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Since I don't see anyone else mentioning this: The geometric mean (6.9) is all that really matters for investors, not the arithmetic mean (8.4) - the arithmeti

Since I don't see anyone else mentioning this: The geometric mean (6.9) is all that really matters for investors, not the arithmetic mean (8.4) - the arithmetic mean under-weights the importance of negative years to long term performance. For example, if the market is down 20% one year and up 20% the next year, the arithmetic mean will be 0%, but you'll be down 4% (0.8*1.2 = 0.96), which is reflected in the geometric mean of (about) -2%.

danuker 4y ago
What really really matters is the Kelly criterion, or expected logarithm of wealth. If you expect returns to be similar to the past, that would be mean(log(1+return) for every year).
- Galanwe 4y ago
  Investors tend to think in terms of 1) volatility 2) exposure/diversification. 1) What _really really_ really matters is the Sharpe Ratio, as in "how much returns you get per unit of volatility". The returns themselves are meaningless if not compared to the volatility to earn them. Also, you want to discount the risk free rate (at least), as your benchmark. 2) The market as a whole is the biggest exposure you can have, you'd want to discount it as being X% of your portfolio
  - danuker 4y ago
    The Sharpe ratio gives you a good score if you lose everything (-100%) and then you gain a 20% return for 10 years. But once you lost everything, there is no capital to invest, so the score should be infinitely bad. Arithmetic averages are dangerous in geometric environments. Use the expected log-return or the geometric mean instead of the arithmetic one in Sharpe's formula. But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility.
    - geysersam 4y ago
      > But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility. Unless you are risk adverse. Which it's probably rational to be.
      - FooBarBizBazz 4y ago
        That's the thing. The Sharpe Ratio looks at a catastrophic situation and says it's ok. It's not appropriately scoring risk! Let's say the risk-free rate of return is 3%. Asset 1: Every year, with 99% probability you get 8% return, and with 1% probability you get -100% return, i.e., you lose everything. This has an expected return of 7%, which is 4% above risk-free; the standard deviation is 0.1; and the Sharpe Ratio is 0.36. But the exponential of the mean log annual multiplier is zero; you will eventually lose everything. Asset 2: With 90% probability you get the risk-free rate of 3%, and with 10% probability, you get a 10,000% return (multiply balance by 101). Yes, this has a good average return of 1,000%, but it also has a giant standard deviation of 30, so its Sharpe Ratio is slightly worse, at 0.33. But, the exponential of the mean log multiplier is 1.62, which means that over time it will have a 62% annual return. Moreover, it literally never goes down; there's no risk. Asset 3: You just take the "risk free rate of return" at 3%. Surely, the best choice is Asset 2. It's literally Asset 3 plus free lottery tickets. But it has the worst Sharpe Ratio of the three. And Asset 1, which has the flavor of some prudent tradeoff, is actually guaranteed to bankrupt you eventually.
        geysersam 4y ago
        > But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility. This is what I'm questioning. We do need to further penalize volatility, if that is our preference. The criteria is optimal in the sense of greatest expected return, in the limit of infinite number of bets. But we don't make infinite numbers of bets, and the variance matters. Any truly optimal strategy has to factor in subjective preferences. Example: We play a game where you are ill and need to pay for medical treatment. At the beginning of the game you obtain a sum of money exactly enough to pay for the treatment. Then you are allowed to place (a finite number of) bets in some gambling, possibly increasing your payoff, or losing part of it. I'd argue that in this scenario the "optimal" strategy is not playing, no matter what criteria is used to select the size of the bets.
- dan-robertson 4y ago
  1. Im not sure that’s what the Kelly criterion is but I didn’t look it up. 2. Arithmetic mean of log returns is the same as the geometric mean of returns. Indeed it’s pretty typical to work with log returns for this reason as adding is easier/better for computers than multiplying. This equivalence is easy to prove: gm(returns) = prod(returns)^(1/N) log(gm(returns)) = 1/N * log(prod(returns)) = 1/N * sum(log(returns)) = mean(log(returns)) gm(returns) = exp(mean(log(returns))) Where returns is a list of the multipliers to go from the values before/after the returns, eg it has 1.01 not 1%.
  - danuker 4y ago
    This is beautiful. I now realize both go to 0 if you lose all money (i.e. any one of "returns" as you define them is 0). Thank you!
  - jasonfarnon 4y ago
    you mean (as your proof shows) "is the same as the [log of] geometric mean of returns."
    - dan-robertson 4y ago
      Yeah, I should have been more clear. The point is that you can convert between them without needing any other information (like the original values that were averaged)
Retric 4y ago
Depends, dollar cost averaging shifts things around. For a typical 401k style investor having down years mid career improves returns at retirement, but then increases risks in retirement.
- zeckalpha 4y ago
  With DCA you have the additional costs of keeping cash around. Unless you mean serial lump sum (investing when you get paid).
  - nonethewiser 4y ago
    That's typically why people DCA. Besides, isnt the opportunity cost is completely independent of the return you're getting from the sp500?
    - zeckalpha 4y ago
      Serial lump sum is not quite DCA. Both involve a series of purchases. The opportunity cost is the inverse of the S&P500 in that case.
- YPCrumble 4y ago
  How would that increase risks in retirement?
  - cj 4y ago
    It shouldn't if you transition to heavier weighting of cash/bonds as you approach retirement (which most people do and most financial planners advise)
  - Retric 4y ago
    When spending down money you get the reverse of cost dollar averaging. In a good year you might sell say 1,000 shares but in a down year you might need to sell twice that to take out the same money. This means more of your shares are sold in down years than good years. This is why people say to increase the bond ratio in retirement, but that also reduces expected returns.
deleted 4y ago
[deleted]
getToTheChopin 4y ago
Agreed. Using an assumption of 5-6% annual real total returns is more reasonable for financial planning.
- xapata 4y ago
  I use a more modest 3.5% real return estimate. I'd rather wind up accidentally rich than accidentally poor.
  - Ntrails 4y ago
    I would describe 3.5% real as pretty reasonable, I would not even call it overly conservative
    - getToTheChopin 4y ago
      The geometric average return of the market is 6.9%, factoring in the re-investment of dividends and inflation (i.e., the real total return). Based on this, I'd consider a 3.5% return assumption over 20+ years to be conservative.
      - Ntrails 4y ago
        First, please repeat the standard mantra: past performance is no guarantee of future success. Then tell me the 95th percentile and the median geometric returns based of fixed periods (say, copy the 20 years.) Let us also grab what an inflation linked gov bond would have given over those same periods. Classically I would always think of pension returns as vs the risk free rate (heh, us gov credit risk) which is essentially an IL bond. Then repeat the analysis on, say, the G8 or G20 countries. Oh, and lets do a variety of stock indexes as well. I am a great believer in diversification - so betting on the US is not my standard behaviour. 6.9% assumed return is mad for any individual. It would be mad for a DB scheme _and they at least have some risk pooling in their favour_. But. I am hella risk averse and see the world through that lens.
        4y ago
fallingfrog 4y ago
Correct, because we're averaging together things that are multiplied, not things that are added. Arithmetic mean is rather meaningless here.
- getToTheChopin 4y ago
  The arithmetic mean gives you a sense of the return you can expect by investing in the market for a single year. When investing over multi-year periods, the geometric average is more relevant. You can see the impact on this chart, where the average return (and volatility) drops over longer time periods: https://themeasureofaplan.com/wp-content/uploads/2023/01/Rolling-Returns-1074.png https://themeasureofaplan.com/wp-content/uploads/2023/01/Rol...
chazeon 4y ago
Interesting, I am looking at (1+x) * (1-x) = 1 - x^2 < 1

