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In my new favorite book 'the missing billionaires' there's a framework for evaluating the value of a lottery ticket. Naively we'd think that a $1.8B payoff and 1:262M chance of winning means expected value is positive and therefore its a good bet but...
Turns out actual value to you depends on your wealth and your risk utility.
Suppose youre willing to bet aggressively with crra of 1. Suppose you only have $1000. So you can buy 500 $2 tickets. Means your chances of not coming away with 0 is 500/262M, which is pretty much zero. You shouldn't spend the money because the impact on your assets of buying the $2 ticket is much more impactful than the upside x chance of winning.
The math for change in your wealth is from this powerball is:
$-2 + (W-$2)(1/262M)ln(1+($1.8B/(W-$2)))
Yields:
Wealth | Dollar Value of Ticket
$10k | $-1.99954
$100k | $-1.99626
$1M | $-1.96915
$20M | $-1.6556
$100M | $-0.87
$250M | $0.0069
$500M | $0.91
$1B | $1.93
$10B | $4.32
$100B | $4.81
$1T | $4.86
So assuming aggressive crra of 1 the value break even point is when your net worth is above $250M. That is where the insignificance of the $2 is so low that the low probability upside starts to balance out.
FWIW I don't believe crra is a good/correct risk utility function but I think its close enough if you determine it for any individual bet size. In reality nobody is a 1 so above break-even numbers are optimistic.
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