- 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
I was playing around with portfolio choice along with a stylized version of an efficient frontier -- both with and without a third asset class that might resemble a trend following allocation of some amount unknown -- in order to see how an economic consumption utility game might play out. I now want to go back and add some more detail.
Some Background "Catch-up"
In the last of the four above I quickly put out a 3D chart of the output for the given assumptions in items 1, 3 and 4. That looked like figure 1 with x being allocation to risk, y being the spend choice, and z1 (blue) being "expected discounted utility of lifetime consumption" without trend, and z2 (color contour) is with some asset like trend-following added...and where utility is evaluated using this model:
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| Figure 1. |
The fake, but illustrative, EFs driving this looked like these:
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| Figure 2. What adding a trend following allocation might do to an eff frontier??? |
Since I had kicked that post out quickly so I could get to my daughter's graduation, I wanted to unpack it a little now because the 3D, while pretty (Dan Egan told me "Looks like the magic carpet from Aladdin...") obscures the detail a little bit and the opacity of the surface chart means you can't see underneath.
Ginning up a "Consumption Frontier"
Consumption frontier is something I am making up but it makes sense in this context. I'll walk it through. If, first, we were to look at the 3D in Figure 1 in profile -- with the spend choice facing us and rendered with the underlying data points I actually calculated with one line for each allocation choice/step -- then in figure 3 the blue plane would be the top chart and the contour (trend) would be the bottom:
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| Figure 3. EDULC for the joint allocation/spend decision |
Here we can see that the EDULC depends on both choices and for any given allocation and spend choice and we are not certain what the max utility is in this mess though we charted some of it out before in the links at the beginning. But here I just want to look at the absolute max EDULC for any choice in order to see the outer bound or "frontier" while also overlaying the frontier from the top and bottom of figure 3 to see the effect of trend-like behavior on the boundary. When I do that, this is what I get which is a little like looking at the top of each plane in Figure 1 edge-wise from the perspective of the y (spend) axis. So same thing as figure 1, just less 3D and a better x-ray of the bones of this thing:
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| Figure 4. Consumption "frontier" |
Observations
- Adding a third asset with the potential to nudge the EF left will raise the lifetime utility (up) while also nudging the potential to spend optimally up (right) a tiny bit.
- Officially the top of the blue line is 3.7% spend (for these fake assumptions, anyway), and the top of red is 3.8, but we can also see in practical terms we are splitting hairs. For blue there is a range somewhere between 3.5 to 3.9 that is more or less the same. For red there is a range from something like 3.7 to 4.1 that is more or less the same. Does move though.
- To the left of max{E[V(c)]} remember that while consumption utility drops off precipitously we are also, by our saving, effectively investing in an option re future net-wealth performance over some timeframe to be used for either upgrades to lifestyle or bequest. We might consider the utility loss as potentially "compensated" for in the future. To the right of max, we have declining utility and even though it is higher consumption, I want to call it "uncompensated' utility loss since there is a lifetime consumption loss due to future wealth depletion that shows up in the slope of the curve. That's an amateur interpretation. I'd have to ask an economist on this uncompensated/comp thing.
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