How Much Investment Is the Right Amount?

On this episode of The Long View, Victor Haghani, author and founder of Elm Wealth, talks investment sizing, retirement spending, and his new book The Missing Billionaires: A Guide to Better Financial Decisions.

Here are a few excerpts from Haghani’s conversation with Morningstar’s Christine Benz and Amy Arnott:

How Much Investment Is the Right Amount?

Christine Benz: We wanted to delve into some specific aspects of the book. One of the topics that you tackle is investment sizing, position size. Most financial advice available today to the average investor focuses almost exclusively on what to invest in rather than the equally critical question of how much investment is the right amount. So why do you think that people so often ignore that question and move straight to what?

Victor Haghani: I think that the what question is where all of the excitement is. What should we buy? Is Nvidia going to keep going up or not? And is Apple going to rebound or whatever? Those are the stories and narratives that are so exciting and attractive to us. And it’s a full contact sport. Stock-picking is probably the most competitive sport on the planet if we count chess as a sport, too. So, I think that that’s where the attention is a lot. And the sizing question just gets left over. Also, to make the sizing decisions you need to have an idea of your own preferences, of your own level of risk aversion, of your own level of subsistence or basic income that you need. And you need to have some assessment of the expected return and riskiness of all the different things you can invest in. So, I think, for a lot of people, they just try to short-circuit the whole thing, or a lot of advisors just try to short-circuit the whole thing and say, OK, if you’re an average person, 60/40 equity fixed income is good for you. If you’re young and aggressive, 90/10 or 50/50 and somehow really jump over all of that. Whereas we think there’s a lot of low-hanging fruit, a lot of low-hanging improvement and welfare that we can get from thinking about sizing. And of course, in the extremes, it’s really important when you could potentially be using leverage or there’s a lot of decisions where this idea of how much is absolutely critical—it’s not just getting an improvement in welfare, it’s really changing your financial situation dramatically by getting it right.

Amy Arnott: I think that’s a good point. And one of the things you write about is that as long as you’re in the right neighborhood of an appropriate allocation size, whether you’re 65% invested in stocks versus 70% is probably not going to make a huge difference, but it’s at the extremes where it can make much more of a difference.

Haghani: Absolutely. It gets really steep the further away you get, but it’s flat around reasonable or optimal places. Exactly. That’s a good thing to take away in general.

The Merton Share

Arnott: One of the foundational concepts you discuss in the book is called the Merton Share, which is a formula for figuring out the optimal amount of wealth to invest in a risky asset or a portfolio. And it actually dates back to a paper Robert Merton wrote back in 1969. But it really never took off outside of academic circles. I’m curious, why do you think that that is?

Haghani: Well, I think there’s a couple of different reasons. One is that in its first incarnation where it came out in—actually in this pair of papers, Bob wrote The Continuous Time Version and Paul Samuelson, his mentor, wrote a discreet version of it. And both papers are pretty inaccessible. But then people worked more on it for the next 10 years or so and made it easier to understand and more realistic. But I think that one thing that happened is that that whole area of research got overtaken or overshadowed by the big breakthroughs and option pricing and derivative pricing and all these things—the whole contingent claims literature that was so exciting because it really was usable in Wall Street that it just eclipsed this earlier work on portfolio choice and lifetime consumption or lifecycle financial decisions. Interestingly, seven or eight Nobel Prizes have been given out for lifecycle investing research to Samuelson, Merton, Modigliani, and others. But it just couldn’t make that jump into the mainstream. And of course, a big part of it is, as soon as you talk about expected utility or talk about a person’s utility curve, people’s eyes can glaze over. And we hope that in our book, we’ve tried to really make all of those ideas more accessible, more practical for people to implement.

What Is Expected Utility?

Benz: Can you discuss when you say expected utility, what does that mean just to make sure that all of our listeners are following along here, too?

Haghani: Well, what we care about is not how much money we have, but we ultimately care about what our money can do for us to make us happy to either be spending it on ourselves, or to be giving it to others as either to our family or to philanthropy. So, these are the three main things we can do with our money. Maybe they’re the only things we can do with our money, because I don’t think that counting it should count as a real use of our wealth.

