Commentary

14 Jan 2010 by Jim Fickett.

*In order to decide whether to take a retirement benefit as a lump sum or an annuity, a good starting point is to look at the inflation-adjusted value of the monthly payments, in aggregate, taking into account how long one expects to live. Varying the inflation rate and possible years of life, within reasonable bounds, can begin to give one a feeling for the plusses and minusses of the two choices.*

It is common for people to have a choice, in receiving a retirement benefit, between a lump sum and an annuity. I am facing such a choice, in which the company involved is offering me a .0056 fraction of the lump sum as a monthly payment for the rest of my life. How to decide if that monthly payment is a reasonable one?

This is, of course, a very complex area – there are people who devote their whole careers to pricing annuities. The unfortunate result of that complexity is that, for the average layperson, the matter is completely opaque. I would guess it is not uncommon for people to choose the lump sum without any real analysis at all, just because they don't fully trust the underwriter and it feels better to have the money under one's own control.

Here I would like to try to answer the question of lump sum or annuity in a way that is much simpler than a full actuarial approach, but hopefully realistic enough to allow one to make an intelligent choice. I will focus on the following very simple question:

**Will the value of the monthly payments, after inflation, be greater or less than the value of the lump sum?**

Even this simple question is not easy to answer, as it requires guessing how long one will live and what inflation will be. However it is simpler than the way the financial press usually treats the issue, which involves bringing in (sometimes very unrealistic) discussions of lifetime changes in risk appetite and investment strategy, and likely overall returns.

Life expectancy tables are easily available. They differ somewhat, but not too much for our purposes. Here is one provided by the Social Security Administration, and here is one from the IRS. My life expectancy is in the mid-20s. I'm interested in thinking about the obvious deviations from the base case:

- I live longer, in which case the annuity option might cause me to win and the underwriter to lose
- I die sooner, in which case the annuity option might cause the underwriter to win and my heirs to lose

So I'll look at the value of the annuity for 20 and 30 years worth of payments. Since many people worry about outliving their savings, I'll also consider the outlier of 40 years worth of payments. (A more sophisticated approach would use probability distributions on the main variables, but I'm more interested in having a very simple, but still plausible, model that I can fully grasp intuitively.)

Next, how do we approach inflation? Inflation has averaged 3.3% (CAGR) over the full record of the Consumer Price Index from 1913 to 2009. But there has been considerable variation. To get a little better idea of what numbers to use, without making things too complex, let's look at CAGR over a moving window of 30 years. (Again, this is somewhat simplified in that it matters whether a lump of high inflation comes at the beginning or end of one's annuity, but this is at least a reasonable place to start.)

(The 30 year window is a trailing one, so the first point on the graph corresponds to the window from Jan 1913 to Jan 1943. Click on graph for larger image. Users of IE may want to disable “automatic image resizing” to obtain a clearer image.)

This analysis suggests that a reasonable range of inflation rates to consider would be 2-5%.

The algebra is pretty straightforward. Let the first payment be P. Suppose inflation runs at 3.5% annually, or 0.29% monthly, compounded. The first payment is paid today and is worth P of today's dollars. The second payment is worth 99.71% of P, in today's dollars, or .9971P. The third payment is worth 99.71% of the second payment, or (.9971)(.9971)P, etc. In general, if the monthly inflation rate is I, the present value of Y years of payments is

- PV = P + (1-I)P + (1-I)
^{2}P + … + (1-I)^{12Y-1}P

If we multiply both sides by (1-I) and subtract, most of the terms cancel and we get:

- PV = P(1 - (1-I)
^{12Y})/I

Here is a table of the PV for the 9 cases where the payout is for 20, 30, or 40 years, and inflation averages 2%, 3.5%, or 5%. In each cell the PV is given as a multiple of the monthly payment P, for the general case, and as an absolute number for my example case, where P is .0056 (in units of the lump sum).

Inflation | 20 years | 30 years | 40 years |
---|---|---|---|

2% | 197P (1.10) | 269P (1.51) | 328P (1.84) |

3.5% | 173P (0.97) | 224P (1.25) | 259P (1.45) |

5% | 153P (0.86) | 188P (1.05) | 210P (1.18) |

From this it would appear that the monthly payment being offered to me is pretty fair. If inflation is somewhere in the normal range and my life expectancy is more or less correct, the PV of the monthly payments is close to the lump sum amount. If high inflation is a danger, however, and if I feel confident that I can invest in a way to at least preserve the real value of my savings (no guarantees there), the PV analysis might suggest taking the lump sum.

Of course there are other considerations. One may want to think about the long-term solvency of the underwriter, about the value of peace of mind concerning income in case of longer life, about whether the Consumer Price Index is an adequate measure of needed cost-of-living adjustments, and about investment opportunities lost if someone else is controlling the money. Having, from the above analysis, some idea of whether the annuity payment is fairly computed, and how the total value of the payout depends on the most basic assumptions, at least gives a good starting point for thinking about these difficult questions.