286 Mathematics for Finance
5.18The expected return on the portfolio can be expressed asKV=w 1 K 1 +···+
wnKnin terms of the expected returns on the individual securities. Because
covariance is linear in each of its arguments,βV=Cov(KV,KM)
σ^2 M=w 1 Cov(K^1 ,KM)
σM^2+···+wnCov(Kn,KM)
σM^2
=w 1 β 1 +···+wnβn.
5.19The equation of the characteristic line isy=βVx+αV, whereβV is the
beta factor of that security andαV=μV−βVμM. In the CAPM the equation
μV=rF+(μM−rF)βVof the security market line holds. Substitution into the
formula forαVgivesαV=rF−rFβV, so the equation of the characteristic line
becomesy=βV(x−rF)+rF. Clearly, the characteristic line of any security
will pass through the point with coordinatesrF,rF.Chapter 6
6.1Yes, there is an arbitrage opportunity. We enter into a long forward contract
and sell short one share, investing 70% of the proceeds at 8% and paying the
remaining 30% as a security deposit to attract interest at 4%.At the time of
delivery the cash investments plus interest will be worth about $18.20, out of
which $18 will need to be paid for one share to close out the short position in
stock. This leaves a $0.20 arbitrage profit.
The ratesdfor the security deposit such that there is no arbitrage oppor-
tunity satisfy 30%× 17 ×ed+ 70%× 17 ×e8%≤18. The highest such rate is
d∼= 0 .1740%.
6.2We take 1 January 2000 to be time 0. By (6.2)F(0, 3 /4) =S(0)e^0.^06 ×(^34)
,F(1/ 4 , 3 /4) = 0. 9 S(0)e^0.^06 ×
(^24)
.
It follows that the forward price drops by
F(0, 3 /4)−F(1/ 4 , 3 /4)
F(0, 3 /4) =
e
(^34) ×6%
− 0 .9e
(^12) ×6%
e^34 ×6%
∼= 11 .34%.
6.3The present value of the dividends is
div 0 =1e−^126 ×12%+2e−^129 ×12%∼= 2. 77
dollars. The right-hand side of (6.4) is equal to
[S(0)−div 0 ]erT∼=(120− 2 .77)e^1012 ×12%∼= 129. 56
dollars, which is less than the quoted forward price of $131. As a result, there
will be an arbitrage opportunity, which can be realised as follows:
- on 1 January 2000 enter into a short forward position and borrow $120 to
buy stock; - on 1 July 2000 collect the first dividend of $1 and invest risk-free;
- on 1 October 2000 collect the second dividend of $2 and invest risk-free;
- on 1 November 2000 close out all positions.