Time Effects of Geometric Returns, Now With Spending

One of the reasons that retirement finance is interesting, in addition to the fact that at least a couple Noble Laureates have called it one of the hardest problems they have faced, is that it is quite a bit different than institutional or accumulation stage investing.  In both of those latter, the impulse is to 1) treat horizons as infinite or at least very long, 2) ignore or suppress the concept of spending or consumption, and 3) to simplify things think of it as a single period investment problem.  That's also why simulators are so common in retirement analysis. They take the concept of finite horizons, multiple periods, and spending, among other things, head on.  Here is another way to look at mutli-period short-horizon geometric returns with a spending element thrown in for flavor.



In a couple of past posts(here and here) I did some geometric return math using the cumulative geometric return of two strategies, one with higher return and higher vol and one with slightly lower return and lower vol, and showed that in retirement time-scales (let's say less than 25 or 30 years) the expected value (not any one real sequence) of the lower return path "could" cross over the higher return path and make more economic sense in real life (on average, in some cases).  Another post showed that the same could be true if one were to consider switching  strategies but are worried about a tax hit; in some circumstances even the tax hit can be overcome.  All of that was relatively abstracted because there was no spending which is the touchstone of most retirement analysis.  Here we'll try to fold in some spending to see what happens in terms of effects within our shorter retirement planning horizons.

This is the formula for geometric return.

Remove the minus 1 and the nth root and you more or less have the formula for the cumulative growth of a dollar over time.  By various derivations done by others (I borrowed this particular one from earlyretirementnow.com but I have seen similar at morningstar and other retirement finance sites) we can get to the final value of the dollar at some time T with withdrawals every period of w by this:

where fv is the final value, w is the withdrawal, and C1 is cumulative return of the dollar if held over the horizon and Ct is the cumulative capital returns (effectively moving backwards from T to t). Being an amateur, I didn't quite get this so I actually had to spreadsheet out a begin-spend-return-end-begin thing to prove that it works and it does... if you get the "moving backwards" thing after enough tries.  

In the past spreadsheet-mini-sim I did I took 20 periods, 1000 randomized returns (I won't confess to the shape of the distribution), cumulative geo returns (for 2 strategies), and then calculated the expected value of the series from period 1 to 20.  This time I'll do the same but use the second equation above to calculate the value of a dollar in each period and then look at the cumulative compound expected return in each period 0 - 20.  The goal is to see what happens when comparing two strategies: 7% return 20% vol and 6% return 10% vol ... what happens inside a retirement planning horizon? 

First of all this might have been easier to do in a simulator than a single excel page, and in fact the effect is really a simulator by any other name, but I wanted to check out the math and do a simple head to head strategy comparison without the theatrics of the more traditional sim and that was easier in a spreadsheet.  Also I will confess to some corrupt data science.  The expected value of the cumulative returns can take on an infinite number of paths around my sim results since I am only randomizing 1000 rows of returns.  That means I had to chug thru quite a few "sims" to get a chart that made my point.  That's not great but I'll do it more perfectly some other day. Here is one of the sim results (many or most of the others don't have this effect of a lagged cross-over but at least some do which will eventually be my point):

Some thoughts:

- Spending is not friendly to returns and can kill outcomes, of course
- Fail rates are higher for higher vol strategies
- At least some paths of the lower vol strategy will perform better than the higher vol strategy over time and sometimes the presence of spending will make it something that will happen sooner rather than later*. 
- None of this is very generalizable the way I did it. 
- The "effect" mostly disappears when a longer sim is run so this is not very persuasive but just points to something that is plausible under some reasonably unrealistic assumptions. 

Postscript - I ran this in R 10k times, same parameters, and came up with this. There is a slight advantage to the lower vol strategy2 earlier as well here but it's small. No inflation.  The blue line ends because I haven't decided what to do yet in R with portfolio losses greater than the inital $. The rising red dotted line I assume is an effect of returns overcoming spending over time but I'm not sure yet.

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*The uptick in cumulative returns in the lower chart can be ignored.  When the value of the dollar fell below zero I set the cum compound return to zero to make the spreadsheet math work. That means that plummeting portfolios on a path stop at zero and the expected value starts to "recover" at some point after I start to coerce the data.