Portfolio Management puzzles, solved step by step

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  1. 029A fund returned 14% with 18% volatility in a year when cash paid 6%. Its benchmark returned 12%, and its tracking error against that benchmark was 4%. What are its Sharpe ratio and its information ratio, and what does each one tell you?Performance measurementWarm upPerformance analysisAsset management

    Try it first

    Which number goes in the denominator of the information ratio?

    Show the worked solution

    The Sharpe ratio is about 0.44 and the information ratio is 0.5. Sharpe divides the excess over cash, 14 minus 6 or 8 points, by total volatility of 18: 0.44. The information ratio divides the excess over the benchmark, 14 minus 12 or 2 points, by the tracking error of 4: 0.5. Sharpe judges the whole portfolio's risk; the information ratio judges only the manager's active bet.

    Why are there two ratios for one fund?

    Picture judging a cook. One question is whether the whole meal was worth its price. Another is whether the chef's changes to the standard recipe made it better. The Sharpe ratio asks whether the fund's total return beat cash by enough to justify all its risk, and the information ratio asks whether the manager's departures from the benchmark earned enough to justify that active risk. An investor choosing an asset mix cares about the first; one who has already chosen the market and is picking a manager cares about the second.

    Two ratios, two questions: reward per unit of which risk?Sharpe ratioIs the whole portfolio worth its risk?Excess over cash14 - 6 = 818Total volatilityRatio8 / 18 = 0.44Information ratioIs the manager's bet worth its risk?Excess over benchmark14 - 12 = 24Tracking errorRatio2 / 4 = 0.50
    The Sharpe ratio divides 8 points of return over cash by 18 points of total volatility and gives 0.44; the information ratio divides 2 points of return over the benchmark by 4 points of tracking error and gives 0.50. Each ratio pairs a reward with the risk that produced it.

    How do you read 0.44 and 0.5 once you have them?

    The tracking errorThe standard deviation of the difference between a fund's return and its benchmark's return, a measure of how far the fund strays. is small against total volatility because most of the fund's ups and downs are the market's, which the benchmark shares. A 0.5 information ratio from a single year is respectable on paper but statistically weak: one year of 2 points against a 4-point tracking error is half of one standard deviation. It would take many years at that rate before anyone could separate skill from luck with confidence. The Sharpe ratio is also best compared with the benchmark's own Sharpe over the same period, not read alone.

    The relationship
    Sharpe=Rp−Rfσp=818=0.44IR=Rp−RbTE=24=0.5\text{Sharpe} = \frac{R_p - R_f}{\sigma_p} = \frac{8}{18} = 0.44 \qquad \text{IR} = \frac{R_p - R_b}{TE} = \frac{2}{4} = 0.5
    R_pthe fund's return, 14%
    R_fthe cash rate, 6%
    \sigma_pthe fund's total volatility, 18%
    R_bthe benchmark return, 12%
    TEthe tracking error, 4%
    What it says in wordsEach ratio is a reward divided by the risk taken to earn it; the two differ in which reward and which risk.

    Say one limitation in the room: both ratios assume returns are roughly normal. A fund that sells insurance-like options can post a smooth, high Sharpe for years and then lose a large amount in one month, which neither ratio sees in advance.

    Where candidates lose it

    The frequent mix-up is putting total volatility under the information ratio, which gives 2 over 18, about 0.11, and makes a reasonable manager look poor. The other is measuring the Sharpe numerator against the benchmark instead of cash.

    Say the pairing aloud before calculating: excess over cash with total risk, excess over benchmark with active risk. Then the numbers take ten seconds.

    What the interviewer asks next

    • The benchmark had 16% volatility. What was its Sharpe ratio, and did the fund beat it on that measure?
    • How many years of a 0.5 information ratio before the excess return is statistically significant?
    • Why can a fund have a higher Sharpe ratio than its benchmark but a negative information ratio?
  2. 052A fund's NAV at six successive year-ends is 100, 130, 91, 120, 84 and 140. What is its maximum drawdown?Performance measurementWarm upPerformance analysisRisk management

    Try it first

    Pick the maximum drawdown before you work it.

    Show the worked solution

    Minus 35.4%, from the peak of 130 to the trough of 84. Track the running peak: 100, then 130, which holds until 140 in the final year. The deepest fall below that peak is 84, and 84 over 130 minus 1 is -35.4%. The earlier dip to 91 was only minus 30%, and the fall from the starting 100 looks like just 16%.

    Measured from where?

    Picture a climber on a ridge. How far she has fallen is measured from the highest point she reached, not from the car park and not from the step she took a moment ago. A drawdown is the fall from the running peak, the highest NAV seen so far, and the peak only resets when the NAV climbs above it. Here the peak jumps to 130 in year 1 and then sits there through the dip to 91, the recovery to 120 and the slide to 84. Every one of those years is inside the same drawdown.

    Drawdown is measured from the running peak, not from the start80100120140Year 0Year 1Year 2Year 3Year 4Year 5100130 peak9112084 trough14084 / 130 - 1= -35.4%dashed green: running peakshaded red: the deepest fall below it
    The running peak holds at 130 from year 1 until the NAV reaches 140 in year 5, and the deepest point under that peak is 84 in year 4, so the maximum drawdown is -35.4%, much deeper than the 16% fall the start value suggests.
    The relationship
    MDD=min⁡t(Vtmax⁡s≤tVs−1)=84130−1=−35.4%\text{MDD} = \min_t \left(\frac{V_t}{\max_{s\le t} V_s} - 1\right) = \frac{84}{130} - 1 = -35.4\%
    V_tthe NAV at year-end t
    max over s up to tthe running peak, the highest NAV seen so far
    What it says in wordsAt each date, compare the NAV with the best level reached so far; the worst of those comparisons is the maximum drawdown.

    Why does the recovery to 120 not end the drawdown?

    Because 120 is still below 130. Anyone who invested at the peak was still under water at 120, and the slide to 84 took them further down. A partial recovery does not reset the peak, so two separate-looking dips can be one long drawdown. Treating 120 as a new start gives a fall of 30% to 84, which is wrong on the question asked, though it is a correct answer to how bad the single year was.

    Say what the number misses as well. Maximum drawdown is one path and one worst point, so it tells you nothing about how long the fund stayed under water, here four years from year 1 to year 5, and a longer history almost always shows a deeper one. Allocators read it next to the recovery time for that reason.

    Where candidates lose it

    The fast answer is 16%, the fall from the starting 100 to 84. It measures the investor who bought at launch, not the worst experience, and interviewers ask this with the path deliberately chosen so the start value misleads.

    The second trap is resetting the peak at 120 and answering 30%. Say the rule out loud before you calculate: the peak only moves when the NAV beats it.

    What the interviewer asks next

    • How long was the fund under water, and why do allocators care about that as much as the depth?
    • What gain was needed from 84 to recover the 130 peak?
    • Why does a longer track record almost always show a deeper maximum drawdown?