## Windham Insights Series

## Factor Methods: Part Two

In the first part of this series, we discussed how to perform factor analysis and the challenges that come with it. Catch up here. CROSS-SECTIONAL REGRESSION ANALYSIS As we learned in the first post, factor analysis reveals covariation in returns, and challenges us to identify the sources of covariation. Cross-sectional regression analysis, on the other hand, requires us to specify the sources of return and challenges us to affirm that these sources correspond to differences in return. We proceed as follows. Based on our intuition and prior research, we hypothesize attributes that we believe correspond to differences in stock returns. For example, we might believe that highly leveraged companies perform differently from companies with low debt, or that performance varies according to industry affiliation. In either case, we are defining an attribute—not a factor. The factor that causes low-debt companies to perform differently from high-debt companies most likely has something to do with interest rates. Industry affiliation, of course, measures sensitivity to factors that affect industry performance (such as military spending or competition). Once we specify a set of attributes that we feel measure sensitivity to the common sources of risk, we perform the following regression. We regress the returns [...]

## Factor Methods: Part One

Financial analysts are concerned with common sources of risk that contribute to changes in security prices, called factors. By identifying these factors, analysts may be able to control a portfolio’s risk more efficiently, and perhaps even improve its return. This post will discuss the first of two common approached used to identify factors. The first, called factor analysis, allows analysts to isolate factors by observing common variations in the returns of different securities. These factors are merely statistical constructs that represent some underlying source of risk (which may or may not be observable). The second approach, called cross-sectional regression analysis, requires that we define a set of attributes that measure exposure to an underlying factor and determine whether or not differences across security returns correspond to differences in these security attributes. We'll get to that next. FACTOR ANALYSIS Let us first begin with an analogy that will highlight the insight behind factor analysis. Suppose we wish to determine whether or not there are common sources of intelligence in students, based on the grades of 100 students in the following nine courses: algebra, biology, calculus, chemistry, composition, French, geometry, literature, and physics. First, we compute the correlation between the algebra grades [...]

## Risk Budgets

For decades, investment managers have evaluated portfolios according to their likelihood of loss, or “value at risk.” Value at risk (VaR) is generally understood to describe the maximum loss an investment could incur at a given confidence level over a specified investment horizon. Below is an example of how to solve for value at risk. Suppose we estimate a portfolio’s expected return and standard deviation to equal 7.10% and 18.20%, and assume that returns are lognormally distributed. It is logical to estimate the probability that this portfolio will suffer a loss of at least 20% in any given year. We begin by converting the periodic expected return and standard deviation into their continuous counterparts (shown below). The continuous expected return and standard deviation equal 5.44% and 16.87% respectively. Next, we convert -20.00% to its continuous counterpart, which equals -22.31% [ln(0.80)], and calculate the area to the left of -22.31% under the normal distribution. We do so by first dividing the distance between -22.31% and 5.44% by the continuous standard deviation (18.67%), which equals 1.645. This value is called the normal deviate, and it means that -22.31% is 1.645 standard deviation units below the continuous expected return of 5.44%. When we [...]

## 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, [...]