The Use of Numerical Simulation in Private Client Investment Management

The complexity of planning and managing investments for private clients has led to increasing use of numerical simulation methods in the financial community. In a variety of areas, techniques such as Monte Carlo simulations have become increasingly popular. Among these areas are asset/liability and spending policy analysis for retirement planning, stress testing in risk assessment, tax deferral strategies, and “portfolio opportunity distribution” methods in performance measurement. Numerical methods are also used to address parameter uncertainty in portfolio optimization exercises and to improve the quality of statistical analysis of markets when higher moments are observable in data samples.

Although numerical methods are growing in popularity, I would like to remind everyone of the big limitation of numerical simulations: The conception of “what if?” is wholly dependent on experience to date. We can’t simulate events we haven’t considered. It should also be remembered that numerical methods are most suitable for situations in which the analytical problem is too complex to allow tractable algebraic solution. Alternative approaches are also available for consideration of this class of problem.

Most simulation methods are based on the Monte Carlo approach, where you assume the parameters of the probability distribution of a variable and then take a series of random draws from that probability distribution. The assumed distribution can be simple. You might assume that the distribution of returns is normal, that it has constant volatility, and that returns are uncorrelated from one time period to another. You could also assume the existence of complex properties, such as skew, kurtosis, time-varying parameters, and even jumps. It is also possible to consider systems of multiple correlated random variables and take random draws that include all the variables in the problem at once. The downside to such procedures is that if there are lots of moving pieces (variables), the number of draws needed to get a sufficient sample is large and hence the process is time consuming. It is thus tempting to overly simplify the problem to cut computation times.

An alternative to Monte Carlo techniques is bootstrap methods. Bootstrap simulations are based on real data samples rather than the parameters of probability distributions. It acknowledges that we have only lived through the path of history that actually happened and lets you ask: What else might have happened even if the distribution of possible events was similar?  

For example, imagine you have 180 monthly observations in a return time series for asset class X. Pick a random number R from 1 to 180 (think of numbers selected in a bingo game). Make the Rth observation from the data series the first observation in a new series. Repeat the random pick another 179 times, each time adding the Rth observation from the original series to the new series.

The order will be different in the new series, and some original observations might be omitted, while others appear more than once, so parameters like mean and standard deviation of the data series will change. Using bootstrap methods, you can create as much simulated data as you want to explore “what might have happened.”

The most popular use of numerical simulations in private client investment management is for asset allocation decisions within 401(k) and other retirement plans. Firms such as Financial Engines and Invesra use numerical simulations to find the asset mix that has the highest likelihood of meeting retirement goals. However, these analyses are often criticized as being insufficient. Often the analysis uses only historical asset class return information, which is certainly not a valid estimate of future return distributions. Such methods also do not really consider by how much you will miss your goals if you do miss. Considering only probability and not magnitude of failure is called a lower partial moment zero approach. Most importantly, these analyses have the implicit assumption that investors’ tolerance for risk is constant, irrespective of wealth, age, or past investment performance.

A different view of the investor life cycle suggests that the key issue in formulating investment policies is how aggressive or conservative an investor should be at each moment in time to maximize his or her long-term wealth subject to a shortfall constraint (a floor on wealth). This approach was originally developed in Wilcox (2003) and was incorporated in a CFA Institute Research Foundation monograph.¹,² One way to express this concept is

where

L is the ratio of total assets to net worth
T* is the effective tax rate
R is the pretax mean return
S  is the pretax return standard deviation

Total assets and liabilities on an investor’s “life balance sheet” can be flexibly defined to include the present value of implied assets (such as lifetime employment savings) and expected expenses (such as college tuition, insurance, and estate taxes).

Taxation, particularly “lot by lot” capital gain taxation, is complex to model, so numerical methods are used to simulate potential after-tax outcomes of strategies. In a recent research study, Wilcox (2008) uses Monte Carlo simulation to estimate the proper adjustments in asset allocation and allows for uncertainty about future capital gain tax rates and the life span of the investor.³ In related work, diBartolomeo (2008) simulates the extent to which opportunities for tax deferral are related to the cross-sectional dispersion of returns of securities within an asset class.⁴

Another popular application of numerical simulation is portfolio optimization under parameter uncertainty. Markowitz efficient frontier analysis assumes that you know with exactness the parameters of the return probability distributions (mean, standard deviation, correlation). In reality, you only have estimates of the future parameters, so you are underestimating the uncertainty of the problem. You are missing the risk of being wrong. One way of addressing parameter uncertainty is to use “resampled” optimization, where you repeatedly draw new parameter estimates from a simulation and calculate the optimal portfolio for each draw. You can then take an average of the many portfolios that proved optimal for a specific instance of data outcomes. Early work on portfolio formation based on bootstrapping was done by Bey, Burgess, and Cook (1990).⁵ Later, Michaud published a parametric approach to this issue using multivariate Monte Carlo methods.⁶ It should be noted that a variety of other techniques have been developed to address parameter uncertainty in portfolio formation, including Bayesian methods and “robust” optimization.

When you make asset allocation and other investment decisions, these choices are based on estimates of the parameters of return distributions. Statistical estimation of return parameters, such as mean, standard deviation, correlation, and beta, can be biased if the observed data are not normally distributed and exhibit time-varying volatility or correlations across time. One approach to estimation in the presence of higher moments is Markov Chain Monte Carlo (MCMC) analysis. In MCMC, you combine Monte Carlo simulations with Bayesian statistics to get more robust parameter estimates. For example, consider doing a simple regression analysis to get a beta coefficient for a stock. In MCMC, you would do the regression many times, each time using input data that are derived from a simulation that extends your original information. For each regression, you would then evaluate whether the estimated beta value seems economically sensible (e.g., 0< beta <5). You would thus ignore beta values you judged to be irrational, and you would then average the beta values that passed your “rationality” test to get a final estimate of beta.⁷

Monte Carlo simulations of “what could go wrong” called “stress testing” are a typical approach to evaluating the downside risk of portfolios. The most common downside risk measure is value-at-risk (VaR) analysis. Investors should be cautioned that VaR is an incoherent measure of risk in that there are some investment situations where VaR will lead investors to clearly incorrect conclusions about the relative risks of different investments. Downside risk methods are most suitable for financial entities with high degrees of leverage where nonsurvival is plausible or where portfolios contain substantial positions in derivative instruments. Some studies have shown that consideration of “downside risk” is relevant only in a limited number of investment situations.⁸

Private client portfolios often have peculiarities that make measuring their performance against conventional benchmark indices less meaningful. Examples are different tax rates, legacy positions, SRI (socially responsible investment) constraints, and differences in investor concern about absolute losses as compared with benchmark-relative underperformance. One alternative is to use Monte Carlo simulations of the possible range of returns from an actively managed portfolio given the same universe and constraint set.⁹ If a portfolio under evaluation ranks highly within the range of simulated possible return outcomes, that is a positive outcome that can be evaluated for statistical significance much more quickly than can traditional performance measures.

For those wishing to pursue “hands-on” use of simulation methods, a good summary of numerical procedures used for investment analysis can be found at various websites.¹⁰ There are also “quasi” Monte Carlo methods that allow you to get a good sample of complex distributions with fewer random draws.¹¹

Monte Carlo simulation and related numerical methods are useful tools for dealing with the complexity of real world investing circumstances but are not substitutes for an understanding of investor needs and preferences. These techniques are most valuable in understanding highly nonlinear relationships where a small change in an input variable may create a large change in the result. A good example is complex derivative instruments, where often no analytical solutions exist. Simulation methods also can be put to good use in testing alternative formulations of closed-form analytical solutions.

Endnotes