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Equation 1.10 displayed the formulation of the problem when the asset owner is concerned with the tracking error between the value of the assets and the value of the liabilities. Similar to Equation 1.18, a general solution for that problem can be obtained as well. Here we present a simple version of it when there is only one risky asset. The covariance between the rate of growth in the liabilities and the growth in assets is denoted by δ, and L is the value of liabilities relative to the size of assets:

(1.20)

It can be seen that if the risky asset is positively correlated with the growth in liabilities (i.e., δ > 0), then the fund will hold more of that risky asset. The reason is that the risky asset will help reduce the risk associated with growth in liabilities. For instance, if the liabilities behaved like bonds, then the fund would invest more in fixed-income instruments, as they would reduce the risk of the fund.

Example: Continuing with the previous example, suppose the covariance between the risky asset and the growth rate in the fund's liabilities is 0.002, and the value of liabilities is 20 % higher than the value of assets. What will be the optimal weight of the equity allocation?

It can be seen that, compared to the previous example, the fund will hold about 14 % more in the risky asset because it can hedge some of the liability risk.

EXHIBIT 1.5 Optimal Weights of Risky Investment and Degree of Risk Aversion

By changing the degree of risk aversion in the first example, we can obtain a set of optimal portfolios, as shown in Exhibits 1.5 and 1.6.

It can be seen that at low degrees of risk aversion (e.g., 4), the investor will be investing more than 100 % in the MSCI World Index, which means a leveraged position will be used. In addition, we can see the full set of expected returns and volatility that the optimal portfolios will assume, which is referred to as the efficient frontier.

The points appearing in Exhibit 1.6 correspond to various degrees of risk aversion. For instance, the optimal risk-return trade-off for an investor with a degree of risk aversion of 4 is represented by a portfolio that is expected to earn 10.5 % with a volatility of about 15.4 %.

EXHIBIT 1.6 Expected Returns and Standard Deviations of Optimal Portfolios