Portfolio Management puzzles, solved step by step

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  1. 027A trader is right on 70% of trades. Each winning trade makes 1% and each losing trade loses 3%. What does the average trade return, and what hit rate would the trader need just to break even?Probability and expected valueWarm upHedge fundsAsset management

    Try it first

    Gut call first: is this trader making money?

    Show the worked solution

    The average trade loses 0.2%, and break-even needs a 75% hit rate. Expected value is 0.7 x 1% minus 0.3 x 3%, which is 0.7% minus 0.9%, or minus 0.2%. To break even the winners must pay for the losers: p x 1 = (1 minus p) x 3, so p is 3 over 4, or 75%. The trader is right more often than wrong and still loses money.

    Why does a 70% hit rate not settle the question?

    A shopkeeper who makes a small profit on seven sales out of ten but sells the other three at a big loss can still close the month in the red. Counting the happy sales tells you nothing until you know how big each one was. Expected return is each outcome's probability times its size, added up, so a hit rate only means something next to the payoff ratio. Here the losers are three times the size of the winners, and that ratio is doing all the damage.

    A 70% hit rate still loses when the losers are three times the winners0+7.07 winnersx 1%-9.03 losersx 3%-2.0Net, 10 trades-0.2 a trade+1%-1%-2%50%60%70%80%90%100%Hit rateExpected return per trade70%: -0.2% a tradebreak-even at 75%= 3 / (1 + 3)
    Over ten trades, seven winners of 1% add 7 points and three losers of 3% remove 9 points, so the trader nets minus 2 points, or minus 0.2% a trade. Expected return crosses zero only at a 75% hit rate, five points above what the trader achieves.

    How do you find the break-even hit rate in one line?

    Set expected value to zero and solve. The break-even hit rate is the loss size divided by the sum of the win and loss sizes: 3 over 1 plus 3, which is 75%. Every point of hit rate is worth 0.04% a trade here, because moving one trade in a hundred from loser to winner swings 4 points in total. So the trader is 5 points of hit rate, or 0.2% a trade, short of break-even.

    The relationship
    E=pW−(1−p)Lp∗=LW+L=31+3=75%E = pW - (1-p)L \qquad p^{*} = \frac{L}{W+L} = \frac{3}{1+3} = 75\%
    pthe hit rate, 70%
    Wthe average win, 1%
    Lthe average loss, 3%
    p^{*}the hit rate at which expected return is zero
    What it says in wordsA strategy breaks even when the chance of losing, times the loss, equals the chance of winning, times the win.

    Then say what you would change. The trader can cut losers sooner, let winners run further, or be more selective; lifting the average win to 1.5% with the same losses moves break-even to 3 over 4.5, about 67%. Averages also hide transaction costs, which push break-even higher still.

    Where candidates lose it

    Candidates hear 70% and say the trader is good, or multiply 70% by 1% and forget the losers entirely. The interviewer built the question so the hit rate looks impressive and the payoff ratio quietly wins.

    The second loss is getting minus 0.2% and stopping. The follow-up is always the break-even hit rate or the break-even payoff ratio, so have the one-line formula ready.

    What the interviewer asks next

    • Keep the 70% hit rate. How big must the average win be to break even?
    • A second trader is right 40% of the time, wins 3% and loses 1%. Who would you rather back?
    • How do trading costs of 0.05% a round trip change the break-even hit rate?