Alternative Investments. Black Keith H.

which all asset classes (except cash) contribute the same amount to the total risk of the portfolio. It is important to note that there is no volatility target, and the total risk of the portfolio is endogenous to the process. That is, the weights are numerically adjusted to create a portfolio in which each asset contributes the same amount to the total risk.

      2.4.2 Creating a Portfolio Using the Risk Parity Approach

This section addresses the central point of the risk parity approach: how to determine portfolio weights. Equations 2.11 and 2.15 demonstrate that, in all cases, the total risk of a portfolio may be expressed as the sum of the marginal contributions of the portfolio's constituent assets. The risk parity approach is the simple prescription that the portfolio's weights should be selected such that the marginal contribution of each asset is equal. Thus, to create a portfolio of N assets using the risk parity approach, the weights need to be adjusted until the marginal contribution to risk for each asset in the portfolio is equal to (1/N) times the total risk of the portfolio. The portfolio weights that equalize all the marginal contributions to risk can be easily found using a trial-and-error approach or an optimization package such as Microsoft Excel's Solver.

Consider the information for three asset classes in Exhibit 2.4.

The data from Exhibit 2.4 can be used to generate portfolio weights using the risk parity approach. A trial-and-error search can lead to the risk parity solutions depicted in Exhibit 2.5. For comparison purposes, the weights associated with other approaches are presented as well.

EXHIBIT 2.4 Properties of Three Asset Classes

      Source: Bloomberg, HFRI, and authors' calculations.

EXHIBIT 2.5 Portfolio Weights and Their Properties

      Source: Bloomberg, HFRI, and authors' calculations.

There are three different portfolios in Exhibit 2.5. The first one is constructed using no optimization or risk parity. The risk parity portfolio is constructed to equalize the risk contributions of the three asset classes. The minimum volatility portfolio is constructed using mean-variance optimization, in which the goal is to use positive weights to create a portfolio with minimum standard deviation regardless of the mean. It can be seen that the risk parity portfolio allocates relatively high weights to bonds and hedge funds. The minimum volatility portfolio has no allocation to equities. The risk contributions of the three asset classes for each of the three portfolios are presented in Exhibit 2.6.

      As expected, in the risk parity portfolio, each asset contributes the same marginal risk (0.53 %), which is 33.3 % of the resulting portfolio's total risk. The process is iterative, because the total risk of the portfolio changes as the allocations are changed.

      Note that in this example risk parity requires a substantial allocation to fixed income. This is because the fixed-income investment exhibits lower total risk; therefore, more of the portfolio can be allocated to fixed income while keeping its marginal contribution to the portfolio's total risk the same as the marginal contributions of equities and hedge funds. The risk parity approach typically prescribes a low-risk portfolio by overweighting low-risk assets relative to the market portfolio.

EXHIBIT 2.6 Risk Contributions of the Three Asset Classes

      * Because of rounding error the columns do not add up to the total.

      Source: Bloomberg, HFRI, and authors' calculations.

      2.4.3 The Primary Economic Rationale for the Risk Parity Approach

      The portfolio in Exhibit 2.5 would have performed very well over the past 20 years because the portfolio was allocated heavily to fixed-income assets, and fixed-income assets outperformed equity assets on a risk-adjusted basis during that period. However, selecting strategies based solely on successful historical performance, even spanning 20 years, runs the risk of unsuccessfully chasing historical performance. Chasing historically superior risk-adjusted performance is futile in an efficient market.

      In theory, it is difficult to find any reason why a risk parity portfolio should be optimal. For example, if asset returns are normally distributed and financial markets are perfect, then there are sound economic reasons why investors should select portfolios that plot on the efficient frontier. In other words, it would be difficult to come up with sound economic reasons supporting the risk parity approach in perfect financial markets. In addition, it is not immediately clear how and why market imperfections should make risk parity portfolios more desirable compared to, say, mean-variance efficient or even equally weighted portfolios. Therefore, the case for the risk parity approach must be built through careful analysis of its properties.

      As was previously shown and as demonstrated by other studies, the risk parity approach creates low volatility portfolios where low-risk assets are overweighted. Under what conditions, then, would low-risk portfolios become optimal for some investors? One obvious answer is that if an investor has a very high degree of risk aversion, then a risk parity portfolio could be optimal for him. However, a low volatility portfolio can also be constructed using the mean-variance framework. For example, one can create a minimum volatility portfolio using the mean-variance approach (see Exhibits 2.5 and 2.6). It turns out that the most compelling arguments that can be put forward in support of the risk parity approach also support the use of low-risk portfolios, including the minimum volatility approach.

The economic rationale for low volatility portfolios is that because of market imperfections, many investors are unable or unwilling to use leverage. This is referred to as leverage aversion. The leverage aversion theory argues that large classes of investors cannot lever up low volatility portfolios to generate attractive returns and that, as a result, low volatility stocks and portfolios are underpriced. While leverage increases risk and excess return by the same factor, under some conditions it might be possible to create portfolios with higher Sharpe ratios by applying leverage to low-risk portfolios. As an example, consider the two portfolios depicted in Exhibit 2.7.

      While the expected rate of return on the RP portfolio may appear unattractively low to some investors, those who are willing and able to acquire funding at attractive rates can lever up the RP portfolio and earn a superior risk-adjusted return compared to the MV portfolio. In particular, a portfolio with 60 % leverage will produce (11.6 % = 1.6 * 8 % − 0.6 * 2 %) return and (16 % = 1.6 * 10 %) volatility.

EXHIBIT 2.7 Two Hypothetical Portfolios

      The important question is why the low volatility portfolio should provide a higher Sharpe ratio in the first place. The leverage aversion argument is that because many investors are not allowed to use leverage, demand for low-risk stocks is low, and therefore they are undervalued. For example, if a mutual fund manager wants to overweight stocks he judges to be undervalued but still track a broad equity benchmark with some degree of accuracy, the manager cannot afford to overweight low volatility stocks. Although low-risk stocks may offer attractive risk-adjusted returns, the mutual fund manager will not be able to lever up their low raw returns to match the overall market return; as a result, the fund will underperform its benchmark in terms of raw returns. The idea that low volatility stocks are underpriced and therefore offer higher expected risk-adjusted returns is known as the volatility anomaly.16 According to this anomaly, portfolios consisting of low volatility