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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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 variance2p=2p2M+X2A2A+X2B2B:
1Applying 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 is2M=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 marketProportionA2420.2
B3600.3
C1550.2
D2100.3
Solution 3
The market beta must satisfy
1 =XAA+XBB+XCC+XDD
=XAAM2M+XBBM
2M+XCCM
2M+XDDM
2M1 = 0:2242
2M+ 0:3360
2M+ 0:2155
2M+ 0:3210
2M soM= 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 deviationSecurity 115.50.92
Security 29.20.89
Market portfolio12112
Risk free asset500
Solution 4
3The 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 DeviationA1001.51515
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 = 150Value ofB:1502 = 300
Total market value= 450
So XA=150450
=13 ; XB=300450 =23Sincer
M=XAr A+XBr B this givesr M=1315 +23
12 = 13
(b)2M=X2A2A+X2B2B+ 2XAXBABAB
2M=13 2 15 2+23 2 92+ 223
13 13159 = 81
So M= 9 (c) By de...nition,A=AM 2M: 4To ...ndAM:
AM=E[(rAr
A)(rMr
M)] but using the de...nition of the return on the marketAM=E[(rAr
A)(XArA+XBrB(XAr
A+XBr B))]Collecting terms
AM=E[(rAr
A)(XA(rAr
A) +XB(rBr
B))] HenceAM=XAE[(rAr
A)(rAr
A)] +XBE[(rAr
A)(rBr
B)]AM=XA2A+XBAB
AM=XA2A+XBABAB
AM=13225 +23
13159 = 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 AArM1A=151:29631311:2963= 6:25
(v) Capital market liner p=rf+r Mrf M pr p= 6:25 + 0:75pSecurity market liner
p=rf+p[r Mrf] 5 r p= 6:25 + 6:75pExercise 6
Consider an economy with three risky assets. The details of these are given below.No. of SharesPriceExpected ReturnStandard DeviationA1004810
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 isW= 1004 + 3006 + 1005 = 2700
The proportions of the assets are
X A=427 ;XB=1827 ;XC=527The expected return on the market portfolio is
rp=4278 +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
182712 1014
+2 427
527
12
1012 + 21827
52712 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
121014 +527
12 1012= 72:593 So