equity) do not have a long-enough history to obtain accurate estimates of their risk exposures and risk premiums. Second, to the degree that alpha was a major source of return for alternative asset classes in the past, the same level of alpha may not be available going forward if there is increased allocation to this asset class by investors. That is, the supply of alpha is limited, and increased competition is bound to reduce it. Third, the alternative investment industry has shown to be quite innovative and adaptive in response to changing economic conditions. Therefore, we should expect to see new classes of alternative assets going forward, with their potential place in investors' strategic asset allocations unknowable at this point.
EXHIBIT 1.4 Hypothetical Strategic Asset Allocation for an Endowment
1.7.3 Developing a Strategic Asset Allocation
Given the risk-return preference of an asset owner and estimates of expected long-term returns from various asset classes, the portfolio manager and the asset owner can develop an SAA. Exhibit 1.4 displays a hypothetical SAA for a U.S. endowment.
A typical IPS contains a strategic asset allocation and describes the circumstances under which the strategic asset allocation could change; for example, due to fundamental changes in the global economy or changes in the circumstances of the asset owner, the SAA could be revised.
1.7.4 A Tactical Asset Allocation Strategy
Related to SAA is tactical asset allocation (TAA), which is a dynamic asset allocation strategy that actively adjusts a portfolio's SAA based on short- to medium-term market forecasts. TAA's objective is to systematically exploit temporary market inefficiencies and divergences in market values of assets from their fundamental values. Long-term performance of a broadly diversified portfolio is driven mostly by its SAA over time. TAA can add value if designed based on rigorous economic analysis of financial data so it can overcome the headwinds created by the costs associated with portfolio turnover and the fact that global financial markets are generally efficient. The next chapter will provide further details about TAA and the more recent developments based on factor allocation and economic regime-driven investment strategies.
1.8 Implementation
After the completion of the IPS, the next step is its implementation. A variety of quantitative and qualitative portfolio construction approaches are available for this stage. We will focus our attention on the mean-variance approach, as it is the best-known approach, and most of the subsequent developments in this area have attempted to improve on its shortcomings. Some of these approaches are discussed in the next chapter.
Earlier, this chapter discussed how the general expected utility approach could be used 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 Ri, 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 wi. Therefore, the rate of return on a portfolio of the N + 1 risky and riskless asset can be expressed as:
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
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:
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:
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