Question

1a. A portfolio manager adds a new stock that has the same standard deviation of return as the existing portfolio but has a correlation coefficient with the existing portfolio that is less than +1. Explain why the standard deviation of the portfolio will decrease?

1b. A portfolio has assets with coefficient of correlation of 0.8. show whether sharpe ratio or Treynor ratio is appropriate for measuring the risk-adjusted return?

Answer 1

1a.  A portfolio manager adds a new stock that has the same standard deviation of return as the existing portfolio but has a correlation coefficient with the existing portfolio that is less than +1

then there will be a chance of diversification so the resultant standard deviation of portfolio will decrease. it can be explained with the below example.

Suppose the existing porfolio consists of two stocks A and B(risk free) with expected return of 12% and 8% respectively,standard deviation of A is 6% and the coefficient of correlation is +0.8 assume the proportions are equal.

by calculating the standard deviation of above portfolio AB is

σ_P = √(w_A² * σ_A² + w_B² * σ_B² + 2 * w_A * w_B * σ_A * σ_B *ρ_AB) (w_{A = proportion of A :} w_{B= proportion of B)}

=√ 0.5*.0.5*0.06*0.06

=0.03

if portfolio C is added with expected return 8% and same standard deviation of portfolio i.e.., 3% then standard deviation is as follows(assume the weights are equal) return of portfolio And B is 12%(r<+1 assume is 0.9

σ_P= √ 0.5*0.5*0.03*0.03+0.5*0.5*0.03*0.03+2*.5*.5*0.03*0.03*0.9

= 2.92

since the r<1 there exists a diversification so portfolio SD decreases

1b.A portfolio has assets with coefficient of correlation of 0.8:

Sharpe Ratio=Rp​−Rf​​ /σp where:Rp​=return of portfolio Rf​=risk-free rate

σp=standard deviation of the portfolio’s excess return​

In the above example

Suppose the existing porfolio consists of two stocks A and B(risk free) with expected return of 12% and 8% respectively,standard deviation of A is 6% and the coefficient of correlation is +0.8 assume the proportions are equal. Rp=12% Rf=8% σp=3%

Sharpe Ratio=Rp​−Rf​​ /σp = 0.12-.0.08/0.03 =1.333

TreynorRatio=AveragePortfolioReturn−AverageRiskFreeRate​/Beta of the Portfolio Beta = 0.9 assume

= 0.012-0.08/0.09 =0.044

sharpe ratio is useful for all portfolios but treynor is useful for well diversified portfolios so treynor is best suitable measure.