QQ 102, assessment

Problem set

Work each one on paper before opening the solution.

Problem 1
A strategy returns +12%, then -8%, then +5%. What is the total return? What are the arithmetic average and the compound (geometric) average per period?
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Total: 1.12 x 0.92 x 1.05 = 1.0819, a gain of 8.19%. Arithmetic average: (12 - 8 + 5) / 3 = 3.00%. Compound average: the cube root of 1.0819, minus 1, which is 2.66%. The compound figure is lower because the returns vary.
Problem 2
An account falls 35%. What gain brings it back?
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0.35 / (1 - 0.35) = 0.538. It needs a gain of about 53.8%.
Problem 3
Daily volatility is 0.8%. Annualise it using 252 trading days. A second series has monthly volatility of 4%. Annualise that too.
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Daily: 0.8 x the square root of 252 = 0.8 x 15.87 = 12.7%. Monthly: 4 x the square root of 12 = 4 x 3.464 = 13.9%.
Problem 4
A fund returned 14% in a year with volatility of 16%. The risk-free rate was 4%. What is its Sharpe ratio? What would a tool that ignores the risk-free rate report?
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(14 - 4) / 16 = 0.625. Ignoring the risk-free rate: 14 / 16 = 0.875. The same fund in the same year reports a number 40% higher.
Problem 5
Equity at the end of each month is 100, 120, 96, 110, 132, 99, 140. What is the maximum drawdown?
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From the peak of 120 down to 96 is 24 / 120 = 20%. From the peak of 132 down to 99 is 33 / 132 = 25%. The maximum drawdown is 25%.
Problem 6
Your account is 50,000 dollars and you risk 0.5% per trade. You buy EUR/USD at 1.1000 with a stop at 1.0975. How many standard lots?
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Amount at risk is 0.5% of 50,000 = 250 dollars. The stop is 25 pips away. One standard lot loses 10 dollars per pip, so 250 dollars per lot over 25 pips. Size is 1 standard lot.
Problem 7
A bet wins 45% of the time and pays 2 to 1. What is the full Kelly fraction, and the half Kelly fraction?
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p - q / b = 0.45 - 0.55 / 2 = 0.45 - 0.275 = 0.175, or 17.5%. Half Kelly is 8.75%.