July 6, 2010
Safe Withdrawal Rate: An "optimum" value
There must be an optimum SWR value. If the SWR is small enough, then the balances remain positive for as long as we like, but in that case our withdrawal amounts likely will be insufficient to support our lifestyle. As the SWR grows, one’s lifestyle may be better supported, but the balances will drop to zero over ever-shorter time horizons. So we seek a SWR that is a useful compromise. The main problem is that future returns and inflation are unpredictable.
Let’s begin to solve the problem by finding the maximum withdrawal rate, such that the account value is zero at the end of the time horizon, assuming the returns earned on investments and inflation are known. I call this rate the Point Withdrawal Fraction, because we still haven’t addressed the variability of returns and inflation.
Any series of returns and inflation can be replaced by an equivalent constant return or an inflation rate for the whole horizon. I lose no generality by assuming that return rates and inflation rates are constant.
The Point Withdrawal Fraction is illustrated in Figure 2.
Figure 2: Point Withdrawal Fraction
This graph is valid for the specific returns and rate of inflation assumed. It plots the withdrawal rate against the arbitrary number of years N in the time horizon.
How can we deal with the randomness of returns and inflation?
The innovation: Apply your own beliefs
I account for random future returns and inflation by capturing the investor’s personal beliefs about his investment returns and inflation. These beliefs dominate all decisions, whether the investor accepts “historical” values, bets on his own, or combines both. The investor can use this framework to make his own best guesses based on his subjective and specific beliefs.
This model structures the assumptions each investor inherently makes and captures them in a useful way. The investor has no need to embrace any particular historical interpretations or predictions based on them, although I recognize that all predictions implicitly rely upon possibilities revealed by history. I simply define a range of possible values of return rates and inflation rates and then specify a probability distribution for those values. Each distribution, by its probability weighting, defines a scenario. A pair of return rate and inflation rate scenarios, as defined by their distributions, combine to form a path.
These paths derive from different beliefs about the rates that will occur. Each path describes a belief about a future state of the world that produces a specific pair of return and inflation probability distributions. The issue here is treating the randomness of those assumptions and not their constancy. A path encapsulates an entire future “world,” as represented by the joint probability distribution of returns and inflation.
Examples of beliefs
To illustrate this idea, assume five possible values for the future returns:
{0%, 2.5%, 5%, 7.5%, 10%}.
Now assume a set of values for inflation. In the model we can choose any number and any values of return rates and inflation rates. Here, I use five values for inflation
{2%, 4%, 6%, 8%, 10%}
I capture the beliefs about the paths under different conditions by assigning subjective probabilities that each scenario value actually occurs over the time horizon. Each particular path, comprised by definition of two scenarios, has a total in this example of 5x5 possible pairs of fixed return rates and inflation rates.
To consider a wider view of our future, I choose three return rate scenarios ("Bad Luck," "Skill," and "Good Luck") and three inflation scenarios ("Minimal," "Observable," and "Runaway"). Then there are nine possible paths, each defined by the specified pair of scenario distributions.
The following six graphs show the investor-generated, subjective probability distributions for return rates (Figure 3) and inflation rates (Figure 4) for all time horizons of interest. Remember, these probabilities capture an investor’s current beliefs about his future performance, for his own idiosyncratic reasons.
The investor can explicitly analyze extremely rare but costly return scenarios simply by including a very negative return value (e.g., -70%) and assigning that value a probability as part of a scenario distribution. Of course, such a negative return would rapidly deplete the account, but it also would occur with very low probability.
Figure 3: Three return rate scenarios
Figure 4:Three inflation rate scenarios
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