Question

Asset A and B have expected returns of 5% and 3% per year respectively. Their annual volatilities are both 20% and the correlation coefficient between the returns of assets A and B is 30%. The risk free rate is 1% per year. 1. Find the weights on A and B in a portfolio with minimal risk. 2. Find the weights on A and B in the optimal risky portfolio that has the maximum Sharpe ratio.

Answer 1

1. Weight of A=0.5, Weight of B= 0.5

2. Asset A has the highest shape ratio. The weight of A and B in the optimal risky portfolio that has the highest shape ratio is:

Weight of A= 0.105, Weight of B= 0.895

Step-by-step explanation:

Expected return of Asset A= 5%Expected return of Asset A= 5%

Expected return of Asset B= 3%

Annual volatilities of Asset A= 20%

Annual votalities of Asset B= 20%

1. Correlation coefficient = 30% = 0.3 < 1

Risk Free Rate = 1% =0.01

1. Weight of A and B in portfolio with minimal risk is:

Weight of A= β^2B - Cov (XAXB) /β^2A + β^2B - 2Cov (XAXB)

Therefore,

CovXAXB = PAB (Volatility of A) (Volatility of B)

= 0.3 × 0.2 × 0.2

= 0.012

Hence,

Weight of A= (0.2)^2 - 0.012 / (0.2)^2 + (0.2)^2 - 2(0.012)

Weight of A= 0.04 - 0.012 / 0.04 + 0.04 - 0.024

= 0.028/ 0.08 - 0.024

= 0.028/ 0.056

=0.5

Weight of A = 0.5

Weight of B= 1 - Weight of A

Weight of B= 1 - 0.5

Weight of B= 0.5

2. Shape ratio of A= RA - Rf / β

= 0.05 - 0.01 / 2

= 0.04/2

= 0.02 =20%

Shape ratio of B= RB - Rf / β

= 0.03 - 0.01/ 2

0.02 / 2

=0.01 = 10%

So, Asset A has the highest shape ratio

Cov (XAXB) = PAB (Volatility of A) (Volatility of B)

= 0.03 × 0.2 × 0.1

= 0.006

Weight of A= β^2B - Cov (XAXB) /β^2A + β^2B - 2Cov (XAXB)

Weight of A = (0.1)^2 - 0.006 / (0.2)^2 + (0.1)^2 - 2(0.006)

= 0.01 - 0.006 / 0.04 +0.01 - 0.012

= 0.004/ 0.05 - 0.012

= 0.004/ 0.038

= 0.105

Weight of A = 0.105

Weight of B= 1 - 0.105

Weight of B= 0.895