Consider a market where two risk factors are sufficient to describe the returns
ID: 2658591 • Letter: C
Question
Consider a market where two risk factors are sufficient to describe the returns on common stock. The following table gives the sensitivities of the stocks of ABC inc. and PQR Inc. to the two factors, as well as their expected returns.
Security
??????
??????
E[Ri]
ABC
0.5
0.8
16.2
PQR
1.5
1.4
.21.6
Riskless
0.0
0.0
10.0
(i) Consider a portfolio, C, made up by selling short $0.50 of security PQR and Purchasing $1.50 of ABC with the proceeded and with $1.00 of your own money in portfolio C. How sensitive will this portfolio be to each of the two factors?
(ii) Consider a portfolio, D, made up by borrowing $1.00 at the risk free rate and investing it with $1.00 0f your own money in portfolio C. How sensitive will this portfolio be to each of the factors?
(iii) What combination of securities ABC, PQR and the riskless security will move on a one-to-one basis with factor 1 and be insensitive to factor 2?
(iv) You have discovered another portfolio P with the following characteristics:
??????
??????
??[????]
P
1
0
0.8
Design an arbitrage to earn your $1M (reference part (iii) above).
Security
??????
??????
E[Ri]
ABC
0.5
0.8
16.2
PQR
1.5
1.4
.21.6
Riskless
0.0
0.0
10.0
Explanation / Answer
Security
B i1
Bi2
E[Ri]
ABC
0.5
0.8
16.2
PQR
1.5
1.4
21.6
Riskless
0
0
10
E(R i ) = Rf + Bi1*Rp1 +Bi2 *Rp2 , where Rp1 and Rp2 are risk premiums and Rf is risk free rate. Bi1 and Bi2 are the sensitivities for factor 1 and factor 2 respectively.
Que (1) Consider a portfolio, C, made up by selling short $0.50 of security PQR and Purchasing $1.50 of ABC with the proceeded and with $1.00 of your own money in portfolio C. How sensitive will this portfolio be to each of the two factors?
Solution:
Weightage of ABC and PQR in portfolio C is 150% in ABC and -50% in PQR due to $1.50 Purchase in ABC and Selling of 0.50$ in PQR.
Sensitivity of Portfolio C to factor 1 (Bi1)
Bc1=Wabc*Bi1 (for ABC) + Wpqr* Bi1 (for PQR)
(Wabc is weightage of ABC in Portfolio C and Wpqr is weightage of PQR in Portfolio C)
Bc1=(1.50/1)*0.5 + (-.50/1)* 1.5=0
Sensitivity of Portfolio C to factor 2(Bi2)
Bc2=Wabc*Bi2 (for ABC) + Wpqr* Bi2 (for PQR
Bc2=(1.5/1)*0.8 +(-.50/1)*1.4=0.50
Question (II) Consider a portfolio, D, made up by borrowing $1.00 at the risk free rate and investing it with $1.00 0f your own money in portfolio C. How sensitive will this portfolio be to each of the factors?
Solution:Since borrowed money is at risk free rate and is not dependent of market, sensitivities for portfolio D for both the factors will be same as sensitivities of Portfolio C.
sensitivities for portfolio D:
Bd1= 0
Bd2=0.5
Question (iii) What combination of securities ABC, PQR and the riskless security will move on a one-to-one basis with factor 1 and be insensitive to factor 2?
Soultion:
This can be worked out by applying the below formula:
For Portfolio Consisting of ABC, PQR and the riskless security
For Factor 1
Wabc*Bi1+Wpqr*Bi1=1 (Equation for portfolio sensitivity of 1)
0.5 *Wabc+1.5*Wpqr=1
For Factor 2
Wabc*Bi2+Wpqr*Bi2=0 (Equation for portfolio sensitivity of 0)
0.8 Wabc+ 1.4 Wpqr=0
After solving both the equations:
Wabc=-2.8
Wpqr=1.6
Wrf=2.2
Security
B i1
Bi2
E[Ri]
ABC
0.5
0.8
16.2
PQR
1.5
1.4
21.6
Riskless
0
0
10
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