Background:
This is a follow up post to The Deferred Annuity (DIA), Not the SPIA, Should Be the Default
In that post I argued that deferred income annuities (DIAs), rather than single premium immediate annuities (SPIAs), should be the default recommendation when considering annuitization. My argument relied on the concept of the "zero-coupon annuity". A 20-year zero-coupon bond, when held to maturity and assuming no default risk, guarantees a specific payment in 20 years. Similarly, a 20-year zero-coupon annuity guarantees a specific payment in 20 years, contingent on the annuitant's survival. Since the payment is contingent on the survival of the annuitant, a fairly priced zero-coupon annuity will be cheaper than a zero-coupon bond of the same maturity and, therefore, will have a higher return. The difference in the returns of the zero-coupon bond and the zero-coupon annuity is often referred to as a "mortality credit". The less likely the annuitant is to survive to the maturity date, the higher the mortality credit, and the higher the return of the zero-coupon annuity relative to the return of the zero-coupon coupon bond.
Annuities, as we typically think of them, are specific bundles of zero-coupon annuities based on the payment dates of the annuity. A SPIA, which begins monthly payments immediately, consists of a series of zero-coupon annuities maturing each month, starting in one month and continuing indefinitely. Similarly, a 10-year DIA, which starts payments after a 10-year deferral, consists of zero-coupon annuities maturing each month from year 10 onward. Essentially, a DIA is just a SPIA without the zero-coupon annuities maturing during the deferral period.
Although more complicated than this, my previous argument was that DIAs are more attractive than SPIAs because zero-coupon annuities with later maturities offer higher mortality credits than those with earlier maturities.
The strongest1 objection to my argument was that investors' spending goals are real, but the annuities actually available are nominal. Additionally, the risk of inflation is increasing over time. So, despite the higher returns, zero-coupon annuities with later maturities are likely less attractive than those which mature earlier.
My response to this objection was that, while the risk of inflation is increasing over time, at some point the benefit of the higher mortality credits outpaces this risk enough to justify the DIA over the SPIA.
My attempt with this post is to give a (very) rough quantification of these two competing effects.
Mean-Variance Optimization of Equities and Zero-Coupon Annuities:
John Campbell and Luis Viceira's The Term Structure of the Risk-Return Tradeoff describes how the risks of stocks and nominal bonds changes depending on the investment horizon. In this case, risk is defined as the variation around the real return at the end of the investment horizon, rather than the volatility. This means that the risk-free rate depends on the investment horizon and is given by the real yield curve derived from TIPS.
Campbell and Viceira produce the graph below which shows the annualized standard deviation of the real-return over different investment horizons for both equities and nominal bonds held to maturity. Keep in mind that, because the bonds are held to maturity and assumed not to default, the entire risk for these bonds is due to inflation.
Notice that the annualized risk for equities starts high and declines as the investment horizon increases. This pattern is representative of the mean reversion of equity returns. This mean reversion suggests that the optimal allocation to equities for spending goals with longer horizons is larger than the equity allocation for spending goals with a shorter horizon.
Turning our attention to nominal bonds held to maturity, we see the annualized risk actually increases with the investment horizon. If we assume the TIPS break-even inflation rate represents expected inflation, then a K-year nominal treasury bond has an expected excess return over the K-year risk-free rate of 0%. Since the real return is risky and there is no expected excess return, there is no reason to hold nominal treasury bonds instead of TIPS.
Although we are assuming that nominal treasuries have no excess return, a fairly priced nominal zero-coupon annuity will have an excess expected return equal to the size of the mortality credit. Thus, adding a zero-coupon annuity to equities may improve the risk-return trade-off for a particular investment horizon, depending on its Sharpe ratio as well as its correlation to equities.
Luckily, Campbell and Viceria also produce the graph below showing how the correlation of the real returns for nominal bonds held to maturity and equities rises and then falls as the investment horizon increases.
Now, the linked paper is 20 years old at this point. It may not be representative of the current market environment. Additionally, it may not capture the long-run and deep risk associated with inflation. However, I still think it is worthwhile to derive the tangency portfolios for nominal zero-coupon annuities and stocks to see how the portfolio weights change based on the horizon. As long as a similar volatility and correlation term structure holds, then it will be informative. To be clear, I am not attempting to derive an optimal annuity purchasing strategy. I'm merely trying to demonstrate that later maturing zero-coupon annuities are likely more attractive than those that mature earlier.
To derive the portfolio weights, we need three things for both equities and zero-coupon annuities. We need their expected excess returns, the standard deviations of those returns, and, finally, the correlation between those returns.
For the expected return of equities, I use Damodoran's current month estimate of 8.91%. To get the expected excess return for a particular horizon, I subtract the nominal treasury rate for the associated maturity as of 1/27/2025 (using linear interpolation and setting the rate for maturities > 30 years to the 30-year rate).
As stated previously, the expected excess return of the zero-coupon annuity is equal to the mortality credit for that particular maturity. I use the 2012 IAM Basic mortality tables with the G2 projection scale applied in order to estimate the mortality credits.
The graph below displays the annualized expected excess returns for equities and zero-coupon annuities across different investment horizons for a 65-year-old couple.
The equity excess return curve is roughly flat due to the relatively flat yield curve and assumed constant return for equities. Meanwhile, there is hardly any excess return for the zero-coupon annuities at the earlier maturities, but the exponential growth in mortality rates causes the mortality credits to grow exponentially as well.
To estimate the annualized standard deviations of equities and nominal bonds held to maturity, I simply eyeball the graph given by Campbell and Viceira. I do the same or the correlation, except I set the minimum correlation to 0.
Similar to the nominal bond held to maturity, the risk of a nominal zero-coupon annuity (contingent on the annuitant being alive at maturity) is entirely due to inflation. Thus, we can calculate the standard deviation of a zero-coupon annuity from the standard deviation of the nominal bond with the same maturity by scaling it by (1 + mortality credit). Additionally, since correlation is scale-invariant, the correlation between equities and the zero-coupon annuity is the same as for the equities and the nominal bond of the same maturity.
The graph below shows the annualized standard deviations of real returns for equities and zero-coupon annuities across different investment horizons for a 65-year-old couple.
The curve for equities has already been discussed while the curve for the zero-coupon annuities looks similar to that of nominal bonds, only it is growing faster due to the scaling by (1 + mortality credit).
Finally, the graph below displays the Sharpe ratios for equities and zero-coupon annuities available to a 65-year-old couple, along with their correlations.
We now have enough information to calculate the optimal equity and zero-coupon annuity portfolio weights for each investment horizon. I plot the weights for each zero-coupon annuity by the investment horizon, assuming no short-selling.
The allocation between equities and zero-coupon annuities follows expected patterns based on their Sharpe ratios. Early-maturity zero-coupon annuities provide little excess return, making them unattractive compared to TIPS, especially given equities' higher expected returns. However, as mortality credits increase, zero-coupon annuities become more appealing despite their rising risk over longer horizons. This trend is even more pronounced for joint annuitants at older ages, as shown in the graph below for an 80-year-old couple.
This looks like a clear win for DIAs over SPIAs. However, a surprising pattern occurs for older individual annuitants. Consider the allocation to zero-coupon annuities for an age 65 male given by the graph below.
Prior to the more familiar pattern on the right of the graph, the allocation to zero-coupon annuities begins at 100% and falls dramatically to 0%. An inspection of the Sharpe ratios and correlation to equities reveals the cause of this pattern.
The mortality credits at the early maturities for individual annuities are large enough to result in a high Sharpe ratio. However, this is mostly due to the low risk associated with them. Moreover, that risk increases fairly quickly with horizon, while the mortality credits take a little more time to gain momentum. This causes a decline in the Sharpe ratio of zero-coupon annuities and the growing Sharpe ratio of equities overtakes it. Since mean-variance optimization is highly sensitive to the inputs, especially when correlations are far from zero, we get the dramatic swings in allocation shown previously.
For even older individual annuitants, the Sharpe ratio of equities never overtakes that of the zero-coupon annuities. Consider the allocation below for a 75 year old male.
While I am surprised by this result, it does not change my belief that DIAs are superior to SPIAs for several reasons. First, I don't think the regions where mean-variance optimization is highly sensitive to the inputs are particularly informative. You'd probably be better off just applying equal weights, that is, assuming your desired risk-return characteristics are met. Which brings me to the second point. Going 100% zero-coupon annuities for earlier maturities likely isn't generating enough excess return to meet an investor's required return. Equities would be needed to get you there. Finally, one of the main benefits of DIAs is that they do not reduce liquidity as much as SPIAs. The mean-variance analysis above does not take this into account. Still, I concede there may be some situations where SPIAs and DIAs are roughly equally attractive.
Overall, I believe the mean-variance analysis above gives strong support to my claim that DIAs are superior to SPIAs, despite inflation risk increasing over time.
1A common objection in the previous thread was to point out that it would be better to delay the annuitization decision rather than purchase a DIA. However, whether or not to delay annuitization is a separate question from whether to choose a SPIA or a DIA as the annuitization method. It may be the case that it is optimal for someone to delay annuitization until the last possible age, at which point a DIA will likely not be available, but that is not an argument for a choosing a SPIA over a DIA at the time of annuitization.