Asset Allocation by Mark Kritzman (Windham Labs)

Source:
https://www.windhamlabs.com/insights/whitepaper-asset-allocation/

• Four steps to asset allocation:
1. identify eligible asset classes
2. estimate their expected returns, volatilities, and correlations
3. isolate the subset of efficient portfolios that offer the highest expected returns for different levels of risk
4. select the specific portfolio that matches our tolerance for risk

• 1. Eligible Asset Classes
• An asset class
• improves our portfolio’s efficiency either by raising its expected return or by lowering its risk.
• has homogeneity among the components of an asset class so that we do not forego opportunities for diversification.
• is sufficiently large to absorb a meaningful fraction of our portfolio. (Low capacity lowers our portfolio’s expected return and increases its risk to the point at which the proposed asset class would no longer improve our portfolio’s efficiency.)

• Asset class examples: US stocks, Foreign stocks, Real estate, US bonds, Commodities, Cash equivalents

• 2. Estimating Expected Returns, Standard Deviations and Correlations
• The standard approach to asset allocation is based on portfolio theory, which requires us to estimate expected returns, standard deviations, and correlations. To estimate expected returns, we start by assuming markets are fairly priced; therefore, expected returns represent fair compensation for the degree of risk each asset class contributes to a broadly diversified market portfolio. These returns are called equilibrium returns, and we estimate them by first calculating the beta of each asset class with respect to a broad market portfolio based on historical standard deviations and correlations. Then we estimate the expected return for the market portfolio and the risk-free return. We calculate the equilibrium return of each asset class as the risk-free return plus its beta times the excess return of the market portfolio. Admittedly, the markets are seldom, if ever, in equilibrium, but the pull in this direction is powerful and persistent. Moreover, we can easily adjust the expected return of each asset class to accord with our views about departures from fair value.
• We can blend our views to derive our expected returns: Expected return = (Equilibrium return) + ((View return) - (Equilibrium return)) * (Confidence)  (Confidence) is 100% if you're 100% sure about your view.
• We also need to estimate the standard deviations of the asset classes as well as the correlations between each pair of asset classes.
• It is important to note that standard deviations and correlations are not always stable through time. It is therefore useful to separate historical returns into those returns associated with normal times and those associated with periods of market turbulence. This separation allows us to estimate these values for each regime and to stress test portfolios by measuring exposure to loss based on risk characteristics that prevail during turbulent periods.

• 3. Efficient Portfolios

• Efficient frontier: Based on our earlier assumptions for expected returns, standard deviations, and correlations, we have derived three specific efficient portfolios: one for a conservative investor, one for an investor with a moderate appetite for risk, and one for an aggressive investor. We used the standard deviations and correlations from the normal periods to derive these portfolios. We use the risk values from both the normal and turbulent periods to measure their exposure to loss.
• The process of employing an equilibrium perspective for estimating expected returns yielded nicely behaved results.

• 4. The Optimal Portfolio
• The final step is to select the portfolio that best suits our tolerance for risk, which we call the optimal portfolio. The theoretical approach for identifying the optimal portfolio is to specify how many units of expected return we are willing to give up to reduce our portfolio’s risk by one unit. If, for example, we are willing to give up ½ unit of expected return to lower portfolio variance (the squared value of standard deviation) by one unit, our risk aversion would equal ½. Risk aversion is the reciprocal of risk tolerance. We would then draw a line with a slope of ½ and find the point of tangency between this line and the efficient frontier (with risk defined as variance rather than standard deviation). The portfolio located at this point of tangency is theoretically optimal because its risk/return tradeoff matches our preference for balancing risk and return.
• In practice, however, we do not know intuitively how many units of return we are willing to sacrifice in order to lower variance by one unit. Therefore, we need to translate combinations of expected return and risk into metrics that are more intuitive.
• Because returns are approximately normally distributed, we can easily estimate the probability that a portfolio with a particular expected return and standard deviation will experience a certain loss over a particular horizon. Alternatively, we can estimate the largest loss a portfolio might experience given a certain level of confidence. We call this measure value at risk. We can also rely on the assumption of normality to estimate the likelihood that a portfolio will grow to a particular value at some future date.
• Investors typically measure exposure to loss at the end of their investment horizon. This view of risk ignores what may happen along the way. Investors should think about risk differently. They should care about exposure to loss throughout their investment horizon and not just at its conclusion. We therefore focus on two additional risk measures to evaluate these portfolios: within-horizon probability of loss and continuous value at risk.
• Within-horizon probability of loss measures the likelihood that an investment will depreciate to a particular level from inception to any point during a specified horizon and not just at the conclusion. Value at risk measured conventionally gives the worst outcome at a chosen probability at the end of an investment horizon. By contrast, continuous value at risk gives the worst outcome at a chosen probability from inception to any time throughout an investment horizon. As we shall soon see, these two risk measures reveal that within-horizon exposure to loss is substantially greater than investors typically assume.
• If exposure to loss were our only consideration, we would choose the conservative portfolio, but by doing so we might forego upside opportunity. One way to assess the upside potential of these portfolios is to simulate the distribution of future wealth associated with investment in each of them by using Monte Carlo simulation.
• Monte Carlo simulation generates possible future outcomes by drawing random numbers from a theoretical distribution such as a normal or lognormal distribution with a pre-specified expected return and standard deviation. If we wish to simulate outcomes that are relatively far into the future, it is important that we recognize the effect of compounding, which leads to a lognormal distribution rather than a normal distribution. Compared to a normal distribution, which is symmetrical, a lognormal distribution has a longer right tail than left tail and an average value that exceeds the median value.
• By mapping the expected returns and standard deviations onto estimates of exposure to loss and the distribution of future wealth and income, we should have a clear idea of the merits and limitations of these portfolios. It is important to keep in mind, though, that there is no universally optimal portfolio; it is specific to each investor. If our focus is to avoid losses, the conservative portfolio might be optimal. If, instead, we believe that we can endure significant losses along the way in exchange for greater opportunity to grow wealth and future income, then we might choose the aggressive portfolio. If our goal is to limit exposure to loss, yet still maintain a reasonable opportunity to grow wealth and income, then perhaps the moderate portfolio would suit us best.

• Asset allocation is a complex process, yet one we cannot ignore. We strongly urge investors to approach asset allocation with discipline, structure, and internal consistency. Moreover, we recommend that investors evaluate portfolio choices based not on statistical abstractions, but rather on intuitive interpretations of their risk and return attributes.