You're invited to a one-on-one coin-flip gambling game. Your opponent has 1 million USD on hand (the max you can bet is 1 million USD). The payouts for flipping heads and tails are as follows:

• Tails: You win 2x your bet.

This game is a one-off gamble; no second chances. How much would you bet?

(This was from a trading interview so it likely involves choosing the right risk-reward profile. Is there a right answer? Or is this a subjective question?)

• sounds like the heads and tails are analogy for limits and stops (with a risk reward ratio of 1:3). "Don't risk more than you can afford to lose" - a rule of thumb is 2% of equity. – xiaomy Sep 21 '17 at 14:05
• I think it's just meant to see how you think and there's no right answer. If I asked that and someone said "use the Kelly criterion cause it maximizes long term growth" I'd receive that much better than, say, "you have an edge; bet it all" but I'd ask followup questions to see if they understood the limitations of the approach and what circumstances would suit it best/worst. I'd be more impressed if the person asked me some questions that showed they understood what issues are pertinent to a risk management decision before answering. – spaceisdarkgreen Sep 21 '17 at 16:26

The Kelly criterion gives the fraction, $f$, of the current bankroll to bet in order to maximize the longterm growth. The criterion is given by $$f = \frac{bp-q}{b},$$ where $b$ is the winnings received on \$1 bet,$p$is the probability of winning, and$q=1-p$is the probability of losing the bet of \$1.
In your case $b=2$, $p=q=0.5$ so the optimal fraction to bet is $$f = \frac{2\cdot0.5-0.5}{2} = 0.25.$$ That is 25% of your bankroll or \$250k. • I guess it's a good answer. The problem for me is that for most "trading" b and p are unknowns, or are interest rates / coupon rates where b = rate and p = 100% less credit risk. So other than credit defaults, trading isn't much to do with probability, it's all about rates, spreads, skews and competition with other traders. – rupweb Sep 25 '17 at 10:46 • @SMeaden Problem sez " you will get 2 + your bet" so$b=2$– noob2 Oct 18 '17 at 15:26 An alternative approach is to size your bet to maximize your expected utility, which is assumed to be given by a function$u(w)$of your total wealth$w$. This could be a better approach than using the Kelly criterion, because the Kelly fraction gives the amount to bet if you want to maximize your long-term growth rate, assuming that you will bet a large number of times, but in this case you are told that you only get one chance to bet. If you bet a fraction$x$of your bankroll, you will have$1+2x$if you win and$1-x$if you lose, so your expected utility is $$\tfrac{1}{2}u(1 + 2x) + \tfrac{1}{2}u(1 - x)$$ Maximizing this is equivalent to maximizing$u(1+2x) + u(1-x)$. In the special case of log utility$u(w)=\log w$you require $$\frac{d}{dx} \left( \log(1+2x) + \log(1-x) \right) = \frac{2}{1+2x} - \frac{1}{1-x} = 0$$ which you can solve to give$x = 1/4$, the same answer as if you used Kelly betting to maximize your long-term growth. Other utility functions will give different results. • As your answers basically says, maximizing expected utility with$\log$utility over terminal wealth is mathematically equivalent to maximizing the expected growth rate (as is done in Kelly criterion betting). Let$\frac{V_t}{V_0} = e^{Rt}$hence$\log V_t - \log V_0 = R t$. Maximizing$E[\log V_t]$or$E[R]\$ are equivalent objectives. Log utility is also a special case of a broader set of constant relative risk aversion utility functions. – Matthew Gunn Sep 26 '17 at 16:36