Sep 6, 2022

Part 6 - Asset Allocation and Portfolio Longevity with a Capital Market Line

This post is part 6 of a series on Portfolio Longevity, a series made up of these links:The point of this post is to:  
  1. Extend the other 5 posts by now adding leverage and a risk free asset. i.e., a capital market line.

  2. Look at the impact of "asset allocation choice" on portfolio longevity, using the same set-up we started with in the first link, and

  3. Compare or contrast the impact of allocation choice along: a) a traditional efficient frontier vs b) the impact of allocation choice along a capital market line.

  4. Try to infer what is going on. Maybe. Sorta.


A friend of mine reminded me the other day, something I've known or should have known for 40 years now, that the choice of portfolio can be separated from the choice of how much risk to take. Basically this is no more or no less (if I have it right) than the capital market line: 1) chose an "optimal" portfolio along the single-period efficient frontier by means of math (say, Sharpe) or some measure of risk (subjective or maybe some kind of utility framework) [1] and then 2) mix that portfolio with cash or borrowing to choose one's risk.  A good primer on this is either a basic finance text or perhaps this article by Cliff Asness: "Why Not 100% Equities, Journal of Portfolio Management Vol 22 Nbr 2 Winter 1996 which I recommend. 

Using the setup in link #1 -- all of which is very arbitrary, in real terms, and only illustrative -- now I will add a risk free asset with a real return of .0095 and zero SD. This is entirely fiction, btw. I picked that Rf for no rational reason other than that gives a max sharpe at ~40% equity which is useful for my post but not particularly realistic. 

When I do that I get an efficient frontier that looks like this with a capital market line in blue: 
Figure 1. CML over EF w tangency at ~40% equity

Note that there isn't a ton of lift here. Maybe another setup would have made my post-point better but whatever, let's go with this and see what happens.  Since I want to normalize this post to the same framework in the last 5 I am going to step up the EF (or CML) in steps equivalent to the SD on the EF so that I can do an apples to apples comparison of Portfolio longevity. That means that the steps will look more like this where in this post we will be stepping up black and blue, not red: 
Figure 2. The EF and CML for my sim

Since it has been a while, let's refresh on the assumptions. Asset 1 is lower risk N[.01, .04] real and asset 2 is higher risk N[.07, .25], correlation is ~ -.20 which is not particularly historical but works here. Again, I remind that this is fake and illustrative only. Iterations are 200k for smoothing some stuff out. spend rate is 3% which is traditionally low but I'm a young retiree so I use that because it makes sense to me for reasons articulated elsewhere in the blog. 

The portfolio longevity is based on a recursive process of W2 = W1*r - spend. Others do it (W1-spend)*r. Whatever. We spend and we earn. I'll compare the two variations later.  The horizon is 100 years which is a proxy for "a very long time" or infinity. Easier to code, too. 

For each portfolio (once with the EF, once with the CML) we run the world 200k times and create a distribution of how long the portfolio lasts (PL) at each step or each portfolio and then report on the 5th thru the 25th percentile of the distributions of PL at each step. When we do this we get the following in figure 3.   
Figure 3. Portfolio Longevity under two methods

where black/grey is walking up the standard EF and red is walking up the CML.

Conclusions?

Maybe. This is a little tricky but I'll take a shot: 
  • This is kinda dumb because I do not really understand borrowing yet.

  • We have not really diversified yet. Me? I prefer adding some trend following for its potential payoff structure which I've touched on a million times but here is at least one example: Lifetime Consumption Utility with addition of trend following-like behavior

  • Under the constraints of this post (see bullet 1) I can say that separating the optimal portfolio from the choice of risk seems to work well, especially at the risk extremes which makes sense

  • One still needs to have a strategic stance, i.e., are we: 1) trying to avoid short lived portfolios or are we 2) attempting to gamble and shoot for longevity. That's a fraught choice and depends perhaps if one is a grey-haired retiree like me or an institution like an underfunded pension fund. TBD. I'll not resolve that here

  • More low risk portfolios are now in play as viable in the red vs the grey but that depends, again, on resolving the last bullet

  • The center-risk portfolios, notwithstanding the strategy question, and because little leverage or lending are involved in the center, still look pretty good

  • Lower risk portfolios, when used in this multi period time world with spending involved have always, on this blog, worked pretty well. Haven't mentioned sequence risk yet but I guess it is implicit 

  • I have not specified the horizon yet. If my horizon is institutionally infinite, obviously (maybe?) I'd lean towards high risk. Alternatively, if I think like me and I am 64 and I might annuitize a bunch of wealth at 80-85 then my horizon is about 16 years (or say 20 max) in which case: a) it might not matter according to figure 3, and b) I'd maybe lean towards lower risk...which I do. 







----- NOTES -----------------------------

1. I will NOT be worrying about how lending costs work here, just working the basic idea for now. Besides I'm not sure how to do it yet.  


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