In the 65 years since Harry Markowitz formalized the theory of asset allocation, academics and practicing investment professionals have sought to master the process, to solve the puzzle that is investment risk, and to find a “fool-proof” way to protect themselves from loss. As one of the most important challenges in the portfolio construction process, the rules and nuances of asset allocation continue to surprise us in an evolving market.
In this guidebook, we begin by going back to the fundamentals. Each chapter focuses on a specific aspect of the portfolio construction process and will explore the most significant concepts and applications. We would like to acknowledge our CEO, Mark Kritzman, for his research and published work in this area. Many of the concepts and applications mentioned in this guidebook were derived from his extensive work and expertise, as well as the efforts of his colleagues. Any referenced concepts can be found at the end of this guidebook.
Our goal is to provide answers to some of the most pertinent questions: What defines something as an asset class? Why should I care about factors? How do I know how much of a portfolio is actually at risk? We hope to enlighten our readers on how these concepts can better prepare them for investing in the modern age. In an often unpredictable market, we aim to give our readers, peers, and clients the tools they need to protect themselves from loss—and provide them with the opportunity for success.
- Relative Independence
- Expected Utility
- Factor Analysis
- Factors in Stock Returns
- Issues of Interpretation
- Cross-Sectional Regression Analysis
- Why Bother with Factors?
- Small-Sample Error
- Independent-Sample Error
- Interval Error
- Stability-Adjusted Optimization
- Monte Carlo Simulation and the Stock Market
Asset allocation is one of the most important decisions faced by investors, however there are no universally accepted criteria that define exactly what an asset class is. Some investments take on the status of an asset class because managers feel that investors are more inclined to allocate funds to products if they are defined as an asset class, rather than merely as an investment strategy. Alternatively, the investment industry tends to overlook investment categories that legitimately qualify as an asset class because investors are reluctant to defy tradition.
What are the real consequences of NOT defining an asset class?
The imprecision about the nature of an asset class reduces the efficiency of the asset allocation process in at least two ways.
1. If dissimilar investments are wrongly grouped together into an asset class, the portfolio will not be diversified efficiently.
2. If an asset class is inappropriately partitioned into redundant components, the investor will be required to deploy resources unproductively to analyze irrelevant expected returns, standard deviations, and correlations.
Furthermore, the investor may waste additional resources in search of relevant investment managers. For these reasons, it is important to establish criteria for the purpose of identifying legitimate asset classes (Kritzman, Toward Defining an Asset Class, 1999).
The list of proposed asset classes is long and diverse. The traditional candidates are:
- Domestic Stocks
- Foreign Stocks
- Real Estate
- Domestic Bonds
- Real Estate
- Domestic Bonds
The stock and bonds are often divided into more specific groups:
- Large Cap Stocks
- Mid Cap Stocks
- Small Cap Stocks
- Growth Stocks
- Value Stocks
- Financial Stocks
- Developed Market Foreign Stocks
- Emerging Market Foreign Stocks
- Long Term Government Bonds
- Long Term Corporate Bonds
- Intermediate Term Government Bonds
- Intermediate Term Corporate Bonds
- High Yield Bonds
- Municipal Bonds
- Developed Market Foreign Bonds
- Emerging Market Foreign Bonds
Finally, there are the so-called alternative investments:
- Hedge Funds
- Managed Futures
- Market Neutral Funds
- Private Equity
- Venture Capital
We propose four criteria for determining asset class status:
1. An asset class should be relatively independent of other classes in the investor’s portfolio
2. An asset class should be expected to raise the utility of the investor’s portfolio without selection skill on the part of the investor
3. An asset class should be comprised of homogeneous investments
4. An asset class should have the capitalization capacity to absorb a meaningful fraction of the investor’s portfolio
Relative independence address whether or not a new asset class will help diversify more efficiently. An asset class will not improve diversification if combinations of asset classes already in the portfolio could duplicate the new asset class’s risk characteristics. It is important to remember that the redundancy does not have to be with a single asset class, but rather with any linear combination of included asset classes.
How can we test for independence?
We can test for independence of a proposed asset class by identifying the combination of asset classes that minimizes tracking error with the proposed new asset class. We call this portfolio a mimicking portfolio. Then, we judge whether the tracking error of the mimicking portfolio is sufficiently large to suggest independence. Tracking error is computer as the square root of the average of the squared differences between the mimicking portfolio’s returns and the returns of the proposed asset class. In other words, it is the standard deviation of the return differences.
