- Forward and Futures Contracts 143
Proposition 6.6
If
N=(βV−a)(1 +rF)V(0)
f(0,T),
thenβV ̃=afor any given numbera.
Proof
We shall compute the beta coefficient from the definition:
βV ̃=Cov(KV ̃,KM)/σ^2 M=Cov(KV,KM)/σ^2 M−1
V(0)
Cov(N(f(1,T)−f(0,T)),KM)/σ^2 M,whereKMis the return on the market portfolio andKV the return on the
portfolio without futures. Since Cov(f(0,T),KM) = 0 and covariance is linear
with respect to each argument,
Cov(N(f(1,T)−f(0,T)),KM)=NCov(f(1,T),KM).Inserting the futures pricef(1,T)=M(1)(1 +rF)T−^1 ,wehave
Cov(f(1,T),KM)=(1+rF)T−^1 Cov(M(1),KM).Again by the linearity of covariance in each argument
Cov(M(1),KM)=M(0)Cov(M(1)−M(0)
M(0)
,KM)=M(0)σ^2 M.Subsequent substitutions give
βV ̃=βV−(1 +rF)T−^1 NM(0)
V(0)=βV−Nf(0,T)
V(0)(1 +rF),
which implies the asserted property.
Corollary 6.7
Ifa=0,thenμV ̃=rF.
Example 6.4
Suppose that the index drops fromM(0) = 890 down toM(1) = 850,that is,
by 4.49% within one time step. Suppose further that the risk-free rate is 1%.