In the past I wrote HARS, a retirement simulator that uses historical stock market and actuarial longevity data. I find HARS useful, but wanted a second opinion. That second opinion is the opinion of mathematics. Mathematics is useful because it provides understanding and insight.
We seek a mathematically tractable model. This means the model will contain a number of simplifications. In order to avoid such simplifications we could use simulation, but this is something we don't want to do. The mathematics of the model are quite tedious (eg. computing the average equity return for every 30 year window over 140 years). We use a computer to help us. We evaluate the complete problem space and avoid the simulation approach of taking a subset of the problem space determined using random number generators.
Where possible our analysis will be performed at the 95th percentile level. We chose this level to give a reasonable level of safety to the result, while also having sufficient stock and bond returns data that the 5th percentile performance shouldn't be swamped by outliers. To perform our analysis we compute life expectancy and poor portfolio performance at the 95th percentile level and then combine them to create an overall analysis. Rigorously speaking, the result will then exceed the 95th percentile level. But it isn't possible to determine by how much both because the goodness of one variable might not be able to compensate for the badness of the other variable and because the two variables might be correlated.
Our analysis assumes that we want the minimum portfolio possible to reach our retirement goals of not running out of money at the 95th or greater percentile level. Or with only a small amount of computational re-arrangement we want to know how much we can withdraw from an existing portfolio and have the portfolio remain on track to provide for our retirement goals. Of course it is possible to have a portfolio larger than what is required. The problem then is how to maximize returns for your legacy while ensuring the retirement goals will still be met. We don't attempt to solve this problem except for suggesting the naive and sub-optimal solution of at least notionally partitioning your money and allocating one part of it to retirement to be managed as described here, and the rest allocated to our legacy and invested in stocks.
We use historical and actuarial data for investment returns and longevity respectively.
We assume fixed annual expenses in order to make the problem tractable.
Calculations are performed in real (inflation adjusted) as opposed to nominal (current) dollars.
The US Government produces the US Life Tables that report the probability of death by age. The analysis here is based on the US Life Tables for 2007.
These tables can with the aid of a computer program be massaged to yield the following data.
Measures of remaining life expectancy at various ages:
--------------- remaining years of life ---------------
age expectation median 90%tile 95%tile 98%tile 99%tile
0 77 81 94 96 99 101
50 30 32 44 47 49 51
55 26 27 39 42 45 46
60 22 23 34 37 40 41
65 18 19 29 32 35 36
70 14 15 25 27 30 32
The 1998 Trinity study attempted to determine the sustainable withdrawal rates for a retiree over periods of 15 to 30 years. As can be seen from the above table these time periods are inadequate at the 95th percentile level for a retiree aged 65, and even more inadequate for a retiree younger that that. Retirement simulators such as HARS are more accurate in that they use the US Life Tables to simulate the probability of a retiree's death.
So far we have only considered an individual. What about the 95th percentile life expectancy for a couple, where we are concerned with whoever lives longest?
95%tile life expectancy for a couple:
------ male ------
female 50 55 60 65 70
50 49 48 48 48 48
55 46 44 43 43 43
60 45 41 39 38 38
65 45 40 36 34 33
70 45 40 35 32 29
Here the rule of thumb is: you will both be dead when the younger of
you reaches 97. There is a little more wiggle room in this rule, but
through age 70 it is still only off by at most two years.
Without loss of generality then, when working at the 95th percentile level, we can treat a couple as very closely equivalent to an individual having the same age as the youngest of them.
Medical and other advances have historically increased life expectancy. However, reviewing old actuarial tables, this happens very slowly. If you wish to account for this occurring in the future you may wish to increase your life expectancy, by 1 year if you are under 70, or 2 years if you are under 55. This will only have a small effect on the results, and we do not consider the matter any further.
The key take away from the above tables is when performing retirement planning you need to be able to provide for substantially more years than your expected life expectancy. How much more? The following table provides the answer.
Length of 95th percentile retirement relative to expected length: age 95%tile / remaining_life_expectation 50 1.55 55 1.62 60 1.70 65 1.80 70 1.92Roughly speaking, to be safe in your retirement because you might outlive your life expectancy, you need to provide funding for 1 1/2 to 2 times as many years as your life expectancy.
Unfortunately, we don't live in normal times. Right now, November 2011, Vanguard's VAIPX TIPS Admiral Fund has a SEC yield of -0.26%. Worst case this might be forced to be held in a non-tax-advantaged account. In the presence of taxes (35% Fed. + 9.3% CA = 44.3%) on the inflated value of the TIPS principle (inflation Sep. 2010 to Sep. 2011 CPI-U: 218.439 to 226.889 = 3.9%) we have a drag of 1.73% (44.3% x 3.9%). Combining the two gives a net return of approximately -2%.
Using the formula given in the section "How did we do?", with a net return of -2%, a person aged 65 would require 45 times their annual expenses to retire safely at the 95th percentile level. This is a lot considering their mean life expectancy is only 18 years. We can only hope that the present times do not continue.
That we receive a net return of 0% (tax free) or -2% (taxed) on TIPS is very real concern when it is remembered we seek returns at the 95th percentile level. We have only had TIPS since 1997, and already we are having a second supposedly "anomalous" period.
We do however assume normal times will at least temporarily return, allowing us to buy a TIPS ladder for retirement that yields 2% if needed.
Using interest rate and CPI data from Shiller ( www.econ.yale.edu/~shiller/data/ie_data.xls ) we construct the real return for a portfolio of constant maturity 10 year bonds by subtracting from the 10 year interest rate the annual inflation rate. We use a sliding window that does not wrap and measure the annual returns for each period. One caution. Due to the limited number of samples, this analysis uses overlaping period windows, so the number of independent observations is less than it might first appear.
US 10 year government bond real return 1872-2010:
--------- annual real return ----------
window length (years) worst 1%tile 2%tile 5%tile 10%tile
1 -15.1% -13.3% -8.9% -6.7% -4.5%
10 -3.2% -3.2% -3.0% -1.9% -1.2%
20 -1.3% -1.1% -1.1% -0.9% -0.3%
30 0.0% 0.0% 0.0% 0.1% 0.4%
139 2.5% 2.5% 2.5% 2.5% 2.5%
One thing that must be mentioned is many of the worst values include the high inflation years of 1946 and 1947. As an experiment we attempted to correct for this by artificially setting the real interest rate for those two years to zero. Performance improved slightly (by 0.2% for the 30 year 1st percentile level), and many of worst values where then caused by high inflation from the years 1916 and 1917. These years exist below the 95th percentile level, and so do not show up in 95th percentile results.
In other words a bet on nominal bonds could be a bet that there won't be any major wars during retirement. This is a risky proposition when judged at the 95th percentile level.
Note that the long run real return on nominal bonds is 2.5%, not much more than the estimated 2% return than can be locked in with inflation indexed bonds in normal times. And certainly not worth the risk in constructing a safe retirement portfolio.
At first, with a 6.5% real return for the period 1872-2010 the picture might seem far more rosy. It must be remembered though that the stock market is highly volatile. It can comfortably go for 10 to 15 years with negative returns. This is the reason that typically no more than a small amount of stocks are recommended in a retirement portfolio. Vanguard's target income fund for instance recommends 30% stocks.
A small computer program can rearrange the stock market data supplied by Shiller to see more precisely how risky a stock only portfolio would be for a safe retirement.
US stock market real return with dividends reinvested 1872-2010:
--------- annual real return ----------
window length (years) worst 1%tile 2%tile 5%tile 10%tile
1 -38.8% -35.4% -33.5% -22.4% -12.0%
10 -3.9% -3.9% -3.5% -2.2% -1.1%
20 0.6% 0.8% 1.2% 2.0% 2.6%
30 3.3% 3.3% 3.7% 3.9% 4.2%
139 6.5% 6.5% 6.5% 6.5% 6.5%
TIPS are relatively new, so the data just isn't there to be able to analyze and answer this question for constant maturity or mutual fund TIPS, but we can answer it for stocks and constant maturity nominal bonds. Besides, for TIPS, it is possible to purchase a TIPS ladder rather than a mutual fund, and lock in the returns you are going to get, so the question isn't as important because the returns you get aren't going to vary.