For example, suppose we allocate our portfolio among U.S. stocks, foreign stocks, U.S. long-term bonds, and U.S. cash equivalents. We want to determine whether or not we should include U.S. intermediate-term bonds in this portfolio. Based on monthly returns from the beginning of 1985 through the end of 1997, a mimicking portfolio consisting of 0.21% U.S. stocks, 0.56% foreign stocks, 48.86% long-term bonds, and 52.38% short-term instruments produces tracking error of only 1.32% with intermediate-term bonds. Compare this value to the tracking error between each of the current asset classes and its mimicking portfolio constructed from other assets in the portfolio, based on the same historical return sample. (Kritzman, Toward Defining an Asset Class, 1999)
Tracking Error of Mimicking Portfolios with Current Asset Classes
U.S. Stocks: 12.15%
Foreign Stocks: 16.34%
Long Term Bonds: 6.46%
Short Term Instruments: 6.94%
It is reasonable to conclude from these tracking errors that immediate-term bonds are redundant to the other asset classes and therefore should not be considered as an asset class in this situation.
The second criterion for asset class status raises two critical distinctions: the distinction between expected utility and expected return, and the distinction between random selection and skillful selection on the part of the investor.
Expected utility refers to happiness or satisfaction, which comes from either the expectation of higher returns or the expectation of less risk. Consider commodities, for example. You may believe that their expected return is insufficient to raise a portfolio’s expected return because advances in technology tend to outpace depletion of scarce resources. However, because commodities offer diversification against financial assets, especially in environments of high unanticipated inflation, their inclusion in a portfolio might lower risk sufficiently enough to more than offset their expected reduction of return.
Expected Return – Risk Aversion X Variance
Suppose our portfolio consists of a single asset with an expected return of 10.00% and a standard deviation of 12.00%. Also assume that our risk aversion equals 1.5, which indicates that we are willing to give up 1.5 units of expected return in order to lower portfolio variance by one unit. With these assumptions, and remembering that variance equals standard deviation squared, we calculate expected utility to equal 7.84%(.10 – 1.5 X .122).
Now let’s assume that we estimate commodities to have an expected return of 9% and a standard deviation of 12%. At first, it does not appear that an allocation to commodities would improve the risk/return profile of our portfolio because commodities have the same risk as our portfolio but less expected return. However, we must not ignore the correlation of commodities within our portfolio. Suppose we estimate the correlation of commodities with our portfolio to equal 5.00%. If we were to shift 38% of our portfolio’s assets to commodities, its expected return would decline from 10.00% to 9.62%, but its standard deviation would fall from 12.00% to 8.92%. Based on risk aversion of 1.5, the expected utility of this new portfolio equals 8.43%(0.0962 – 1.5 X 0.08922). Given our willingness to exchange expected return for risk reduction, we would be happier to allocate some of our portfolio to commodities even though this shift reduces the expected return we expect to achieve. The point is that expected return, by itself, is insufficient for gauging an asset’s impact on investor satisfaction. (Kritzman, Toward Defining an Asset Class, 1999)
The distinction between random selection and skillful selection is subtle, yet important. An asset class should raise a portfolio’s expected utility not because the investor is skillful in identifying superior portfolio managers within that asset class. Rather the investor should expect the asset class to raise utility even if managers within an asset class are selected randomly. It does not follow, however, that an asset class is disqualified if improvement in expected utility requires skillful managers within the asset class.
Assume that passive exposure to domestic stocks is expected to raise a portfolio’s utility. Domestic stocks therefore might qualify as an asset class. Let’s also assume that the top quartile growth stock managers would also be expected to raise utility. If growth stock managers on average are note expected to raise utility, and only a skillful investor would be able to identify top quartile growth stock managers before the fact, than top quarter growth stock managers would not qualify as an asset class.
Finally, let’s consider hedge funds. Let us first acknowledge the conjecture that hedge funds would raise a portfolio’s utility because their managers, on average, perform better than do non-hedge fund investors who invest in the same assets. In other words, random compositions of the assets that typically constitute hedge funds are not expected to raise utility, but a random selection of hedge fund managers would be. This unusual conjecture arises from the fact that hedge funds typically require a lock-up period; that is, their investors are precluded from withdrawing funds for a pre-specified period of time. Therefore, hedge fund managers are in a position to collect a liquidity premium, which may explain why their performance exceeds the average performance of the assets in which they invest. If this conjecture were indeed true, a naïve or random selection of hedge fund managers, requiring no skill selection on the part of the investor, would be expected utility because hedge fund managers as a group are skillful in extracting a liquidity premium. Hence, hedge funds might qualify as a legitimate asset class, even if their constituent investments do not.
