2014 part ii ec solutions

7/26/2019 2014 Part II EC Solutions

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CHAPTER 5: RISK AND RETURN: PAST AND PROLOGUE

1. i and ii. The standard deviation is non-negative.

11. a. The holding period returns for the three scenarios are:

Boom: (35 25 + 1)/25 = 0.44 = 44.00%

Normal: (25 25 + 0.50)/25 = 0.02 = 2.00%

Recession: (10 25 + 0.25)/25 = 0.59 = 59.00%

E(HPR) = [(1/3) 44%] + [(1/3) 2%] + [(1/3) (59.00%)] = 4.33%

2

(HPR) = [(1/3) (44 (

4.33))]2

+ [(1/3) (2 (

4.33))]2

+ (1/3) [(59

(

4.33))]2

= 1788.22

= 22.1788 = 42.29%

b. E(r) = (0.5 4.33%) + (0.5 4%) = 0.165%

= 0.5 42.29% = 21.145%

13. a. Mean of portfolio = (1 y)rf+ y rP= rf+ (rP rf)y = 6 + 9y

If the expected rate of return for the portfolio is 12%, then, solving for y:

12 = 6 + 9y y =9

612 = 0.667

Therefore, in order to achieve an expected rate of return of 12%, the client

must invest 66.7% of total funds in the risky portfolio and 33.3% in T-bills.

b.

Security

Investment

Proportions

T-Bills 33.3%

Stock A 0.667 27% = 18.0%

Stock B 0.667 33% = 22.0%

Stock C 0.667 40% = 26.7%

c. P= 0.667 25% = 16.7% per year

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14. a. Portfolio standard deviation = P= y 25%

If the client wants a standard deviation of 20%, then:

y = (20%/25%) = 0.80 = 80.0% in the risky portfolio.

b. Expected rate of return = 6 + 9y = 6 + (0.80 9) = 13.20%

15. a. Slope of the CML =24

614 = 0.333

b. My fund allows an investor to achieve a higher expected rate of return for any

given standard deviation than would a passive strategy, i.e., a higher expected

return for any given level of risk.

28.

Average Rate of Return, Standard Deviation

and Reward-to-Variability Ratio of the Risk Premiums

for Small Common Stocksover ne !onth "ills

for #$%&-%''( and Various Sub-Periods

Risk Premium(%) Reward-to-

Mean SD aria!i"it# Ratio

1926-194$ 18.4 6. .$66

1944-196$ 17.84 29.86 .&97&

1964-198$ 1$.61 $&.$ .$8&&

1984-2$ 9.8 2&.69 .$&$&

1926-2$ 14.64 $8.72 .$78

Risk Premium(%) Reward-to- SD aria!i"it# Ratio

1926-194$ 7.1& 29.7 .246

1944-196$ 14.74 18.$8 .818

1964-198$ 2.76 17.29 .1&96

198$-2$ 9.8 16.79 .&48

1926-2$ 8.46 2.8 .471

a. For the entire period (1926-2003), small stocks had a lower reward-to-

variability ratio (0.3780) than large stocks (0.4071). However, in two of thefour sub-periods, small stocks performed better than large stocks.

b. In general, yes.

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CHAPTER 6: EFFICIENT DIVERSIFICATION

1. E(rP) = (0.5 16%) + (0.4 10%) + (0.10 6%) = 12.6%

13.

Year Annualized %

Large Stocks L-t T-Bonds

1984 6.46 15.9

1985 !."" !.68

1986 18.4" !.96 Large Stocks L-t T-Bonds

198# 5.!4 -.65 Large Stocks 1

1988 16.86 8.4" L-t T-Bonds ".848989 1

1989 !1.!4 19.49

199" -!." #.1!

1991 !".66 18.!9

199 #.#1 #.#9199! 9.8# 15.48

1994 1.9 -#.18

1995 !#.#1 !1.6#

1996 !."# -".81

199# !!.1# 15."8

1998 8.58 1!.5

1999 1."4 -8.#4

""" -9.1" ".#

""1 -11.89 4.1

"" -.1" 16.#9

""! 8.69 .!8

A$erage 14.!" 11.66Std. de$iation 1#.1 11.61

CALCULATION OF INVESTMENT OPPORTUNITY SET:

&ort'olio &ro(ortions &ort'olio

AB) *Y+ ,ean Std.e$.

"."" 1."" 11.66 11.61

".1" ".9" 11.9 11."6

"." ".8" 1.19 1".#8

".!" ".#" 1.45 1".#8

".4" ".6" 1.#1 11."8".5" ".5" 1.98 11.6!

".6" ".4" 1!.4 1.4

".#" ".!" 1!.5" 1!.4"

".8" "." 1!.## 14.5!

