Portfolio Management puzzles, solved step by step

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  1. 025An asset has an expected return of 10% and volatility of 12%, and the risk-free rate is 6%. If you lever it 1.5 times, borrowing at the risk-free rate, what are the expected return, the volatility and the Sharpe ratio?Portfolio risk mathsCoreMulti-assetHedge funds

    Try it first

    What happens to the Sharpe ratio when you lever at the risk-free rate?

    Show the worked solution

    Expected return 12%, volatility 18% and the same Sharpe ratio of 0.33. With 1.5 in the asset and minus 0.5 in cash, the return is 6% plus 1.5 times the 4% excess, which is 12%, and the volatility is 1.5 x 12%, which is 18%. The Sharpe ratio is (12 minus 6) / 18, unchanged at 0.33. Leverage at the risk-free rate moves you along the line without changing its slope.

    Why does leverage leave the Sharpe ratio alone?

    Think of a recipe scaled up by half: every ingredient rises by 1.5 times, so the taste, which depends on the proportions, is the same. Borrowing at the risk-free rate multiplies both the excess return and the volatility by the leverage, so their ratio, the Sharpe ratio, does not change. The asset earns 4 points over cash with 12 points of volatility, 0.33 per unit of risk. At 1.5 times it earns 6 points over cash with 18 points of volatility: still 0.33. The borrowed cash itself has no volatility, so it adds none.

    Leverage at the risk-free rate slides you along the line; the slope, the Sharpe ratio, stays put6%8%10%12%14%0%6%12%18%24%VolatilityExpected returnrisk-free 6%asset: 12% vol, 10%1.5x: 18% vol, 12%borrow at 8%: 11%Sharpe, unlevered(10 - 6) / 12 = 0.33Sharpe, levered(12 - 6) / 18 = 0.33Borrowing at 8%(11 - 6) / 18 = 0.28
    The asset at 12% volatility and 10% return and the 1.5 times levered portfolio at 18% and 12% sit on the same line from the 6% risk-free rate, so both have a Sharpe ratio of 0.33, while borrowing at 8% instead drops the levered point to 11% and a Sharpe ratio of 0.28.
    The relationship
    E[rL]=rf+L(E[r]−rf)=6+1.5(4)=12%σL=Lσ=18%S=618=412≈0.33E[r_L]=r_f+L(E[r]-r_f)=6+1.5(4)=12\% \qquad \sigma_L=L\sigma=18\% \qquad S=\frac{6}{18}=\frac{4}{12}\approx 0.33
    Lthe leverage, 1.5 times
    r_fthe risk-free rate, 6%, also the borrowing rate here
    Sthe Sharpe ratio, excess return over volatility
    What it says in wordsLeverage scales the excess return and the volatility by the same factor, so the Sharpe ratio is unchanged.

    When does leverage lower the Sharpe ratio?

    When borrowing costs more than the risk-free rate, which is the normal case for anyone but a government. At a borrowing rate of 8%, the borrowed half costs 2 points more than cash earns, so the levered return falls to 11% and the Sharpe ratio to 0.28. The line bends down to the right of the asset. Say the other limitations: volatility is not the only risk that scales, because a levered portfolio can be forced to sell after a large loss, and the higher volatility drags down compound growth even when the Sharpe ratio is unchanged. This is why a manager with a high Sharpe ratio, low-volatility strategy can lever it, while the same move on a volatile asset is far more dangerous.

    Where candidates lose it

    The common answer is that leverage raises the Sharpe ratio because it raises return, or lowers it because it raises risk. Both miss that the two rise together.

    The arithmetic slip is levering the whole 10% return, answering 15%, instead of levering the 4% excess and paying 6% on the borrowed half. Write return as the risk-free rate plus leverage times the excess.

    What the interviewer asks next

    • What leverage gives an expected return of 14%, and what is its volatility?
    • Why do investors who cannot borrow tend to hold riskier assets instead?
    • How does the answer change if the borrowing rate is 8%?
  2. 026A portfolio has an expected return of 10% a year and a volatility of 15%, and yearly returns are roughly normal and independent. What is the chance of losing money in any one year, and the chance that its average annual return over ten years is below zero?Portfolio risk mathsCoreWealth managementMulti-asset

    Try it first

    Before you calculate: roughly how likely is a losing ten-year average?

    Show the worked solution

    About 25% for one year and about 1.75% for the ten-year average. One year: zero is 10 over 15, or 0.67 standard deviations below the mean, and the normal table gives 25.2%. The ten-year average has a standard deviation of 15 over the square root of 10, which is 4.74, so zero is 2.11 standard deviations away and the chance falls to 1.75%.

    Why is a losing year so common when the portfolio expects 10%?

    Think of a bus that is due every 10 minutes but can be 15 minutes early or late on a normal day. Being late is not rare; it happens about one day in four. A 10% expected return with 15% volatility means zero sits only two thirds of a standard deviation below the average, and about a quarter of any normal distribution lies further out than that. So a client holding this portfolio should expect a losing year roughly one year in four, even if nothing is wrong.

    Same portfolio, two questions: a losing year against a losing decade-40%-30%-20%-10%0%10%20%30%40%50%60%Return, per cent a yearTen-year average: sd 4.74below zero: 1.75%One year: sd 15below zero: 25.2%ten-year tail 1.75%But the ten-year total spreads outsd of total: 15 x √10 = 47 ptsagainst 15 points for one year
    Both curves are centred on 10%, but the one-year curve is wide and 25.2% of it lies below zero, while the ten-year average curve is narrow and only 1.75% of it lies below zero. The total ten-year return still spreads out, with a standard deviation of about 47 points.