The requirement for homogeneity among the components of an asset ensures that we do not ignore opportunities for diversification. If an asset class comprises dissimilar components, then by investing in that asset class we implicitly impose the unnecessary and potentially harmful constraint that the components must be held in the same relative proportions as their weights in the asset class. It is likely that we could achieve a more efficient portfolio if we partitioned the dissimilar components into multiple asset classes.
For example, foreign stocks are typically viewed as a single asset class. For this example, let’s define foreign stocks as a 40% allocation to German stocks, a 30% allocation to UK stocks, and a 30% allocation to Japanese stocks—all unhedged. This construction is not significantly different in its risk profile from the EAFE index. Based on the returns, standard deviations, and correlations from the beginning of 1980 through the end of 1997, a portfolio comprised of 25% U.S. stocks, 50% U.S. bonds, and 25% foreign stocks, has an expected return of 14.54% and a standard deviation of 9.20%. (Kritzman, Toward Defining an Asset Class, 1999)
Now suppose we partition foreign stocks into six components: hedged and unhedged German stocks, hedged and unhedged UK stocks, and hedged and unhedged Japanese stocks. If we retain the same weights in U.S. stocks, U.S. bonds, and foreign stocks as a group, but allow the allocations within foreign stocks to vary among the six components, we could achieve a more efficient portfolio. By recognizing that foreign stocks are not homogeneous, we can segment this asset class into several homogeneous components and improve our portfolio’s expected return and risk significantly.
The final criterion for asset class status—that it has to be sufficiently large to absorb a meaningful fraction of our portfolio—is self-evident. If we were to invest in an asset class with inadequate capacity, we would likely drive up the cost of investment and reduce our portfolio’s liquidity. The consequence might be to lower our portfolio’s expected return and increase its risk to the point at which the proposed asset class would no longer be expected to improve our portfolio’s utility.
The process of selecting a portfolio may be divided into two stages. The first stage starts with observation and experience and ends with beliefs about the future performance of available securities. The second stage starts with relevant beliefs about future performances and ends with choice of portfolio.
– Harry Markowitz, “Portfolio Selection” Journal of Finance, March 1952
In order to effectively construct and evaluate portfolios, one must first establish future estimates of an asset class’s performance through the lenses of both risk and return.
The risk of an asset class in a portfolio is defined by its standard deviation and correlation to the other assets within the portfolio. When building an asset allocation model, standard deviation is most often annualized. Let us consider several methodologies for estimating the future risk of an asset class within the portfolio.
HISTORICAL risk is, understandably, the amount of risk that an asset class has had prior and can be used to anticipate how an asset may perform.
EXPONENTIAL risk is calculated using the exponentially weighted moving average. Exponential risk has two main benefits over historical risk:
- Recent time series observations carry greater importance relative to observations in the past. This allows exponentially weighted risk estimates (covariance) to react faster to recent shocks in markets.
- Risk estimates in this model have a shorter memory following a shock. The risk estimates decline smoothly and rapidly as the significance of shock observations decreases through time. In contrast, shocks observed by the equally weighted historical risk model will increase risk estimates for the full observation period, and will cause an abrupt shift when they fall out of the observation window.
QUIET & TURBULENT REGIMES – It is known that standard deviations and correlations are not static throughout time. By identifying periods of unusual behavior, we can estimate the future risk in periods of financial turbulence. There periods can be divided into quiet and turbulent regimes. Quiet risk periods are statistically predictable asset movements within a certain date range. Turbulent risk periods are statistically unusual asset movements within a specific date range, identified by calculating the multivariate vector distance. Turbulent periods are typically marked by large asset movements (volatility) and/or unusual correlation. When looking at the history of an asset or portfolio, it is important to remember the state of the market during that investment horizon.
HISTORICAL returns are often annualized. We have all read (and probability written) “Past performance is no guarantee of future results” at some point in our career. However, historical returns can often be a useful tool acting as a benchmark in the process of determining future expectations for the return of a particular asset class.
CAPITAL ASSET PRICING MODEL The Capital Asset Pricing Mode (CAPM) is a theory of market equilibrium which partitions risk into two sources: systematic risk and unsystematic risk. An asset’s systematic risk is equal to its beta squared multiplied by the market portfolio’s variance. The CAPM implies that investors should incur only systematic risk because they are not compensated for bearing unsystematic risk. (Kritzman, Kinlaw, & Turkington, A Practitioner’s Guide to Asset Allocation, 2017)
BLACK-LITTERMAN The Black-Litterman model is a mathematical model for portfolio allocation that combines the theories of modern portfolio theory, the CAPM, and mean-variance optimization (which we will discuss in chapter 6). The Black-Litterman model is a method for calculating optimal portfolio weights based on the given inputs.
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.
There are two common approaches 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.
