I recently had the pleasure of featuring on **Bloomberg’s Odd Lots** podcast, with Joe Weisenthal and Tracy Alloway. They asked me some great questions about mathematics in investing.

You can listen to the full 25 minutes here, or alternatively read a transcript of our discussion below.

**Tracy**: Let’s bring in Victor Haghani. Like I said, he was at LTCM, but he is now the CEO of Elm Partners, which is basically a portfolio of low cost index and exchange traded funds. Victor, thanks so much for joining us.

**Victor**: Thank you very much for having me.

**Tracy**: Victor, we actually brought you on after reading a paper that you did basically about what coin tossing and the probabilities involved in coin tossing can teach us about investing. Can you tell us about that paper?

**Victor**: It came out of an experiment that I did with a colleague of mine who I’d worked with at Elm Partners, Rich Dewey. We had heard about some research that had been done involving coin flipping and how people managed situations where they were given a favourable odds investment opportunity. Sometimes you can’t quite remember where the ideas come from, but we decided to do this experiment where we would give subjects real money and allow them to flip a coin that was biased to be 60% likely to come up heads, 40% tails, and we told them that to begin with, and we gave them half an hour to flip, to bet as much of their starting $25 as they wanted, and at the end, however much money they had left in their bank, we would pay them up to a maximum amount of $250.

What we found was that our participants, who were pretty quantitatively trained young men and women, didn’t do very well and they didn’t get some of the basic concepts of decision making under uncertainty or they didn’t quite get the independent nature of the flips and the fact that it just made sense to keep betting heads, to bet some modest constant proportion of how much they had in their bank at any point in time on heads. It was really interesting to think about how people were having trouble with that and to give us some ideas for trying to help with education, as well, on that topic.

**Joe**: Explain really quickly the exact mechanics. They had $25, and they were supposed to bet what? Explain to us what the nature of the bet is. Then what did the lesson show about mistakes that people might or might not make when they invest?

**Victor**: Sure. The exact mechanics of it were that we told the people to come for a lecture, and then we asked them to get out their laptops and to play this game. We gave them $25. That turned up on their screen in their bank accounts or their bankroll. Then they could bet up to the $25, or however much they had in their bank, on the flip of a coin and they could do it repeatedly. Some people flipped the coin 300 times in the 30 minutes that they had. If they won the flip, then their bankroll would go up and vice versa, and however much they were left with at the end, we told them, and we did pay them out as a check or cash, which was especially for a bunch of college students, which were the majority of our subjects, was very welcome.

We told them that there was a maximum payout to begin with, but it was only when they got to a point where they could reach the $250, for example if they had $225 in their bank accounts and they were betting $30 on heads, we would say, “By the way, the most we’ll pay you is 250, so you might want to reduce your bet from $30 to $25 because there’s no point in winning $255. We won’t pay you that.”

The most surprising thing in a way was the fact that people would relatively frequently bet on tails. Even though we told them it was 60% likely to be heads, even though in general heads was coming up more frequently for most people after they had flipped it a number of times, and particularly after a string of heads like four heads in a row, they were then more likely to bet on tails. Not everybody.

**Joe**: That seems like a deep failure of numeracy to ever bet on tails, even to think that the past streak of flips has any bearing on the next flip.

**Victor**: It is, but it’s like this deep-seated need that we have to see a story in random things. Given that half of the subjects at some point bet on tails, and 30% of them bet on tails a fair amount of the time, there’s something deep-seated in there. I had my mom do the experiment, and we talked about it afterwards, and she said to me, “I know that I should have never bet on tails, but I just couldn’t resist.” She knew it. She knew it didn’t make any sense, but she just couldn’t resist. It was interesting.

We did another experiment following up on the famous interview question about family planning, that if everybody in a society wants to have a girl, and so each family keeps having children until they have a girl, does that change the expected number of boys and girls? Most people feel that it does even though when thought of as a coin flip, you can see that they’re independent and there’s really nothing much you can do to change the expected number of boys being equal to the expected number of girls to any finite horizon.

