Should You Buy an Annuity?

The Formula

When does it make sense to buy an annuity? The mathematics ends up with a formula that is identical to the Safe Withdrawal Rate formula that Gummy derived. I express the Safe Withdrawal Rate formula in these terms:

[Balance at year N/Initial Balance] = Gain_Product_Term*(1 - w/wfail(N)), where the Gain_Product_Term is the portfolio’s balance when there are no withdrawals (and no deposits either), w is the withdrawal rate and wfail(N) is the withdrawal rate that would have produced a balance of exactly zero dollars at year N for this particular sequence of investment gains and losses.

When used for calculating Safe Withdrawal Rates, this formula is restricted to withdrawal amounts less than or equal to wfail(N).

When this same formula shows up in the annuity equation, negative balances are OK. The annuity increases your withdrawal amount by the negative Safe Withdrawal Rate formula times the fraction of you initial investment that you use for buying the annuity. That is, the annuity increases your income at year N if it provides you with a withdrawal rate bigger than wfail(N).

Applying the Formula

In terms of your planning, you will not be interested in a withdrawal amount that lasts exactly as long as you do. You will want a cushion. If you were 65 years old, you would want to plan for your portfolio to last at least 30 years. And 40 years would be better. But an insurance company would base your payout on your life expectancy, which would be in the neighborhood of 20 years.

My experience from using Vanguard’s calculator (see below) is that today’s annuity amounts, when they match inflation, are approximately equal to inflation-matched cash (at zero percent real interest) spread over your life expectancy. If you are 65, your annuity amount will be about 5% (plus inflation) of your investment.

[I am using a coarse approximation to illustrate a point. Use the calculator and work with real numbers.]

We can do better if we invest for exactly 20 years. If we invest for 40 years, however, your withdrawals fall to 4% (plus inflation) of your initial balance. [It took us a lot of effort to bring this number up to 4% in today’s market.] With 2% TIPS, the withdrawal rate is 4.46% (plus inflation) for 30 years.

The insurance company makes its money on the spread. In essence, it is collecting the interest on investments similar to a TIPS-only portfolio to compensate if you live beyond your life expectancy.

[The actual calculations are much more sophisticated. These are rules of thumb to help you understand what is behind the numbers.]

What if you are a 40-year old retiree? Your life expectancy is around 40 to 50 years. Inflation-matched cash would provide payments of 2.0% to 2.5% (plus inflation) of your initial balance. In your case, you would do much better with 2% TIPS. They would provide an income stream of 2.88% (plus inflation) for 60 years.

Summary

Annuities can make a lot of sense for traditional retirees. They make much less sense for younger retirees.

Note: There are lots of real-world side issues. One important issue is how well you will be able to handle your finances in your later years.

Vanguard’s Annuity Calculator

For those considering annuities, I recommend that you visit the Vanguard web site.

Vanguard Web Site
Vanguard Annuities

Locate this section: An annuity that provides retirement income. Click on the words: Vanguard Lifetime Income Program. Read what Vanguard has to say. Then, click on the Request a Quote listing in the left-hand column. Use an amount of $100000 to answer what-if questions. If you put in a large enough amount, you will be told to call them (to talk to a salesman).

Annuities as Part of a Retirement Portfolio

For the mathematics of annuities, visit Gummy’s site. [Gummy is a highly respected participant at several discussion boards. His real name is Peter Ponzo. He is a retired mathematics professor.]

Gummy's (Peter Ponzo's) Web Site
Gummy on Annuities
Gummy on Annuity, yes or no?
Gummy on Buying an Annuity

Gummy derived the equations when an annuity forms a fraction of your portfolio and you withdraw varying amounts from the combined portfolio. It turns out that the amount that you withdraw does not influence whether investing in the annuity is a good idea. Everything depends upon the annuity itself.

Gummy’s equations are based upon the traditional annuity, which provides a constant income stream without any inflation adjustments. It does not matter whether you adjust your withdrawals from the combined portfolio to match inflation.

What if you select an annuity that has an inflation adjustment? Everything still centers around the annuity itself. But now, you must adjust the investment gain terms to match the payout. For example, if the annuity payout matches inflation, the gain terms should be in terms of real dollar amounts. If the annuity payout produces constant nominal dollars (without an inflation adjustment), the gain terms should be in terms of nominal dollar amounts.

Again, what you withdraw from the combined portfolio has no bearing on whether the annuity increases your portfolio balance. Everything depends on the annuity itself and the manner in which the annuity makes payments.

Have fun.

John Walter Russell
August 24, 2005