Back to the question at hand. It is relatively easy to combine the returns of nominal bonds and stocks into returns for a variety of combined portfolios. In making these calculations it is assumed the portfolio is continuously rebalanced. In practice rebalancing is only necessary when the portfolio drifts significantly from its target allocations.
There is one adjustment we need to make. For stocks we will assume a net return of 5%. This is what we consider the long term forward looking real return on stocks before taxes. This is slightly below the 6.5% real return recorded by US stocks 1872-2010 primarily because part of this 6.5% return is due to P/E growth, and possibly also because US GDP is projected to slow on account of a decline in population growth.
So far we have only considered stocks and nominal bonds, not inflation indexed bonds. If we truly are in extraordinary times, and if it will once again be possible to purchase a TIPS ladder returning 2%, the previous graph would look as follows.
The portfolio transition from stocks to nominal bonds as time horizons decrease occurs quite a bit later and quicker than the commonly cited rule of thumb "age in bonds". However, we are not aware of any studies justifying this rule.
Now look at stocks / inflation indexed bonds. In retirement at age 50 through 80, the minimum safe portfolio switches from a pure stock portfolio to a pure TIPS ladder portfolio.
Just because the composition of the minimal safe portfolio changes over time doesn't mean you should automatically switch portfolios over time. You may now have more assets than are needed to fund your retirement at the 99th percentile level. In such a case the additional assets provide a buffer against hitting the 99th percentile level of risk again. Increasing withdrawal rates or taking any extra money off the table each year would be like rolling the same 100 sided die over and over again.
Now, turning to the returns of these two portfolios:
5th percentile safest adjusted portfolio annual return: age TIPS portfolio nominal portfolio 50 3.51% 3.50% 55 3.16% 3.11% 60 2.95% 2.89% 65 2.79% 2.66% 70 2.53% 2.37%
A challenge to the validity of our results is known as Merton's portfolio problem as described in "Measuring and Controlling Shortfall Risk in Retirement", Smith & Gold (2005):
Samuelson (1963, 1969) and Merton (1969) show that a rational risk-averse investor's optimal bond-stock allocation for a fixed horizon of length T does not depend on the value of T. Samuelson (1994) writes that "it is an exact theorem that investment horizons have no effect on your portfolio proportions." Nonetheless, many financial advisors believe that investors should hold more bonds as they grow older ...The most likely explanation for our seemingly anomalous result is Merton's portfolios will over time typically become substantially larger than the 95th percentile worst case portfolios, which provides a buffer, allowing them to remain more aggressive. By way of contrast our portfolio calculations keep the portfolio as small as possible every step of the way, forcing the portfolio to become more conservative as its duration decreases. We lack memory of previous portfolio success, and thus can't use this memory to relax the portfolio requirements for the future. In fact we pay no attention to what might be the typical portfolio success outcomes, only to failure. We are concerned with minimal portfolios at the beginning of each time period rather than the expected portfolio value beginning at the first time period.
It should be clear that our approach is in this regard more conservative than is required.
This could also shed light on the popularity of age in bonds and other time dependent portfolios that appear anomalous. Unable to calculate the optimal time independent portfolio, either because of a lack of mathematical sophistication, or because the portfolio is a target retirement mutual fund and the assets and expenses of the customers unknown, they fall back to computing the minimal safe portfolio at each age.
Let p be the net present value needed for a safe retirement. Let a be the initial annual expenses. Let s be the initial sustainable withdrawal rate. Let d be (1 / (1 + annual_rate_of_return)). Let n be the number of years we need to provide income for a safe retirement.
