Optimal Purchasing of Deferred Income Annuities -- Huang, Milevsky, Young 2015

I was reading today Optimal Purchasing of Deferred Income Annuities When Payout Yields are Mean-Reverting by Huang, Milevsky and Young 2015.  Or, rather, let's say "trying" to read. Over the last couple years I have vastly improved my ability to read and interpret quant-finance notation but me trying to follow these guys, which I attempted for maybe the first 9 pages, is like a dog chasing a UFO: charmingly and endearingly naive, odd, and impossible.

Since there is nothing I can interpret myself and nothing I can plausibly (yet) replicate in an amateur-hack to see how it's done the least I can do is regurgitate some of what they said since the underlying topic is both interesting and important.  Personally I believe, and have proved to at least myself via simulation, that: a) hedging out superannuation risk (say 85+) with a capital allocation shift to DIAs to cover a minimum late age lifestyle can pay off (for me) in about a 15% increase in capacity for lifestyle (consumption) now, ceteris paribus, and b) when it comes to SPIAs (I didn't look at DIAs) there probably is option value to waiting before irreversibly annuitizing. The paper here makes an entirely different point than my a and b but then again it is also a "wait and see" conclusion which is useful to know and adds something to my annuity toolkit.

Here is what I went back and found that I had highlighted or annotated:

• Even though the pricing methodology underlying (new, popular) DIAs is the same as for (old, less popular) SPIAs, the fact that income payments are delayed for ten, twenty, and possibly thirty years implies that pricing becomes extremely sensitive to the far-end of the term structure of interest rates. The duration of cash flows in a DIA is obviously much higher relative to a SPIA. And, whereas in early 2015 global interest rates are at historical lows, industry participants express a valid concern that this particular environment might not be the best time to purchase any long-term fixed-income instrument, including a DIA.

• We remark that this [DIA term] surface changes from day to day and prices move as frequently as any bond traded in the market. … , we estimate the annualized volatility of πt(68, 88) [DIA payout yield] for individual companies to be 3% to 12% per year

• we confirm practitioners’ intuition that one should wait to purchase until payout yields revert to (close to) normal for someone who is solely concerned with maximizing expected retirement income at some fixed age.

• The continuation region in which the consumer does not purchase any more DIAs and continues to hold the money in cash is both time and wealth dependent. DIAs are purchased by risk-averse individuals when payout yields increase beyond some threshold level, but the buying process is then suspended if payout yields decline. Buying resumes as rates increase until the entire (pre-allocated) budget is exhausted…In fact, this is a type of dollar-cost averaging (DCA), which although has been long-dismissed in the investment literature, might have some merit when it comes to DIA purchases.

• The pricing discount rate rk and the present value factor (1 + rk)^k are not necessarily based on the risk-free (government) curve. The rk is company specific, generally higher than the risk-free rate and includes a default-risk premium. See, for example, the recent work by Charupat et al. (2015) for evidence that annuity prices do not move in lockstep with government bond yields and are (more) highly correlated with 30-year mortgage rates, albeit with a lag.

• The free-boundary for the risk-averse buyer approaches the free-boundary for the risk-neutral buyer as wealth approaches zero, that is, π ∗ (0, t) = π ∗ (t) for all γ ≥ 0.

• According to data from CANNEX Financial Exchange, age the median purchase age for (income) annuities is approximation age 68, and the T = 20 deferral period takes us to age y = 88 which is the latest age at which all companies are willing to still quote DIAs. [I was surprised at the median of 68. Also, I could have intuited  that there is max quote age but I didn't know that it was specifically 88]

• Our main qualitative result is that when payout yields are mean-reverting, a risk neutral consumer who wishes to maximize her expected retirement income should wait until yields reach a threshold – which lies above historical averages – and, then, purchase the DIA in one lump sum. In contrast, we also showed that a risk-averse consumer, who is concerned payout yields will remain below average for an extended period and who worries about losing mortality credits while waiting should employ a barrier purchasing strategy. This is quite similar to optimal behavior in the portfolio problem under transaction costs. The optimal behavior of a risk-averse consumer resembles an asymmetric dollar-cost averaging strategy, and a portion of the “budget” for DIAs is spent even while payout rates are below historical averages.