danuker 4y ago

What really really matters is the Kelly criterion, or expected logarithm of wealth. If you expect returns to be similar to the past, that would be mean(log(1+return) for every year).

Galanwe 4y ago
Investors tend to think in terms of 1) volatility 2) exposure/diversification. 1) What _really really_ really matters is the Sharpe Ratio, as in "how much returns you get per unit of volatility". The returns themselves are meaningless if not compared to the volatility to earn them. Also, you want to discount the risk free rate (at least), as your benchmark. 2) The market as a whole is the biggest exposure you can have, you'd want to discount it as being X% of your portfolio
- danuker 4y ago
  The Sharpe ratio gives you a good score if you lose everything (-100%) and then you gain a 20% return for 10 years. But once you lost everything, there is no capital to invest, so the score should be infinitely bad. Arithmetic averages are dangerous in geometric environments. Use the expected log-return or the geometric mean instead of the arithmetic one in Sharpe's formula. But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility.
  - geysersam 4y ago
    > But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility. Unless you are risk adverse. Which it's probably rational to be.
    - FooBarBizBazz 4y ago
      That's the thing. The Sharpe Ratio looks at a catastrophic situation and says it's ok. It's not appropriately scoring risk! Let's say the risk-free rate of return is 3%. Asset 1: Every year, with 99% probability you get 8% return, and with 1% probability you get -100% return, i.e., you lose everything. This has an expected return of 7%, which is 4% above risk-free; the standard deviation is 0.1; and the Sharpe Ratio is 0.36. But the exponential of the mean log annual multiplier is zero; you will eventually lose everything. Asset 2: With 90% probability you get the risk-free rate of 3%, and with 10% probability, you get a 10,000% return (multiply balance by 101). Yes, this has a good average return of 1,000%, but it also has a giant standard deviation of 30, so its Sharpe Ratio is slightly worse, at 0.33. But, the exponential of the mean log multiplier is 1.62, which means that over time it will have a 62% annual return. Moreover, it literally never goes down; there's no risk. Asset 3: You just take the "risk free rate of return" at 3%. Surely, the best choice is Asset 2. It's literally Asset 3 plus free lottery tickets. But it has the worst Sharpe Ratio of the three. And Asset 1, which has the flavor of some prudent tradeoff, is actually guaranteed to bankrupt you eventually.
      - geysersam 4y ago
        > But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility. This is what I'm questioning. We do need to further penalize volatility, if that is our preference. The criteria is optimal in the sense of greatest expected return, in the limit of infinite number of bets. But we don't make infinite numbers of bets, and the variance matters. Any truly optimal strategy has to factor in subjective preferences. Example: We play a game where you are ill and need to pay for medical treatment. At the beginning of the game you obtain a sum of money exactly enough to pay for the treatment. Then you are allowed to place (a finite number of) bets in some gambling, possibly increasing your payoff, or losing part of it. I'd argue that in this scenario the "optimal" strategy is not playing, no matter what criteria is used to select the size of the bets.
dan-robertson 4y ago
1. Im not sure that’s what the Kelly criterion is but I didn’t look it up. 2. Arithmetic mean of log returns is the same as the geometric mean of returns. Indeed it’s pretty typical to work with log returns for this reason as adding is easier/better for computers than multiplying. This equivalence is easy to prove: gm(returns) = prod(returns)^(1/N) log(gm(returns)) = 1/N * log(prod(returns)) = 1/N * sum(log(returns)) = mean(log(returns)) gm(returns) = exp(mean(log(returns))) Where returns is a list of the multipliers to go from the values before/after the returns, eg it has 1.01 not 1%.
- danuker 4y ago
  This is beautiful. I now realize both go to 0 if you lose all money (i.e. any one of "returns" as you define them is 0). Thank you!
- jasonfarnon 4y ago
  you mean (as your proof shows) "is the same as the [log of] geometric mean of returns."
  - dan-robertson 4y ago
    Yeah, I should have been more clear. The point is that you can convert between them without needing any other information (like the original values that were averaged)

