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The market portfolio is composed of four securities Given the following data, calculate the market portfolio's standard deviation Security Covariance with market 



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[PDF] Solution 7 - peopleexeteracuk

The market portfolio is composed of four securities Given the following data, calculate the market portfolio's standard deviation Security Covariance with market 

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Exercise Sheet 7

Exercise 1

Assume there are two stocks,AandB, withA= 1:4andB= 0:8. Assume also that the CAPM model applies. (i) If the mean return on the market portfolio is 10% and the risk-free rate of return is 5%, calculate the mean return of the portfolios consisting of: a. 75% of stockAand 25% of stockB, b. 50% of stockAand 50% of stockB, c. 25% of stockAand 75% of stockB. (ii) If the idiosyncratic variations of the stocks areA= 4;B= 2and the variance of the market portfolio is2M= 12, calculate the variance of the portfolios in (a), (b), (c). (iii) What are the mean return and variance of the portfolios if they are 50% ...nanced by borrowing?

Solution 1

(i) The security market line can be used to write rA=rf+A(rMrf) = 5 + 1:4(105) = 12; and rB= 5 + 0:8(105) = 9:

For the portfolios

a:rp=XArA+XBrB = 0:7512 + 0:259 = 11:25; b:rp= 0:512 + 0:59 = 10:5; c:rp= 0:2512 + 0:759 = 9:75: (ii) The beta of a portfolio is found using p=XAA+XBB; and the variance

2p=2p2M+X2A2A+X2B2B:

1

Applying these results

a: p= 0:751:4 + 0:250:8 = 1:25:

2p= 1:25212 +0:75216 + 0:2524

= 28: b: p= 0:51:4 + 0:50:8 = 1:1;

2p= 1:1212 +0:5216 + 0:524

= 19:52: c: p= 0:251:4 + 0:750:8 = 0:95;

2p= 0:95212 +0:25216 + 0:7524

= 14:08: (iii) If 50% ...nanced by borrowing the portfolio proportions areXp= 2and X f=1. So the expected return and variance are r=Xprp+Xfrf; 2=X2p2p:

Evaluating for the individual portfolios

a:r= 211:2515 = 17:5; 2= 2228 = 112: b:r= 210:515 = 16; 2= 2219:52 = 78:08: c:r= 29:7515 = 14:5; 2= 2214:08 = 56:32:

Exercise 2

Assume there are just two risky securities in the market portfolio. Security A, which constitutes 40% of this portfolio, has an expected return of 10% and a standard deviation of 20%. SecurityBhas an expected return of 15% and a standard deviation of 28%. If the correlation between the assets is 0.3 and the risk free rate 5%, calculate the capital market line.

Solution 2

The expected return on the market is

rM=XArA+XBrB = 0:410 + 0:615 = 13; 2 and the variance of the market is

2M=X2A2A+X2B2B+ 2XAXBABAB

= 0:42202+ 0:62282+ 20:40:60:32028 = 426:88: The standard deviation of the market portfolio follows asM=p426:88 =

20:661:Hence the capital market line is

rp=rf+rMrf M p = 5 +

13520:661

p = 5 +

820:661

p:

Exercise 3

The market portfolio is composed of four securities. Given the following data, calculate the market portfolio"s standard deviation.SecurityCovariance with marketProportion

A2420.2

B3600.3

C1550.2

D2100.3

Solution 3

The market beta must satisfy

1 =XAA+XBB+XCC+XDD

=XAAM

2M+XBBM

2M+XCCM

2M+XDDM

2M

1 = 0:2242

2M+ 0:3360

2M+ 0:2155

2M+ 0:3210

2M so

M= 15:824

Exercise 4

Given the following data, calculate the security market line and the betas of the two securities.Expected returnCorrelation with market portfolioStandard deviation

Security 115.50.92

Security 29.20.89

Market portfolio12112

Risk free asset500

Solution 4

3

The security market line is

ri=rf+i(rMrf) = 5 +i(125):

For security 1

15:5 = 5 +1(125); 1= 1:5:

For security 2

9:2 = 5 +2(125); 2= 0:6:

Exercise 5

Consider an economy with just two assets. The details of these are given below.Number of SharesPriceExpected ReturnStandard Deviation

A1001.51515

B1502129

The correlation coe¢ cient between the returns on the two assets is1=3and there is also a risk free asset. Assume the CAPM model is satis...ed. (i) What is the expected rate of return on the market portfolio? (ii) What is the standard deviation of the market portfolio? (iii) What is the beta of stockA? (iv) What is the risk free rate of return? (vi) Construct the capital market line and the security market line.

Solution 5

(a) Value ofA:1001:5 = 150

Value ofB:1502 = 300

Total market value= 450

So X

A=150450

=13 ; XB=300450 =23

Sincer

M=XAr A+XBr B this givesr M=13

15 +23

12 = 13

(b)

2M=X2A2A+X2B2B+ 2XAXBABAB

2M=13 2 15 2+23 2 9

2+ 223

13 13

159 = 81

So M= 9 (c) By de...nition,A=AM 2M: 4

To ...ndAM:

AM=E[(rAr

A)(rMr

M)] but using the de...nition of the return on the market

AM=E[(rAr

A)(XArA+XBrB(XAr

A+XBr B))]

Collecting terms

AM=E[(rAr

A)(XA(rAr

A) +XB(rBr

B))] Hence

AM=XAE[(rAr

A)(rAr

A)] +XBE[(rAr

A)(rBr

B)]

AM=XA2A+XBAB

AM=XA2A+XBABAB

AM=13

225 +23

13

159 = 105

Therefore

A=10581

= 1:2963 (iv) The risk-free return is derived from the either the capital market line or the security market line.

The security market line givesr

A=rf+A[r

Mrf] So r f=r AAr

M1A=151:29631311:2963= 6:25

(v) Capital market liner p=rf+r Mrf M pr p= 6:25 + 0:75p

Security market liner

p=rf+p[r Mrf] 5 r p= 6:25 + 6:75p

Exercise 6

Consider an economy with three risky assets. The details of these are given below.No. of SharesPriceExpected ReturnStandard Deviation

A1004810

B30061214

C10051012

The correlation coe¢ cient between the returns on any pair of assets is 1/2 and there is also a risk free asset. Assume the CAPM model is satis...ed. (i) Calculate the expected rate of return and standard deviation of the mar- ket portfolio. (ii) Calculate the betas of the three assets. (iii) Use solution to (ii) to ...nd the beta of the market portfolio. (iv) What is the risk-free rate of return implied by these returns? (v) Describe how this model could be used to price a new asset,D.

Solution 6

(i) Total value of risk assets is

W= 1004 + 3006 + 1005 = 2700

The proportions of the assets are

X A=427 ;XB=1827 ;XC=527

The expected return on the market portfolio is

rp=427

8 +1827

12 +527

10 = 11:037

The standard deviation of the market portfolio is

2p=427

2 (10)

2+1827

2 (14) 2+527 2 (12)

2+ 2427

1827
12 1014
+2 427
527
12

1012 + 21827

527
12 1412
= 132:1 Hence p=p132:1 = 11:493 6 (ii) This follow from

AM=Cov(rA;rM)

=E((rArA)(rMrM)) =E((rArA)(XA(rArA) +XB(rBrB) +XC(rCrC))) =XA2A+XBAB+XCAC =427 (10)

2+1827

12

1014 +527

12 1012
= 72:593 So

A=72:593132:1= 0:54953

For the other two

BM=XAAB+XB2B+XCBC

= 156:59

B=156:59132:1= 1:1854

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