Comm 371 Marked Problem Set 2
Sauder School of Business
Finance Division
COMM 371 Sep-Dec 2011
Gonzalo Morales
Marked Problem Set 2 - Solution Notes
1. First, compute the correlation coefficient between assets A and B ρ(RA , RB ) =
Cov (RA , RB )
−0.0322
=
= −1. 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 α= 0.23
= 0.622.
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.
Finance Division
COMM 371 Sep-Dec 2011
Gonzalo Morales
Marked Problem Set 2 - Solution Notes
1. First, compute the correlation coefficient between assets A and B ρ(RA , RB ) =
Cov (RA , RB )
−0.0322
=
= −1. 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 α= 0.23
= 0.622.
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.