So, then we have to think, well, how happy, how much satisfaction, how much welfare does spending our money on these different objectives bring to us? What we generally find is that this utility curve that’s mapping wealth or spending into happiness, that that curve is concave. In other words, it flattens out the more wealth or the more spending you do, because we have a decreasing marginal improvement in our welfare with higher spending or wealth. In the book we talk about, well, if you like gummy bears, and you have a couple of them, that’s great. The next two are going to give you less enjoyment than the first two and the 50th and 51st one are going to give you really a lot less satisfaction than the first five or 10. And that decreasing or diminishing marginal utility of consumption or wealth is what makes us risk-averse. And it’s what we have to take into account when we’re making these financial decisions under uncertainty. So that’s really all that’s meant by a utility curve is that we’re risk-averse because making more money or spending more money generally brings us less of an improvement in our welfare or happiness than the same amount of decrease would decrease our happiness by.

How to Make Investment-Sizing Decisions

Arnott: Another theme that runs through the entire book is the fact that making sizing decisions with the goal of maximizing expected wealth rather than expected utility can really lead to bad results and disastrous results in some cases. Can you walk us through why that is and how the variance of the investment and your personal degree of risk aversion should also come into play?

Haghani: Absolutely. This is an observation that goes back almost 400 years to Daniel Bernoulli and the St. Petersburg Paradox, a game which was a paradox because it has an expected value. If you were allowed to play the game, the expected value is infinite, but nobody would pay all of their wealth in order to play it. It’s a coin-flipping game and I won’t go into it here, but for interested listeners, it’s fun to look it up on Wikipedia, the St. Petersburg Paradox.

We like to illustrate the idea with a more reasonable, still abstract a little bit, but more reasonable thing. Imagine that you’re allowed to flip a coin, which you know has a 60% chance of landing on heads. It’s not a physical coin, but like a computer digital coin. Every time you flip it, it has a 60% chance of landing on heads, 40 on tails. You can invest as much of your wealth as you want to on each one of these coin flips and you get to do it, say, 25 times. Well, the question is what strategy should you follow, and should you have an objective of trying to maximize your expected wealth? Well, if you work through the math of what you should do if you want to maximize your expected wealth, you’ll find that betting as much of your wealth as you can on each flip is the optimal strategy to maximize your expected wealth. But clearly, betting 100% of your wealth on every flip for 25 flips is almost definitely going to wipe out all of your wealth and nobody would want to do that because all you need to flip is one tails and now you don’t have any money left, but that gives you the highest expected wealth.

So, as soon as we realize that, we see that, oh, that’s not a good objective, that maximizing expected wealth, the expected value of our wealth, the probability of each outcome times how much wealth you’d have in that outcome, that can’t be the right thing to do. And what we find is that just by exploring this coin-flipping toy example is that there’s going to be a fraction of your wealth that you would want to bet that maximizes your comfort with the range of outcomes. And that is how you can calibrate your degree of risk aversion. If you had no risk aversion at all, you would want to bet all of your money on every flip every time. And that’s why we don’t think that really anybody should have or does have zero risk aversion, although the famous case of Sam Bankman-Fried claiming that he was risk neutral and had no risk aversion at all, well, it seems that maybe he did behave that way. But if you do behave as if as though you have no risk aversion, you will very likely go bust with a very, very high probability at a very short period of time.

So, as we think about what’s the optimal amount that we want to bet of our wealth on each one of these coin flips, which helps us to calibrate our degree of risk aversion. And what we find is that the optimal amount that we want to bet on some opportunity is proportional to how attractive it is, what the odds are in our favor, but it also is proportional to the risk that’s involved in that investment or that gamble. And what we also find is that as the attractiveness goes up, we want to bet more and more in a proportional way. But as the risk goes up, we want to bet less and less, but in a quadratic way, in a squared way. So, increases in risk bite more quickly than increases in expected return.