Supplement to "optimizing spend rates in a WDT model"

In optimizing spend rates in a WDT model I was doing some casual play with a prototype of a Wealth Depletion Time model.  Nothing huge to report there; I was just goofing around.  In that post I had a hard coded end age and was doing a simple average of utility to that age and then backing into a certainty equivalent that said something about lifetime consumption utility.

In this post there is also nothing to report...still just goofing around.  I kept everything the same, except for:

1. I changed the interval from 60:(hard coded T) to 60:120 because now we can probability weight
2. I discounted the utility by weighting it with conditional survival probability at each age
3. I added a small subjective discount for a little bit of a symmetrical time preference effect
4. I changed the form of the utility function by adding a constant.



A. The utility function is now expressed like this

I'm sure there is a neat rationale in economics. For me this is (a) a common form for CRRA utility I see a lot and (b) it has the benefit of being > 0 over the interval of interest.

B. The Value function is a discrete form I have seen a few places in the lit. The continuous versions and extensions were in past posts by folks like Yaari, Leung, Milevesky, Lachance, etc. Here I am borrowing and tweaking the notation from this (Graduate Macroeconomics I ) online lecture note link (yes I read it but just the parts I needed):

V(c) is the discounted utility of lifetime consumption in its discrete form. tPx is the probability of being alive at time t for an x year old. g[c(t)] is consumption utility as above and the theta term is a subjective discount term that I keep small in this post (.005) based on some always appreciated advice from Gordon Irlam.


C. The assumptions are the same as before with some mods

age 60 t=0 to T = 120-60 (i.e., the full lifecycle because we can probability weight)
1$M starting endowment
Spend rate = .04 to .03 in .01 increments
Spend grows nominally at .03 real = 0
Returns are .055 nominal .025 real with a 1% penalty for whatever needs to be penalized
A dumb sequence risk filter is applied: -.01 over 0:(T(h)-60)/2 and +.01 over (T(h)-60)/2:(T(h)-60)
   this is the hard coded T(h) here to age 104 btw...geometric return over horizon is neutral fwiw
Soc Security of 12k (FV) is started at age 70 and inflated
No annuitization of wealth over horizon
Risk aversion coefficient = 2

Importantly: as before, consumption is forced to available income at wealth depletion time (note the triple emphasis)


D. The initial state looks like before and is artificial for play and personal learning not for realism or advice.

E. The outcome is intuitive if you read the linked prior post.

The optimum in this one junky, deterministic example is lower than 4% due to the drop in late age utility during wealth depletion for which precautionary savings is needed...after which is a long slide into self denial (given the assumptions anyway).  But note that in simulation (or parallel lives) there would be scenarios with optima even lower than this chart.  The optimum here is much closer to 4% than in the last post because we have strongly preferenced the near future due to longevity probabilities and a point of view about how we want to consume over time (gimme now).   We saw the same shift left in the last post when we changed the hardcoded end of life from 104 to 96.  There the survival probabilities were binary: 100% to T and 0 thereafter.  Either way, the late age utiles are discounted one way or another as we bring in the extreme assumptions about longevity.

F. Conclusions?

Nope. This is just goofing around and supplementing the last post.  I think there will be more conclusions when I next look at probability weighted, discounted utility in WDT for combinations of spend rates and annuities.  I'm hoping something cool comes out of that.  Think 3D surface if it looks good.

G. Fun facts to know and tell

As an amateur I just assumed that one would discount consumption and then calculate utility rather than the opposite.  It's the terribly dated MBA in me.  I was instructed on proper terminology and method by Professor Milevsky thusly:

When I'm in a very precise and careful mood, I write that V(c) is the Expected Discounted Utility of Lifetime Consumption (EDULC). [RH comment: this is for the continuous context where T is a random variable so V(c) is random as well] 

If everything in life were nice and linear then the order wouldn't really matter. But because of the concavity (see also Jensen's inequality) of the utility function, the order does matter. So, the reason you have to do utility first is that the discounting function applies to UTILES and and not money. So, like an exchange rate, you have to convert to Canadian Dollars and then computed the PV using a Canadian interest rate. I hope this makes sense.