Retric 4y ago

Depends, dollar cost averaging shifts things around. For a typical 401k style investor having down years mid career improves returns at retirement, but then increases risks in retirement.

zeckalpha 4y ago
With DCA you have the additional costs of keeping cash around. Unless you mean serial lump sum (investing when you get paid).
- nonethewiser 4y ago
  That's typically why people DCA. Besides, isnt the opportunity cost is completely independent of the return you're getting from the sp500?
  - zeckalpha 4y ago
    Serial lump sum is not quite DCA. Both involve a series of purchases. The opportunity cost is the inverse of the S&P500 in that case.
YPCrumble 4y ago
How would that increase risks in retirement?
- cj 4y ago
  It shouldn't if you transition to heavier weighting of cash/bonds as you approach retirement (which most people do and most financial planners advise)
- Retric 4y ago
  When spending down money you get the reverse of cost dollar averaging. In a good year you might sell say 1,000 shares but in a down year you might need to sell twice that to take out the same money. This means more of your shares are sold in down years than good years. This is why people say to increase the bond ratio in retirement, but that also reduces expected returns.

deleted 4y ago

[deleted]

getToTheChopin 4y ago

Agreed. Using an assumption of 5-6% annual real total returns is more reasonable for financial planning.

xapata 4y ago
I use a more modest 3.5% real return estimate. I'd rather wind up accidentally rich than accidentally poor.
- Ntrails 4y ago
  I would describe 3.5% real as pretty reasonable, I would not even call it overly conservative
  - getToTheChopin 4y ago
    The geometric average return of the market is 6.9%, factoring in the re-investment of dividends and inflation (i.e., the real total return). Based on this, I'd consider a 3.5% return assumption over 20+ years to be conservative.
    - Ntrails 4y ago
      First, please repeat the standard mantra: past performance is no guarantee of future success. Then tell me the 95th percentile and the median geometric returns based of fixed periods (say, copy the 20 years.) Let us also grab what an inflation linked gov bond would have given over those same periods. Classically I would always think of pension returns as vs the risk free rate (heh, us gov credit risk) which is essentially an IL bond. Then repeat the analysis on, say, the G8 or G20 countries. Oh, and lets do a variety of stock indexes as well. I am a great believer in diversification - so betting on the US is not my standard behaviour. 6.9% assumed return is mad for any individual. It would be mad for a DB scheme _and they at least have some risk pooling in their favour_. But. I am hella risk averse and see the world through that lens.
      - 4y ago

fallingfrog 4y ago

Correct, because we're averaging together things that are multiplied, not things that are added. Arithmetic mean is rather meaningless here.

getToTheChopin 4y ago
The arithmetic mean gives you a sense of the return you can expect by investing in the market for a single year. When investing over multi-year periods, the geometric average is more relevant. You can see the impact on this chart, where the average return (and volatility) drops over longer time periods: https://themeasureofaplan.com/wp-content/uploads/2023/01/Rolling-Returns-1074.png https://themeasureofaplan.com/wp-content/uploads/2023/01/Rol...

chazeon 4y ago

Interesting, I am looking at (1+x) * (1-x) = 1 - x^2 < 1