In the old year, I wrote a piece about ARKK, Bitcoin and co. and showed how dangerous it is for long-term investment success to experience deep drawdowns like the ones we saw since early 2022. There, I said that the long-term result of investments doesn’t depend on returns but on the ratio between returns and volatility. Apparently, there were some readers who were surprised by this, so with apologies to all the people who know this by heart, let me spend today discussing the difference between arithmetic and geometric returns and why understanding this relationship is crucial for long-term investment success.
Very thought provoking. However, .157-(.5*.37^2) = 8.9%, not 8.4%. Have I misunderstood?
Also, to play devil's advocate, the conclusion doesn't seem to follow for a randomly selected stock, although it does follow for the average stock in the index. This is because the volatility in the equation is directionally neutral. This neutrality smuggles in the assumption that there are no stocks above the capital market line.
Thank you, Joachim. A question on a closely related topic: once you start thinking about geometric returns, you soon come across the Kelly criterion. If I remember correclty, the Kelly ratio is equal to the arithmetic mean return (minus the risk-free rate) divided by the variance of the returns. One problem is that, not knowing either the returns or their variance in advance, you have to play safe by investing only, say, half Kelly. In his book „Safe Haven“, Mark Spitznagel argues that both diversification and Kelly-betting cost too much in terms of returns. Instead, he promotes the idea of insurance, which I interpret as options. Spitznagel shows, by way of some examples, that buying insurance improves your geometric returns even if you pay a premium that is „too high“. I lack the mathematical confidence to support or contradict Spitznagel‘s arguments, and I wonder if you have an opinion.
Totally. Fortunately, I *think* most professional managers understand this; it would be unusual to characterize multi-period returns by the average rather than the geo for exactly the reason you state. An initial $100 that finishes at $100 is 0%, from the perspective that matters! At least that's what we teach in CFA & FRM. The decision is usually between so-called time-weighted (i.e., geometric) and dollar-weighted.
Nonsense. This article would be useful if anyone actually confused arithmetic and geometric returns. No one does. No investment calculator, no retirement site, no RIA, no CFP, no investment book, etc.
I don't think volatility is the issue. If you buy a stock at $10 and sell it at $20, you double your money. The path it takes to get there, whether that be gradual and linear, or volatile and erratic, makes absolutely no difference, if the total time is the same. It's the two end points that matter, averaging the points along the way is a pointless exercise.
The fact that you buy a stock at $10, it drops to $5, then goes back to $10, and that apparently gives you an average 25% return, just means that averages are meaningless when applied to the situation.
Another example. A King has a billion dollars, he has 1000 subjects who have $1 each. Apparently the kingdom has an average wealth per capita of $1,000,001. Nice kingdom, on average, everybody is a millionaire, right? Mathematically true, but unhelpful in describing the actual situation.
An eye-opener, at least for me, who has been investing in a highly diversified „world“ portfolio for over a decade. I find it astonishing that the promoters of such world portfolios and of index-tracking ETFs do not make use of this simple but powerful explanation. I read e.g. most books by your compatriot Gerd Kommer but do not remember him mentioning that at all.
Vastly presumptuous argument here. As others have said, volatility is not a one way street. Bitcoin is known to crash and recover, it’s also known to surge and then correct. The premise that people are “confused” about arithmetic vs. geometric returns is wildly unsupported. If I buy bitcoin at 40000 and it falls to 30000, I am not going to sell it. However, I don’t dread the fact that I now need a 33.3% return to recover from the 25% loss to get back to breakeven price; instead, I am confident enough in my investment to trust that the volatility and market correction will take care of that naturally. Crypto’s volatility is no more a curse than it is a blessing. Just because your argument is rooted in a “volatility bad” basis, it’s no surprise that you would end up skeptical of cryptocurrency. And…FYI, the current BEAR case for BTC by 2030 is annualized 30% returns…so if that sounds ridiculous to you, know that that is the opinion held by those LEAST confident in the coin…
A great explanation of a really important & often poorly understood issue
Very thought provoking. However, .157-(.5*.37^2) = 8.9%, not 8.4%. Have I misunderstood?