For our calculation based model, we allocate d^i initial resources to cover year i, then the sustainable withdrawal rate is computed as:
s = a / p = a / (a . (1 + d + .. d^(n-1))) = a / (a . (1 - d^n) / (1 - d)) = (1 - d) / (1 - d^n)
Using the 5th percentile annual real return of the safest portfolio for d, and the 95th percentile life expectancy for n, allows use to compute s. This is not an exact science, especially since our rate of return is variable and we are using a fixed rate formula.
For the HARS retirement simulator, we assume a retired male, with a 5% geometric mean return on stocks, a 2% fixed return on inflation indexed bonds, the 2003 US Life Tables, and simulation years from 1872-2008 in looped sequential order, the HARS simulator can be used to compute the 95th percentile sustainable withdrawal rate.
Sustainable withdrawal rates for simulator and mathematical model: age simulator SWR model TIPS SWR model nominal SWR 50 4.05% 4.22% 4.21% 55 4.29% 4.18% 4.15% 60 4.55% 4.32% 4.28% 65 4.95% 4.58% 4.50% 70 5.47% 4.92% 4.83%The y axis on the following graph has been blown up to show detail:
On the other hand the model is consistently predicting sustainable withdrawal rates that are about 0.5% above or below those of the simulator. This is not insignificant. One possible explanation for this is the model assumes all years are the same as far as returns are concerned, while the simulator may experience bad years followed by good years. It is the early years that count the most as far as portfolio success is concerned.
How many times our annual expenses would we need when we retire? Or equivalently what is 1 divided by this value, the sustainable withdrawal rate, when we retire? We will pretend all of our expenses need to be funded though savings, and we don't have a defined benefit pension, annuity, or Social Security, although the same analysis would apply if it was only some fraction of our expenses we needed to provide. Assume we seek 95% certainty of not running out of money.
We previously saw the sustainable withdrawal rate for a person aged 65 using a stock / inflation indexed bonds portfolio was 4.58%. Thus, a person aged 65 would need 21.83 (1/4.58%) times annual expenses for a safe retirement.
One word of caution. Reversing this calculation, and computing how much you can spend each year is dangerous. Repeating the calculation and taking that amount out every year would prevent your portfolio building up a buffer of good years to offset the inevitable bad times.
Some 65 year olds will die early and some will die late, so if there was a way to pool the "risk" of living to the 95th percentile age, the amount we need would go down. To evaluate all of this a small computer program is needed to massage the US Life Tables. Extra precision is computed to support later calculations.
Discounted life expectancy:
remaining years of life
age 0% disc. 2% disc. 2.79% disc. 3% disc. 5% disc.
50 30 22 - 19 14
55 26 19 - 17 13
60 22 17 - 15 12
65 18 14 13.72 13 11.27
70 14 12 - 11 9
And we can do better than this. In present case risk pooling is occurring only between other people of the same age, generational risk pooling. We can also conceivably allow risk pooling to occur over time, intergenerational risk pooling. This allows us to invest in a pure stock portfolio returning 5%. In this case a person aged 65 needs to have 11.27 times their annual expenses to meet their retirement goals. Whether it makes sense to do this is a policy issue.
Such risk pooling is clearly desirable. It results in a 37% ((21.83 - 13.72) / 21.83) to 48% ((21.83 - 11.27) / 21.83) reduction in the amount needed for a safe retirement at age 65. The question is how to obtain such risk pooling.
Defined benefit plans are out of fashion.
We could try and purchase an immediate annuity to pool our risk. Unfortunately there doesn't appear to be a single provider of inflation indexed immediate variable annuities. I believe Vanguard used to provide them through AIG but then AIG went bankrupt, and the multitude of variable annuity providers only let you index your returns to the returns of nominal bonds or stocks. I am not sure why this is the case. But as we have seen in general neither nominal bonds nor purely stocks appear suitable for guaranteeing a retirement portfolio at the 99th percentile level. Nominal bonds have inflation risk. Stocks have volatility risk except over large time periods.
And unfortunately, apart from Social Security, risk pooling is becoming increasingly difficult to obtain.