Education: the path from cocky ignorance to miserable uncertainty.
Mark Twain
I wrote a small blog post introducing the new Monte Carlo simulation app on the quantamental platform at the end of September. Since then, I have implemented a couple of upgrades, and the application now supports the simulation of simple Glide Paths and withdrawal plans.
An investor's time horizon often depends on the ultimate goal and is thus subject to change. This is especially important in the case of retirement savings. Young investors have a long time horizon which ceteris paribus justifies a relatively aggressive asset allocation. However, the number of people who want to risk losing 50% of their retirement savings at 65 is probably pretty low. Thus, as investors approach retirement age, their time horizon decreases, and a more defensive asset allocation becomes more adequate.
Glide Paths consider this and automatically adjust the asset allocation according to the changing time horizon (usually by gradually constructing more defensive portfolios). It is, therefore, a pretty intuitive concept, easily understandable also for private end clients. Moreover, the adoption of such an approach and its impact on expected returns and the distribution of ending wealth can now be modeled straightforwardly. The user thereby provides expected return and volatility for the first and last year of the investment period. Then, the program linearly interpolates those values based on the chosen number of years the projection is supposed to cover.
One factor determining the risk/return profile of a Glide Path is the algorithm selected to change the allocation over time and investors can choose between a lot of paths in the market. This publication by MPI Research, for instance, compared the trajectory of the equity quota in various Target Date funds. Most products studied by MPO Research seem to follow a three-staged approach. During the first period, the allocation is kept between 80% and 100%. Then, the actual Glide Path is adopted in the second period. During this time, the allocation is reduced linearly to a value between 30% and 50%. Finally, the third period keeps the equity quote relatively stable again. However, there is also a product that follows a strictly linear approach throughout the whole time.
I tackle the topic slightly differently and, in the Monte Carlo application, utilize three different algorithms. This includes a simple linear interpolation between the start and end risk/return, a logarithmic interpolation, whereby risk is reduced faster at the beginning, and an exponential interpolation whereby risk drops more quickly towards the end of the investment horizon. As I couldn't find a package or function for non-linear two-point interpolations on the internet, the attached PDF also provides a simple R implementation of the three methods and a visualization of the resulting paths.
I used the application to simulate a savings plan following these three Glide Paths and a scenario where the initial (high) risk is kept throughout. The following illustration shows the expected median ending wealth compared to the values expected with 95% and 99% confidence. The expected median return is much higher if the higher initial risk is kept throughout. However, interestingly, the riskier allocation still achieves a higher ending wealth at the 95% confidence level (upper range). Thus, the Glide Paths only fare better when the confidence level is increased to 99%. In other words, the Glide Path decreases tail risk, but due to the long time horizon, a constant, more aggressive allocation will yield at least the same result with a pretty high degree of confidence. Of course, this only holds a priori in time t=0 and is based on the simplifying assumption of normally distributed returns. I elaborate on the latter problem in the second part of this blog post which illustrates withdrawal plans and uses realized S&P 500 returns.
Another interesting observation that makes sense intuitively is that the exponential algorithm results in significantly higher expected ending wealth (median) while featuring similar tail risk as measured by the 99% confidence interval.
The following two charts, produced with the Monte Carlo application, show the projected development of the same savings plan presented above, assuming constant risk and an exponential Glide Path.
The simulation of savings plans can be helpful for investors interested in building and growing their wealth. However, when dealing with High Net Worth Individuals, we are more frequently confronted with the opposite problem. Investors have accumulated a certain amount of wealth and attempt to live off it. Back in the glorious days of risk-free interest income, this could be achieved relatively easily if expectations weren't too high. However, nowadays, if you want to maintain your purchase power, there is no way around taking some volatility.
This makes it more important to understand the behavior of withdrawal plans under risk. Therefore, the new withdrawal function runs the same Monte Carlo simulation while deducting a particular cash flow based on user specifications every month. Beyond that, I also ran an analysis on the S&P 500 between 1926 and 2021 again. Finally, I simulated the distribution of ending wealth of an investor who retrieves 3% of initial or contemporary wealth every year.
The results are pretty interesting and illustrate the magnitude of uncertainty investors who attempt to live on their savings face in the presence of portfolio volatility. Chart 1 and Chart 2 show the distribution of ending wealth for both scenarios based on all rolling 25 years time horizons between 1926 and 2021 (monthly returns). As the S&P 500 returned way more than 3% on average, it is not surprising that the mean and median ending wealth after withdrawals is still significantly higher than the initial investment. However, the dispersion is vast even for scenario one and becomes even more significant if withdrawals are fixed as a percentage of initial wealth. As can be seen in Chart 2, in the worst cases, an investor who took out only 3% of initial wealth per year over 25 years eventually faced bankruptcy.
As this looks almost too extreme to believe, the following chart illustrates what happened to investors during the World Economic Crisis. Sometimes it's good to recall that back then, the S&P 500 experienced a maximum drawdown of roughly 80% and only recovered in the mid-50s. It may be easy to brush this off as a singular event during a time of mass unemployment followed by a World War. Nevertheless, this chart could easily be replicated using some Japanese data from the 90s and 2000s.
The first and most important lesson to learn from this is that it is not a good idea to calculate with a fixed cash flow based on the initial investment in the presence of risk (I recommend this article for more empirical findings on the historical performance of withdrawal plans)
If a flexible plan based on contemporary wealth is chosen, the Monte Carlo application, therefore, models the trajectory of the investor's wealth and the expected cash flow. The following two charts produced by the app show this for an example that assumes 5% annual return and 10% volatility, which is in line with the performance of a Balanced Multi-Asset portfolio denominated in EUR since the early 2000s.
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In this case, I assumed an initial investment of 1m. Again, we face a pretty wide dispersion. In comparison, an investor can, on average, expect his cash flow to grow from 30.000 to a median expected value of 37.000. However, the cash flow drops to less than 16.000 with a probability of 5%, and the minimum value achieved with 99% confidence is only 11.000, two-third below the initial cash flow.
Finally, to illustrate the impact of volatility on median cash flow projections, I reran this example, assuming a slightly higher withdrawal rate of 4%. This means the withdrawal rate is still 1% lower than the expected return, and if we invested in risk-free assets, we would expect the portfolio to grow by 1% per year, as illustrated in the first chart below.
However, as the following chart illustrates, this 1% buffer essentially gets eaten up by volatility. In our example, the expected median ending wealth is slightly below the investor's initial investment. If the investor out the entire 5% or 100% of expected return, the predicted median ending wealth would drop to 0.776m or 25% less than the initial investment.
The small example illustrates another problem investors face in the low or negative interest rate environment. Even if investors generate the same expected return by taking on more risk, expected returns in the presence of volatility are essentially worth much less. As a result, investors who attempt to live on volatile assets need to build in a sizable buffer if they want to maintain a certain level of wealth over time with a reasonably high level of confidence.
Bad news for the new breed of #vanlife frugalistas and their magic formula ("if you keep your pot of money invested in stocks and shares, you can live off the "safe withdrawal rate" of 4% a year, and your money will never run out") ... unfortunately this idea comes from a paper published in 1998.
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