BMAN20072 Investment Analysis 2021/22 Semester 2 Week 3 Problem Set

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BMAN20072 Investment Analysis 2021/22 Semester 2

Week 3 Problem Set

Topic: Index Models and Security Analysis

Part B Solutions

Part B. Numerical questions

3.B.1

You are considering to form a risky portfolio consists of three stocks, A, B and C. You are assuming a single index model and estimated alpha, beta, standard deviation of firm specific factor,  market  risk  premium,  and  standard  deviation  of the  macroeconomic  factor.  The estimates are summarised in below tables.

Alpha, beta, and standard deviations of firm specific factor

 

alpha

beta

std.dev.

Stock A

0.012

0.8

0.03

Stock B

0.018

1.2

0.04

Stock C

0.020

2.0

0.05

 

 

Market risk premium

0.01

Std. dev. of macro factor

0.02

Suppose you are considering below two portfolios X and Y.

Weights

X

Y

A

0.5

0.2

B

0.3

0.3

C

0.2

0.5

Which one is the better portfolio? X or Y?

Solution

Expected excess return of Stock i: E(Ri ) = ai  + FE(RM )

Expected excess return of Stock A: 0.012 + 0.8(0.01) = 0.02

Expected excess return of Stock B: 0.018 + 1.2(0.01) = 0.03

Expected excess return of Stock C: 0.02 + 2(0.01) = 0.04

Expected excess return of portfolio X: 0.5 × 0.02 + 0.3 × 0.03 + 0.2 × 0.04 = 0.027 Expected excess return of portfolio Y: 0.2 × 0.02 + 0.3 × 0.03 + 0.5 × 0.04 = 0.033

Standard deviation of a portfolio: GP    = 1 ∑j(n)=1 wi wj Cov(ri , rj )

If i ≠ j, Cov (ri , rj ) = Fi Fj GM(2) , otherwise, Cov(ri , rj ) = Var(ri ) = Fi(2)GM(2)  + G 2 (ei).

Cov(rA , rA ) = Var(rA ) = 0.82  × 0.022  + 0.032  = 0.0012

Cov(rA , rB ) = 0.8 × 1.2 × 0.022  = 0.0004

Cov(rA , rC ) = 0.8 × 2 × 0.022  = 0.0006

Cov(rB , rB ) = Var(rB ) = 1.22  × 0.022  + 0.042  = 0.0022

Cov(rB , rC ) = 1.2 × 2 × 0.022  = 0.0010

Cov(rC , rC ) = Var(rC ) = 22  × 0.022  + 0.052  = 0.0041

Standard deviation of X:

0.5 × (0.5 × 0.0012 + 0.3 × 0.0004 + 0.2 × 0.0006) = 0.000411   0.3 × (0.5 × 0.0004 + 0.3 × 0.0022 + 0.2 × 0.0010) = 0.000311

0.2 × (0.5 × 0.0006 + 0.3 × 0.0010 + 0.2 × 0.0041) = 0.000286

GX    = √0.000411 + 0.000311 + 0.000286 = 0.0317

Standard deviation of Y:

0.2 × (0.2 × 0.0012 + 0.3 × 0.0004 + 0.5 × 0.0006) = 0.000133 0.3 × (0.2 × 0.0004 + 0.3 × 0.0022 + 0.5 × 0.0010) = 0.000363 0.5 × (0.2 × 0.0006 + 0.3 × 0.0010 + 0.5 × 0.0041) = 0.001233

GY    = √0.000133 + 0.000363 + 0.001233 = 0.0416

Sharpe ratio of X: 0.027 / 0.0317 = 0.8507.

Sharpe ratio of Y: 0.033 / 0.0416 = 0.7936.

Sharpe ratio of portfolio X is higher than Y. X is the better portfolio.

3.B.2

a) After a careful fundamental analysis of a company, you assume its stock will have dividend growth of 5% indefinitely. Suppose that the next dividend payment is £1. If the required rate of return is 10%, find the intrinsic value of the stock.

b) Suppose the above stock is currently traded at £16. You believe that the price will converge to the intrinsic value in a year. What is your expected annual return of this stock?

c) Suppose the annual risk free rate is 0.5%. You estimate the above stock’s beta is 1.2 and expected annual return of the FTSE ALL Index is 10.5%. Applying single index model as the benchmark, what is the expected alpha generated from your fundamental analysis?

Solution

a)  Let V be the intrinsic value of the stock, D1 be the next dividend payment, k be the required return, and g be the growth. Constant dividend growth model: V = D1/(k-g).  V = 1/(0. 1-0.05) = £20.

b)  (V – P) / P = (20 – 16) / 16 = 0.25. E(r) = 25%.

c)   E(T) − Tf  = a + F(E(TM ) − Tf )

25% - 0.5% = a + 1.2(10.5% – 0.5%).

a = 24.5% − 12.5% = 12.5%















2022-07-18