**Tracy**: The point of those types of experiments is essentially that the optimal investment strategy is dictated by maths, and yet, we choose to ignore it for whatever reason because we instinctively don’t understand probabilities or there’s some emotional thing going on.

**Victor:** I think people understand it. Our subjects were really quantitatively trained. They understood all of this. Some of them were even mathematicians at one of the universities where we did it, and some of the subjects were also investment professionals that had a lot of math and econ and finance training, so they understand it, but I think there’s something deep-seated that comes up and steers us off the path. It’s quite a lot of statistics training is probably what’s needed to get people to be disciplined. To be disciplined is not a lot of fun. Think about you’re sitting there flipping a coin for a half an hour and you’re just trying to bet 15% of your bankroll on it and keep betting on heads.

**Joe**: It reminds me of reading about professional poker players who know that they can make a steady profit playing limit poker, which is a very mathematical, very little bluffing version of the game of poker, but they’re just bored out of their minds when they play it. No limit is more fun. It’s more exciting. It’s a little less mathematical and more based on emotion. They are more inclined to lose. These sure things are not very enjoyable practises.

**Victor**: Think about index investing. The most boring thing you could do is take all of your savings and to put it into two index funds. Very few people really do that, and very few people do that and stick to it. People will do it, and then they will come back and feel that they need to change it because there was an election or there was a change in interest rates or something. It’s fighting that urge to leave it alone. Fighting the urge to be active is difficult in a lot of different contexts.

**Tracy**: We’re all suckers for a sense of control.

**Joe**: Let’s talk about a different mathematical concept that’s incredibly important to investing, and that is compounding. I forget who said it. Maybe it was Einstein. Someone famous said something about compounding being one of the most powerful forces on Earth.

**Victor**: I think that they say Einstein may have said something like that, as strange as it may be.

**Joe**: I don’t know why he would have been talking about it, but I think he did say something about it for whatever reason. What is it people don’t understand? Why is compounding such an important concept to understand? What do people get wrong about this?

**Victor**: I think that in these investing things or math things, in general, one of the things that really gets us is non-linearities, things that are not proportional. Compounding is one of those things, that the growth of your money doesn’t go up in a straight line. It goes up in this exponential line. It starts off growing slowly, and then as it gets bigger, it’s growing faster in terms of the amount of money by which it’s growing. The rate of growth, let’s say, stays the same. When you start to look at relatively long periods of time, which are the kinds of periods of time that are relevant to us in terms of building savings for retirement or personal security, the effects become large.

Those long-term horizons are important, and compounding of small effects really magnify out there. The one that we hear a lot about is the effect of fees, that if you’re compounding at a 5% return, because you’re paying 2% fees, compared to if you’re compounding at a 7% return with low fees, that what you wind up with at the end is not proportional to 7 over 5, but rather that 7% winds up giving you a lot more at the end because it’s 1.07 being raised to a power divided by 1.05 being raised to a power. Everything gets magnified by compounding.

Another thing that’s similar to fees is taxes. If we can invest in a way where we don’t pay tax until the end of our investment horizon, we wind up with a lot more money than if we’re paying the same rate of tax on our growth every year as we go along. An example of that would be, let’s say that you have an investment that has an 8% rate of return, and let’s say tax rates are 50%, just to make the math simple. After 30 years, if you’re paying tax every year, then your 8% return is only a 4% after tax return. If you have $100,000 and you’re investing it, then after tax, that $100,000 has grown to $324,000 after 30 years at this 4% rate of growth, half of the 8%. But if instead you’re deferring your tax to the end, then you’re growing at 8% because you’re not paying any tax on it, but at the end, you have to pay 50% tax on all your gain. When you do that, you wind up with close to double the money after 30 years. You wind up with $550,000-

**Joe**: Wow.

**Victor**: And almost a 6% instead of a 4% rate of return. That stuff really kicks in over these long horizons and is important. Small differences wind up being big differences because of compounding.

