"But if you take Samuelson’s argument seriously, then Vanguard’s advice is… wrong?"
No, the key difference is that the results of the coin flips are independent of each other, while the performance of stocks show high degrees of temporal correlation. If there is a recession right before your retirement, it doesn't matter if you hold 100 different stocks (or an index fund) - it's going to severely derail your retirement plans.
That being said, I agree with the rest of the essay - you should take the bet for one coin flip if you would take it for 100 coin flips given the positive expected value, unless if $100 represents a huge fraction of your overall wealth such that the diminishing marginal returns to wealth result in negative expected utility for the bet.
"performance of stocks show high degrees of temporal correlation"
is not really correct, though. If stock returns were highly correlated, you could make lots of money very easily. So, the i.i.d. assumption for stock returns is a pretty good first approximation.
My understanding is that the argument for reallocating one's capital to a higher share of bonds over time is that this is essentially done to mitigate the downside risk of retiring during a recession, a time during which one's entire portfolio of stocks is likely to see a substantial reduction in value. So you are essentially "locking in" a larger and larger share of your capital gains as you approach retirement age, at the expense of reducing your upside potential from exposure to further market gains.
The degree to which individual stocks are correlated during a bull market is a bit orthogonal to this logic, which is driven by the strong desire to retire at a very particular point in time, which in turn is dictated by a combination of social convention and tax/retirement policy. That being said, I think that a lot of people probably rebalance their portfolio away from stocks too early.
"But if you take Samuelson’s argument seriously, then Vanguard’s advice is… wrong?"
No, the key difference is that the results of the coin flips are independent of each other, while the performance of stocks show high degrees of temporal correlation. If there is a recession right before your retirement, it doesn't matter if you hold 100 different stocks (or an index fund) - it's going to severely derail your retirement plans.
That being said, I agree with the rest of the essay - you should take the bet for one coin flip if you would take it for 100 coin flips given the positive expected value, unless if $100 represents a huge fraction of your overall wealth such that the diminishing marginal returns to wealth result in negative expected utility for the bet.
Thanks for your comment! This part:
"performance of stocks show high degrees of temporal correlation"
is not really correct, though. If stock returns were highly correlated, you could make lots of money very easily. So, the i.i.d. assumption for stock returns is a pretty good first approximation.
It is true, though, that there is some predictability (e.g., mean reversals), but it's not so obvious if that this changes the logic of the argument. See, e.g., this paper: https://www.tandfonline.com/doi/abs/10.2469/faj.v51.n3.1901
My understanding is that the argument for reallocating one's capital to a higher share of bonds over time is that this is essentially done to mitigate the downside risk of retiring during a recession, a time during which one's entire portfolio of stocks is likely to see a substantial reduction in value. So you are essentially "locking in" a larger and larger share of your capital gains as you approach retirement age, at the expense of reducing your upside potential from exposure to further market gains.
The degree to which individual stocks are correlated during a bull market is a bit orthogonal to this logic, which is driven by the strong desire to retire at a very particular point in time, which in turn is dictated by a combination of social convention and tax/retirement policy. That being said, I think that a lot of people probably rebalance their portfolio away from stocks too early.
>In his paper, Samuelson proved that his MIT colleague was irrational.
ROTFL. Samuelsons math does not take into account diminishing marginal utility. Nor actuarials science: https://en.wikipedia.org/wiki/Ruin_theory