In Part I, I introduced Monte Carlo simulations for personal finance. These use the past behavior of investment classes, perhaps nudged according to a view of the future, to create a range of probabilities for likely outcomes. Their main value is that they show what may happen in about two-thirds of the cases. This corresponds to cumulative probabilities of about 20% to 80%.

I emphasized that we do not understand the actual probabilities of low-probability outcomes. These are those whose cumulative probabilities are below about 20% in such simulations. The reason is that we do not have enough evidence to really know the “tails” of the distribution of annual returns, where unlikely results occur. In my view, telling a retiree there is a 90% probability that his portfolio will last 30 or 40 years is not sensible. One does not know the odds that well.

I also emphasized that is a mistake to imagine that Monte Carlo simulations can produce guaranteed success in a retirement plan. One way to guarantee success is to have enough money when you retire. A commenter on one of my past articles suggested that $10M is the right number. This article is not aimed at folks with that much money, whom I congratulate. Another way to guarantee success is to fit one’s spending within the scope of an inflation-adjusted annuity, while providing for potential health care costs late in life by insurance or other means. If this fits your circumstances, congratulations! But why aren’t you out on the lake or the golf course instead of reading this?

Instead, this article is aimed at people like me who fall short of the $10M and for whom a successful retirement could only be absolutely guaranteed by dropping down to a standard of living I am hoping to avoid. For us, guaranteed success, in some sense, can only be obtained by flexible spending approaches that reduce spending if and when things have not gone well enough.

If this is also your context, Monte Carlo simulations *can *help you choose an approach, consistent with your needs, that has the best potential to meet or exceed those needs. One way or another, though, you must be prepared to adjust if you prove to be unlucky. Wade Pfau points (p.52) out that the fixed rate of real return that corresponds to the worst historical sequence of returns, for a traditional 50/50 portfolio of stocks and bonds, is about 1.3%. This corresponds to the experience of someone who retired in 1966. With this return, a portfolio will last 30 years if one withdraws 4% per year. This is the origin of the “4% rule” identified by Bengen. Here we will look at the statistical performance of various portfolios under several withdrawal rules.

Average performance of the historical markets would have enabled one to withdraw far more than 4%. The Monte Carlo simulations shown below illustrate approaches that can enable more spending or added security, depending how you use them.

**The investments modeled here **

Here I consider potential investments of several types, with properties shown in Table 1. The first is historical US stocks, as represented by the S&P 500 index. The second is historical US bonds. The returns of these two are correlated quite weakly, historically.

Table 1. Investment properties for modeling

Investment | Average real return | Standard Deviation |

US Stocks | 8.2% | 17.5% |

US Bonds | 2.5% | 6.8% |

US Historical TIPs | 2.2% | 6% |

Real Estate (TIAA see text) | 6.4% | 8% |

Non-US Developed Country Stocks | 6% | 20% |

Emerging Market Stocks | 12% | 30.2% |

Low-Yield US Stocks | 6% | 17.5% |

Low-Yield TIPs | 1% | 1% |

In some models, I used historical Treasury Inflation Protected securities (OTC:TIPS), rather than bonds. I modeled the TIPs either based on their historic yield or for a low-yield universe, as shown in the table. Considering my view that coming decades will see a return of inflation, I am skeptical about the value of a traditional portfolio of stocks and intermediate-term bonds. It makes more sense to me, if one wants to include bonds, to include them as TIPs in a bond ladder. This has the added psychological benefit, for typical investors, of securing some years of guaranteed funding to support spending.

For a third type of investment I used the TIAA Real Estate Fund, for which I have the data. Its returns have the statistical properties shown and have been weakly correlated with stocks and bonds. For applications to your own case, the need is for a third category of investment that has moderate yield and is weakly correlated to the others.

