3 ms·

How does this work? By binomial theorem, the expectation at time t is ((1.5 + 0.6)^n)/2^n) = 1.05^n so the "annualized" return per unit time is 1.05, exactly t

by gowld 7y ago

How does this work? By binomial theorem, the expectation at time t is ((1.5 + 0.6)^n)/2^n) = 1.05^n so the "annualized" return per unit time is 1.05, exactly the same as the expectation. What is "time average growth rate"? Web search turns up lots of references do it by Peters, but no definitions.

yellowstuff 7y ago
You have the wrong model. You're betting a different amount each time. Say you start with $1 and get a heads then a tails: ($1 * 1.5) * .6 = $.90 What if you get a tails then a heads? ($1 * .6) * 1.5 = $.90 Doesn't seem like such a great game now, does it? The Kelly Criterion sets a limit for how much of your total wealth you should bet when the odds favor you. If you bet more than that limit you increase your odds of losing everything without improving your expected return. So imagine a better game where you could bet any amount, and you still got paid $5 for every $4 you risked and had a 50% chance of winning. The correct amount to bet is 10% of your bankroll. If you bet more than that in the long run you will go broke.
- kgwgk 7y ago
  > If you bet more than that in the long run you will go broke. The optimal bet is not the same as the critical bet separating the positive growth and negative growth regions.
  - yellowstuff 7y ago
    You're correct. Betting more than double the Kelly amount results in negative expected growth: https://wizardofodds.com/gambling/kelly-criterion/ https://wizardofodds.com/gambling/kelly-criterion/
- gowld 7y ago
  You are ignoring half of the possible outcomes. If you get HH or TT, that's $2.25 and 0.36, which averages to $1.25. 50% of ($1.25 + $.90) for a $1 bet seems like a great game to play.
  - zawerf 7y ago
    After many rounds, you expect to see half heads and half tails. In that case your outcome at n rounds would be around 1.5^n/2 * 0.6^n/2 = 0.9^n/2 which trends towards zero. You would need around 26% more heads than tails just break even (0.6*(1.5^1.26)=1). But even if you were lucky at first, you will always "regress toward the means" in terms of head to tails ratio if you kept playing.
  - yellowstuff 7y ago
    This is actually an awesome illustration of why a lot of famous results in academic finance were wrong in the 70s and 80s. Returns don't average, they multiply. If you get HHTT or TTHH you're in terrible shape: 2.25 * .36 = .36 * 2.25 = .81 You started with $1 and now have 81 cents. I recently learned that a lot of early computer-era academic finance actually made the same error of averaging returns, as described in Finding Alpha: http://falkenblog.blogspot.com/2016/08/finding-alpha-pdf.html http://falkenblog.blogspot.com/2016/08/finding-alpha-pdf.htm...
deleted 7y ago
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mjollnir 7y ago
This is assuming additive growth, not multiplicative growth. The expectation is relevant when you have additive growth, but multiplicative growth is the relevant model here because you are increasing by 50% of your wealth vs. decreasing by 40% of your wealth (not adding fixed amounts, e.g., +$0.50 vs. -$0.40). Multiplicative growth also returns 0.95 per unit time, whereas the (irrelevant) expectation value is 1.05 (hence the violation of most people's intuition). Here is some R code demonstrating it: set.seed(123) # simulating multiplicative growth n <- 1e4 init_wealth <- 1 x <- sample(c(0.6, 1.5), n, replace=TRUE) init_wealth * prod(x) # analytic model n1 <- length(x[x==0.6])/n n2 <- length(x[x==1.5])/n rate <- (0.6^n1) * (1.5^n2) init_wealth * (rate^n) (Edit: formatting, and wording for clarity)

by gowld 7y ago

yellowstuff 7y ago

You have the wrong model. You're betting a different amount each time. Say you start with $1 and get a heads then a tails: ($1 * 1.5) * .6 = $.90 What if you get a tails then a heads? ($1 * .6) * 1.5 = $.90 Doesn't seem like such a great game now, does it? The Kelly Criterion sets a limit for how much of your total wealth you should bet when the odds favor you. If you bet more than that limit you increase your odds of losing everything without improving your expected return. So imagine a better game where you could bet any amount, and you still got paid $5 for every $4 you risked and had a 50% chance of winning. The correct amount to bet is 10% of your bankroll. If you bet more than that in the long run you will go broke.

kgwgk 7y ago
> If you bet more than that in the long run you will go broke. The optimal bet is not the same as the critical bet separating the positive growth and negative growth regions.
- yellowstuff 7y ago
  You're correct. Betting more than double the Kelly amount results in negative expected growth: https://wizardofodds.com/gambling/kelly-criterion/ https://wizardofodds.com/gambling/kelly-criterion/
gowld 7y ago
You are ignoring half of the possible outcomes. If you get HH or TT, that's $2.25 and 0.36, which averages to $1.25. 50% of ($1.25 + $.90) for a $1 bet seems like a great game to play.
- zawerf 7y ago
  After many rounds, you expect to see half heads and half tails. In that case your outcome at n rounds would be around 1.5^n/2 * 0.6^n/2 = 0.9^n/2 which trends towards zero. You would need around 26% more heads than tails just break even (0.6*(1.5^1.26)=1). But even if you were lucky at first, you will always "regress toward the means" in terms of head to tails ratio if you kept playing.
- yellowstuff 7y ago
  This is actually an awesome illustration of why a lot of famous results in academic finance were wrong in the 70s and 80s. Returns don't average, they multiply. If you get HHTT or TTHH you're in terrible shape: 2.25 * .36 = .36 * 2.25 = .81 You started with $1 and now have 81 cents. I recently learned that a lot of early computer-era academic finance actually made the same error of averaging returns, as described in Finding Alpha: http://falkenblog.blogspot.com/2016/08/finding-alpha-pdf.html http://falkenblog.blogspot.com/2016/08/finding-alpha-pdf.htm...

deleted 7y ago

[deleted]

mjollnir 7y ago

This is assuming additive growth, not multiplicative growth. The expectation is relevant when you have additive growth, but multiplicative growth is the relevant model here because you are increasing by 50% of your wealth vs. decreasing by 40% of your wealth (not adding fixed amounts, e.g., +$0.50 vs. -$0.40). Multiplicative growth also returns 0.95 per unit time, whereas the (irrelevant) expectation value is 1.05 (hence the violation of most people's intuition). Here is some R code demonstrating it: set.seed(123) # simulating multiplicative growth n <- 1e4 init_wealth <- 1 x <- sample(c(0.6, 1.5), n, replace=TRUE) init_wealth * prod(x) # analytic model n1 <- length(x[x==0.6])/n n2 <- length(x[x==1.5])/n rate <- (0.6^n1) * (1.5^n2) init_wealth * (rate^n) (Edit: formatting, and wording for clarity)