COMM 371 Sep-Dec 2011
Marked Problem Set 2 - Solution Notes
1. First, compute the correlation coeﬃcient between assets A and B ρ(RA , RB ) =
Cov (RA , RB )
σ (RA )σ (RB )
0.14 × 0.23
The assets are perfectly negatively correlated. Consider portfolio P formed from assets A and B such that you invest α fraction of your wealth into A and (1 − α) fraction into B. The variance of such portfolio is
σ (RP )2 =
α2 σ (RA )2 + (1 − α)2 σ (RB )2 + 2α(1 − α)Cov (RA , RB ) α2 σ (RA )2 + (1 − α)2 σ (RB )2 + 2α(1 − α)σ (RA )σ (RB )ρ(RA , RB ) α2 σ (RA )2 + (1 − α)2 σ (RB )2 − 2α(1 − α)σ (RA )σ (RB ) [ασ (RA ) − (1 − α)σ (RB )]2 .
Therefore, the standard deviation of portfolio P is
σ (RP ) = ασ (RA ) − (1 − α)σ (RB ).
As assets A and B are perfectly negatively correlated, we can construct portfolio P such that its standard deviation is 0. The weights of such portfolio are 0 = ασ (RA ) − (1 − α)σ (RB )
= 0.14 × α − 0.23 × (1 − α).
Solving the above equation for α gives
0.14 + 0.23
Portfolio P with standard deviation zero has weight 0.622 on asset A and weight 0.378 on asset B. The expected return of this portfolio (equal to the actual return as the portfolio is riskless) is
E [RP ] = RP = 0.622 × 0.08 + 0.378 × 0.11 = 0.091.
The arbitrage trade per $1 invested is as follows: (i) Borrow $1 at the riskless rate 5%; (ii) Buy portfolio P with standard deviation zero whose return is 9.1% using the borrowed $1; (iii) In one year, liquidate the portfolio getting $1.091; and repay $1.05 on your loan. The diﬀerence $0.041 is the arbitrage proﬁt per $1 trade. 1
2. Given that we are only considering risk, the better investment is the one contributing the least to portfolio variance. The contribution of an asset to portfolio variance is measured by the covariance between the asset’s return and the return on the bank’s portfolio. The return on the bank’s portfolio is
R = 0.5 × Rs + 0.2 × Rb + 0.3 × Rd ,
where Rs is the return on the stock portfolio, Rb the return on the bond portfolio, and Rd the return on the derivatives. The covariance between the return RP on Polynesia and return R on the bank’s portfolio is
Cov (RP , R) = 0.5 × Cov (RP , Rs ) + 0.2 × Cov (RP , Rb ) + 0.3 × Cov (RP , Rd ). We have
Cov (RP , Rs ) = ρ(RP , Rs )σ (RP )σ (Rs ) = 0.4 × 30% × 25% = 0.03, and similarly
Cov (RP , Rb ) = 0.0081
Cov (RP , Rd ) = 0.03.
Cov (RP , R) = 0.02562.
We similarly ﬁnd
Cov (RM , R) = 0.02513
for Micronesia, and
Cov (RN , R) = 0.03881
for New Caledonia. Therefore, Micronesia is the better investment. Notice that this is in spite of the fact that Micronesia has the highest standard deviation. 3. Consider portfolio P = wX X + wY Y + wZ Z constructed from stocks X, Y, and Z. Under the assumption that the variance of returns is the same for all three stocks, V (RX ) = V (RY ) = V (RZ ) = σ 2 , and the correlation of returns between all pairs of stocks is the same, ρX,Y = ρX,Z = ρY,Z = ρ, the variance of portfolio P is 2
V (RP ) = wX + wY + wZ σ 2 + 2 × (wX wY + wX wZ + wY wZ ) ρσ 2 . 1
When P is the equally weighted portfolio, wX = wY = wZ = 3 , the variance of portfolio P simpliﬁes to
V (RP ) = 3 ×
σ2 + 2 × 3 ×
ρσ 2 .
(a) When σ = 19% and ρ = 1.0, the standard deviation of portfolio P is σ (RP ) =
V (RP ) = 19%.
In this case, the three stocks are perfectly positively correlated and the standard deviation cannot be reduced by mixing the stocks in a portfolio. There are no beneﬁts of diversiﬁcation.
(b) When σ = 19% and ρ = 0.5, the standard deviation of portfolio P is σ (RP ) =
V (RP ) = 15.51%.
(c) When σ = 19% and ρ = 0.0, the standard deviation of portfolio P is σ (RP ) =
V (RP ) = 10.97%.
(d) When σ = 19% and ρ = −0.5, the standard deviation of portfolio P is σ (RP ) =
V (RP ) = 0%.
(e) The equally weighted portfolio...