Trend Following and Portfolio Longevity

In a previous post (Trial run: effects on "Perfect Withdrawal Rates" from allocations to trend following) I tried to make a case that trend following, if an allocation to that particular alt-risk could actually trim some left-tail return risk, might be able to expand spending capacity over a lifetime. I didn't say it at the time but that is more or less also saying that trend following, given the same caveat but with the added constraint that spending stays the same (and constant...yes I know, an unfortunate assumption), can enhance portfolio longevity. This would look like a non trivial outcome if one were to happen to be at the end of one's portfolio and one realizes that one forgot about portfolio design and alternative risk. I guess that's kind of obvious but the purpose of this post is to take a closer look at this idea.
Let's set up a fake world in the RH lab (tm) to test it out.  We'll start with some cagey steps into the assumptions that will let us look at this, not all of which even I buy. 


1. Observed data shows some fat tails in market returns. 

In the market data I can see, the data looks a little fat. At this point it does not matter if it is a normal distribution with some outliers or a naturally occurring asymmetric leptokurtic process. What matters is that either way the effect we see will impinge on spending capacity and/or portfolio longevity. Here is a distribution of S&P returns using Shiller data thrown up against a normal distribution using the same mean and standard deviation. This assumes some myopia around anything other than the first two moments of the distribution but let's roll...

Figure 1. S&P returns v normal distribution

2. We can create a fake version of something like this in the lab.

In order to liberate ourselves from market data and have some more flexibility, we can create something similar to test out more varieties of scenarios. Ignore for the moment that at higher aggregation units, say years, some of the leptokurtosis will fade out. We just want to force a fat tail for testing and since I work in years rather than months we'll force it. Here below is a synthetic attempt to replicate the shape of the above.  We take two distributions, one with 6.5% return and 10% standard dev, another with -2% return and 25% standard dev and we mix them 83% dist 1 and 17% dist 2. The fat outcome (red line) is a combined distribution with 5% return and 14% standard dev. This is compared to a normal distribution of 5%/14% (black line). Again, there is some myopia around anything other than the first two moments.  Note that the return and var are arbitrary and not meaningful except in the sense that it is not wildly implausible for a mixed portfolio.  When we do the above it looks like this which, at least visually, does not look too dissimilar to the graphic above.

Figure 2. fabricated Gaussian mix v normal distribution

3. Let's then assert, with no real blog-post rigor, that adding a trend following allocation or overlay to a retirement portfolio can nudge a fat left-tailed distribution towards a more normal one all else being equal. (from red line to black, in other words)

This is a leap of faith since we don't have a ton of analysis to show an actual transformation from skewed, kurtotic to something more normal by way of the addition of trend following nor do we have much on the magnitude and stats of the changes involved. That's ok, though, because we are still just illustrating a point.  For this point I'll go back to what I used in "Trend Following in Retirement Portfolios - another quick look" where I used some work by Newfound Research that animated the approximate transition like this

Figure 3. Newfound Research animation of the potential
effect of trend following on a return distribution

and about which they said: 

"By combining different asset classes and payoff functions, we may be able to create a higher quality of portfolio return. For example, when we overlay a naive trend strategy on top of U.S. equities, the result converges towards a distribution where we simply miss the best and worst years. However, because the worst years tend to be worse than the best years are good, it leads to a less skewed distribution. "

I'll also point out that the dotted line from Newfound would correlate to the red lines and columns in Figures 1 and 2 above. 

The point here is that the idea is that trend following can potentially, over a long enough horizon, mitigate at least some tail risk and hopefully partially unskew a return distribution. That's a hypothesis in this post. If that hypothesis were to be true then we could play with the return distributions in figure 2 to see what might happen if it came to pass.

4. If we model a simple portfolio longevity (PL) simulation with two distributions, one fat tailed (red) and one normally distributed (black), and if we keep spending the same, we should see some enhancement of portfolio longevity all else being equal. 

Effectively this proposition is likely an artifact of mitigating sequence risk a little bit where sequence risk does not necessarily have to be a first-then-last sequence because of the presence of spending.  To do the modeling I'll borrow one half of the simulation for lifetime probability of ruin, which I illustrated in the Five Processes - Continuous Monitoring and Portfolio Longevity.  LPR is a construct that weights or discounts a probability distribution representing portfolio longevity in years with a conditional survival probability and where the result is the likelihood of ruin over a stochastic lifetime.  The formulation is like this

Eq 1. Lifetime Probability of ruin

The right side of the right side is the function for PL and is a mini sim that evolves a net wealth process like this w’ = µw - 1 (where w is wealth in a prior period in units and 1 is 1 unit of consumption, µ is return and the variance is introduced in the simulation). PL in this approach is unconstrained and is allowed to run to infinity (I stop at 200 years); the lifetime constraint comes later (left side of the right side in Eq 1).

If we animate g[w](t) with return and variance from Figure 2 above, and spending of 3.5%, we should be able to see some differences in how long a portfolio lasts and in fact we do. I use the parameters mentioned in point 2 above. It looks like this where we are using the cumulative distribution of portfolio longevity in years. The y axis is effectively the likelihood of failure at any give year t (x axis). Note that at a long horizon some percent of the portfolios will last forever and the fail rate limits out.
  

Figure 4 - cumulative portfolio longevity, fat and normal returns

We can see that the mitigation of the left tail, all else equal, pushes the cum PL probability down and right. Overall risk is lower and portfolios last longer which is kind of the same thing kind of not.  Using the artificial RH Lab assumptions in Figure 2, this is approximately "how much longer" for some given fail threshold: 

Just by clipping the tail (assuming that is possible) we have added something around three years.  That doesn't sound like much but it would be solid gold if you were alive in a year where you could have engineered three more but didn't.  Spending changes could have solved this too but that's not today's game. 

5. There is likely a non-trivial attenuation of ruin risk too when we "constrain" PL and view things on a human scale.  

If we animate Eq1 in full with both return distributions and use a constraint or kill-time based on a 65 year old, a gompertz equation, modal life of 90 (dispersion of 8.5), and a spend rate of 3.5% we can see the effect on general risk as measured via LPR (ignore the fallacies of ruin risk evaluation, I can hear Dirk Cotton whispering in my ear "there's no such thing as ruin").  Charted out it's like this

While the actual value of ruin is more or less meaningless the information value of the change, the 2nd derivative if you will, is meaningful. A 33% change is not nothing. Of course there are an infinity of other parameterizations, this is a good illustration of the point.

Conclusions

If trend following in portfolio design (and other strategies?) works as advertised and it contributes to left tail risk mitigation, it is likely that it can, all else being equal, enhance portfolio longevity to some degree. Personally I'd put the planning around spending and it's various trajectories, along with spending control, ahead of asset allocation to alternative risk but this is a rich and interesting area to explore nonetheless.