".9" ".1" 14."! 15.#8

1."" "."" 14.!" 1#.1

Minimum Variance Portfolio 0!"#$ 0%$&$ &!'& &0%"

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18. The probability distribution is:

Probability Rate of Return

0.6 100%0.4 -50%

Expected return = (0.6 100%) + 0.4 (50%) = 40%Variance = [0.6 (100 40)2] + [0.4 (50 40)2] = 5400

Standard deviation = &4%%= 73.48%

19. a. The risk of the diversified portfolio consists primarily of systematic risk. Beta

measures systematic risk, which is the slope of the security characteristic line

(SCL). The two figures depict the stocks' SCLs. Stock B's SCL is steeper, and

hence Stock B's systematic risk is greater. The slope of the SCL, and hence the

systematic risk, of Stock A is lower. Thus, for this investor, stock B is the riskiest.

b. The undiversified investor is exposed primarily to firm-specific risk. Stock A

has higher firm-specific risk because the deviations of the observations from the

SCL are larger for Stock A than for Stock B. Deviations are measured by the

vertical distance of each observation from the SCL. Stock A is therefore riskiest

to this investor.

20. Monthly rates of return, excess returns and means for the five-year period May

2000 through April 2005 are shown in the table on the next page. The calculation

of beta for GM is shown on the following page.

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Mont(l) rate* of return E+ce** Return*

,ont /, S0&5"" T-ills /, S0&5""

,a -"" -4.5# -.19 ".5" -5."# -.693un-"" -1#.#9 .!9 ".49 -18.8 1.91

3ul-"" -1.94 -1.6! ".51 -.45 -.15Aug-"" .94 6."# ".5 .4 5.55Se -"" -#.14 -5.!5 ".5 -#.66 -5.86ct-"" -4.4 -".49 ".5 -4.95 -1."o$-"" -".! -8."1 ".5! -".85 -8.54ec-"" .9" ".41 ".5" .41 -"."93an-"1 5.4 !.46 ".44 4.98 !."e-"1 -".#1 -9.! ".4 -1.1! -9.65,ar-"1 -.#6 -6.4 ".!8 -!.14 -6.8"A r-"1 5.#1 #.68 ".!! 5.!8 #.!5,a2-"1 !.81 ".51 ".!1 !.5" "."3un-"1 1!."9 -.5" ".!" 1.8" -.8"3ul-"1 -1.1# -1."# ".!" -1.46 -1.!#

Au -"1 -1!.9 -6.41 ".9 -14." -6.#"Se -"1 -1.64 -8.1# ". -1.8# -8.4"

ct-"1 -!.68 1.81 ".18 -!.8# 1.6!o$-"1 ".8 #.5 ".16 ".1 #.!6ec-"1 -.1 ".#6 ".14 -.!6 ".613an-" 5.! -1.56 ".14 5."9 -1.#"e-" !.6" -."8 ".15 !.45 -.,ar-" 14.1" !.6# ".15 1!.95 !.5A r-" 6.1 -6.14 ".15 5.9# -6.9,a -" -!.1 -".91 ".15 -!.6 -1."53un-" -14."" -#.5 ".14 -14.14 -#.!93ul-" -1.91 -#.9" ".14 -1!."5 -8."4

Aug-" .81 ".49 ".14 .68 ".!5Se(-" -18.# -11."" ".14 -18.86 -11.14ct-" -14.5 8.64 ".1! -14.66 8.51o$-" 19.4" 5.#1 ".1" 19.9 5.6"ec-" -#.15 -6."! ".1" -#.5 -6.1!

3an-"! -1.44 -.#4 ".1" -1.54 -.84e-"! -#."5 -1.#" ".1" -#.15 -1.8",ar-"! -".44 ".84 ".1" -".54 ".#4A r-"! #.! 8.1" ".1" #.1! 8."1,a2-"! -."" 5."9 "."9 -."9 5.""3un-"! 1.9" 1.1! "."8 1.8 1."53ul-"! !.9# 1.6 "."8 !.9" 1.55

Au -"! 9.8" 1.#9 "."8 9.# 1.#1Se -"! -".41 -1.19 "."8 -".49 -1.#ct-"! 4.5 5.5" "."8 4.1# 5.4o$-"! ".6 ".#1 "."8 ".18 ".6!ec-"! 4.8 5."8 "."8 4.#5 5.""3an-"4 -6.9# 1.#! "."8 -#."4 1.65e-"4 -!.14 1. "."8 -!. 1.14,ar-"4 -1.81 -1.64 "."8 -1.89 -1.#

A(r-"4 ".!6 -1.68 "."8 ".8 -1.#6,a -"4 -4.8 1.1 "."9 -4.!# 1.13un-"4 .64 1.8" ".11 .54 1.693ul-"4 -#.41 -!.4! ".11 -#.5 -!.54

Au -"4 -4.4 ".! ".1! -4.!# ".1"Se -"4 .8! ".94 ".14 .69 ".8"ct-"4 -9.5 1.4" ".15 -9.4" 1.5o$-"4 ".1" !.86 ".18 -"."# !.68ec-"4 !.81 !.5 ".19 !.6 !."63an-"5 -8.11 -.5! "." -8.!1 -.#!e-"5 -!.15 1.89 ". -!.!# 1.68

