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To gain a better understanding of the solution, consider the case of only one risky asset and a riskless asset. In this case, the optimal weight of the risky asset using Equation 1.18 will be:

(1.19)

The optimal weight of the risky asset is proportional to its expected excess rate of return, E[R − R₀], divided by its variance, σ². Again, the higher the degree of risk aversion, the lower the weight of the risky asset.

For example, with an excess return of 10 %, a degree of risk aversion (λ) of 3, and a variance of 0.05, the optimal portfolio weight is 0.67. This is found as (1/3) × (0.10/0.05). Note that Equation 1.19 may be used to solve for any of the variables, given the values of the remaining variables.

APPLICATION 1.8.2

Consider the case of mean-variance optimization with one risky asset and a riskless asset. Suppose the expected rate of return on the risky asset is 9 % per year. The annual standard deviation of the index is estimated to be 13 % per year. If the riskless rate is 1 %, what is the optimal investment in the risky asset for an investor with a risk-aversion degree of 10?

The solution is:

That is, this investor will invest 47.3 % in the risky asset and 52.7 % in the riskless asset. By varying the degree of risk aversion, we can obtain the full set of optimal portfolios.