Adding a DIA to the last post that was looking at a consumption utility "frontier"

The Point

Lifetime consumption utility is, inter alia, a function of: allocation (return, vol), spending, longevity, risk aversion, and the presence of lifetime income. In the last post "Lifetime Consumption Utility "Frontier" with both Trend-following (fake) and Partial Annuitization (also fake)" I looked at the impact of partial annuitization (assuming a relatively fair annuity) by consuming 10 or 20% of initial wealth to buy a nominal cash flow by way of a modeled SPIA-like concept.

The point of this post is to compare the "10% of wealth" immediate SPIA with a "10% of wealth" purchase of a cash flow that starts at 85 i.e., a "deferred income annuity" or DIA or DIA-like thing, rather.


Assumptions

The link above (or here if you dread moving your cursor) will hold all but the DIA assumptions since I won't re-type it here.  I weary of typing right now.

The DIA Price

10% of initial wealth is used to purchase a cash flow that starts at 85 rather than now at 61. The cf is nominal so will degrade over time in the presence of inflation but that's ok for a look-see. The price, as before is calculated in model with this:

Annuity calc

Where x is age 61, l is a load (I think I put in 5%, should be higher), tPx is survival prob for a 61 year old using SOA annuitant data, and R is the discount. I'd have to go and look for what I used for R and I'm not in the mood. Either way I normed my price to immediateannuities.com (indirectly) and AAcalc.com (directly) and was quite close.  I used Excel solver to determine the cash flow that made the DIA = 10% x W.

Wealth is decremented by the price of the DIA prior to model run and all allocation choices are made afterwards in the model.  20% of wealth was not run this time. 10 was enough to show a glimmer of the principle at stake.

The Output

1. Expected Utility value function (y axis) for different allocations (lines) and spend rates (dots along the lines -- x axis).

Figure 1.

2. Max{E[V(c)]} at each spend rate, and in the context of the output of the last post or two. The consumption "frontier" of the simulation using the DIA-like-thing is in green. EF, if you missed the series, means efficient frontier in an MVO sense.

Figure 2.

3. Asset Allocation Effects.  This is the max E[V(c)] for each allocation step looking across all spend rates. It is Figure 2 turned to the side 90 degrees.

Figure 3.

Discussion

In figure 2 and 3 we are primarily interested in the green (DIA at 85 purchased with 10% of wealth) and the black (SPIA) lines.

1. Green dominates black for most of its arc when either looking at the spend rate perspective or the asset allocation perspective. The exception is in the higher risk allocations and spend rates.

2. The in-model optimal spend is not changed but the DIA achieves slightly higher lifetime utility relative to the max for the SPIA within an interval that goes from about 4.1 to 5.1% spend.

3. The in-model optimal allocation for SPIA can tolerate quite a bit more risk than the DIA in relative terms though the DIA dominates in absolute terms between interval 2 and 8 which would correspond to a 2-asset allocation of between 10% to 70% to risk assets.

Discussion 2

High risk allocations and high spend rates are not friendly to the DIA approach. This might or might not be counter-intuitive. After all the DIA is "cheaper," right? and it pre-buys a larger life-income at 85 than the SPIA would have delivered since it is spread out over more years.

The main intuition, or "tells," on this should be that the DIA starts at 85 and that this is a consumption utility model.  With high spend and/or high risk allocations there is a non zero risk that there is a wealth depletion interval prior to 85. Since the utility math is a power function, the crash in spending, which snaps to available income (say Social security) when wealth depletes, is heavily penalized over a lifetime even though it is heavily discounted by survival probabilities and the subjective discount.

Since visuals help, here is the intuition by way of looking at the first 21 iterations of a run I did with 100% allocation to risk and a high spend rate (I think this was from the 6% spend):

Figure 4.

In figure 4, spending is fine and constant as long as wealth supports it. After wealth depletes spending snaps to income: SS before age 85 and the DIA+SS thereafter. In this illustration of the sim iterations (first 50) we can see some spend paths crashing (in real terms here) before (and after) the DIA kicks in. It is this that drives the relative utility dominance of the SPIA in the high risk/high spend scenarios. At least something more than SS is there for the retiree during the full depletion period before the DIA.

This, of course, might beg the question of what age makes a good DIA start. I have no idea. Creating a robust answer would take me forever. Just working with my specific case as in the last 5 or 6 posts it would still take forever.  My guess is that those vertical crash lines in figure 4 have some kind of weird distribution and that moving the age forward enough to scoop up the risk in that distribution would make some sense. At some point we'd end up with the SPIA.  All that is for another post, if ever.