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,ar-"5 -1#.56 -1.91 ".! -1#.#9 -.15A r-"5 -1.15 -!.65 ".4 -1.!8 -!.88

A,era-e: .&$# .0'& 0!& .&%/ .0$&

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COVARIANCE MATRI:

GM S&P500

/, 1"#.98659#

S0&5"" 6.#881#9"6 ".4856691

SUMMARY OUTPUT OF ECEL RE1RESSION:

S7,,AY 7T&7T

Regression Statistics

,ulti(le ".5#88#"9

Suare ".!81996

Ad:. Suare ".!16616855

Standard ;rror 8.668"94!8

ser$ations 6"

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CHAPTER 7: CAPITAL ASSET PRICING

ANDARBITRAGE PRICING THEORY

1. a, c and d.

2. a. E(rX) = 5% + 0.8(14% 5%) = 12.2%

X= 14% 12.2% = 1.8%

E(rY) = 5% + 1.5(14% 5%) = 18.5%

Y= 17% 18.5% = 1.5%

b. i. For an investor who wants to add this stock to a well-diversified equity

portfolio, Kay should recommend Stock X because of its positive alpha, whileStock Y has a negative alpha. In graphical terms, Stock Xs expected

return/risk profile plots above the SML, while Stock Ys profile plots below

the SML. Also, depending on the individual risk preferences of Kays clients,

Stock Xs lower beta may have a beneficial impact on overall portfolio risk.

ii. For an investor who wants to hold this stock as a single-stock portfolio,

Kay should recommend Stock Y, because it has higher forecasted return and

lower standard deviation than Stock X. Stock Ys Sharpe ratio is:

(0.17 0.05)/0.25 = 0.48

Stock Xs Sharpe ratio is only:

(0.14 0.05)/0.36 = 0.25

The market index has an even more attractive Sharpe ratio:

(0.14 0.05)/0.15 = 0.60

However, given the choice between Stock X and Y, Y is superior. When a

stock is held in isolation, standard deviation is the relevant risk measure. For

assets held in isolation, beta as a measure of risk is irrelevant. Although

holding a single asset in isolation is not typically a recommended investment

strategy, some investors may hold what is essentially a single-asset portfolio

(e.g., the stock of their employer company). For such investors, the relevanceof standard deviation versus beta is an important issue.

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3. a. The beta is the sensitivity of the stock's return to the market return. Call the

aggressive stock Aand the defensive stock D. Then beta is the change in the

stock return per unit change in the market return. We compute each stock's

beta by calculating the difference in its return across the two scenarios divided

by the difference in market return.

%%.22%4

$$1'( =

=

2&.%2%4

1%6'D =

=

b. [Note: Part (b) of this problem should read: What is the expected return on

each stock if the market return is equally likely to be 4% or 20%?]

With the two scenarios equal likely, the expected rate of return is an average

of the two possible outcomes:

E(rA) = 0.5 (1% + 33%) = 17.0%

E(rD) = 0.5 (6% + 10%) = 8.0%

c.The SML is determined by the following: T-bill rate = 6% with a beta equal to zero,

beta for the market is 1.0, and the expected rate of return for the market is:

0.5 (20% + 4%) = 12.0%

The equation for the security market line is:

E(r) = 6% + (12% 6%)

See the following graph.

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D

A

E(r)

&%02

&!02

#02

%$2

302

1.0 2.0

SML

M

.25

d. The aggressive stock has a fair expected rate of return of:

E(rA) = 6% + 2.0(12% 6%) = 18.0%

The security analysts estimate of the expected rate of return is 17%. Thus

the alpha for the aggressive stock is:

A= actual expected return required return predicted by CAPM

= 17% 18% = 1.0%

Similarly, the required return for the defensive stock is:

E(rD) = 6% + 0.25(12% 6%) = 7.5%

The security analysts estimate of the expected return for D is 8%, and hence:D= 8% 7.5% = 0.50%

The points for each stock are plotted on the graph above.

e. The hurdle rate is determined by the project beta (i.e., 0..25) not by the firms

beta. The correct discount rate is therefore 7.50%, the fair rate of return on

stock D.

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22. a. Since the market portfolio, by definition, has a beta of 1.0, its expected rate

of return is 12%.

b. = 0 means the stock has no systematic risk. Hence, the portfolio'sexpected rate of return is the risk-free rate, 5%.

c. Using the SML, the fairrate of return for a stock with = 0.5 is:

E(r) = 5% + (0.5) (12% 5%) = 1.5%

The expectedrate of return, using the expected price and dividend for next year:

E(r) = ($44/$40) 1 = 0.10 = 10%

Because the expected return exceeds the fair return, the stock must be

under-priced.

23. a. Both the CAPM and APT require a mean-variance efficient market

portfolio. This statement is incorrect. The CAPM requires the mean-

variance efficient portfolio, but APT does not.

b. The CAPM assumes that one specific factor explains security returns but

APT does not. This statement is correct.

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2014 part ii ec solutions

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