Question

Suppose we have 3 assets: Expected returns = [0.1 0.15 0.12] Standard déviations = [0.2 0.25 0.18] Correlations = [1 0.8 0.4 0.8 1 0.3 0.4 0.3 1] Find all possible pairwise two-asset portfolios and plot on a backround of random portfolios of all three assets. Comment on the efficient frontier.

Answer 1

sol:

In two asset portfolio, there are two risky assets and accordingly portfolio risk, returns are taken into consideration. Now we will calculate all possible portfolios by calculating portfolio risk and return.

Assuming investment in two asset portfolio is equal as no information is given regarding weight of assets in portfolio.

Correlations are as follows:

Asset 1 Asset 2 Asset 3

Asset 1	1	0.8	0.4
Asset 2	0.8	1	0.3
Asset 3	0.4	0.3	1

1)

Expected return of asset 1= 0.1=10%

Expected return of asset 2 = 0.15=15%

Standard deviation of asset 1 = 0.2

Standard deviation of asset 2 = 0.25

Correlation from above table for asset 1 & 2 = 0.8

Calculation of portfolio return

Asset	Expected return	Weights	Weight*return
1	10%	0.50	10%*0.50=5%
2	15%	0.50	15%*0.50=7.5%

Therefore portfolio return = total of weight*return= 5%+7.5%= 12.5%

Portfolio risk=

=

=

=  

Portfolio Risk = 0.1887 i.e. 18.87%

2)

	Asset 2	Asset 3
Expected return	0.15=15%	0.12=12%
Standard deviation	0.25	0.18
Correlation between 2 & 3	0.3

Calculation of portfolio return

Asset	Expected return	weight	weight* return
2	15%	0.50	0.50*0.15=7.5%
3	12%	0.50	0.50*0.12=6%

Therefore, Portfolio return = 7.5%+6%=13.5%

Portfolio Risk =

=  

=

Portfolio risk= 0.1646 i.e. 16.46%

3)

	Asset 3	Asset 1
Expected return	0.12=12%	0.1=10%
Standard deviation	0.18	0.2
Correlation between Asset 3 & 1	0.4

Calculation of Portfolio return

Asset	Expected return	weight	weight*return
3	12%	0.50	0.50*12%=6%
1	10%	0.50	0.50*10%=5%

Therefore, Portfolio return= 6%+5%=11%

Portfolio risk=

=

=

Portfolio Risk= 0.1473 i.e. 14.73%

Statement of selection of securities:

There are 3 rules for selection of securities as:

• Higher return at same risk
• Lower risk for same return
• Higher return for lower risk

Asset   Return   Risk Dominated by Reason

1 & 2	12.5%	18.87%	Portfolio 2 & 3	Higher return but higher risk
2 & 3	13.5%	16.46%	None	Higher return as compared to other
3 & 1	11%	14.73%	Portfolio 2 & 3, Portfolio 1 & 2	lower risk but lower return

Efficient frontier:

In the efficient frontier, only those securities are selected which are dominated by others.

In this case portfolio 2 & 3 is not dominated by any other portfolio. Hence this portfolio is efficient frontier.

And securities lying on this portfolio will not be dominated by any other security or portfolio.