ABSTRACT
We can identify further properties of the zero-beta portfolio for the example considered here by looking at portfolios that involve the following proportionate investments across the three risky assets:
=
⎛⎜⎜⎜⎜⎝ 5
−3 2 0
⎞⎟⎟⎟⎟⎠+ψ ⎛⎝ 1−2
⎞⎠
where ψ is a parameter that can take on any numerical value. Thus, the portfolios defined here will have a proportionate investment of w1 = 52 + ψ in the first asset, a proportionate investment of w2 = − 32 − 2ψ in the second asset and a proportionate investment of w3 = ψ in the third asset. Note that
w1 +w2 +w3 = (
2 +ψ
) + ( −3
2 − 2ψ
) +ψ = 1
that is, the total of the proportionate investments is unity, as required (and this will be so irrespective of the value assumed by the parameter ψ). Moreover, we can use the expected returns vector given in §2-13 above to determine the expected return on portfolios comprised of these proportionate investments:
E(Rz) = w1E(R1)+w2E(R2)+w3E(R3) = (
2 +ψ
) 1
10 + ( −3
2 − 2ψ
) 3
20 +ψ 1
Collecting terms in the above expression shows
E(Rz) = (
20 − 9
) + (
10 − 6
20 + 1
) ψ = 1
40 = 0.025
or that the expected return on portfolios comprised of these proportionate investments will be E(Rz) = 212 per cent (per annum), and this will be so irrespective of the value assumed by the parameter ψ . Moreover, using the betas calculated in §2-13 for the three risky assets, we can also show that the beta, βz, for these zero-beta portfolios will have to be
βz = w1β1 +w2β2 +w3β3 = (
2 +ψ
) 3
5 + ( −3
2 − 2ψ
) 1+ψ 7
= (
2 − 3
) + (
5 − 2+ 7
) ψ = 0
That is, these portfolios will all have a beta of zero, and again this will be so irrespective of the specific value assumed by the parameter ψ . This will mean that the returns on these portfolioswill have no systematic relationshipwith the return on themarket portfolio. Finally, the variance of the return for these portfolios can be computed from the formula
σ 2z = w21σ 21 +w22σ 22 +w23σ 23 + 2w1w2σ12 + 2w1w3σ13 + 2w2w3σ23 where wj is the proportionate investment in the j = 1, 2, 3 risky assets, σ 2j is the variance of the return on the jth asset and σjk is the covariance of the return on the jth asset with the return
on the kth asset. Substituting the relevant proportionate investments and the variances and covariances from the variance-covariance matrix , defined in §2-13 above, into the above expression, we have
σ 2z = (
2 +ψ
)2 1 5
+ ( −3
2 − 2ψ
)2 2 5
+ψ21
+ 2 (
2 +ψ
)( −3
2 − 2ψ
) 1
10 + 2ψ
( 5
2 +ψ
) 1
5 + 2ψ
( −3
2 − 2ψ
) 1
or
σ 2z = 12
5 ψ2 + 14
5 ψ + 7
Now one can differentiate through the expression for σ 2z and thereby determine the zerobeta portfolio with an expected return of E(Rz) = 212 per cent that has the smallest possible variance in the rate of return:
dσ 2z dψ
= 24 5
ψ + 14 5
= 0
or ψ = − 712 . This in turn implies that the portfolio with the proportionate weights given by
=
⎛⎜⎜⎜⎜⎝ 5
−3 2
⎞⎟⎟⎟⎟⎠+ψ ⎛⎝ 1−2
⎞⎠= ⎛⎜⎜⎜⎜⎝
−3 2
⎞⎟⎟⎟⎟⎠− 712 ⎛⎝ 1−2
⎞⎠= ⎛⎜⎜⎜⎜⎜⎜⎝
−1 3
− 7 12
⎞⎟⎟⎟⎟⎟⎟⎠ will have a return whose variance is
σ 2z = 12
5 ψ2 + 14
5 ψ + 7
5 = 12
( − 7
)2 + 14
( − 7
) + 7
5 = 7
and this variance will be lower than the variance of the return on any other portfolio with an expected return of E(Rz) = 212 per cent (per annum). Hence, if an investor was obliged to hold one of these zero-beta portfolios, they would opt to hold the zero-beta portfolio with the above proportionate investments, since this minimizes the risk they would have to take in earning the expected return of E(Rz) = 212 per cent (per annum) that it promises.
