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Question 15.1: You have two asset classes, 1 and 2. Asset class 1 has an ex...

You have two asset classes, 1 and 2. Asset class 1 has an expected return of 8% per annum with a standard deviation of 15%, whilst asset class 2 has an expected return of 5% with a standard devation of 6.5%. The correlation between the asset classes is –20%. Find the expected risk and return for a portfolio consisting of 20% of asset 1 and 80% of asset 2, and show that this is the minimum risk portfolio.

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The expected return for this portfolio is:

μ=∑n=12ωnμn=(0.2×0.08)+(0.8×0.05)=0.05600,\mu =\sum\limits_{n=1}^{2}{\omega _n\mu _n}=(0.2×0.08)+(0.8×0.05) =0.05600,

or 5.600%.

The variance of the portfolio is given by:

σ2=∑m=12∑n=12ωmωnσmσnρm,n=(0.2×0.2×0.15×0.15×1)+(0.2×0.8×0.15×0.065×−0.2)+(0.8×0.2×0.065×0.15×−0.2)+(0.8×0.8×0.065×0.065×1)=(0.2×0.15)2+(0.8×0.065)2+(2×0.2×0.8×0.15×0.065×−0.2)=0.00298,\sigma ^2=\sum\limits_{m=1}^{2}{\sum\limits_{n=1}^{2}{\omega _m\omega _n\sigma _m\sigma _n\rho _{m,n}}} =(0.2×0.2×0.15×0.15×1) +(0.2×0.8×0.15×0.065×−0.2) +(0.8×0.2×0.065×0.15×−0.2) +(0.8×0.8×0.065×0.065×1) =(0.2×0.15)^2+(0.8×0.065)^2+(2×0.2×0.8×0.15×0.065×−0.2) =0.00298,

or 0.298%. The standard deviation is the square root of this amount, 5.459%.

Reworking this calculation with asset allocations of 19% for asset class 1 and 81% for asset class 2 gives a standard deviation of 5.463%; using asset allocations of 21% and 79% gives a standard deviation of 5.461%. Since both of these are higher than 5.459%, the allocation of 20% to asset class 1 and 80% to asset class 2 is the minimum risk asset allocation.