Also, to play devil's advocate, the conclusion doesn't seem to follow for a randomly selected stock, although it does follow for the average stock in the index. This is because the volatility in the equation is directionally neutral. This neutrality smuggles in the assumption that there are no stocks above the capital market line.
Thank you, Joachim. A question on a closely related topic: once you start thinking about geometric returns, you soon come across the Kelly criterion. If I remember correclty, the Kelly ratio is equal to the arithmetic mean return (minus the risk-free rate) divided by the variance of the returns. One problem is that, not knowing either the returns or their variance in advance, you have to play safe by investing only, say, half Kelly. In his book „Safe Haven“, Mark Spitznagel argues that both diversification and Kelly-betting cost too much in terms of returns. Instead, he promotes the idea of insurance, which I interpret as options. Spitznagel shows, by way of some examples, that buying insurance improves your geometric returns even if you pay a premium that is „too high“. I lack the mathematical confidence to support or contradict Spitznagel‘s arguments, and I wonder if you have an opinion.
Excellent post. You concisely relate the key relationship between risk and reward / returns and how this impacts overall profitability. Cheers.
Hi Klement,
Great article! Not to detract from your explanation and conclusions, which I agree with, but the formula you give is only an approximation.
I give the full formula in the appendix of this paper: https://jii.pm-research.com/content/10/3/58.
I can't post the article but can email folks a copy on request.
Totally. Fortunately, I *think* most professional managers understand this; it would be unusual to characterize multi-period returns by the average rather than the geo for exactly the reason you state. An initial $100 that finishes at $100 is 0%, from the perspective that matters! At least that's what we teach in CFA & FRM. The decision is usually between so-called time-weighted (i.e., geometric) and dollar-weighted.
I am skeptical of this argument. Stocks have high vol when they have fallen a lot. If you are buying bitcoin you expect the vol to fall.
Nonsense. This article would be useful if anyone actually confused arithmetic and geometric returns. No one does. No investment calculator, no retirement site, no RIA, no CFP, no investment book, etc.
This assumes volatility of that security (bitcoin in this place) remains constant and high
I don't think volatility is the issue. If you buy a stock at $10 and sell it at $20, you double your money. The path it takes to get there, whether that be gradual and linear, or volatile and erratic, makes absolutely no difference, if the total time is the same. It's the two end points that matter, averaging the points along the way is a pointless exercise.
The fact that you buy a stock at $10, it drops to $5, then goes back to $10, and that apparently gives you an average 25% return, just means that averages are meaningless when applied to the situation.
Another example. A King has a billion dollars, he has 1000 subjects who have $1 each. Apparently the kingdom has an average wealth per capita of $1,000,001. Nice kingdom, on average, everybody is a millionaire, right? Mathematically true, but unhelpful in describing the actual situation.
An eye-opener, at least for me, who has been investing in a highly diversified „world“ portfolio for over a decade. I find it astonishing that the promoters of such world portfolios and of index-tracking ETFs do not make use of this simple but powerful explanation. I read e.g. most books by your compatriot Gerd Kommer but do not remember him mentioning that at all.
Vastly presumptuous argument here. As others have said, volatility is not a one way street. Bitcoin is known to crash and recover, it’s also known to surge and then correct. The premise that people are “confused” about arithmetic vs. geometric returns is wildly unsupported. If I buy bitcoin at 40000 and it falls to 30000, I am not going to sell it. However, I don’t dread the fact that I now need a 33.3% return to recover from the 25% loss to get back to breakeven price; instead, I am confident enough in my investment to trust that the volatility and market correction will take care of that naturally. Crypto’s volatility is no more a curse than it is a blessing. Just because your argument is rooted in a “volatility bad” basis, it’s no surprise that you would end up skeptical of cryptocurrency. And…FYI, the current BEAR case for BTC by 2030 is annualized 30% returns…so if that sounds ridiculous to you, know that that is the opinion held by those LEAST confident in the coin…