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 aptitude in students, based on the grades of 100 students in the following nine courses: algebra, biology, calculus, chemistry, composition, French, geometry, literature, and physics. (Kritzman, What Practitioner’s Need to Know… About Factor Methods, 1993)
- First, we compute the correlation between the algebra grades of all 100 students and their grades in each of the other eight courses
- Next, we compute the correlations between the biology grades of all 100 students and their grades in each of the other seven courses
- We continue until we have computed the correlations between the grades of every pair of correlations – 36 in all (shown in Table 1, below).
That all of these correlations are positive suggests the presence of a pervasive factor (probably related to study habits). In addition to this factor, there appear to be three other factors or commonalities in performance.
First, the variation in algebra grads is highly correlated with the variation on calculus and geometry grades. Moreover, performance in calculus is highly correlated with performance in geometry. The three grades, however, are not nearly as highly correlated with the grades in ANY of the other six courses. Therefore, we conclude that there is a common aptitude that underlies performance in these three courses.
Second, performance in biology is highly correlated with performance in chemistry and physics, and performance in chemistry is highly correlated with performance in physics. Again, performance in these courses does not correspond as closely with performance in any other course. We may therefore again conclude that there is a common source of aptitude associated with biology, chemistry, and physics.
Finally, the grades in composition, French, and literature are all highly correlated with each other, but not with the grades of any other courses. This leads us to deduce the presence of a third factor.
Our next task is to identify these factors, which is where our intuition comes into play. We may reasonable conclude that one of the common sources of scholastic aptitude is skill in mathematics or quantitative methods, because we observe high correlations between the three math courses. Aptitude in science appears to be another common factor, given the high correlations in the three science courses. Finally, verbal aptitude seems to be another common factor, due to the high correlations in French, composition, and literature.
We do not actually observe the underlying factors; we merely observe that a student who performs well in algebra is more likely to perform well in geometry or calculus than in French. From this observation, we infer that there is a particular aptitude that helps to explain performance in algebra, calculus, and geometry—but not in French. The aptitude is the factor. (Kritzman, What Practitioner’s Need to Know… About Factor Methods, 1993)
We should note that these results do not imply that performance in a certain course is explained by a single factor. If such were the case, we would only observe correlations of 1 and 0. This point is underscored by the fact that the variation in physics grades (science) is more highly correlated with performance in math courses than it is with French, literature, or composition. This result is intuitively pleasing in that physics depends more on mathematics than French, literature, or composition. We may therefore conclude that performance in physics is primarily explained by aptitude in science, but that it is also somewhat dependent on aptitude in math as well.
FACTORS IN STOCK RETURNS
Ok, so how do we apply this thought process to the stock market? Let’s assume instead that we wish to determine the factors that underlie performance in the stock market. We begin by calculating the daily returns of a representative sample of stocks during some period. (In this study, the stocks are analogous to courses, the days in the period are analogous to students, and the returns are analogous to grades!).
To isolate the factors that underlie stock market performance, we begin by computing the correlations between the daily returns of each stock and the returns on every other stock. Then, we seek out groups consisting of stocks that are highly correlated with each other, but not with stocks outside the group.
For example, we might observe that stock 1’s returns are highly correlated with the stocks of 12, 21, 39, 47, 55, 70, and 92, and that the returns of all the other stocks in this group are all highly correlated with each other. From this observation, we may conclude that the returns of these stocks are explained, at least in part, by a common factor. We proceed to isolate groups of stocks whose returns are highly correlated with each other, until we isolate all the groups that seem to respond to a common source of risk. (Kritzman, What Practitioner’s Need to Know… About Factor Methods, 1993)
Our next task is to identify the underlying source of risk for each group. Suppose that a particular group consists of utility companies, financial companies, and a few other companies that come from miscellaneous industries but that all have especially high debt-to-equity ratios. We might reasonably conclude that interest rate risk is a common source of variation in the returns of this group of stocks. Another group might be dominated by stocks whose earnings depend on the level of energy prices; we may thus deduce that the price of energy is another source of risk. Yet another group might include companies across many different industries that all derive a large fraction of their earnings from foreign operations; we might conclude that exchange risk is another factor. (Kritzman, What Practitioner’s Need to Know… About Factor Methods, 1993)
We must first rely on our intuition to identify the factor that underlies the common variation in returns among the member stocks. Then, we can test our intuition as follows:
- We define a variable that serves as a proxy for the unanticipated change in the factor value
- We regress the returns of stocks that seem to depend on our hypothesized factor with the unanticipated component of the factor value
It is important that we isolate the unanticipated component of the factor value, because stock prices should not respond to an anticipated change in a factor. It is new information that causes investors to reappraise the prospects of a company.
