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  1. 065A stock rises 50% or falls 40% each year with equal probability. Its expected return is positive, but what happens to a buy-and-hold investor over time? And what fraction of wealth should sit in the stock if the rest is held in cash and the mix is rebalanced every year?Betting and sizingHardMulti-manager platformsProp and quant trading firms

    Try it first

    Over many years, what happens to the typical buy-and-hold investor?

    Show the worked solution

    The typical buy-and-hold investor loses about 5.1% a year, yet a 25% stake rebalanced yearly grows about 0.6% a year. The average year returns +5%, but a good year and a bad year multiply wealth by 1.5 x 0.6 = 0.9. With 25% in the stock the two years multiply wealth by 1.125 x 0.9 = 1.0125. That 25% is the Kelly fraction, the stake that maximises the average log return.

    How can a positive average return shrink your wealth?

    Imagine a shop whose sales rise 50% in a good year and fall 40% in a bad one. After one of each it is at 90% of where it began, whatever the order. Wealth compounds by multiplying, so over many years what matters is the typical growth factor, the square root of 1.5 x 0.6, about 0.949, not the average return of +5%. The average is real, but it is carried by rare paths with long lucky streaks. After 20 years the typical investor holds about 0.35 of the starting money while the average across all paths is 2.65 times it.

    The relationship
    g(f)=12ln⁡(1+0.5f)+12ln⁡(1−0.4f),g′(f)=0  ⇒  f∗=0.25g(f) = \tfrac12\ln(1 + 0.5f) + \tfrac12\ln(1 - 0.4f), \qquad g'(f) = 0 \;\Rightarrow\; f^{*} = 0.25
    fthe fraction of wealth held in the stock, the rest in cash
    g(f)the expected log growth per year of the rebalanced mix
    f*the stake that maximises it
    What it says in wordsPick the stake that makes the average log return per year as large as possible; here that is a quarter of your wealth.
    The average path climbs while the typical path shrinks0.51.01.52.02.5yr 0yr 5yr 10yr 15yr 20mean 2.6525% mix 1.13stock 0.35average over all stock pathsall in the stock, typical path25% stock, rebalanced yearly
    Over 20 alternating years the all-stock investor falls to about 0.35 of the starting wealth while the average across all paths climbs to 2.65, and a 25% stake rebalanced every year grows to about 1.13, because growth depends on the log return, not the average return.

    Why does holding less of the stock help?

    Rebalancing to a fixed mix sells after gains and buys after losses, and a smaller stake shrinks the swings. With a fraction f in the stock, a good year multiplies wealth by 1 + 0.5f and a bad year by 1 - 0.4f; typical growth peaks where 0.5/(1 + 0.5f) equals 0.4/(1 - 0.4f), which gives f = 25%. At 25% a pair of years gives 1.125 x 0.9 = 1.0125, about 0.6% a year. The quick check is return over variance: 0.05 divided by 0.45 squared is 0.247.

    What is the limit of this answer?

    The 25% rests on knowing both outcomes and their odds exactly, on cash earning nothing and on free rebalancing. Change any of those and the fraction moves, and because a real edge is only an estimate, desks size well below the full Kelly number. The lesson to lead with in the room is the gap itself: a positive average return is not a positive growth rate, and position size decides which one you earn. That gap is called volatility dragThe shortfall of the compound growth rate below the average return, roughly half the variance of returns..

    Where candidates lose it

    Most candidates answer that the investor earns 5% a year, because that is the average. The interviewer built the numbers so the average and the typical outcome point in opposite directions, and wants to see you notice.

    The second loss is concluding the stock is simply bad and putting nothing in it. Zero earns nothing; the point is that a small, rebalanced stake turns the same gamble into positive growth.

    What the interviewer asks next

    • Cash now earns 3% a year. How does the best stake change?
    • What changes if you rebalance every two years instead of every year?
    • Two such stocks move independently. What happens if you hold half in each and rebalance yearly?