    What does the square root of ten do, and what does it not do?

    Averaging independent years cancels part of the noise: good and bad years offset. The standard deviation of an average of n years is the one-year figure divided by the square root of n, so ten years takes 15 down to 4.74. The chance of a losing average falls sharply with time, but the spread of the total amount you end up with keeps growing. The total ten-year return has a standard deviation of 15 times the square root of 10, about 47 points, against 15 for one year. A longer horizon makes a loss less likely, not smaller when it comes.

    The relationship
    z=0−μσ/n=−1015/10=−2.11P(Rˉ<0)=Φ(−2.11)≈1.75%z = \frac{0 - \mu}{\sigma / \sqrt{n}} = \frac{-10}{15/\sqrt{10}} = -2.11 \qquad P(\bar{R} < 0) = \Phi(-2.11) \approx 1.75\%
    \muthe expected annual return, 10%
    \sigmathe annual volatility, 15%
    nthe number of independent years averaged, 10
    \Phithe standard normal cumulative probability
    What it says in wordsMeasure how many standard deviations of the average zero sits below the mean, then read the tail off the normal table.

    Name the assumptions when you give the number. Real yearly returns have fatter tails than a normal curve and are not fully independent; losing years tend to cluster. Both make the true ten-year figure higher than 1.75%, so treat it as a floor, not a promise.

    Where candidates lose it

    The common slip is to say the risk disappears with time, or to divide the 25% by ten. Neither is how averages behave: the spread of the average shrinks with the square root of the number of years, not with the number itself.

    The subtler loss is stopping at 1.75% and calling long horizons safe. Say the second half: the total outcome still spreads out with time, so a patient investor faces fewer losing decades but not smaller losses when one arrives.

    What the interviewer asks next

    • How many years until the chance of a losing average drops below 1%?
    • What happens to both answers if yearly returns have fat tails?
    • Is the chance of losing money over ten years the same as the chance of a negative average annual return?
  3. 071Over the last 100 trading days, portfolio A's five worst daily losses were Rs 5.2, 5.5, 6.0, 6.3 and 7.0 crore. Portfolio B's were Rs 5.2, 6, 9, 15 and 25 crore. Both report the same one-day 95% value at risk of Rs 5 crore. What is each portfolio's expected shortfall?Portfolio risk mathsCoreRisk managementInstitutional asset management

    Try it first

    How do the two portfolios compare once you look past the value at risk?

    Show the worked solution

    A's expected shortfall is Rs 6.0 crore; B's is about Rs 12.0 crore. With 100 days, the worst 5% are the five worst days, and expected shortfall is their average. A's five add to 30, an average of 6.0. B's add to 60.2, an average of 12.04. Both portfolios cross Rs 5 crore on the same number of days, but when B has a bad day it is, on average, twice as bad.

    What does value at risk leave out?

    Think of a river's flood mark. Saying the river tops the bank five days a year tells you nothing about whether it rises a hand's width or floods the town. Value at risk tells you the loss that is exceeded on the worst 5% of days, and nothing about how big those losses are. Both portfolios breach Rs 5 crore on five days in a hundred, so their VaR is identical. What happens beyond that line is where they differ, and it is exactly the part the measure does not describe.

    Same value at risk, very different tails: the five worst days, Rs crorePortfolio A5.25.566.37Portfolio B5.2691525expected shortfall 6.0expected shortfall 12.0value at risk, Rs 5 crore, the same for bothexpected shortfall, the average of the five
    Both portfolios have the same Rs 5 crore value at risk, but A's five worst days average Rs 6.0 crore while B's average Rs 12.0 crore, because B's tail stretches out to a Rs 25 crore day.
    The relationship
    ES95=15∑i=15L(i)A:305=6.0B:60.25=12.04ES_{95} = \frac{1}{5}\sum_{i=1}^{5} L_{(i)} \qquad A: \frac{30}{5} = 6.0 \qquad B: \frac{60.2}{5} = 12.04
    L_(i)the i-th worst daily loss, Rs crore
    5the number of days in the worst 5% of a 100-day sample
    What it says in wordsExpected shortfall is the average loss on the days worse than the value at risk.

    Why do risk managers prefer expected shortfall?

    Because it cannot be gamed as easily and it adds up sensibly. A trader can sell deep out-of-the-money options, which earn small premiums almost every day and lose heavily now and then. That book can show a low VaR while hiding a large tail. Expected shortfall looks inside the tail, so strategies that pile risk just past the cut-off show up in it. It also behaves well when books are combined: diversification never makes it worse, which VaR cannot promise.

    Be honest about the limits. Five observations are a thin sample for estimating anything, and one Rs 25 crore day dominates B's figure. A longer history, or a model of the tail, would be needed before setting limits on it. The comparison is still the right first read: same VaR, and one portfolio's bad days are twice as painful.

    Where candidates lose it

    The trap is saying the two portfolios carry the same risk because the VaR is the same. The question is built to show that VaR stops at the edge of the tail.

    The second miss is quoting B's worst day, Rs 25 crore, as its risk. That is a single observation; expected shortfall averages the whole tail, which is the fair comparison with A.

    What the interviewer asks next

    • Why does VaR sometimes rise when two books are combined, and why can that not happen with expected shortfall?
    • What kind of strategy produces a tail like portfolio B's?
    • How would you estimate expected shortfall with only 100 days of data?