305pelusa wrote: ↑Sun Mar 10, 2019 12:49 pm
Ben Mathew wrote: ↑Wed Mar 06, 2019 10:13 am
Yes, it's not widely recognized, but lifecycle investing and the sequence of return risk problem are really one and the same.
I've given this further thought and I can talk a bit more intelligently about it. I've gone through some of the math from the Samuelson and Merton papers and here are my thoughts:
It's true that borrowing X money, investing it as a lump sum, and then not touching it at all for the rest of your life has no sequence of return risk. But this isn't quite what lifecycle investing is.
Lifecycle investing assumes a constant, relative
risk aversion (RRA) (like Merton and Samuelson). That means the proportion
invested in the risky asset does not depend on wealth or time. But the dollar amount will certainly change.
To determine that proportion, they use equation 25 from the Merton paper "LIFETIME PORTFOLIO SELECTION UNDER UNCERTAINTY: THE CONTINUOUS-TIME CASE". That equation tells the optimal proportion of wealth to be invested in the risky asset as a function of the risky asset premium over the riskless asset, volatility and relative risk aversion.
The RRA is obtained by first finding out your own personalized utility function (done very elegantly and with all the math in the background, via the "New Job" question of the book), and RRA just flows out of that personalized utility function. This isn't relevant to this post but thought I should mention it.
Here's why it matters and why I have the feeling that this method is superior to the lump sum:
None of these equations are dependent on the past. Equation 25 doesn't have a term to the extent of "did you hit your allocation correctly before? Then in that case, do not modify buy/sell any stocks".
From this I gather that perhaps investing a lump sum is the perfect time diversification as measured by sequence of return risk (since it has none of it)
. But lifecycle investing does not do that; it rebalances the dollar amount in stocks to hit the optimal stock proportion as calculated by Merton. In this sense, they are not "one and the same" IMO. It's close and along the same veins, but they fundamentally do different things. Here's an issue of nomenclature: If you define "time diversification" your way, the lump sum method wins. If you define it as "always having the Samuelson share exposure", then obviously lifecycle investing, by definition, wins.
Having gone through the math I gotta say I feel more attracted to the lifecycle investing model. It feels to me like it's the lump sum model, but improved because you make decisions after every result of your bet, rebalancing back to the optimal proportion exposure. I think the lump sum model is ideal if you're not allowed to make decisions based on the results year after year (i.e. you're only allowed to buy/sell X dollars and you make that decision years in advance). But if you are allowed to see the results (which you are in real life) and make modifications, instinct is telling me that the approach that rebalances back to the optimal proportion is more effective.
You shouldn't equate lifecycle investing with the constant relative risk aversion (CRRA) assumption. CRRA is a strong assumption about preferences that leads to a nice closed form solution. But its results should not be used as a prescription. CRRA assumes that you are equally willing to bet half your wealth on a gamble whether you have $100K or $100 million in wealth. Your preferences might well be different. Mine certainly are. Our attitudes towards risk is a complicated thing. Assuming a simple functional form for our utility will lead to results which, while illuminating in some ways, can't be taken literally.
Under CRRA, you're right that if you lose some money in stocks, you'll want to rebalance back into stocks to maintain the same fraction of wealth allocated to stocks. And if stocks gain, you'll rebalance back into bonds. But that's simply because you've assumed this utility function that says you'll want to do that. Let me give you an example where that won't be the case. A common way to think about the stock/bond AA is to say you'll hold bonds to cover your basic necessities and stocks to cover your the luxuries you can live without. Suppose you're 50 and stocks crash. Should you convert a large fraction of your bonds to stocks to maintain the same proportion of your wealth in stocks? No, because you still need those bonds to cover your basic necessities. Replenishing stocks will put your funding of basic necessities at risk, and that's not what you want to do. Such a person does not have CRRA preferences, and should not rebalance to maintain a constant proportion of wealth in stocks.
I think you also misunderstand how I'm suggesting the lump sum benchmark strategy be used--maybe because I wasn't clear about it. I don't think people should put $X lump sum in stocks in at age 25 then stick to it rigidly. Everything is financial planning is "until new information comes in." If you get a raise, or an inheritance, or become disabled, have an expensive divorce, or your investments do unexpectedly well or unexpectedly badly, of course, reevaluate what you should do in light of that new information. Presumably, there was a process whereby at age 25 you said, "Knowing what I know now, I think I should put $X in stocks and let it ride unless something unexpected happens" Well, at age 26, ask yourself that question again. Something unexpected probably did happen in the last twelve months. So now the answer is "Knowing what I know now, I think I should actually put $Y in stocks and let it ride, unless, of course, something unexpected happens again" Yes, that will mean extra contributions or withdrawals from the stock account. But you have to respond to new information. You can't leave something as-is just because that's what you decided when you were 25. The lump sum benchmark is not meant to be an inflexible strategy. It's just a benchmark to help you figure out how to act given what you know now.
If adopting the "stocks first" strategy of investing 100% stocks till age N, and sending all further contributions to 100% bonds, age N is subject to change. Early years should clearly be stocks. But towards the middle, it will be hard to know when to make the switch over to bonds. A person might decide at age 40 that it's time to switch to bonds, but then realize at age 41 that that was too early, and switch those contributions back to stocks. Maybe he or she switches again at age 43. But then realize that was too late, and change some existing stocks to bonds. I expect some back and forth like this to happen for a few years in the "in between" zone until one is clearly into the bond phase.
It's important to note also that risk minimization is only one of the important considerations of asset allocation. The other big consideration is that you want to find out the results of your bet early rather than late because that will give you more time to respond (work more and consume less if the portfolio performs poorly; loosen the belt and take vacations if portfolio performs well). This means you'll prefer to place your bets early, meaning more than even stock exposure when young.
So what I'm really saying is use the "lump sum and let it ride" version as a benchmark strategy to help you think about how much you should have in stocks. Re-evaluate every year (or two or three), and change your allocations if you need to. If you have CRRA preferences, yes, you will end up keeping a constant fraction of your wealth in stocks. Every year you will say, "Given that I know that I have this much wealth at this time, I will put 40% of it into stocks." But if you don't have those preferences, then you don't need to do that. Look at the distribution of returns, and decide what looks good to you.