Suppose we identify inflation as a factor. If the Consumer Price Index is expected to rise 0.5% in a given month, and it rises precisely by that amount, the prices of inflation-sensitive stocks should not change in response. If, however, the CPI rises 1.5%, then the prices of these stocks should change in response. In order to test whether or not a particular time series represents a factor, we must model the unanticipated component of its changes. (Kritzman, What Practitioner’s Need to Know… About Factor Methods, 1993)
A reasonable approach for modeling the unanticipated component of inflation is to regress inflation on its prior values under the assumption that the market’s outlook is conditioned by past experience. The errors, or residuals, from this regression represent the unanticipated component of inflation. We thus regress these residuals on the returns of the stocks we believe to be dependent on an inflation factor to determine is inflation is indeed a factor.
This approach is heuristic, designed to expose factors by identifying groups of stocks with common price variations. Its intuitive appeal is offset by the fact that it produces factors that explain only part of the variation in returns. Moreover, these factors are not necessarily independent of each other.
ISSUES OF INTERPRETATION
Factors derived through factor analysis, whether we employ the heuristic approach or the more formal approach, are not always amenable to interpretation. It may be that a particular factor cannot be proxied by a measurable economic or financial variable. Instead, the factor may reflect a combination of several influences, some perhaps offsetting, that came together in a particular way unique to the selected measurement period and the chosen sample of securities. In fact, factors may not be definable.
We thus face the following trade-off with factor analysis. Although we can account for nearly all of a sample’s common variation in return with independent factors, we may not be able to assign meaning to these factors, or even know if they represent the same sources of risk from period to period or sample to sample. Next, we’ll consider an alternative procedure called cross-sectional regression analysis.
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 across a large sample of stocks during a given period—say a month—on the attribute values of the stocks as of the beginning of that month. Then, we repeat this regression over many different periods. If the coefficients of the attribute values are not zero and are significant in a sufficiently high number of the regressions, we conclude that differences in return across the stocks relate to the differences in their attribute values.
FACTOR ANALYSIS VS. CROSS-SECTIONAL REGRESSION ANALYSIS: which is better?
There are pros and cons to both factor methods. Through factor analysis, we can isolate independent sources of common variation in returns that explain nearly all of a portfolio’s risk. It is not always possible, however, to attach meaning to these sources of risk. They may represent accidental and temporary confluences of myriad factors. Because we cannot precisely define these factors, it is difficult to know whether they are stable or simply an artifact of the chosen measurement period or sample.
As an alternative to factor analysis, we can define a set of security attributes we know are observable and readily measureable and, through cross-sectional regression analysis, test them to determine if they help explain differences in returns across securities. With this approach we know the identity of the attributes, but we are limited in the amount of return variation we are able to explain. Moreover, because the attributes are typically codependent, it is difficult to understand the true relationship between each attribute and the return. Which approach is more appropriate depends on the importance we attach to the identity of the factors versus the amount of return variation we hope to explain with independent factors.
Why Bother With Factors?
At this point, you may be questioning why we bother to search for factors or attributes in the first place. Why not address risk by considering the entire covariance matrix instead?
There are two reasons why we might prefer to address risk through a limited number of factors. A security’s sensitivity to a common source of risk may be more stable than its sensitivity to the returns of all the other securities in the portfolio. If this is true, then we can control a portfolio’s risk more reliably by managing its exposure to these common sources.
The second reason has to do with parsimony. If we can limit the number of sources of risk, we might find that it is easier to control risk and to improve return simply because we are faced with fewer parameters to estimate.
Optimization is a process by which we determine the most favorable tradeoff between competing interests, given the constraints we face. Within the con-text of portfolio management, the competing interests are risk reduction and return enhancement. Asset allocation is one form of optimization. We use an optimizer to identify the asset weights that produce the lowest level of risk for various levels of expected return. Optimization is also used to construct portfolios of securities that minimize risk in terms of tracking error relative to a bench-mark portfolio. In these applications, we are usually faced with the constraint that the asset weights must sum to one.
We can also employ optimization techniques to manage strategies that call for offsetting long and short positions. Suppose, for ex-ample, that we wish to purchase currencies expected to yield high returns and to sell currencies expected to yield low returns, with the net result that we are neither long nor short the local currency. In this case, we would impose a constraint that the currency exposures sum to zero.
One of the most popular optimization methods is known as mean-variance optimization. Mean-variance optimization is a robust portfolio construction technique that identifies combinations of assets that offer the highest expected return for a given level of risk. It assumes the investor is indifferent to a portfolio’s tracking error against a benchmark. This is the framework that Harry Markowitz introduced.
