Q

Use the single index model to compute the covariance between stocks G and H.

Home, - Calculate beta for stock G

Question - Yearly data on returns are presented below for stock G and the market index (M).

Year

Stock G

Market Index (M)

2014

-1

-2

2015

2

3

2016

6

4

2017

4

-1

2018

0

2

2019

1

3

(i) Calculate beta for stock G. Consider stock H with a beta equal to 1.2. Use the single index model to compute the covariance between stocks G and H.

Answer -

 

Stock G

Market Index (M)

 

mean return

2.00

1.50

=(-2+3+4-1+2+3)/6

 

return variance

6.80

5.90

=((-2-1.5)^2+(3-1.5)^2+(4-1.5)^2+(-1-1.5)^2+(2-1.5)^2+(3-1.5)^2)/6

return stdev

2.61

2.43

=(5.9)^(1/2)

Correlation

0.41

 

 

The covariance between G and the market is

2.60 =0.41*6.8*5.9 =((-2-1.5)(-1-2)+(3-1.5)(2-2)+(4-1.5)(6-2)+(-1-1.5)(4-2)+(2-1.5)(0-2)+(3-1.5)(1-2))/6

The beta of G is 0.44 =2.6/5.9

The beta of H is 1.2

The covariance between G and H is 3.12 =5.9*1.2*0.44

(ii) Consider the following:

- a stock Awith expected return 10%and beta 1.0;

- a stock B with expected return 14% and beta 1.4;

- a stock D with expected return 16% and beta 1.2;

- a stock E with expected return 7% and beta 1.4.

Describe how to exploit the arbitrage opportunities.

Answer - If we consider another portfolio, say portfolio C, which is formed by half of portfolio A and half of portfolio B, the characteristics of this portfolio C, in terms of expected return and beta would be:

- Expected return = 0.5×10 + 0.5×14 = 12;

- Beta = 0.5×1.0 + 0.5×1.4 = 1.2.

Now D has a higher return (16%) than C, but the same beta (risk) of C (1.2). D cannot exist in the market for a long time. This is because an investor could sell short, for example, £100 of portfolio C and buy £100 of portfolio D.

In doing this, the investor would set up an arbitrage Portfolio with the following characteristics:

- The investment cash Invested is zero;

- Beta equal to 0;

- Expected return of 16%-12%=4%.

This arbitrage portfolio gives:

-a positive profit on average,

-with a zero systematic risk,

-and a zero net investment.

An investor will engage in this arbitrage and in doing this he/she will bring down the expected return on D to the equilibrium level of 12%.

Then E has a lower expected return (7%) than B, but the same beta as B (1.4). Also E cannot exist for a long time:

- an investor could sell short, for example, £100 of E and buy £100 of B.

In doing this, the investor would set up an arbitrage Portfolio with the following characteristics:

- The investment cash Invested is zero;

- Beta equal to 0;

- Expected return of 14%-7%=7%.

This arbitrage portfolio gives:

-a positive profit on average,

-with a zero systematic risk,

-and a zero net investment.

An investor will engage in this arbitrage and in doing this he/she will bring up the expected return on E to the equilibrium level of 14%.

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