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results are important. The lognormal model can underestimate CVaR by as much as 5.6 percentage points for an aggressive portfolio. That’s a huge margin, and it can mislead advisors as they estimate the downside risk of clients’ portfolios. Impact of Fat Tails on Wealth Accumulation 83 years. The estimate from the log-TLF model (two times in 80 years) is much closer to the historical record than that from the lognormal model (one time in 100 years). In year six at the first percentile, the moderate portfolio under the log-TLF distribution model can lose as much as 33%; in the lognormal distribution model, the highest loss is 28% in the first six years. These results are particularly important for the wealth accumulation of investors who are six years away from retirement. Such an investor holding a moderate portfolio has a 1% probability of losing one third of his or her total wealth. With this knowledge, advisors could decide to hedge against this extreme downside risk by using a portfolio insurance product—such as an appropriately priced equity-linked certificate of deposit with a maturity of six years or an insurance product that includes guaranteed minimum withdrawal benefits. Conclusion would be prudent to add a principal hedge against this downside risk for investors nearing their retirement. K James X. Xiong, Ph.D, CFA, is a senior research consultant at Ibbotson Associates, a Morningstar company. The author thanks Peng Chen, Thomas Idzorek, and Paul Kaplan at Morningstar for their helpful comments. To study the impact of fat tails on the portfolios’ wealth accumulation, we next ran two sets of Monte Carlo simulations for each of the three portfolios. The first simulation assumes a lognormal distribution, the second a log-TLF distribution. Each simulation contains 10,000 30-year return scenarios. The simulated results are similar for the three portfolios, so we only will report the results for the moderate portfolio. Both the log-TLF and lognormal distributions have almost the same wealth at the 50th percentile, but the difference in wealth at the first percentile— the worst-case outcomes—is significant. This makes sense because the log-TLF distribution has a fatter tail and, thus, larger downside risk. At the first percentile, the moderate portfolio under the log-TLF model can lose 27.5% of its total value in year one. (In other words, the log-TLF model says that in one out of 100 years the moderate portfolio will lose 27.5% of its value.) The lognormal model at the first percentile predicts that the moderate portfolio can lose only 20.1% in one year—a significant difference of 7.4 percentage points. Put slightly differently, the Monte Carlo simulations show that under a log-TLF model it takes the moderate portfolio 40 years to suffer a 20% one-year loss. Under a lognormal model, it takes about 100 years for the moderate portfolio to lose 20% in one year. To test these results, we observed the returns that the moderate portfolio would have earned since 1926. The portfolio would have lost more than 20% in three calendar years: 1931, 1937, and 2008. Thus, the likelihood of the portfolio losing 20% in one year is about three times in Footnote 1 Stocks are represented by the S&P 500. Bonds are represented by the BarCap Aggregate Bond, which is backfilled with U.S. intermediate government bonds from 1926 to 1975. References Bachelier, Louis, 1900. Théorie de la Spéculation. Doctoral dissertation. Annales Scientifiques de l’École Normale Supéieure (ii) 17, 21-86. Translation: Cootner 1964. Cootner, Paul H., ed. 1964. The Random Character of Stock Market Prices. MIT Press, Cambridge, Mass. Blattberg, R.C., and N.J. Gonedes, 1974. “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices.” The Journal of Business, Vol. 47, pp. 244-280. Clark, P.K., 1973. “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices.” Econometrica, Vol. 41, pp. 135–155. Fama, E. F., 1965. “The Behavior of Stock-Market Prices.” The Journal of Business, Vol. 38, pp. 34-105. Kaplan, P.D., 2009. “Déjà Vu All Over Again.” Morningstar Advisor, February/March 2009, pp. 29-33. Lévy, P., 1925. Calcul des probabilités (Gauthier-Villars, Paris). Mandelbrot, B., 1963. “The Variation of Certain Speculative Prices.” The Journal of Business, vol. 36, pp. 392–417. Mantegna, R.N., H.E. Stanley, 1999. An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge. Rockafellar, R.T. and S.Uryasev, 2000. “Optimization of Conditional Value-At-Risk.” The Journal of Risk, Vol. 2, No. 3, 2000, pp. 21-41. Xiong, J.X., 2009. “Using Truncated Lévy Flight to Estimate Downside Risk.” Working paper. We show that returns models that use a lognormal distribution underestimate the downside risk of a portfolio. Models using a log-TLF distribution are superior, as evidenced by the fact that log-TLF models fit well the entire distribution of historical monthly returns. These fat tails have further impact on a portfolio’s downside risk and wealth accumulation. In general, a diversified portfolio’s annualized CVaRs under the log-TLF distribution model are 3.5 to 5.6 percentage points higher than that under the lognormal distribution model. As a result, the lognormal model can mislead advisors and investors when they are considering the risks of their portfolios. Finally, Monte Carlo simulations using a log-TLF distribution model indicate that investors in a moderate portfolio have a 1% probability that they will lose one third of their portfolio’s total value in six years. Therefore, advisors MorningstarAdvisor.com 27

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