Question

Assume the correlation between the two stocks equals 1 exactly.Graph the investment opportunity set.

Now assume the correlation between the two stocks equals exactly –1. Again, graph the investment opportunity set.

If you would like, you can perform on the same graph.

Answer 1

Let Wa, Wb be the weights of two stocks A & B respectively in the portfolio. Let expected return of A and B be Ra and Rb respectively and the standard deviation be σa and σb. Let Rp and σp be the expected return and standard deviation of the retun of the portfolio.

Rp = WaRa + WbRb

σp² = (Waσa)² + (Wbσb)² + 2ρ(WaσaWbσb)

When ρ = coefficient of correlation = 1 then

σp² = (Waσa)² + (Wbσb)² + 2(WaσaWbσb) = (Waσa + Wbσb)²​​​​​​​

Hence, σp = Waσa + Wbσb

Please see the table below:

Wa	Wb	Ra	Rb	σa	σb	σp	Rp
0%	100%	10%	20%	15%	25%	25.00%	20.0%
10.0%	90%	10%	20%	15%	25%	24.00%	19.0%
20.0%	80%	10%	20%	15%	25%	23.00%	18.0%
30.0%	70%	10%	20%	15%	25%	22.00%	17.0%
40.0%	60%	10%	20%	15%	25%	21.00%	16.0%
50.0%	50%	10%	20%	15%	25%	20.00%	15.0%
60.0%	40%	10%	20%	15%	25%	19.00%	14.0%
70.0%	30%	10%	20%	15%	25%	18.00%	13.0%
80.0%	20%	10%	20%	15%	25%	17.00%	12.0%
90.0%	10%	10%	20%	15%	25%	16.00%	11.0%
100.0%	0%	10%	20%	15%	25%	15.00%	10.0%

And the plot of the investment opportunity set is :

When ρ = coefficient of correlation = -1 then

σp² = (Waσa)² + (Wbσb)²​​​​​​​ - 2(WaσaWbσb) = (Waσa - Wbσb)²​​​​​​​

Hence, σp = |Waσa - Wbσb|

Please see the table below:

Wa	Wb	Ra	Rb	σa	σb	σp	Rp
0%	100%	10%	20%	15%	25%	25.00%	20.0%
10.0%	90%	10%	20%	15%	25%	21.00%	19.0%
20.0%	80%	10%	20%	15%	25%	17.00%	18.0%
30.0%	70%	10%	20%	15%	25%	13.00%	17.0%
40.0%	60%	10%	20%	15%	25%	9.00%	16.0%
50.0%	50%	10%	20%	15%	25%	5.00%	15.0%
60.0%	40%	10%	20%	15%	25%	1.00%	14.0%
70.0%	30%	10%	20%	15%	25%	3.00%	13.0%
80.0%	20%	10%	20%	15%	25%	7.00%	12.0%
90.0%	10%	10%	20%	15%	25%	11.00%	11.0%
100.0%	0%	10%	20%	15%	25%	15.00%	10.0%

And the investment opportunity frontier is as shown below: