. Factors 2 and 3 have different signs in the two equations.

- 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.

A positive HML factor loading indicates a value stock or portfolio. Portfolio B has a positive loading on this factor, and therefore is the more likely candidate for the value oriented portfolio. A negative HML factor loading indicates a growth stock or portfolio. Portfolio A has a negative loading on this factor, and therefore is more likely to be a growth-oriented portfolio.

- 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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