Lifetime Consumption Utility "Frontier" with both Trend-following (fake) and Partial Annuitization (also fake)

The Point

In my previous several posts I have been toying around with

 - Simple abstracted 2-asset portfolio choice, along with a jointly made
 - Spend choice (using constant spend for now), and then
 - Adding a highly stylized third asset that might look like trend-following

to see how it plays out using a simulator that evaluates the expected discounted utility of lifetime consumption (EDULC) across the different combinations.

The point of this post is to now add some partial annuitization to the mix to see how it affects EDULC, especially when it is stacked on top of the other things we are working with before.


Prior Post Links

The prior posts are the background for this and hold most of the assumptions necessary to explain what I am trying to do.

- Self-Evaluation of My Own Lifetime Consumption Utility
- Self-Evaluation of My Own Lifetime Consumption Utility, Part 2
- Having Some Fun with Portfolio Choice vs. Lifetime Consumption Utility
- Lifetime Consumption Utility with addition of trend following-like behavior
- An attempt at a stylized lifetime consumption utility "frontier"

- A Wealth Depletion Time Model 

Consolidated Assumptions - General

My assumptions have evolved a bit so here is a cursory summary of what I am working with: 

• The models and questions and data are roughly tuned to my personal situation so I can practice my blog on myself for a change. Note that all of this is illustrative and very fake. This is not an empirical study just a mind-game, 
• The post/model has a lifetime consumption utility focus with the utility model described formally here,
• Age = 61,
• Risk aversion coefficients = 2; see the links for why “2,”
• Social Security that is realized with full probability over the entire life interval starts at age 70,
• Available assets are net of liabilities and include only what is reasonably monetizable over the life-cycle,
• High-risk asset nominal arithmetic return is .08 with standard deviation of .18,
• Low-risk asset is nominal arithmetic .035 with standard dev of .04,
• Correlation coefficient for standard EF is -.10,
• Portfolio is combined in 11 steps from 0% risk asset to 100%
• Addition of a “stylized” (i.e., fake) third asset class is considered. In this case it just means the vol of the portfolio is reduced between 0 and 3 points between 0 and 100% risk asset with the addition. The precise allocation to the third is implicit and not known. EF with the third asset roughly mimics CME representations for addition of of managed futures to a portfolio,
• Inflation = 3% where needed; discount rates, if needed are 3%, subjective discount on real consumption utiles in the utility model is .005.
• Spending is constant with all the flaws of that assumption,
• Every parameter is (ahem) assumed stable over time with no shocks,
• There were 10,000 iterations per data-point,
• Survival probabilities (conditioned on my age of 61) were extracted from an actuarial table for annuitants (SOA),
• Initial wealth is redacted but in “unit terms” would range from ~17 to 50 depending on spend, 
• I abbreviated runs of the sim to reduce manual effort 
• I’m probably missing a few others… 

Assumptions - Annuity Purchase 

This post assumes that we purchase a private cash flow using either 10% or 20% of initial wealth. This purchase buys a lifetime cashflow appropriate for the age and the interest environment that we have inside the model. The cash flow is not inflation adjusted and so will lose real value over time. We just want to see what happens.  

The price is not real, of course, it is a Rivershedge "fake price" based on this formula

where l is the load, cf is the cash flow, tPx is the conditional survival probability and R is the annuity discount, so a(t,x) is the price of an annuity at t=0 for a person aged x. This purchase is executed before the model is run, wealth is decremented by the amount of the purchase, taxes are ignored, and an income flow starting at 61 is added to the process of the model. There is a load of 5% added though that may be low.  The price was validated against immediateannuities.com and was within a few dollars. The wtd average annuity discount rate worked out to be .038 as of today. All allocation choice is made after the annuity purchase.

The Model Output

1. EDULC for different spend rates along each allocation with an annuity purchase = 10% x W.  Each line is one allocation step (say 40% risk) and each dot is a spend choice along that vector.

Figure 1. Annuitize 10% of W

2. EDULC for different spend rates along each allocation with an annuity purchase = 20% x W.  Each line is one allocation step and each dot is a spend choice along that vector.

Figure 2. Annuitize 20% of W

3. Putting it together with past posts... Since each chart would be a little bit of a dome if allocation were its own axis and this were 3D, then if, for each spend rate, we take the max spend rate across all the allocations (for both figure 1 and figure 2) we get what I was calling "consumption frontiers." Not sure if that name is valid or makes sense...yet. Then if we overlay those maxima/frontiers on the chart from the last post we can see a little bit of what partial annuitization adds to the mix.

Figure 3. Frontier wrt spending

4. A quick look at asset allocation effects... In the past post we had a 3D chart with x axis as the allocation steps or choice, y as the spend choice, and z was the expected value of the value function used in the sim.  In that context, Figure 3 would be looking straight at the y axis and choosing the max values. Figure 4 is looking straight on at the x axis and also choosing the max value. 

Figure 4. Optimal allocation with partial annuitization

5. This is a placeholder for the 3D chart if I get to it which I am not prepared to do for this post yet...

Discussion

There is a lot going on up there but I'll keep it brief. 

1. Exercises like this always amaze me in the sense that I so rarely see discussion with advisors about the legitimate contribution of annuities. Yes they are vilified and yes there appears to be an annuity-paradox behavioral constraint for people buying them and yes they can be expensive. But the "pool" and the life-income that comes from it can create uniquely valuable programs for retirees. Fwiw, I do not yet own annuities. 

2. For spending under around 3.5% the annuity strategy generally is less compelling but if the gameplan is not to spend more then the annuity may not be the right choice anyway.

3. Over around 4% it looks like the utility of lifetime consumption is enhanced via partial annuitization. This is both in higher utility as well as a potentially higher spend rate. By annuitizing 20% of wealth up-front using the imperfect/fake methods in this post, let's say spending can go from 3.7% to as high as 6% (or higher). Unless I missed something that's about a 60% gain in lifestyle.  Not that simple in real life but in the model it at least looks like a good opportunity to ponder.

4. With the presence of lifetime income, the remaining portfolio can take on more risk. This idea is common in the literature. It is corroborated in figure 4 where it looks like one would have a reasonable case to have a much higher allocation to risky assets over the lifecycle with some degree of lifetime income.  By the way, there is no discussion here of glidepaths or other considerations related to portfolio design. We are just ruminating within the limited confines of the model only.

5. I don't want to ding all the super-bright portfolio managers and engineers out there but when I look at a model like this I see large impact coming from spending and annuitization before I start to think too much about fine tuning asset allocations and taking on esoteric strategies.  This, spending and annuities, in my 40+ years as a client, is rarely discussed. But then again I am still young.