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INTRODUCTION TO ACTUARIAL SCIENCE 2013-14/S1: Tutorial 6 – Portfolio & Capital Market Theory

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SOLUTION TUTORIAL 6 – PORTFOLIO AND CAPITAL MARKET THEORY

Q1 BKM7Q4-10

(i). The parameters of the opportunity set are:

E(r S) = 20%, E(r B) = 12%, σS = 30%, σB = 15%, ρ = 0.10

From the standard deviations and the correlation coefficient we generate the covariancematrix [note that ( , )S B S BCov r r ]:

Bonds Stocks

Bonds 225 45Stocks 45 900

The minimum-variance portfolio is computed as follows:

wMin(S) = 1739.0)452(225900

45225

)r ,r (Cov2

)r ,r (Cov

BS

2

B

2

S

BS

2

B

wMin(B) = 1 0.1739 = 0.8261

The minimum variance portfolio mean and standard deviation are:

E(r Min) = (0.1739 × .20) + (0.8261 × .12) = .1339 = 13.39%

σMin

= 2/1

BSBS

2

B

2

B

2

S

2

S )]r ,r (Covww2ww[

= [(0.17392 900) + (0.82612 225) + (2 0.1739 0.8261 45)]1/2

= 13.92%

(ii).Proportion

in stock fund

Proportion

in bond fund

Expected

return

Standard

Deviation

0.00% 100.00% 12.00% 15.00%

17.39% 82.61% 13.39% 13.92% minimum variance

20.00% 80.00% 13.60% 13.94%

40.00% 60.00% 15.20% 15.70%

45.16% 54.84% 15.61% 16.54% tangency portfolio

60.00% 40.00% 16.80% 19.53%

80.00% 20.00% 18.40% 24.48%

100.00% 0.00% 20.00% 30.00%

0.00

5.00

10.00

15.00

20.00

25.00

0.00 5.00 10.00 15.00 20.00 25.00 30.00

TangencyPortfolio

MinimumVariancePortfolio

Efficient frontier of risky assets

CML

INVESTMENT OPPORTUNITY SET

f = 8.00

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(vii). Using only the stock and bond funds to achieve a portfolio expected return of 14%, we

must find the appropriate proportion in the stock fund (wS) and the appropriate

proportion in the bond fund (wB = 1 − wS) as follows:

.14 = .20 × wS + .12 × (1 − wS) = .12 + .08 × wS wS = 0.25

So the proportions are 25% invested in the stock fund and 75% in the bond fund. The

standard deviation of this portfolio will be:

σP = [(0.252 900) + (0.752 225) + (2 0.25 0.75 45)]1/2 = 14.13%

This is considerably greater than the standard deviation of 13.04% achieved using T-

bills and the optimal portfolio.

Q2 BKM 7 Q12

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be

created and the rate of return for this portfolio, in equilibrium, will be the risk-free rate.

To find the proportions of this portfolio [with the proportion wA invested in Stock A

and wB = (1 – wA ) invested in Stock B], set the standard deviation equal to zero. With

perfect negative correlation, the portfolio standard deviation is:σP = Absolute value [wAσA wBσB]

0 = 5 × wA − [10 (1 – wA)] wA = 0.6667

The expected rate of return for this risk-free portfolio is:

E(r) = (0.6667 × 10) + (0.3333 × 15) = 11.667%

Therefore, the risk-free rate is: 11.667%

Q3 BKM 7Q15.

The probability distribution is:Probability Rate of Return

0.7 100%

0.3 −50%

Mean = [0.7 × 100%] + [0.3 × (-50%)] = 55%

Variance = [0.7 × (100 − 55)2] + [0.3 × (-50 − 55)2] = 4725

Standard deviation = 47251/2 = 68.74%

Q4 BKM 7 Q16

σP = 30 = y × σ = 40 × y y = 0.75

E(r P) = 12 + 0.75(30 − 12) = 25.5%

Q5 BKM 8Q6

a. The standard deviation of each individual stock is given by: 2/1

i

22

M

2

ii )]e([

Since βA = 0.8, βB = 1.2, σ(eA ) = 30%, σ(eB ) = 40%, and σM = 22%, we get:

σA = (0.82 × 222 + 302 )1/2 = 34.78%

σB = (1.22 × 222 + 402 )1/2 = 47.93%

b. The expected rate of return on a portfolio is the weighted average of the expected

returns of the individual securities:

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E(r P ) = wA × E(r A ) + wB × E(r B ) + wf × r f

E(r P ) = (0.30 × 13%) + (0.45 × 18%) + (0.25 × 8%) = 14%

The beta of a portfolio is similarly a weighted average of the betas of the

individual securities:

βP = wA × βA + wB × βB + wf × β f

βP = (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78

The variance of this portfolio is:

)e( P

22

M

2

P

2

P

where 2

M

2

P is the systematic component and )e( P

2 is the nonsystematic

component. Since the residuals (ei ) are uncorrelated, the non-systematic variance

is:

2 2 2 2 2 2 2( ) ( ) ( ) ( ) P A A B B f f e w e w e w e

= (0.302 × 302 ) + (0.452 × 402 ) + (0.252 × 0) = 405

where σ2(eA ) and σ2(eB ) are the firm-specific (nonsystematic) variances of Stocks

A and B, and σ2(e f ), the nonsystematic variance of T-bills, is zero. The residual

standard deviation of the portfolio is thus:

σ(eP ) = (405)1/2 = 20.12%

The total variance of the portfolio is then:

47.699405)2278.0( 222

P

The total standard deviation is 26.45%.

Q6 BKM 8Q7

a. The two figures depict the stocks’ security characteristic lines (SCL). 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.

b. Beta is the slope of the SCL, which is the measure of systematic risk. The SCL for

Stock B is steeper; hence Stock B’s systematic risk is greater.

c. The R 2 (or squared correlation coefficient) of the SCL is the ratio of the explained

variance of the stock’s return to total variance, and the total variance is the sum o f theexplained variance plus the unexplained variance (the stock’s residual variance) :

)(eσσβ

σβR

i

22

M

2

i

2

M

2

i2

Since the explained variance for Stock B is greater than for Stock A (the explained

variance is 2

M

2

B , which is greater since its beta is higher), and its residual variance2 ( )

Be is smaller, its R 2 is higher than Stock A’s.

d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a small

positive alpha whereas Stock B has a negative alpha; hence, Stock A’s alpha is larger.

e. The correlation coefficient is simply the square root of R 2, so Stock B’s correlation with

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the market is higher.

Q7 BKM 8Q8

a. Firm-specific risk is measured by the residual standard deviation. Thus, stock A

has more firm-specific risk: 10.3% > 9.1%

b. Market risk is measured by beta, the slope coefficient of the regression. A has a

larger beta coefficient: 1.2 > 0.8

c. R 2 measures the fraction of total variance of return explained by the market return.

A’s R 2 is larger than B’s: 0.576 > 0.436

d. Rewriting the SCL equation in terms of total return (r) rather than excess return

(R):

( )

(1 )

A f M f

A f M

r r r r

r r r

The intercept is now equal to:

(1 ) 1% (1 1.2) f f r r

Since r f = 6%, the intercept would be: 1% 6%(1 1.2) 1% 1.2% 0.2%

Q8 BKM 8CFA2

The R 2 of the regression is: 0.702 = 0.49

Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk.

Q9 BKM 9Q3

a. False. β = 0 implies E(r) = rf , not zero.

b. False. Investors require a risk premium only for bearing systematic (undiversifiable or

market) risk. Total volatility includes diversifiable risk.

c. False. Your portfolio should be invested 75% in the market portfolio and 25% in T-bills.

Then:

Q10 BKM 9Q20

r 1 = 19%; r 2 = 16%; β1 = 1.5; β2 = 1

a. To determine which investor was a better selector of individual stocks we look at

abnormal return, which is the ex-post alpha; that is, the abnormal return is the

difference between the actual return and that predicted by the SML. Without

information about the parameters of this equation (risk-free rate and market rate of

return) we cannot determine which investor was more accurate.

b. If r f = 6% and r M = 14%, then (using the notation alpha for the abnormal return):

α1 = .19 – [.06 + 1.5 × (.14 – .06)] = .19 – .18 = 1%

α 2 = .16 – [.06 + 1 × (.14 – .06)] = .16 – .14 = 2%

Here, the second investor has the larger abnormal return and thus appears to be thesuperior stock selector. By making better predictions, the second investor appears

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to have tilted his portfolio toward underpriced stocks.

c. If r f = 3% and r M = 15%, then:

α1 = .19 – [.03 + 1.5 × (.15 – .03)] = .19 – .21 = – 2%

α2 = .16 – [.03+ 1 × (.15 – .03)] = .16 – .15 = 1%

Here, not only does the second investor appear to be the superior stock selector, but the first investor’s predictions appear valueless (or worse).

Q11 BKM 9Q21

a. Since the market portfolio, by definition, has a beta of 1, its expected rate of return is

12%.

b. β = 0 means no systematic risk. Hence, the stock’s expected rate of return in market

equilibrium is the risk-free rate, 5%.

c. Using the SML, the fair expected rate of return for a stock with β = – 0.5 is:( ) 0.05 [( 0.5) (0.12 0.05)] 1.5% E r

The actually expected rate of return, using the expected price and dividend for next year

is:

$41 $3( ) 1 0.10 10%

$40 E r

Because the actually expected return exceeds the fair return, the stock is underpriced.

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