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2021 EXAMINATIONS

PART II (SECOND AND FINAL YEAR)

ACCOUNTING AND FINANCE

AcF 321 INVESTMENTS

QUESTION 1

ANSWER ALL PARTS OF THE QUESTION

a.    Explain how the CAPM beta of a stock is estimated.

[3 marks]

b.   Suppose the index model for stocks A and B is estimated from excess returns with the following results:

!  = 3% + 0.7"  + !

#  = − 2% + 1.2"  + #

"  = 20%; _!  = 0.20; _#  = 0.12

i.      What is the standard deviation of each stock?                                          [4 marks]

ii.      Break down the variance of each stock into its systematic and idiosyncratic           components.                                                                                              [4 marks]

iii.      What are the covariance and the correlation coefficient between the two stocks?  [4 marks]

iv.      What is the covariance between each stock and the market index?         [4 marks]

v.      Consider portfolio P with investment propositions of 0.60 in A and 0.40 in B. What is the covariance between portfolio P and the market index?                   [6 marks]

vi.      Now, consider another portfolio Q with investment propositions of 0.50 in A, 0.30  in the market, and the remaining T-Bills. What is the covariance between portfolio Q and the market index?                                                                           [8 marks]

Solution:

i.

The standard deviation of each stock can be derived from the following equation for R2 :

2 β i(2)σ M(2) Explained variance

σ i

Therefore:

s2  = bA2sM(2) = 0.72 ´.202 = .098

A R2                        0.20

s  = .3130 = 31.30%

For stock B:

s2  = bB2sM(2) = 1.22 ´.202 = .048

B R2                        0.12

s  = .6928 = 69.28%

ii.

The systematic risk for A is: b ´ sM(2)   = 0.702  ´202  = 196

The firm-specific risk of A (the residual variance) is the difference between A’s total

risk and its systematic risk: 980 – 196 = 784

The systematic risk for B is:

b ´ sM(2)   = 1.202 ´202  = 576

B’s firm-specific risk (residual variance) is:

4,800 – 576 = 4,224 or 0.0422

iii

The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated):

Cov(rA , rB ) = bAbBsM(2)  = 0.70´1.20´.04 = .0336

The correlation coefficient between the returns of A and B is:

Cov(rArB ) .0336 .1549

A B

iv

Note that the correlation is the square root of R2 : ρ =

Cov(rA, rM ) = rsAsM = 0.201/2 ´31.30´20 = 280

Cov(rB , rM ) = rsBsM = 0.121/2 ´69.28´20 = 480

v

Cov(rP,rM ) = Cov(0.6rA + 0.4rB, rM ) = 0.6 × Cov(rA, rM ) + 0.4 × Cov(rB,rM ) = (0.6 × 280) + (0.4 × 480) = 360

vi

Note that the variance of T-bills is zero, and the covariance of T-bills with any asset is zero. Therefore, for portfolio Q:

σ Q = [wP(2)σ P(2)  + wM(2)σ M(2)  + 2´ wP ´ wM ´Cov(rP , rM )]1/ 2

= [(0.52 ´1,282.08) + (0.32 ´ 400) + (2´0.5´0.3´360)]1/ 2  = 21.55%

bQ = wP bP + wM bM = (0.5 ´ 0.90) + (0.3 ´ 1) + (0.20 ´ 0) = 0.75 σ 2 (eQ ) = σ Q(2)  - β Q(2)σ M(2)  = 464.52 - (0.752  ´400) = 239.52

Cov(rQ , rM ) = β Q σ M(2)  = 0.75´400 = 300

QUESTION 2

ANSWER ALL PARTS OF THE QUESTION

a.   If a substantial body of investors decided to exclude “sin stocks” (eg tobacco, armaments) from their portfolios, would both the Arbitrage Pricing Theory (APT) and the Capital Asset Pricing Model (CAPM) would continue to hold? Explain.                                        [6 marks]

b.   An analyst is interested in testing whether liquidity risk (LIQ) is priced in addition to the market factor (Market) that is considered in CAPM. The analyst has decided to employ the Fama-McBeth two-stage procedure.

i.      Explain the first-stage of the regression analysis you would employ.         [6 marks]

ii.      What are the dependent and independent variables in the second stage of the       analysis?                                                                                                     [6 marks]

iii.      What would the results be in a CAPM world?                                             [6 marks]

a.   Consider the following data for a one-factor economy. All portfolios are well diversified.

Suppose that another portfolio, portfolio E, is well diversified with a beta of 0.6 and expected return of 7%. Would an arbitrage opportunity exist?  If so, what would be the arbitrage

opportunity?                                                                                                                  [9 marks]

Solution:

[1 mark]The expected return for Portfolio F equals the risk-free rate since its beta equals 0.