Another method, known as mean-variance-tracking error optimization (or “multi-goal” optimization), combines mean-variance analysis with tracking error. 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. (Chow, 1995)
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. (Kritzman & Rich, The Mismeasurement of Risk, 2002)
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 frequently in reality than is implied by a lognormal distribution. These estimation errors often cause investors to underestimate the probability of loss, and to overestimate the probability of gain.
Rather than focusing simply on addressing these issues (though we do address them), we focus on what we believe to be a more fundamental cause of financial failure: Investors’ wealth is affected by risk throughout a period in which it is invested, but risk is generally measured only for the termination of the period.
Figure 1 demonstrates the distinction between risk based on ending outcomes and risk based on outcomes that may occur along the way. Each line represents the path of a hypothetical investment through 5 possible paths. The horizontal line at 90% represents a loss threshold (which equals 10% here). It is clear here than
only one of the five paths beaches the loss threshold at the end of the horizon, which may cause some to believe that the likelihood of a 10% loss is 20%. However, four of the five paths breach that loss threshold throughout the investment horizon. So, if we care about the investment’s performance along the way, we will conclude that likelihood of a 10% loss is not 20%… but a whopping 80%. (Kritzman & Rich, The Mismeasurement of Risk, 2002)
Why should we care about interim risk?
Investors care about exposure to loss throughout the investment horizon because there are often thresholds that cannot be breached if the investment is to survive to the end of the horizon. If survival is not the main concern, investors may be motivated to pay attention to within-horizon risk because they could be penalized for breaching a barrier. Consider the following:
- Asset management. A client has a portfolio with a provision that it should not depreciate more than 10% over a 5-year investment horizon. Should the asset manager assume that the client will only review performance at the end of the investment period? Not likely. The client will review performance throughout the investment horizon… and terminate the manager if the portfolio dips below 90% of its value at inception. To limit the likelihood of termination, the manager should consider within-horizon risk.
- Hedge-fund solvency. A hedge-fund manager who believes that the likelihood of significant loss at the end of the investment horizon is slim, leverages the portfolio to increase expected return. However, a significant decline from the value of the underlying assets from inception to any point throughout the investment horizon is much more likely than the likelihood implied by the ending distribution of a hedge-fund’s assets. Additionally, significant interim loss could trigger withdrawals that might impair the hedge-fund’s solvency.
- Loan agreement. A borrower is required to maintain a particular level of reserves as a condition of a loan. If the reserves fall below the required balance, the loan is called.
- Securities lending. Many institutional investors lend their securities to others who engage in short selling. These investors are required to deposit collateral with the custodian of the securities. The required collateral is typically adjusted on a daily basis, to offset changes in the values of the securities. Suppose the investor wishes to estimate the amount of additional collateral that might be required at a given probability for the duration of the loan. This value depends on the distribution of the securities’ values throughout the term of the loan.
- Regulatory requirements. A bank is required to maintain a capital account equal to a certain fraction of its loan portfolio. A breach in this requirement will result in a fine. The probability that the bank will need to replenish the capital account to avoid breaching depends on the distribution of the ratio of the capital account to the loan portfolio throughout the planning horizon, not at the end of the horizon or a finite period within the horizon.
These examples are only a few of the many circumstances in which investors should pay attention to probability distributions that span the duration of their investment horizons.
How do you measure within-horizon exposure loss?
To estimate probability of loss:
- Calculate the difference between the cumulative percentage loss and the cumulative expected return
- Divide this difference by the cumulative standard return
- Apply the normal distribution function to convert this standardized distance from the mean to a probability estimate
To capture within-horizon variability, we use a statistic called “first-passage time probability,” which measures the probability of a first occurrence of an event within a finite horizon.
How can I apply within-horizon probability of loss and continuous VaR?
- Currency Hedging. Suppose we allocate a portfolio equally to Japanese stocks and bonds, represented by the MSCI Japan Index and the Solomon Brothers Japanese Government Bond Index. Table 1 shows, based on monthly returns from January 1995 to December 1999, the standard deviations and correlations of these indexes together with the risk parameters of the Japanese yen from a U.S. dollar perspective.
Let us assume further that the underlying portfolio has an expected return of 7.50 percent, hedging costs equal 0.10 percent, and our risk aversion equals 1.00. Based on these assumptions, the optimal exposure to a Japanese yen forward contract is -87.72 percent. The expected return and risk of the unhedged and hedged portfolios are shown in table 2.
Now, let us estimate the probability of loss for the unhedged and hedged portfolios. Table 3 shows the likelihood of a 10 percent or greater loss over a 10-year horizon at the end of the horizon and at any point from inception throughout the horizon for an unhedged and optimally hedged portfolio of Japanese stocks and bonds.
