Adding some lifetime income to the machine learning model

Premise

The original set up, with the references and links to other posts, is here:

• Machine Spending

In this post I added 15k  (real) in lifetime income starting at age 70 to the other parameters. This can be viewed as exogenous income like Social Security or some other external pension or annuity.

Something to keep in mind is that:

a) This move to add income with no other changes is more or less like adding new wealth to the balance sheet since the probability weighted present value of that stream at 60 is something like 226k, money that we didn't have before. 

b) the income really isn't like a static wealth PV since it is a "flow" that, in the model, is set up to last forever. I mean except that at some age the survival probability goes to zero...which moots the forever aspect.

The game is still the same as before:  recommend to the machine that it spend 4% but also let it learn, via some randomizing and an evaluative reinforcing value function, what might be better given random returns and lifetime, now this time with life income present.

The Learning Model

This is from the link above and is here just for context and without explanation.

Figure 0. Reinforcement Model

Parameters

The parameters of note, if you'll recall, are:

• $1M endowment
• 40k initial expected spend is "proposed" to the machine
• Risk aversion coefficient = 2
• Return is 4% real with 12% std dev
• Longevity is defined in the machine by Gompertz model, mode =88, dispersion=9
• Longevity in the benchmark is defined by an annuitant mortality table
• 15k permanent income starts at 70
• The range of machine spend exploration is +/- 1%
• If wealth fails, spending snaps to income
• Absent income, spending snaps to subsistence level
• Interval of interest for now is age 60 to 80 so 21 years
• Machine value function used is in the link above

Benchmarks

In this post, since I only have one tool in the box that can handle utility evaluation in the presence of income, I used that. I used my lifetime consumption utility simulator to evaluate, for a comparable portfolio, a range of spend rates and select (more or less manually) the spend with max expected discounted life consumption utility. In general I tried to parameterize apples to apples. Got close but it's not perfect. See discussion note 5 below.


The Output

Figure 1. Policy Spend rates suggested by machine for $1M at age vs benchmark

Figure 2. Mean (expected) real spend across all iterations and wealth levels inside machine 

Discussion

1. Looks like we are mostly hitting benchmarks in Figure 1. There is some divergence in the income scenario. Not sure why other than some mismatch in parameterization and maybe some under-training.

2. One would expect the wealth shift (adding a new resource to the personal balance sheet) to produce a higher spend but probably not to the degree we see here. Most of the changes in consumption policy come from the perpetual nature of the income flow. In some other run I will make the balance sheet "even" by consuming from starting wealth to purchase income, like an annuity.

3. In figure 1 we see uniformly higher spending policy recommendations across ages. The inference without getting into the math is one of higher utility from later life income even at earlier ages.

The large divergence at the earliest ages in figure2 (mean spend by machine) shows, I think, the inter-temporal transfer or reserve of capital for late age spending risk in a world without income, a risk that one would expect to abate as longevity expectations come in or income becomes available. In other words, at 60 I would not have to hold back capital from being used for spending now in order to hedge uncertain-to-unlikely spending at 105.  Even without the economics, it's easy to see the desirability of that.

4. These charts do not have the classic shape of economic lifecycle consumption models but then we are only looking at decumulation and only at the relatively short interval 60 to 80.  Close though. Also the way it was parameterized no doubt has an effect. Figure 3 is a representation of a classic model I saw in a paper by M LaChance.

Figure 3 - Stylized optimal spend in the lifecycle model

5. The obvious question, as it was in the last post in case you missed it, is "if you can create the benchmark, why not just use that?" Yep. Precisely. I would. But, I'm playing with the machine just to see...