## Windham Insights Series

## Mismeasurement of Risk

Investors tend to consider risk as an outcome—how much could be lost at the end of an investment period? Risk is typically measured as the probability of a given loss or the amount that can be lost with a given probability at the end of their investment horizon. This perspective considers only the result at the end of the investment horizon, ignoring what may happen within the portfolio along the way. We argue that exposure to loss throughout an investment horizon is important to investors, and propose two new ways of measuring risk: within-horizon probability of loss and continuous value at risk (VaR). Using these risk measures, we reveal that exposure to loss is often substantially greater than investors assume. Where is the danger in measuring risk at the end of an investment period? Financial analysts worry that means and variances used in portfolio construction techniques are estimated with error. These errors bias the resultant portfolio towards asset for which the mean is over-estimated and variance is underestimated, which may lead analysts to invest in the wrong portfolio. Additionally, financial analysts worry that higher moments, such as skewness and kurtosis, are misestimated. In that case, extreme returns occur more [...]

## The Cost of Socially Responsible Investing

As we evolve into a more political and environmentally conscious society, the concept of “socially responsible investing” is becoming an increasingly popular investment option. A socially responsible investor is one who chooses not to invest in certain companies whose behavior they judge to be at odds with the social good as they perceive it. Some proponents of socially responsible investing claim that “good” companies – those in harmony with the social good – perform as well or better than “bad” companies. They conclude, therefore, that socially responsible investing is without cost and may even enhance performance. However, one may argue that if the motive to invest is due to higher expected returns, it is not in fact “socially responsible” investing. As with any method of investing, it is important to be aware of the costs. Suppose, for example, that exclusion of tobacco companies reduces return. Some may argue that this cost is worth incurring because it focuses attention on the issue, or because it may influence company behavior. Others may argue that it would be more effective to collect the return of the tobacco investment, and deploy it directly towards policies designed to curb smoking or treat lung cancer. Without [...]

## Within-Horizon Risk Measures

Financial analysts and investors often focus on the distribution of outcomes at the end of their investment horizons, while ignoring the fact that their investments are exposed to risk throughout the investment period as well. This approach to risk management ignores the losses that might occur, either as an accumulation of many small losses or from a significant loss that later– too late, perhaps—would otherwise recover. So, how can we take into account the exposure to loss throughout an investment period? Two innovative, related risk measures capture the nature of investment risk more realistically than traditional, end-of-period measures. Within-horizon probability of loss and continuous value at risk (CVaR) give a more accurate and realistic view of risk than current methods. They also reveal that exposure to risk is substantially greater than investors typically assume. Within-horizon probability of loss is the probability that an investment will depreciate to a particular value from inception to any time during an investment period. Continuous value at risk (CVaR) gives the worst outcome of an investment at a chosen probability of loss from inception to any time during an investment period. By focusing on end-of-horizon risk measures, investors are under the false impression that exposure to loss diminishes over time. However, within-horizon [...]

## Monte Carlo Simulation

Fortune tellers, palm readers, the Farmer’s Almanac, financial analysts. What do these things have in common? They all attempt to anticipate the future. While some use a crystal ball, the ridges of their subject’s palm, or sunspots, financial analysts use numeric models and mathematical techniques to generate simulations of their client’s financial future. There are two kinds of forecasting models: deterministic models and stochastic models. Deterministic models assume a fixed relationship between the inputs and the output, while stochastic models depends on inputs that are influenced by chance. Deterministic models can be solved analytically via mathematical formulas, while stochastic models require numerical solutions. To solve stochastic models numerically, one must try various values for the model’s parameters and variables. When these variables come from a sequence of random numbers, the solution is called Monte Carlo simulation. Monte Carlo simulation was originally introduced by financial analysts John Von Neumann and Stanislaw Ulam while working on the Manhattan Project at the Los Alamos National Laboratory. They invented a procedure of substituting a random sequence of numbers into equations to solve problems regarding the physics of nuclear explosions. The term Monte Carlo was inspired by the gambling casinos in Monaco. Monte Carlo simulation [...]

## Asset Allocation in Taxable Portfolios

Taxes can consume a substantial portion of returns in an individual’s portfolio, and it is important to consider assets on an after-tax basis. It allows us to find an individual’s optimal portfolio— which may vary significantly from person to person. Considering assets on an after-tax basis also allows us to estimate the future value of a portfolio. In this post, we will outline the steps to convert pre-tax return and risk into after-tax values. Next, we will identify optimal portfolios on a pre- and after-tax basis. Finally, we will simulate future wealth on a pre- and after-tax basis. Calculating after-tax return and risk For US based investors, not all investments are created equal when it comes to taxes. Realized gains and income are classified and taxed at different rates. After-tax expected return and risk should be used when making asset allocation decisions. The effective tax rate is the average annual percentage of total return that will be paid in taxes. Total returns include both income and capital appreciation. In a taxable account, income can be taxed at a qualified dividend rate or an individual’s marginal tax rate. Capital appreciation can be taxed at short-term capital gains rates or long-term [...]