**Tracy**: What’s your favourite financial formula for investing if you had to choose one?

**Victor**: I don’t know. I guess one of the simplest ones, one that’s been on my mind lately …… I think if I had more time to think of it, I’d find a better one, but it’s been on my mind a bit, is what’s known as Sharpe’s Equality from a paper that William Sharpe, the Nobel Prize winner, wrote in the early 1990s. I think the paper was called The Arithmetic of Active Investing. In that, he just made the very simple statement that the return on the average actively managed dollar has to equal the return of the market minus fees on the active stuff, and that comes about because the market return must equal a weighted average of the returns of the passive and active segments of the market. If the total market return is the same as the index return of the passive part, it’s like saying 2 equals 1 plus 1, then 2 minus 1 equals 1.

It’s a very simple…It’s like in physics, the idea of the conservation of energy-

**Joe**: What are the practical ramifications of that from an investor standpoint? It sounds like an identity essentially. How does that manifest itself practically in terms of making investing decisions?

**Victor**: It just helps us a lot in terms of thinking about what we’re doing when we choose active strategies, for an active strategy to be working for us, that we have to believe that there’s some other active strategy that’s losing money, and we have to be able to identify why and who that’s likely to be, that if we think that we’re going to make money, we really have to be sure of who we’re making the money from.

**Tracy**: It’s a zero sum game essentially.

**Victor**: Yes, within that space. I think that it’s a good first approximation. It’s a valid identity. There’s some caveats and so on that people would bring into it, but I like that it’s simple. It reminds us of Bill Sharpe who is a really cool guy. I think it’s a really useful one to remember.

**Tracy**: I promised a potential LTCM question. I guess one of the other things we’ve observed in markets recently is the rise of smart Beta, but also risk parity strategies, and some people have likened risk parity to the old Black-Scholes portfolio insurance of the 1980s, and some people have connected LTCM’s collapse with Black-Scholes, so I guess I’m just curious how you feel about risk parity and how you feel about the down-sides of mathematics in finance.

**Victor**: For me, the really short answer is leverage…My LTCM experiences just made me not want to use leverage explicitly in any sort of investment strategy for myself or anybody that I would be trying to help. Leverage has its place in our financial system. It has its place perhaps within the investment community, but personally, that was the primary cause of the problems at LTCM, so for me anyway, I know the arguments for risk parity may well be that the aversion to leverage by people like me is what makes using a moderate amount of leverage a good idea. That’s what some people who are proponents of risk parity would argue, that it’s an inefficiency that a bunch of people like me now are averse to using leverage, but I’m averse to using it.

I’m not a fan of risk parity because I just don’t feel that I need to use leverage to get better quality returns. I think that the returns afforded by the marketplace without using leverage and the risks attached thereto, are all sufficient for me, and then I can go to sleep and not worry about having to reduce exposures because my leverage is causing me to do that.

**Tracy**: What about financial formulas, in general, and maths in investing? What are the down-sides?

**Victor**: Models used in investing are very useful. One of my colleagues once said to just think about the yield to maturity of a bond. Think about that as a model. At some point in time, yield to maturity wasn’t really used. People used to talk about the price of a bond. They used to talk about the current yield, the coupon divided by the price, and then people started yield to maturity. Yield is just a much more useful thing to use in thinking about comparing different bonds with each other; implied volatility is a more useful way of comparing stock options to each other.

There’s nothing magical. It doesn’t tell you what to do, but it’s just a more useful lens. These models are a useful way of decomposing things into more intuitive quantities that we can use in our decision making. I think that math in finance is useful, for sure. There’s no doubt about that, but when we start to try to optimise things too much using math, trying to become too optimal and following narrow mathematical rigour too far is extremely dangerous. You come up with a whole portfolio of different investments and you look at an optimization of that and it tells you to do things that common-sense would tell you probably don’t make sense to do.

Taken to an extreme, I think that mathematical models can lead us to dangerous places sometimes. That’s a great question. I wish I had more time to think about it and give you a better answer to it.

**Joe**: That’s a great answer, and Victor Haghani of Elm Funds, really appreciate you coming on. Fascinating conversation. Looking forward to reading and learning more about some of these concepts, and I think our listeners will have learned a lot from this.

**Tracy**: Thank you.

**Victor**: Thank you very much. It was a pleasure.