Here is the derivation of the withdrawal rate of a retirement portfolio. The value of the original portfolio is P, the initial withdrawal amount is x, the inflation rate is fi, the real return rate is ri, and the term is n. The withdrawal amount is adjusted according to the inflation rate.
The withdrawal rate is x/P. From the denominator, we can seen the effect of the real return, r1 > r2 > ... > rn.
Withdrawal Rate
3/08/2012
Kelly Betting and Logarithmic Utility
proportional betting factor, l, betting n times, (W,L) probability (p,q).
From the derivation, we can see that there is no need to set n to infinity.
Kelly Betting and Logarithmic Utility
From the derivation, we can see that there is no need to set n to infinity.
Kelly Betting and Logarithmic Utility
3/06/2012
Kelly Criterion and Betting
Kelly Criterion is used as a betting method mentioned in a lot of articles. One of the famous proponents is Ed Thorp, he wrote several articles about how to use Kelly criterion in betting and stock investing.
But does Kelly betting really work or is it just an illusion?
The major problem with Kelly betting is that the volatility is large, so sometimes people would recommend half Kelly. Not many articles discuss about the analysis of the "Risk and Rewards" of Kelly Criterion. Usually they use Monte Carlo simulation to show the results of Kelly betting, half Kelly betting, or 0.25 Kelly betting.
I don't intend to challenge the correctness of Kelly's formula, I just want to use the simple probability concept to show the relationship between fractional betting and Kelly betting.
Let's say, there are two betting outcome W and L with probability (p,q). p + q = 1. We make N bets.
If N=2, the possible outcomes would be (WW,WL,LW,LL) with probability (pp,pq,qp,qq). If the sequence doesn't matter, the outcomes would be (WW, WL, LL) with probability (pp,2pq,qq).
What's the Expected value of the outcomes? It's the payoff of each outcome multiplied by its probability.
What is Kelly Criterion trying to do anyway?
Let's say, N=1,000, p=60%, q=40%. So Kelly Criterion is trying to maximize the payoff of the outcome of (W*600, L*400) and then there comes the factor f for the proportional betting.
But the truth is that there are 1001 different outcomes from W*1000, (W*999, L),..., to (W, L*999) and L*1000. The real expected value of the N=1000 betting with factor f would not be like what Kelly Criterion shows. That's why the result of the Monte Carlo simulation always cannot be explained well by the author. You can easily spot something wrong within the data.
If Wp+Lq is favorable, it's always best to choose f=1 to maximize the expected value of the proportional betting, just the volatility will be very large as well. So the key is to choose the right f to give the appropriate risks and rewards.
But does Kelly betting really work or is it just an illusion?
The major problem with Kelly betting is that the volatility is large, so sometimes people would recommend half Kelly. Not many articles discuss about the analysis of the "Risk and Rewards" of Kelly Criterion. Usually they use Monte Carlo simulation to show the results of Kelly betting, half Kelly betting, or 0.25 Kelly betting.
I don't intend to challenge the correctness of Kelly's formula, I just want to use the simple probability concept to show the relationship between fractional betting and Kelly betting.
Let's say, there are two betting outcome W and L with probability (p,q). p + q = 1. We make N bets.
If N=2, the possible outcomes would be (WW,WL,LW,LL) with probability (pp,pq,qp,qq). If the sequence doesn't matter, the outcomes would be (WW, WL, LL) with probability (pp,2pq,qq).
What's the Expected value of the outcomes? It's the payoff of each outcome multiplied by its probability.
What is Kelly Criterion trying to do anyway?
Let's say, N=1,000, p=60%, q=40%. So Kelly Criterion is trying to maximize the payoff of the outcome of (W*600, L*400) and then there comes the factor f for the proportional betting.
But the truth is that there are 1001 different outcomes from W*1000, (W*999, L),..., to (W, L*999) and L*1000. The real expected value of the N=1000 betting with factor f would not be like what Kelly Criterion shows. That's why the result of the Monte Carlo simulation always cannot be explained well by the author. You can easily spot something wrong within the data.
If Wp+Lq is favorable, it's always best to choose f=1 to maximize the expected value of the proportional betting, just the volatility will be very large as well. So the key is to choose the right f to give the appropriate risks and rewards.
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