## Multi-goal Optimization

Investors are often must decide whether to consider absolute return and risk, or relative return and risk. Sophisticated investors, however, employ optimization in an attempt to be sensitive to both. Many address this duel concern of absolute and relative return and risk by constraining the asset weights in the optimization process, which is time-consuming and produces sub-optimal portfolios. Imagine for a moment, a new approach. An approach that allows investors to simultaneously consider absolute and relative performance, yields a higher expected return, in a more efficient process. Here at Windham Labs, we offer an innovative optimization technique: Multi-goal Optimization. Multi-goal Optimization combines mean-variance analysis, which characterizes assets and portfolios by their expected absolute return and standard deviation, with tracking error—which gives a measurement of the volatility of relative returns. This technique allows investors to identify efficient allocations that consider both absolute and relative performance. Rather than producing an efficient frontier in two dimensions, Multi-goal Optimization produces an efficient surface in three dimensions: expected return, standard deviation, and tracking error. Benefits of Multi-goal Optimization: Multi-goal optimization typically yields an expected result that is superior to constrained mean-variance optimization. For a given expected return, multi-goal optimization produces [...]

## Whitepaper: Asset Allocation

Asset allocation is one of the most important and difficult challenges we face as investors. Thanks to Harry Markowitz, we have an elegant and widely accepted theory to guide us, though implementation in the face of real world complexities is less straightforward than theory might suggest. In this white paper, we describe how to determine allocation to broad asset classes given the complexities of the real world. There are four steps to asset allocation (Figure 1). We must first identify eligible asset classes. Then we need to estimate their expected returns, volatilities, and correlations. Next we must isolate the subset of efficient portfolios that offer the highest expected returns for different levels of risk. Finally we need to select the specific portfolio that matches our tolerance for risk. Figure 1: Four steps to asset allocation 1. Eligible Asset Classes What constitutes an asset class? First, we should expect an asset class to improve our portfolio’s efficiency either by raising its expected return or by lowering its risk. Consider commodities, for example. We might believe that their expected return is insufficient to raise our portfolio’s expected return because advances in technology tend to outpace depletion of resources, thereby lowering commodity prices. However, because commodities offer diversification against financial assets, [...]

## Rethinking Exposure to Loss

Investors typically measure risk as the probability of a given loss or the amount that can be lost with a given probability at the end of their investment horizon, ignoring what might happen along the way. Moreover, they base these risk estimates on return histories that fail to distinguish between calm environments, when losses are unlikely, and turbulent environments, when losses occur more commonly. We propose modifying exposure to loss to account for within-horizon losses as well as the regime-dependent nature of large drawdowns. Because value at risk and probability of loss are two sides of the same coin, we focus our analysis on value at risk. Conventional Value at Risk Simply stated, value at risk is equal to a portfolio’s initial wealth multiplied by a quantity equal to expected return over a stated horizon minus the portfolio’s volatility multiplied by the standard normal variable[1] associated with a chosen probability. Unfortunately, this simple description ignores an important complexity. Asset returns are not normally distributed. Because compounding causes positive cumulative returns to drift further above the mean than the distance negative cumulative returns drift below the mean, returns tend to be lognormally distributed.[2] This means that logarithmic returns, also called continuous [...]

## Understanding Estimation Error

When investors build portfolios, they begin with a long history of returns of the assets to be included in the portfolio. They use these historical returns to compute volatilities and correlations, which they typically extrapolate to estimate future volatilities and correlations. Although they might also use these historical returns to guide their estimates of expected returns, it is not common to extrapolate historical means. More often, investors rely on fundamental analysis or other information to estimate expected returns.[1] Sources of Error Investors base their risk estimates on histories that are typically decades-long; however, the investment horizon they are attempting to characterize usually ranges from one to five years. This estimation process, therefore, exposes investors to three sources of error: small-sample error, independent-sample error, and interval error. Small-sample error arises because the realization of volatilities and correlations from a small sample of returns will differ from the volatilities and correlations of the large sample from which it is selected. But investors are not concerned with a small sample within a large sample; rather, they care about the volatilities and correlations of a future small sample that is independent of the large historical sample. Therefore, investors also face independent-sample error because the [...]