The TIAA fund, in particular, is unique. It features direct ownership of Real Estate, like a REIT, but is like a growth stock in that it has no distributions. Since its inception, its correlation with stocks is quite small. David Swenson writes quite favorably of it, in his book Unconventional Success. This fund has underperformed the REIT sector in the present century, but REITs have been remarkably successful by historic standards, in their own brief existence. I find myself hopeful that this can last, but skeptical that it will. I can’t quite bring myself to plug REIT performance into a model like this. It seems like cheating. Maybe owning REITs is cheating successfully, but maybe the other shoe will somehow drop. Personally, I own some REIT investments and some of the fund mentioned.

Finally, in some cases I used the historical behavior of non-US, developed-country stocks and of emerging-market stocks for some models. I treated these as completely uncorrelated from US stocks, which is overly optimistic. They serve to make the point that adding additional weakly correlated investments to your entire portfolio improves its security.

## The** Model Portfolios **

Below I show results from several model portfolios, for various possible withdrawal rules. Table 2 shows the portfolios. I start with a portfolio that is entirely in US stocks. Then I consider a 60/40 mix of stocks and bonds. The next portfolio adds the real estate investment as a source of diversification and uses TIPs rather than 10-year bonds. I also consider diversifying into international stocks; and I consider cases where the stocks and also TIPs have average yields well below their historical values. The withdrawal amount and strategy varies as discussed below.

Table 2. Model Portfolios

Name | US Stocks | US Bonds | US TIPs | Real Estate | Developed Stocks | Emerging Market |

Historical Stocks | 100% | |||||

Historical Stocks and Bonds | 60% | 40% | ||||

Stocks, TIPs, and Real Estate | 50% | 0% | 25% | 25% | ||

Diverse stocks, TIPs, and Real Estate | 30% | 0% | 25% | 25% | 10% | 10% |

Real Estate and Low-yield Stocks and TIPs | 50% | 0% | 25% | 25% | ||

Real Estate, International Stocks, and Low-yield Stocks and TIPs | 30% | 0% | 25% | 25% | 10% | 10% |

**Results for fixed, constant-dollar withdrawal rates **

The figures in this article are in the format introduced in Part I. Let’s review, with an eye on Figure 1. All the simulations shown are for a 40-year period. The left panel shows, for each portfolio value shown on the horizontal axis, the probability that the portfolio is no larger than that amount after 40 years. Thus, the intercept of a given curve with the left axis shows the fraction of cases in which blindly following the initial withdrawal rule would exhaust the portfolio.

For the cases when this would occur, the right panel shows when it occurs, also as a cumulative distribution. The intercept of a given curve with the right axis here gives the same fraction as one finds in the left panel. If the portfolio never approached becoming exhausted, then one would not need to adjust one’s plans, so the left axis of the right panel shows the chance one would need to adjust one’s plans. In reality, of course, one would change one’s plans before funds were exhausted, and the probability of this lies some distance above the curves.

Figure 1 uses a 4% withdrawal rule with annual rebalancing. The real value of the annual withdrawal is taken to equal 4% of the initial value of the portfolio. The nominal value is adjusted for inflation, so the real value does not change. The black curve shows a portfolio composed of only US Stocks, based on historical returns for the S&P 500. As discussed in Part 1, what is meaningful about these curves is where they cross 20% and where their median is. If future stock returns resemble historical ones, one has a 4-in-5 chance of reaching 40 years with more than half the portfolio value remaining. The most likely portfolio value is somewhat more than four times its initial value.

Figure 1. Performance of a 4% withdrawal rate. Models and plots by author. Values below 0.2 on such plots are produced by the highly uncertain tails of the probability distributions. Because of this, my view is that one does not really know the chance that any of the portfolios plotted in this figure would run out of money, if the withdrawal rules were blindly followed. The black curve on the right panel in Figure 1 is informative about the time when a portfolio of stocks alone would do poorly enough to force one to change plans. It suggests that one would need to adjust in less than 20 years in some cases. Once again, though, these numbers are too small to be considered to be accurately known.

The blue curves in Figure 1 show results for a 60/40 portfolio of historical US stocks and historical US bonds. There is no significant difference in the amount one would have left with 20% probability after 40 years. The average size of the remaining portfolio would be roughly half as large as for the portfolio of stocks only. From this perspective, including the bonds seems to do little.

