Geometric mean, simulation, short horizons and portfolio choice

I don't think this post is all that innovative. We see this kind of stuff in thousands of papers because this is basically just simulation but without the spending and fancy parameters one ususally sees. I just wanted to see how some stuff works here. 

In this post I'll look at 2 strategies -- 1) high return, high vol & 2) lower return lower vol -- to see what it all looks like. We know that we can use deterministic N-period geometric return formulas to estimate the N-per geo mean, something typically and probably incorrectly evaluated at infinity, for evaluating portfolios -- especially for the possibility of a "crossover point" where one strategy should dominate another. We also know that we can also construct a geometric efficient frontier in order to try to limit the portfolio choice interval to the one from risk free to the Kelly optimal (growth optimal) portfolio given believable inputs. Even Markowitz says that. But that latter method again often evaluates at infinity. What is missing is shorter horizons. Hence the post.

Simulation methods are prone to trial and error evaluative methods but MC sim also helps visualize what is going on inside a modeled process. Here I am trying to combine the two - geo and sim. Let's take a look.  Here are the deterministic formulas for geo return and for it's variance from Michaud 2003. 

       (1)

 (2)

The simulation is the simulation. Geometric returns are a chained product of the randomized arithmetic return input over 100 periods done 10000 times where the end game mean(G) approaches E(log(1+r)) if I have it right. Here is the geo formula

 (3)

Net Wealth is a a recursive chain as well over the same interval and iterations where c and i are zero here today.

   W(t) = W(t-1)*(1+r) - c + i           (4)

 The two portfolios are for convenience and illustration: arbitrary, normally distributed, and parameterized like this

1) High return high vol  -- N[.08, .25] 

grey paths, blue: mean (returns) median (wealth)

2) Low return low vol  --  N[.06, .10] 

red paths, black: mean (returns) median (wealth)

For our internal pedagogy, let's say these two are two different portfolios on an efficient frontier. We could choose based on the expected geometric return at infinity but we don't live to infinity. We need some other sense making apparatus. We could of course use (1) above for this task but it is hard to visualize the risk.  We could use (2) for the risk but a) it takes a little while to engineer the math while a sim is almost easier sometimes, and b) why do something simple when we can make it hard.  Heh. 

Figure 1. Geo return and net wealth for two strategies
mean return, median wealth in blue and black

We can perhaps see:

• The lower vol lower return dominates at the 30 year right side, which is not infinity but might stand in for now. 
• The cross over happens pretty early in the life cycle
• The relative dispersion of outcomes of geo return and wealth should be apparent for both the high and low vol strategies. 
• It is hard to see the early years at this resolution. 

So, given the last bullet, let's zoom in on the first 10 years and a foreshortened Y scale...

Figure 2 -- zooming in on figure 1

Discussion

• Given this highly stylized example: blue (high return high vol) "wins," sort of, over a short horizon, maybe 6-8 years in wealth terms (remember, no spending here)
• Black (lower return lower vol) wins thereafter...forever.
• The individual paths are pretty gnarly.  The outliers to the downside I don't want. 
• Given the dispersion of the paths, especially the grey ones (high vol), the difference between blue and black at the median seem trivial. 
• There is some embedded optionality in strategy 1 but there is no downside limit like in a call option.
• Given the hard dispersion to the downside, the short-horizon cross over, the minor differences at the median, and the ultimate dominance of strategy 2, I'd probably go just dive into strategy 2 at the outset. Behavioral finance and all ya know...