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Summary This article refines a previously-presented method for qualifying investment portfolios as suitable for retirement. It uses simple formulas for the effect of taxes on returns and volatilities, which leads to a surprising result: an investor in a higher tax bracket can accept a lower volatility. The method also extends the previous analysis to cover more volatile portfolios, such as those trading XIV and VXX. Introduction A previous article introduced a method for comparing investment portfolios based on back-test results. It considered a recently-retired person who: – Invests an initial amount at the start of retirement, – withdraws a percentage of the initial amount each year, adjusted for inflation, and – holds a portfolio with an expected volatility and return for the duration of their retirement. The previous article showed how to make a go/no-go decision about investing in a portfolio, based on its expected after-tax annualized return, after-tax annualized volatility of returns, and historical inflation. However, back-tests provide pre-tax returns and volatilities, not after-tax figures, and the current level of inflation remains below the mean historical level. To improve the usefulness of the method, this new article shows how to decide whether to invest in a portfolio based on its expected pre-tax returns and volatilities, and based on other-than-historical inflation rates. As before, this article defines risk as a number with direct impact on the retiree, the chance of running out of money during retirement; rather than as a more abstract number, the annualized volatility of returns. A prudent retiree would first seek to reduce risk, the chance of running out of money, to a negligible level. That ensured, the retiree would next seek to increase the portfolio’s balance at the end of retirement to leave a legacy. Simulation method As in the previous article, this analysis uses a Monte Carlo simulation tool at portfoliovisualizer.com to test the risk of a portfolio with a given volatility and return. Table 1 shows the input parameters for the simulation. For each volatility shown in the table, the analysis tried various values of expected return until the simulation output showed a 99% probability of success. This means that at the preset annual withdrawal and volatility settings, 99% of Monte Carlo trials showed a positive balance at the end of retirement. In other words, the retiree did not go broke. The expected return setting that yields 99% probability of success represents the average annualized return necessary throughout retirement to reduce risk to a negligible level at the given settings for annual withdrawal and volatility. Defining negligible risk as 99% probability of success (1% risk) seems appropriate considering the severity of the consequences of running out of money. The simulation tool also provides a median end balance, the retiree’s legacy at the end of retirement in 50% of Monte Carlo trials at the given withdrawal rate and volatility settings, and at the expected return necessary for 99% probability of success at those settings. The simulator shows median end balance discounted for inflation, and therefore expressed in the same dollars as the initial invested amount at the start of retirement. This procedure yielded (volatility, return) pairs at 1% risk of going broke for withdrawing an inflation-adjusted fixed amount annually, equal to 3% of the initial amount. It also provided the median end balance at this volatility, return, and withdrawal rate. Simulation results The simulation tool provided the results in Table 2, where: “Median annual return” = (Median end balance / Initial amount)^(1/30)-1. This gives the median annual rate of return during retirement after inflation and withdrawals at the selected withdrawal rate, the selected volatility, and the rate of return required to reduce risk to 1%. Consider, for example, a portfolio with 15% volatility – similar to the historical volatility of the S&P 500 index. Suppose inflation remains near zero. Table 2 shows that a retiree would need an average annual return of 12% in this portfolio for an acceptable risk of going broke. If the portfolio in fact delivers this 12% return, year after year, then the investor will benefit from a median return after withdrawals of 9%, and the original investment of $1M will rise to a median legacy of $13M at the end of retirement. While this median performance seems more than adequate, remember that there remains a 1% chance of leaving no legacy at all. Each row in Table 2 represents a hypothetical portfolio. Each portfolio has the same 1% risk of going broke, but the portfolios with higher volatility require higher annual returns to reduce risk to that level, and as a consequence, investors benefit from higher median annual returns, and their heirs should benefit from greater legacies. An investor who chooses a higher-volatility portfolio at the same level of risk should expect to experience a jumpier account balance and to leave a greater legacy. Effect of inflation Chart 1, graphed from Table 2, shows how annual return required for 99% success probability increases with volatility. A portfolio with annual return on or above the line has acceptable risk. The lines in Chart 1 can be considered “lines of equal risk,” or in this case, “lines of 1% risk.” The difference between the two lines in Chart 1 is close to the mean historical inflation rate (4.18%). Over the range studied here, the annual return required for 99% success probability can reasonably be estimated as the zero-inflation annual return (lower line in Chart 1), plus the expected inflation rate. For simplicity, the remainder of this article assumes zero inflation, which is close to the situation today. Chart 2, also graphed from Table 2, shows how median annual return (and therefore the investor’s legacy) also increases with volatility. As explained above, each row in Table 2 gives returns for a different volatility, but all rows have the same 1% risk. Similarly, all points on the same line in Chart 2 have the same 1% risk. For these curves, annual return was selected to reduce the worst-case risk to 1% at a given volatility and withdrawal rate. Chart 2 shows that for two portfolios with equal risk, an investor leaves a larger legacy by selecting the portfolio with higher volatility, provided that it delivers the required higher return. Chart 2 also shows, like Chart 1, that the difference between the two curves is close to the mean historical inflation rate (4.18%). Over the range studied here, the median annual return with inflation can reasonably be estimated as the zero-inflation median annual return (lower line in Chart 2), plus the expected inflation rate. Required pre-tax return Until now, the analysis has not considered the effect of