The benefit from the bonds is stronger at 30 years, which was the most common planning horizon 20 years ago. One can see this in the right-hand panel. The difference between the black and blue curves in the right panel is large at 30 years after retirement. This may have driven a lot of investment decisions. My view is that the difference is not accurately known. Pfau finds that wide variations in the fraction of bonds often make small differences. Being a skeptic myself about where we sit in the long-term cycle of interest rates, I find it unlikely that such a portfolio will do as well as shown over coming decades.

The magenta curves in Figure 1 show results for a portfolio that would make more sense to me, having 50% US stocks, 25% TIPs, and 25% real estate. As discussed above, this is modeled on the performance of the TIAA Real Estate Fund, but it really represents any uncorrelated investment with a moderate yield. The role of the TIPs in such a portfolio is not to provide yield. Instead, held as a ladder they provide an inflation-protected outramp. This is of great psychological benefit for many investors. In any event, such a diversified portfolio provides an 80% chance that the portfolio will have grown in real value over 40 years, with it being likely to have reached more than three times its initial size.

My own view is that adopting a 4% rule is extremely conservative, embracing limited spending to achieve gains that are too small to predict with confidence. Adopting a 6% rule involves embracing measurably higher risk, as Figure 2 illustrates. Here the portfolios are the same as in Figure 1, the colors of the curves mean the same thing, and the model uses annual rebalancing. The only difference is that the annual withdrawal is 6% of the initial real value of the portfolio.

Figure 2. Performance of a 6% withdrawal rate. Models and plots by author. What is really interesting is that, for such aggressive withdrawal rates, the notionally safer portfolios perform worse in some sense. Specifically, the fraction of cases in which one would have to change one’s plans is larger. This is an effect I first learned of from Wade Pfau. One can get a better idea of what happens by focusing on the right panel in Figure 2. The earliest cases in which the portfolio sometimes becomes too small are those involving stocks only. But while the portfolios, including bonds, TIPs, and real estate become increasingly risky at a steady rate with advancing years, the stock-only portfolio benefits statistically from the larger average return of stocks.

One could look at Figure 2, decide to put money only in stocks, adopt a 6% rule, and figure that one has a 50% chance to end up with a lot more money than one started with. Then one could go fishing for five years. If at that time the portfolio had not grown, one could revisit the plan. To me, this seems a fine way to spend one’s early, most-energetic years after one graduates from working.

**Spend More! Use Designed Declining Withdrawals **

There is a modification of the withdrawal plan used in Figure 2 that has a much smaller chance of running low on funds. Studies show that spending tends to decrease by (roughly) 1% per year in retirement, as the go-go years transition to the go-slow years and then the no-go years. If one’s income floor, from social security, pensions, and annuities, provides one-third of one’s desired spending, this implies that the amount of money one takes from the portfolio would decrease by 1.5% per year. (That is, it would be multiplied by a factor of 0.985.)

Figure 3 shows the results of withdrawing 6% of the portfolio value in the first year, and then reducing the withdrawal (in constant dollars) by a factor of 0.985 in each subsequent year. This reduces the withdrawal to 3.3% of the initial value in year 40. The portfolios are the same as those used for Figures 1 and 2. The portfolio with three distinct investments (US stocks, TIPs, and real estate in this case) has roughly an 80% chance of preserving or growing one’s initial funds. This declining-withdrawal approach is likely to produce significant portfolio growth.

Figure 3. Performance of an initial 6% withdrawal rate that then is reduced each year by a factor of 0.985. Models and plots by author.The declining-withdrawal approach, with the diverse portfolio, also calculates to have very little chance that the portfolio will be substantially depleted within 40 years. This result is not quantitatively accurate, but from the small probabilities of depletion one knows the following. One would have to receive a sequence of extraordinarily small or negative returns to end up needing to change one’s plans.

One of the things I especially like about declining-withdrawal strategies is that they make it easy to respond to better returns than those representing the unlikely worse cases. Once the total portfolio has increased by some factor that one can choose, there is no reason to reduce subsequent withdrawals. One may even have options after 10 or 15 years of living more grandly, in some way, or of planning for a larger legacy.

