Optimizing Spend Rates in a simple WDT model

I thought I'd see what happens to spend rates in a simple Wealth Depletion Time WDT model in terms of the certainty equivalent consumption over an (unrealistic) fixed utility horizon.

The Setup (illustrative but not very realistic)

age 60 to age T in [104,96] so lifetime is not random except for the simple toggle here
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-60)/2 and +.01 over  (T-60)/2:(T-60)
  (geometric return over horizon is neutral)
Soc Security of 10k (FV) is started at age 70 and inflated
No annuitization of wealth over horizon
Risk aversion coefficient = 2
Importantly: Spending is forced to SS income at wealth depletion




CRRA Utility

Utility for consumption in time t is calculated with RA coefficient (gamma) = 2, in this form

Utility is averaged over horizon and the certainty equivalent consumption "average" is backed into by a transformation of g[c(t)].  There are no subjective discounts (yet) and there is no longevity probability weighting (yet). And the utility of bequest is ignored.    This is all recalculated for each spend rate increment as well as for the change of T in going from Case A (104) to case B (96). I'm pretty sure this below is the right way to express the value function from which the CE is extracted. Or at least it is close for a super simple model like this. The subjective discount is not used.  I'm going to think about this some more to see if missed the boat.

The Initial State

This is the initial state for planning horizon T = 104. Utility is calculated against the blue bars in both its high (before WDT) and low state (after WDT) until age 104 (and we then do it again for age 96).

The Results

Case A - T = 104

Blue (left) is the CE of mean utility. Red (right) is the nominal value of  bequest at T just for reference. X axis is the spend rate in % going from 4% on the left to 3% on the right.

Case B - T = 96

Blue (left) is the CE of mean utility. Red (right) is the nominal value of  bequest at T just for reference. X axis is the spend rate in % going from 4% on the left to 3% on the right.

Conclusions

Any grad level textbook in macro econ will explain the potential for the utility optimizing value of precautionary savings over a lifecycle for a given set of assumptions and analytic expressions.  This just shows the same thing in practice in a simple Excel model.

The conclusion (here), if one can be made, is that if you think you are going to live a long time the range of precautionary saving cuts, in utility terms, to get to something optimal is going to be relatively deep. If, on the other hand you think you are going to live shorter, it still may pay to cut spending up to a point depending on expectations about T.  Utility, when wealth is >= 0 at and after T, starts to fall because we are spending less than we need to to get to T unless we consider bequest which we said we were going to ignore.  Some of this is intuitive and does not need Excel or CRRA utility for us to figure it out.

More realistic terms for SS and some annuitization of wealth will change the optima, probably by a lot. I don't know yet how to get a handle on that yet.   Also, randomizing T is the real game.  BTW, the financial economics on "optimal C" has formal expressions not shown here. Note also that the literature shows that the possible optima in the no bequest case, if any, have to be inside the 60:T range when there is income at T...where my optima are at T which probably means the model is overly simple (or broken) and therefore maybe misleading...or basically an overly complicated no-bequest almost-no-income model...if I have understood this correctly...which may not be the case.  I don't think the exercise was completely un-useful though for me trying to understand the underlying processes in play.  Time will tell.