# The Arithmetic of Prudence, Part 2

In the second installment of this series I want to further expound on the point made at the end of Part 1: that when projecting into the future, the expected value is the mean which for a normal distribution is the same as the median or the 50 percentile value. A more comprehensive approach is to examine the distribution of all possible outcomes. Keeping in mind the asymmetry between the extra pain from missing the target number and the more subdued joy from exceeding it, a prudent planner may want to optimize a lower percentile outcome, e.g. 10 or 25 percentile.

To illustrate the point I present the probability density functions (PDFs) of the ending values of two hypothetical portfolios . For simplicity, the outcomes are assumed to follow standard Gaussian distributions [the actual parameters are mean = 115 and standard deviation (SD) = 20 for Portfolio 1, mean = 100 and SD = 5 for Portfolio 2]. The target portfolio value is indicated by the red dash line. I intentionally left out the context where for this exercise to make it as general as possible. It could be about reaching the number at the end of accumulation phase to support a certain retirement lifestyle; or it could be about the end of the distribution phase where success means portfolio value is positive.

The majority outcome from both portfolios are to the right of the red dash line: both portfolios are likely to meet the financial goals, a very good thing. Now let’s delve a little deeper. The expected ending values are at the center of the respective distributions and clearly Portfolio 1 has a higher value. On the other hand, Portfolio 1 has a wider spread or larger standard deviation. Note that the areas below the two curves both equals 1 by definition. The areas under the curves to the left of the red dash line represent the probability of failure. Portfolio 1, despite having a higher expected value, actually has a higher probability of failure! As you can see from the diagram, the two PDFs cross at some point, the exact location depends on the relative mean and SDs. Mathematically, when we examine the left side of the distribution we’re looking at lower percentile outcomes. For example, in calculating withdrawal scenarios we regularly look at the 95% confidence level, in other words, the 5 percentile outcome. It is my contention that a more conservative planner should pay more attention to the lower percentile outcomes (the “sure thing”) than the mean expected value.

While it is often the case that a portfolio with higher expected rated of return will also have a larger standard deviation, the two PDFs above were specifically chosen to illustrate a point. We should rightly ask how real portfolios behave. To that end, we can turn to the excellent tools available at Portfoliovisualizer.com.

#### Step 1. Determine sample portfolio CAGR and SD from backtesting

First we establish three sample portfolios: portfolio 1, 100% equities with a 50/50 US/international split; portfolio 2, a 60/40 3-fund portfolio, again with a 50/50 US/international split in equities, and the total bond market for the fixed income portion; and lastly the Permanent portfolio with equal weights in US equities, long and short term treasuries, and gold. Note that the backtest was from 1987, the earliest time data on all asset classes involved was available. The CAGRs were 7.79%, 7.53% and 7.11% respectively. The SDs were 15.53%, 9.35% and 7.11% respectively. There was a clear reduction in volatility from adding bonds, while the Permanent Portfolio exhibited even less volatility as expected. The difference in CAGR appeared minor but would be appreciable over a multi-decade compounding period. My personal view in portfolio construction is that the closer to the end of the projection period the greater impact of volatility and lesser of CAGR.

#### Step 2. Use Monte Carlo simulation to project portfolio value at the end of distribution phase

For illustration purposes I’ll use Porfoliovisualizer’s Monte Carlo simulation tool for a standard 30-year withdrawal at 4% of portfolio value per year, matching the recommendations from the famous Trinity study. The starting portfolio value is \$1MM, the parameters of the Monte Carlo study was set up as follows: statistical returns from 1987-2015 was used (only slightly different from the numbers from 1987-2016), assume returns follow normal distributions, and a 3% inflation with a volatility of 1.5%.

The screen capture above shows the simulation the 100% equity portfolio. Results are for 25/50/75 percentile ending portfolio values. The same simulations were run for the 60/40 3-fund portfolio and the Permanent portfolio and summarized below.

For both the 50 and 75 percentile results, the 100% equity portfolio has by far the highest expected value, followed by the 60/40 portfolio and then the permanent portfolio. However, when looking at the 25 percentile results, the order is exactly reversed with the Permanent Portfolio on top and the 100% equity portfolio on the bottom. Here we have a situation similar to the very first, entirely hypothetical diagram: the portfolio with lower expected value but tighter spread wins when examining lower percentile outcomes. So the take-away is that for conservative planners who want to adopt a maximin approach (maximize the guaranteed minimum) looking at the expected value alone can give misleading results.

A couple footnotes to the study above: 1) the start date of 1987 missed one of the biggest bull markets in gold so should have hurt the Permanent Portfolio, 2) in the Monte Carlo simulations, normal (Gaussian) return distributions was assumed instead of fat-tailed distributions which probably helped equity-rich portfolios, 3) my assumption of 3% inflation was based on current economic conditions and didn’t seem to make a big impact on the results, 4) note the study is NOT a forecast, backtesting was used to establish historically realistic CAGRs and SDs which under a standard withdrawal plan provided historically realistic portfolio value probability distributions. These probability distributions proved that the PDFs shown in the first diagram were qualitatively correct.

Edit: added the assumption that the distribution is normal, thus the mean and median are the same.