Many commentators believe that the future will be a period of reduced returns by both US stocks and US bonds. They might be right. Commentators frequently have reasons that seem compelling. Despite this, most commentators are almost always wrong. Even so, it is worthwhile to examine the consequences of a lower-yield world.

Figure 4 illustrates how things might play out in a lower-yield world. The figure uses the low-yield cases for US Stocks and TIPs shown in Table 1. This example uses an initial 6% withdrawal rate, reduced (in constant dollars) by a factor of 0.99 each year. This reduces the withdrawal to 4% of the initial value in year 40.

Figure 4. Studies of models with reduced stock and bond yields, or wide diversification. Performance is shown for an initial 6% withdrawal rate that then is reduced each year by a factor of 0.99. Most cases shown have reduced average yields. Details are discussed in the text. Models and plots by author. One can see from the black curves that a 60/40 portfolio of stocks and TIPs does quite poorly, having nearly a 45% chance of forcing a change in plans. It also rarely produces a large increase in portfolio value. Even so, any funding shortfall is more likely to occur in the fourth decade of retirement and is not the most likely outcome. Those who begin retirement well into their 60s and don’t expect to live past 100 could well be fine if they take this path, even if yields are low.

The blue curve adds real estate, using the comparatively low historical rates of return of the TIAA fund. This level of diversification both reduces and delays the chances that one would need to adjust one’s plans.

Astute investors, though, are likely to seek further diversification beyond just three categories of investment. The magenta curve in Figure 4 addresses one such case. One might split the stock investment into sectors, about which one finds many articles on Seeking Alpha. Here I used international stocks as a proxy for increased diversification, dividing the 50% portion of the portfolio that is in stocks so that it is 30% low-yield US stocks and 10% each in developed-country stocks and emerging market stocks. The international stocks have their historic yields, as shown in Table 1.

I personally would be comfortable with this case, shown by the magenta curve, subject to the discussion of Black Swans below. This level of diversification makes it very likely that the portfolio will not shrink, even if yields are low. For comparison with the magenta curve, the red curve shows results for the same portfolio with historic yields. Of course, this case is much better.

For reference below, the approximate average gain associated with the four curves in Figure 4 can be calculated as follows. Take the value of the portfolio where the CDF=0.5 to the power of 0.025, which is 1/40. Subtract 1. Add 5%. For the four curves in Figure 4, from left to right, this gives 3.3%, 5.5%, 7.1%, and 8.5%. The entire reason one needs somehow to think about the distribution of possible returns (and sequence of returns risk), and to somehow prepare for possible disappointments, is that these averages are not guarantees.

**Allow for Black Swans **

What is missing from the above models is any consideration of Black Swans – unlikely events that can break their assumptions. One of these is unplanned costs of health care. One must address this. Options include buying long-term-care insurance, establishing an independent health savings fund, or making a decision to reduce spending, if necessary, to keep one’s portfolio above some threshold value. Beyond that, it is a personal decision how much to prepare for the potential of war, hyperinflation, or a prolonged deflationary recession. Many investors hold some gold as protection.

Many readers use planning spreadsheets that project spending forward in time, with whatever level of complexity they choose to include. This might include reductions in spending with age. A simple spreadsheet would include only an investment return on a portfolio and would track everything in nominal dollars. A more complex one might model price inflation going forward and include the real investment return.

An approach I use as I assess my options is to adjust the average investment return in the spreadsheet until my funds would become exhausted the year I reach age 100. I then compare this to the average return of the various Monte Carlo models, including models of my own specific investment plan. The larger the difference between the return that would exhaust my funds and the average from the model, the safer the plan is.

The main takeaway is this. You can spend a lot more money than conservative spending rules would conclude. Just allowing for a decrease in spending with age takes you a long way. Diversification, ideally beyond three investment categories, significantly reduces risk. The psychological key to enable such spending is to replace fear of very unlikely outcomes with a plan for responding to them, if they do happen to occur.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure: **I am not a financial adviser or a tax advisor, but am an independent investor. Any securities or classes of securities mentioned are not recommendations.