If we were concerned only with the portfolio’s performance at the end of the investment horizon, we might not be impressed by the advantage offered by hedging. But, if we instead focus on what might happen along the way to the end of the horizon, the advantage of hedging is much more apparent.
Even with the foreknowledge that we are more likely than not at some point to experience a 10 percent cumulative loss, we may consider such a loss tolerable. But what about a loss of 25 percent or greater? Again, calculating the probabilities indicates that, although the impact of hedging on end-of-period outcomes is unremarkable, it vastly reduces the probability of a 25 percent or greater loss during the investment horizon. Although many investment programs might be resilient to a 10 percent depreciation, they are less likely to experience a decline of 25 percent or more without consequences.
Now, let us compare VaR measured conventionally with continuous VaR for the hedged and unhedged portfolios. Table 4 reveals that the improvement from hedging is substantial whether VaR is measured conventionally or continuously. For example, measured conventionally, hedging improves VaR from a 5 percent chance of no worse than a 14.68 percent loss, to a 5 percent chance of no worse than a 26.52 percent gain. More important, however, is the substantial difference between VaR measured conventionally and VaR measured continuously. Continuous VaR is more than twice as high as conventional VaR for the unhedged portfolio, and when the portfolio is hedged, continuous VaR shows a substantial loss compared with a substantial gain when it is measured conventionally.
2. Leveraged Hedge Fund. Now consider the implications of these risk measures on a hedge fund’s exposure to loss. Supposed we are interested in a hedge fund that uses an overlay strategy, which has an expected incremental standard deviation of 5 percent. This hedge fund also leverages the overlay strategy. Table 5 shows the expected returns and risks of the hedge fund and its components for varying degrees of leverage.
The data in Table 5 assume that the underlying asset is a government note with a maturity equal to the specified three-year investment horizon and that its returns are uncorrelated with the overlay returns. Managers sometimes have a false sense of security because they view risk as an annualized volatility, which diminishes with the duration of the investment horizon, but as we have noted, the fund’s assets may depreciate significantly during the investment horizon. Figure 2 compares the likelihood of a 10 percent loss at the end of the three-year horizon with its likelihood at some point within the three-year horizon for various leverage factors (e.g., 2 to 1). Figure 2 reveals that the chance of a 10 percent loss at the end of the horizon is low, but there is a much higher probability that the fund will experience such a loss at some point along the way, which could trigger withdrawals and threaten the fund’s solvency.
The same issue applies if exposure to loss is perceived as VaR. Figure 3 shows the hedge fund’s VaR for various leverage factors measured conventionally and continuously. Whereas conventional VaR for leverage factors less than 6 to 1 is negative (a gain) and still very low for leverage factors up to 10 to 1, continuous VaR rangers from approximately 10 percent of the portfolio’s value to approximately 40 percent of its value.
Investors measure risk incorrectly if they focus exclusively on the distribution of outcomes at the end of their investment horizons. This approach to risk measurement ignored intolerable losses that might occur throughout an investment period, either as the result of the accumulation of many small losses or from a significant loss that later (possibly, too late) recovers. To address this shortcoming, we have introduced two new approaches to measuring risk—within-horizon probability of loss ad continuous VaR. Our applications of these measures in reasonable scenarios illustrates vividly that investors are exposed to far greater risk throughout their expected investment periods than end-of-horizon risk measures indicate.
When building a portfolio, investors begin with a long history of returns of the assets to be included. They use these historical returns to compute volatilities and correlations, which they typically extrapolate to estimate future volatilities and correlations. Investors base their risk estimates on histories that are typically decades long, but the investment horizon that they are attempting to characterize usually ranges from one to five years. This process therefore exposes investors to three sources of error: small-sample error, independent-sample error, and interval error. (Kritzman & Turkington, Stability-Adjusted Portfolios, 2016)
Small-sample error arises because the realization of volatilities and correlations from a small sample of returns will differ from those of the large sample from which it was selected.
Independent-sample error arises because the volatilities and correlations of a future sample will differ from those of a prior historical sample, regardless of the sample size.
Interval error arises because investors usually base their calculations of volatilities and correlations on monthly returns and then extend these estimates to annual or multi-year horizons. By doing so, they are assuming that returns are independent and identically distributed. They are also implying that correlations are invariant to the return interval used to estimate them.
Investors typically address estimation error by blending individual estimates with their cross-sectional average, or some other prior belief. This approach is called Bayesian shrinkage. Another common approach to estimation error is called resampling, a process by which optimal weights are generated many times from a distribution of inputs, and then averaged to determine the final portfolio. Both Bayesian shrinkage and resampling cause the portfolios along the efficient frontier to be more self-similar.
