On Adding a Time-Preference Discount

The idea of perpetuities is cool and all but the perpetuality, if real, would demand a certain degree of stability in government, culture, taxation, markets, law, policy, institutions, civilization, etc over very very long timeframes. Me? I'm not so sanguine on that whole set of stability assumptions these days over even short horizons. In a recent paper by Barton Waring, he limits the interval of evaluation of endowments (otherwise a type of perpetuity we might say) to 50 years. His rationale for 50 goes like this: 

In generating our forecast distributions, we’ll use 50 years as our simulation horizon, but that number is arbitrary—we felt it to be a horizon that should represent three to five “generations” of board members or trustees, and one that is also long enough to show the long term trend as time marches on towards the endowment’s hoped-for immortality.

50 is arbitrary so right there he is pitching us an ever so slight preference for the near future over infinity. And in fact in most of the consumption utility math I've ever seen there is a factor or discount for biasing us towards the present a bit. LaChance, following Yarri, presents the evaluative goal like this in continuous form:  

Eq1. Value Function from LaChance 2012

where f(t) is some combo of both longevity and time preference weighting, u(c) is a CRRA utility function, and w is "long age" which is often set up as 120 if not infinity, though here it's 100. I usually don't include the time preference because it is a small factor that can distract from some of the other points I am investigating. I always kinda thought over the long haul that maybe it should be zero. But others use it so I'll throw it in today and see how it moves at least one portfolio parameterization (4/12) of what I have done recently. Haghani (2021) says that the discount can be as high as 5% though he himself settles on 2%. Gordon Irlam, whom I trust, told me in private correspondence that it should be very small, on the order of .5% or less.   

Because I am using a longevity weighting in this post I am not picking a hard horizon like 30 or 50 or 60 years like Waring. I am letting the conditional survival probabilities do all the heavy lifting here.  But I am now also feathering in a time preference using 0, .5, 2 and 5% just to see what happens. The framing here, except for the new discount, is very similar to what I did here:  

• Long Horizon Spending (con't.)  

except that I am using age 63 and an annuitant life expectancy distribution -- not age 50 and a freakishly long-lived longevity distribution. Speaking of assumptions. let's put out the core ones here. Might've missed a couple: 

• Age - 63 (me)
• Iterations - 20000
• Portfolio - N 4/12 (arbitrary)
• Risk aversion - 2 (arbitrary)
• Spend - 2 to 6% in .1 increments
• Constant spend not % of portfolio or rule
• Gompertz Longevity with m=90, b=8.5
• There is a consumption/utility floor
• Endowment is 100 or you can think 1M
• Horizon is max 100 years
• Returns and spend are real which might be a tough way to model in 2022, idk

The scenarios are the above param set done 4 ways: 

• Scenario 1 - time preference of .000, 
• Scenario 2 - time preference of .005, 
• Scenario 3 - time preference of .020, 
• Scenario 4 - time preference of .050, 

I am using Irlam and Haghani for the boundaries between .005 and .05; 0 is what I did before. 

I am not going to throw a lot of charts out here but when I take scenario 1 and I run the 41 spend rates, using a RA coeff of 2, I get the following  Figure1 where the X axis is the spend rate and Y axis is the summed, discounted utility "score." The axis titles of "hack" is just me hacking around with my code and I was too lazy to crop and edit.   

Figure 1. Expected discounted utility of lifetime consumption | scenario1 params

Using the max objective in Eq1, I can call a spend rate of 3.0% a winner (max in Figure1) under this particular set of parameters in scenario 1. Now, I want to add a time preference discount to the longevity weighting/discount we apply to the utility calc. Without much fanfare it looks like this[1]

Table 1

I refuse tonight to look at other parameterizations but in general I more or less got what I came for. I can say that in goosing the time-preference towards a steeper discount I do in fact preference the present over the future and I spend more as I expected. Not a ton more but at least a little bit. Looks kinda linear but probably isn't. The impact is not huge for the lower discounts I would personally use. Nothing too wow here imo. 

------------ references ----------------------------

Haghani, Victor, Elm Wealth (2021), Spending Like You’ll Live Forever https://elmwealth.com/spending-like-youll-live-forever/ 

LaChance, E. M. (2012), Optimal onset and exhaustion of retirement savings in a life-cycle model, Journal of Pension Economics and Finance, Vol. 11(1), pp. 21-52 

Waring M B, Siegel L B. “Where’s Tobin? Protecting Intergenerational Equity for Endowments, A New Benchmarking Approach.” SSRN 2022 

---------- Notes ---------------------------------------

[1] I did this whole thing twice. The first time it came out 2.9, 3.1, 3.1, 3.3. This, I assume, was a sim sampling thing since I maybe should have done more than 20k iterations. My guess is that if I did this 100 times and averaged thing out or ran a zillion iterations it would sort out like Table 1. Idk. I was getting a little antsy to get out of the code and the blog. Just pointing out the instability and that my conclusion is resting on a soft foundation.