Alternative Investments. Black Keith H.

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СКАЧАТЬ to represent preferences in terms of moments of a portfolio's return distribution. In particular, we noted that optimal portfolios could be constructed by selecting the weights such that the following function is maximized:

      (1.12)

      where μ is the expected return on the portfolio, λ is a parameter that represents the risk-aversion of the asset owner, and σ² is the variance of the portfolio's return. The next section provides a more detailed description of this portfolio construction technique and examines the solution under some specific conditions. Later sections will discuss some of the problems associated with this portfolio optimization technique and offer some of the solutions that have been proposed by academic and industry researchers.

      1.8.1 Mean-Variance Optimization

      The portfolio construction problem discussed in this section is the simplest form of mean-variance optimization. The universe of risky investments available to the portfolio manager consists of N asset classes. The single-period total rate of return on the risky asset i is denoted by R_i, for i = 1, … N. We assume that asset zero is riskless, and its rate of return is given by R₀. The weight of asset i in the portfolio is given by w_i. Therefore, the rate of return on a portfolio of the N + 1 risky and riskless asset can be expressed as:

(1.13)

(1.14)

      For now, we do not impose any short-sale restriction, and therefore the weights could assume negative values.

From Equation 1.14, we can see that

. If this is substituted in Equation 1.13 and terms are collected, the rate of return on the portfolio can be expressed as:

(1.15)

The advantage of writing the portfolio's rate of return in this form is that we no longer need to be concerned that the weights appearing in Equation 1.15 will add up to one. Once the weights of the risky assets are determined, the weight of the riskless asset will be such that all the weights would add up to one.

      Next, we need to consider the risk of this portfolio. Suppose the covariance between asset i and asset j is given by σij. Using this, the variance-covariance of the N risky assets is given by:

      (1.16)

      The portfolio problem can be written in this form, where the weights are selected to maximize the objective function:

      (1.17)

      This turns out to have a simple and well-known solution:

(1.18)

      The solution requires one to obtain an estimate of the variance-covariance matrix of returns on risky assets. Then the inverse of this matrix will be multiplied into a vector of expected excess returns on the N risky assets. It is instructive to notice the role of the degree of risk aversion. As the level of risk aversion (λ) increases, the portfolio weights of risky assets decline. In addition, those assets with large expected excess returns tend to have the largest weights in the portfolio.

      1.8.2 Mean-Variance Optimization with a Risky and Riskless Asset

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.

      1.8.3 Mean-Variance Optimization with Growing Liabilities

      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.

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