We believe that investors should measure the relative stability of volatilities and correlations and treat this information as a distinct component of risk. We call this approach stability-adjusted optimization.
We begin by taking a large sample of historical returns for the assets to be included in our portfolio. We then select all possible overlapping sub samples of a size equivalent to our investment horizon, and compute covariance matrices for all of these small samples using return intervals equal to the duration of our investment horizon.
Next, we subtract the covariances in all of the sub samples from the covariances of that part of the original sample that does not include the respective sub samples. We use the heuristics describes earlier to convert the covariances of each sub sample’s complementary sample to the same interval or our investment horizon. This leaves us with error matrices for all of the sub samples.
These error matrices reflect small-sample error because the sub samples are smaller than the complementary portion of the large sample. They reflect independent-sample error because the small samples are independent of their large-sample complements. They also reflect interval error because the small-sample covariances are estimated from longer interval returns that account for lagged correlations, whereas the complementary-sample covariances are estimated from shorter-interval returns and extended to reflect longer-interval returns by using the heuristics described earlier.
We then select a base case small sample, such as the median sub sample, and we add the error matrices to the covariance matrix of the base case sub sample. Then, assuming normality, we generate return samples from all of the error adjusted covariance matrices, and we combine them into a new large sample of returns, which by virtue of this process reflects the relative stability of the asset covariances. (Kritzman & Turkington, Stability-Adjusted Portfolios, 2016)
This process for generating a stability-adjusted return distribution yields a distribution that has fatter tails than a normal distribution, because the variances of the error-adjusted small samples will differ from each other. These fatter tails should not present a problem to mean-variance optimization so long as we can reasonably describe investor preferences with mean and variance.
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. (Kritzman, What Practitioner’s Need To Know… About Monte Carlo Simulation, 1993)
Monte Carlo simulation is performed with a sequence of numbers that are distributed uniformly, are independently of each other, and are random. Random sequences can be generated using mathematical techniques such as the mid-square method, but most spreadsheet software includes random-number generators. It is important to note that most financial analysis applications generate random variables that are not distributed uniformly. In order to perform a Monte Carlo simulation, the sequence of uniformly distributed random numbers must be transformed into a sequence of normally distributed random numbers.
Applying Monte Carlo Simulation to the Stock Market
Imagine we invest $100,000 to an S&P 500 index fund in which the dividends are then reinvested, and we want to predict the value of that investment 10 years from now. As previously mentioned, we start by generating a series of 10 random numbers that are uniformly distributed. However, we assume that the S&P’s returns are normally distributed and we therefore need to transform our sequence. This transformation can be accomplished easily by applying the Central Limit Theorem.
It is clear to see in Table 1 that while neither X nor Y are normally distributed on their own, their average beings to approach a normal distribution. Therefore, we can create a sequence of random numbers that is normally distributed by taking the averages of many sequences of random, uniformly distributed numbers (Table 2). (Kritzman, What Practitioner’s Need To Know… About Monte Carlo Simulation, 1993)
Figure 1 shows the relative frequency of both sequences, and we can see from this graph that the average of the 30 uniformly distributed sequences approached a normal distribution.
The next step in Monte Carlo simulation is to scale the normally distributed sequence so it has a standard deviation of one and a mean of zero. By dividing each observation by 0.05 (the theoretical standard deviation), and then subtracting 10 (the theoretical mean of this sequence) from each other its observations, we can achieve this transformation.
Next, we must rescale our sequence to reflect our assumptions about the mean return and standard deviation of the S&P 500. For this exercise, let us believe that the average return of the S&P 500 is 12% and its standard deviation is 20%. We rescale our standardized normal distribution by multiplying each observation by our assumption of 20% for the standard deviation, and adding to this value our assumption of 12% for its average return. Now we have a sequence of returns that we can then use to simulate our investments performance (shown in Table 4). (Kritzman, What Practitioner’s Need To Know… About Monte Carlo Simulation, 1993)
To carry out the Monte Carlo simulation, we then link the sequence of random returns and multiply the result by 100,000 (our investment) to derive an estimate of its value in 10 years. For example, the simulated returns in Table 4 yield a value of $333,810.
We repeat the entire process, beginning with the generation and averaging of 30 random sequences. We proceed until we generate a sufficiently large quantity of estimates. The distribution of these estimates is the solution to our problem. The figure below shows the frequency distribution of the terminal value of $100,000 over 10 years, resulting from 100 simulations.
While many problems can be solved using an analytical solution, Monte Carlo simulation is immensely more simple for problems that are too complex to be described by equations. However, in order for Monte Carlo simulation to obtain a reliable result, it must be repeated enough times.