Having Some Fun with Portfolio Choice vs. Lifetime Consumption Utility

Point of the Post

The question is:

• What combination of portfolio choice and spend rate works the best given a particular set of limiting and probably naive and unstable-in-real-life assumptions 

• for me at age 61 

• using a lifetime consumption utility model?

I know I've done something like this before (e.g., here) but

  - I didn't do it specifically for me, and
  - The 3D surfaces were cool and all, but were also distracting and washed out some of the nuance

So the point here is to now personalize it (with some redactions) and limit it to some finite, reasonable set of parameters just for illustration (and don't do 3D). This will not be too exhaustive or necessarily rigorous or "scientific." I just wanted to see what impact different choices along efficient frontier have on (my) consumption utility for different spend rates over (my) remaining lifetime and then compare it to where I am today. It'll go like this:

1. Create an efficient frontier in 11 allocation increments from 0% risk asset to 100%
2. Go up the frontier one allocation increment at a time
3. At each increment, seek out the spend rate that maxes a lifetime-consumption value function
4. Then: chart it and step back to behold the results...and maybe make some observations




The Model(s)

1. The Utility Value Function. The lifetime consumption utility model is in the link above as well as, more formally, here. The value function can be framed as the following with the terms defined in the link:

Lifetime consumption utility
value function in a simulation context

The key thing to note is that (a) g is a CRRA utility function, and (b) consumption "c" snaps to available income when wealth depletes. The implication, not shown or otherwise discussed here, due to the convexity of CRRA, is that with high lifetime income (and no bequest) it is often optimal to deplete wealth well before death while with low or no lifetime income it is often better to to land wealth softly near or at the terminal date. This is a "low" lifetime income post.

2. The Portfolio Model.  The efficient frontier generated here in this post is entirely fake, an illustration, maybe not representative of future expectations in 2019, but then again is not totally implausible, either. It is a 2-asset construct, by the way. While I do not personally have a 2-asset portfolio (more like 35/35/30), I have to admit modeling 2-asset portfolios is awfully easy. Also, I already know the effects I want, and plan to see, by adding complimentary payoffs (say trend-following) from alternative assets. We saw a glimmer of this here.

Illustrative asset class returns, volatility and correlation[1]:

- Risk-high = 8% arithmetic return and 18% standard deviation
- Risk-low = 3.5% arithmetic return and 4% standard deviation
- correlation -.10 (in our real world this is highly variable over time) 

- just to be clear these are expected arithmetic forthcoming returns where we are ignoring parameter uncertainty, parameter stability and the effects of time over many periods...for now.

Figure 1. EF in 11 steps

        R          SD

1  0.0350 - 0.0400

2  0.0395 - 0.0386

3  0.0440 - 0.0457

4  0.0485 - 0.0583

5  0.0530 - 0.0736

6  0.0575 - 0.0902

7  0.0620 - 0.1076

8  0.0665 - 0.1254

9  0.0710 - 0.1434

10 0.0755 - 0.1616

11 0.0800 - 0.1800

Some Assumptions

See these links for more insight and discussion:

Self-Evaluation of My Own Lifetime Consumption Utility, and 

Self-Evaluation of My Own Lifetime Consumption Utility, Part 2

but basically it goes like this in an abbreviated form... 

The assumptions in this post are perfunctory and kind-of shoot-from-the-hip but will do for now. They include the following, noting that personal wealth info is redacted: age = 61, a risk aversion coefficient 2* is used, max inflation-adjusted 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, returns and standard deviation are as described above, inflation = 3%, spending is constant with all the flaws of that assumption, conveniently for me: every parameter (including correlation coefficient) is stable over time with no shocks, and there were 10,000 iterations per data-point. Survival probabilities (conditioned on my age of 61) were extracted from an actuarial table for individual annuitants.

* I assert with no proof, or plausible way of proving, that a coefficient of risk aversion = 2 is the same as me and my own behaviors.