taxes. The required return as a function of volatility in Chart 1 must apply to after-tax returns and volatilities, because those are what affect the balance in the retiree’s account. This begs a question, what are the corresponding pre-tax volatilities and returns? Define “Rtn” as the required annual after-tax return for a given after-tax volatility (“Vol”), that is, the annual return required for 99% probability for reaching the end of a 30-year retirement, making 3% annual withdrawals, and assuming zero inflation. At a marginal tax rate “Tax,” the after-tax return: Rtn = (1-Tax)*PreRtn, where PreRtn is the pre-tax return (Equation 1). The after-tax volatility is reduced by the same ratio: Vol = (1-Tax)*PreVol, where PreVol is the pre-tax volatility (Equation 2). Equation 2 holds true for volatility because volatility is a standard deviation (“σ”), and for a random variable X and a constant m: σ(m*X) = m*σ(X). For example, at a tax rate of Tax = 50%, for a portfolio to provide an after-tax volatility of Vol = 15% and an after-tax return of Rtn = 12%, it must have a pre-tax return of PreRtn = Rtn/(1-Tax) = 24%, but it can have a pre-tax volatility as high as PreVol = Vol/(1-Tax) = 30%. Table 3 and Chart 3 show after-tax and pre-tax (volatility, return) pairs for 1% risk. The after-tax volatilities and returns come from Table 2, and the pre-tax volatilities and returns come from applying the simple equations in the preceding paragraph to the after-tax figures. Table 3 and Chart 3 provide pre-tax figures for 50% and 25% marginal tax rates: For example, in Chart 3, portfolio “K” has 45% after-tax volatility, which, from Chart 1, requires 67% after-tax return for 1% risk. With 25% tax, this corresponds to pre-tax volatility of 60% and pre-tax return of 89%. With 50% tax, this corresponds to pre-tax volatility of 90% and pre-tax return of 133%. Back-test results are pre-tax. By the way, these stratospheric volatilities and back-test returns are included here for exceptional strategies, such as those trading derivatives of derivatives (XIV and VXX). Charts 3b and 3c show an expanded view of more usual volatilities and returns. Consequently, Charts 3, 3b, and 3c provide an investor with a way to qualify a portfolio for retirement – it must fall above the line in these charts that corresponds to investors’ marginal tax bracket. If an investor used the lines in the previous article (which were after-tax lines) to qualify a portfolio based on back-tested volatility and return (which are pre-tax figures), this would have been too stringent a qualification test. In effect, the investor would have required a return above the green line in Chart 3, when a return above the yellow or red line would have sufficed. To take inflation into account, the investor needs to shift the curves in Chart 3, 3b, or 3c upward by the expected inflation rate. Chart 3b shows an expanded view of the low-volatility part of Chart 3: Chart 3c shows an expanded view of the midrange of Chart 3: Charts 3, 3b, and 3c show that at a given back-test volatility – which is a pre-tax volatility – the required back-test return – which is a pre-tax return – is lower for a higher tax rate. This non-intuitive result occurs because taxes not only reduce returns, but also reduce volatility. When an investor does poorly, so does the tax collector. Effectively, the tax collector shares the investor’s risk along with the investor’s returns. This analysis has other interesting (and perhaps non-intuitive) consequences: Consider a strategy with back-tested (pre-tax) average annual return of 25% and volatility of 40%. Row F in Table 3 shows that this has acceptable risk for an investor in the 50% tax bracket, but row H in Table 3 shows that it is too risky for an investor in the 25% tax bracket. This investor needs the tax collector to share more of the risk. Now, consider a strategy with a back-tested (pre-tax) average annual return of 20% and volatility of 40%. Rows F and H in Table 3 show that this is too risky for an investor in either tax bracket. However, if that investor keeps 25% of the retirement account in that portfolio and 75% in cash at zero return and zero volatility, the account would have a pre-tax return of 25% * 20% = 5% and a pre-tax volatility of 25% * 40% = 10%. Rows B and C in Table 3 show that this is enough return at this volatility to reduce risk to an acceptable value for an investor in either tax bracket. Discussion and conclusion Investors could use this method to qualify portfolios for retirement investments, based on back-tested returns and volatilities, and taking taxes and inflation into account. The method extends to cover unusually volatile portfolios: even those with 50% volatility can provide acceptable risk after taxes and inflation, provided they maintain acceptable returns. This opens a door toward including non-traditional portfolios – such as those trading VXX and XIV – in a prudent retiree’s account. This method is subject to the classical limitation of back-tests: they do not consistently predict future results. Most investors will want to maintain a mix of qualified portfolios, including a traditional core. Acknowledgement: The author thanks Dr. Toma Hentea for reviewing and clarifying the article. Appendix: Alternative calculations with a pseudo-Sharpe ratio Although Charts 3, 3b, and 3c provide enough information to make a go/no-go decision about investing in a portfolio, there is another method for looking at the data. Both methods reach the same decision in the same situation. For the second method, portfolio back-tests provide not only (volatility, return) pairs, but they also provide a ratio of annualized return to annualized volatility. This is similar to a Sharpe ratio, except it assumes a risk-free return of zero (close to the situation today). Table 4 and Chart 4 show the required return/volatility for 1% risk, using the data from Table 3. Chart 4 shows that the required return/volatility ratio (“pseudo-Sharpe ratio”) for 1% risk increases with volatility over the range studied. It also shows that the pseudo-Sharpe ratio required for a given portfolio (“A” through “L”) does not change with the investor’s tax situation. This follows directly from equations 1 and 2, because volatility and required return change by the same proportion when changing tax situations. Like Chart 3, Chart 4 provides an investor with a method to qualify a portfolio – its pseudo-Sharpe ratio must fall above the curve in Chart 4 for that investor’s marginal tax bracket. Chart 4b provides an expanded view of the lower-volatility part of Chart 4: Charts 4 and 4b show that at a given back-test volatility, the required back-test pseudo-Sharpe ratio for 1% risk is lower for a higher tax rate. As in Charts 3, 3b, and 3c, this occurs because the tax collector shares the investor’s risk along with the investor’s returns.