[2 marks] For Portfolio A, the ratio of risk premium to beta is: (12 − 6)/1.2 = 5 For Portfolio E, the ratio is lower at: (7 – 6)/0.6 = 1.67

This implies that an arbitrage opportunity exists. For instance, you can create a Portfolio G with beta equal to 0.6 (the same as E’s) by combining Portfolio A and Portfolio F in equal weights. [3 marks]

The expected return and beta for Portfolio G are then:

[1 mark] E(rG ) = (0.5 × 12%) + (0.5 × 6%) = 9%

[1 mark] βG  = (0.5 × 1.2) + (0.5 × 0%) = 0.6

Comparing Portfolio G to Portfolio E, G has the same beta and higher return.  Therefore, an arbitrage opportunity exists by buying Portfolio G and selling an equal amount of Portfolio E. The profit for this arbitrage will be:

[1 marks] rG rE =[9% + (0.6 × F)] - [7% + (0.6 × F)] = 2%

QUESTION 3

ANSWER ALL PARTS OF THE QUESTION

a.   A manager buys three shares of stock today and then sells one of those shares each year for the next three years. His actions and price history are summarized below. The stock pays no dividends.

Time

Price

Action

0

1

2

3

£90

£100

£100

£100

Buy 3 Shares Sell 1 Share Sell 1 Share Sell 1 Share

i.       Calculate the time-weighted geometric average return for the manager          [3 marks]

ii.        Calculate the time-weighted arithmetic average return for the manager           [3 marks]

iii.      Explain if you would expect the dollar-weighted average would be higher or lower than the time-weighted geometric average? (Do not have to calculate the dollar-weighted average)                                                                                                                 [2 marks]

Solution:

Time               Cash Flow               Holding Period Return

0

1

2

3

i.

3×(–$90) = –$270

$100

$100

$100

(100–90)/90 = 11.11%

0%

0%

Time-weighted geometric average rate of return =

(1.1111 × 1.0 × 1.0)1/3 – 1 = 0.0357 = 3.57%

ii.

Time-weighted arithmetic average rate of return = (11.11% + 0 + 0)/3 = 3.70%                           The arithmetic average is always greater than or equal to the geometric average; the greater the dispersion, the greater the difference.

iii.

Dollar-weighted  average  rate  of  return  would  exceed  the  other  averages  because  the investment fund was the largest when the highest return occurred.

b.   Consider the following the information regarding the performance of a money manager in a recent month. The table represents the actual return of each sector of the manager’s portfolio in column 1, the fraction of the portfolio allocated to each sector in column 2, the benchmark or neutral sector allocation in column in 3 and the returns of sector indices in column 4.

Actual Return

Actual Weight

Benchmark

Weight

Index Return

Equity

Bonds

Cash

2%

1%

0.50%

0.7

0.2

0.1

0.6 0.3 0.1

2.5% (S&P 500)

1.2 (Salomon Index)

0.5

(i)   Comment on the managers performance

[3 marks] (ii)  What was the contribution of security selection to the manager’s performance?

[5 marks] (iii)  Confirm that the sum of selection and allocation contributions equals her total “excess”

return.

[8 marks]

Solution:

i.

Benchmark:(0.60 × 2.5%) + (0.30 × 1.2%) + (0.10 × 0.5%) = 1.91% [1 mark][1 mark]

[1 mark]

ii.

Security Selection: [5 marks] (1)

Differential return

within market

(Manager index)

(2)

Manager's    portfolio weight

(3) = (1) × (2)

Contribution to

performance

Equity

Bonds

Cash

0.5%

–0.2%

0.0%

0.70 −0.35%

0.20 –0.04%

0.10                         0.00%

Contribution of security selection:            −0.39%

iii.

Asset Allocation: [5 marks]

(1)

Excess weight

Market

(Manager benchmark)

(2)

Index

Return

(3) = (1) × (2)

Contribution to

performance

Equity

0.10%

2.5%

0.25%

Bonds

0.10%

1.2%

0.12%

Cash

0.00%

0.5%

0.00%

Contribution of asset allocation:

Summary: [3 marks]

Security selection–0.39%

Asset allocation 0.13%

Excess performance –0.26%


c.   The following figure shows the pattern of cumulative abnormal returns (CARs) for insider       sales and purchases around insider trading days. Explain whether this evidence is consistent with the Grossman-Stiglitz paradox?

[9 marks]

QUESTION 4

ANSWER ALL PARTS OF THE QUESTION

a.   You observe the following term structure:

Time to Maturity

Annual YTM

1-year Zero Coupon Bond 2-year Zero Coupon Bond 3-year Zero Coupon Bond 4-year Zero Coupon Bond

6.10%

6.20%

6.30%

6.40%

i.      If you believe that the term structure next year will be the same as today’s, calculate the  return on the1-year zero coupon bond and the 4-year zero coupon bond.           [4 marks]

ii.      Which zero coupon bond provides the greatest expected 1-year return?             [3 marks]

iii.      Redo your answers to (i) and (ii) if you believe in the expectations hypothesis    [5 marks]

Solution:

i.

The return on the one-year zero-coupon bond will be 6.1%. [1 mark]

The price of the 4-year zero today is:

$1,000/1.0644  = $780.25


Next year, if the yield curve is unchanged, today’s 4-year zero coupon bond will have a 3-year maturity, a YTM of 6.3%, and therefore the price will be:

$1,000/1.0633  = $832.53

The resulting one-year rate of return will be: 6.70% [3 marks]

ii.

Therefore, in this case, the longer-term bond is expected to provide the higher return because its YTM is expected to decline during the holding period.