A Note on Method

This was more manual in some way than I really wanted so to reduce my effort I did a two pass approach:

1. at each EF "step" I ran spend rates in .01 increments between .02 and .06, then

2. for a range near the max I re-ran the model in .001 increments to get more granularity near top

3. to compensate for the gaps, chart 1 below uses a 4th-order poly trend line to visually fill in as well as to partly compensate for variation due to the nature of the simulation.

This probably wouldn't fly elsewhere but it works for my own look-see.

The Model Output

Here are four ways of looking at the same thing...

1. Value Function vs Spend Rates (allocations are the colored lines, individual dots are spend rates along that allocation). I have not labeled the allocation lines because I couldn't figure out a good way to do it. Maybe by color.

2. Table of the max Utiles (shaded green) for different spend rates (abbreviated interval) and allocation choices. Caution, the highest spend rate achieved does not have the highest value. Blanks were combos not run but presumed to be not-max.

3. The max value for allocation and spend (abbrev.). Note: this is a 3D representation of just the green shaded values in the previous table.

4. Value function against allocation choice. Spend rates that generate the optima at some allocation step are implicit in the color coding. This, again, is an alternative rendering of the green shaded cells in the table in #2.

5. You KNEW I couldn't resist going 3D. It makes it easier to understand the implications of portfolio choice as the spend rate goes into less rational territory. Had to interpolate missing data but I don't think it hurts the picture.

Observations

Dare I? Ok, fine, I'll step in.

1. In chart 1, allocation choice does not move the needle in a good direction, in terms of life consumption utility, nearly as much as the spend choice does. For "moderate allocations" there is some general consensus for this set-up only in the 3 1/2 to 4% range.

2. In a neat trick I wasn't expecting, the highest spend rate achieved via allocation does not necessarily generate the highest utiles.  Allocating to about 50/50 achieves the highest spend rate in terms of max utiles but the highest overall utility comes a little lower at 40/60 or even 30/70. Splitting hairs, probably, but there it is.

3. Adding a third asset-class with a convex payoff like trend following would likely have an effect but is not illustrated here. It would be interesting to see what happens with the allocations at the extremes (100% or 0% risk). We already saw in a past post that reducing vol for equal return will more than likely nudge the curves in chart 1 up and to the right a bit. Not a lot but maybe enough. 

4. Underspending appears to have broad "consensus" when it comes to utility evaluation at different allocations. The opposite is not true. For over-spending: it depends.  Since this is a rough, dirty illustration I can't make any hard conclusions. For this run and parameterization only, if we were to look, say, at the 6% spend rate and then sort the allocations by utiles, the order would come out like this:

risk%	E[V(c )]
60	23.267815
70	23.267813
50	23.267811
40	23.267794
80	23.267780
30	23.267771
20	23.267747
10	23.267719
90	23.267696
0	23.267693
100	23.267579

This is where a 3d surface map might be better. The middle (40-70, a range I've mentioned before) wins here in over-spend mode while the extremes (0, 10, 100, 90) lose.  But that's only at the 6% "over spend." The order might be materially different (isn't much) at a different spend rate.

5. Moderate allocations -- given the presence of consumption, multiple periods, a consumption utility context (if you believe this stuff), and this particular parameterization -- seem to be more friendly than not to a retiree.  

6. I don't have a 2-asset allocation but this exercise confirmed, in pretty general terms, my own joint spending and allocation choices that were made in the past by more naive methods.

7. The sub-4% muted spend suggestions also probably have an option-like payoff under some parameterizations at some hypothetical tenure. I started to look at this before but it is otherwise undiscussed here. 

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[1] These return parameters are not really intended to reflect my personal expectations about forward forthcoming returns in an era when the foot of the Fed has been heavy, past returns have been weird, valuations high, and interest rates low. In general, they may be viewed as a kind of average or composite of capital market expectations recently published by JP Morgan in 2018 and so represent a type of not-totally-insane reflection of broad market expectations. For now they are a placeholder. Note that commentary in the financial press in ~mid-2018 argues for lower expectations almost across the board.