Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

Showing 1–10 of 37 · filtered from 100Clear filters

005Which is the better bet: at least one six in four rolls of a single die, or at least one double six in twenty-four rolls of a pair of dice?Probability and expected valueCoreQuantitative asset managementHedge funds
Try it first
Which bet has the better odds?
Show the worked solution
The single six in four rolls, which wins 51.8% of the time against 49.1% for the double six in 24. Work each through its complement. No six in four rolls has chance (5/6) to the 4th, 48.2%. No double six in 24 rolls has chance (35/36) to the 24th, 50.9%. The naive count, trials times chance, gives two thirds for both and is wrong for both.
Why does multiplying the trials by six not keep the odds the same?
A double six is six times rarer than a six, so 24 rolls look like a fair swap for four. The intuition treats the chance of success as growing in a straight line with the number of tries. It does not. What compounds is the chance of failing every time, and a rare event's failure chance, raised to a high power, falls more slowly than the straight line suggests. Think of looking for a friend in a crowd: glancing four times at a small crowd and twenty-four times at a crowd six times the size are not the same search, because each extra glance adds less as the misses pile up.
The chance of no six in four rolls is 48.2%, so that bet wins 51.8%, while the chance of no double six in 24 rolls is 50.9%, so that bet wins only 49.1%, even though the naive count gives two thirds for both.
How many rolls would the double six bet need to be favourable?
Solve for the number of rolls where the chance of no double six drops below a half. (35/36) to the 24th is 0.5086, still above a half; (35/36) to the 25th is 0.4945, just below. So 25 rolls, not 24, is where the double six bet turns favourable, a gap of a single roll between a losing and a winning bet. That is the second way to show you understand the question: the naive rule is off by a small amount, and in a repeated game a small edge is everything.
The relationship
$1-\left(\tfrac{5}{6}\right)^{4}\approx 0.518 \qquad 1-\left(\tfrac{35}{36}\right)^{24}\approx 0.491$
5/6 the chance a single roll is not a six
35/36 the chance a roll of two dice is not a double six
What it says in wordsEach bet wins with one minus the chance of missing on every roll.
On a desk the lesson is the difference between a rough count and an exact one: a strategy that looks equivalent on a back-of-envelope scaling can sit on the wrong side of break-even once you do the compounding properly.
Where candidates lose it
The trap is the naive count. Four sixths and twenty-four thirty-sixths are both two thirds, and a candidate who answers "the same" has fallen for the exact error the puzzle was built to catch.
The other way to lose it is getting the complements right but rounding both to about 50% and calling it a tie. The whole answer lives in the gap between 51.8% and 49.1%, so say both to one decimal.
What the interviewer asks next
How many rolls of one die give better than even odds of at least one six?
What is the expected number of sixes in four rolls, and why is it not the same as the chance of at least one?
If you win Rs 100 on the double six bet and lose Rs 100 otherwise, what is your expected value over 24 rolls?
007A company trades at 20 times earnings. It uses cash that was earning 3% after tax to buy back 10% of its shares at the market price. Is the buyback accretive to earnings per share, and by how much?Valuation riddlesCoreFundamental asset managementAsset management
Try it first
Which comparison decides whether the buyback lifts earnings per share?
Show the worked solution
Yes, it is accretive, by about 4.4%. Take Rs 50 crore of earnings on 10 crore shares at Rs 100, so EPS is Rs 5.00. The buyback spends Rs 100 crore, which was earning Rs 3 crore, so earnings fall to Rs 47 crore while the share count falls to 9 crore. EPS becomes Rs 5.22. It is accretive because the 5% earnings yield beats the 3% return on cash.
What is the company actually swapping?
Suppose you hold a fixed deposit paying 3% after tax and use it to buy out a partner's share of a shop that earns 5% on its price. Your income goes up, because you replaced a 3% asset with a 5% one. A buyback is the same trade. The company gives up the after-tax return on its cash and gets back a slice of its own earnings, priced at the earnings yield, which is one over the P/E. At 20 times earnings that yield is 5%, so the swap raises earnings per share. At 33.3 times earnings the yield would be 3% and the buyback would leave EPS unchanged.
The buyback gives up cash earning 3% after tax and buys shares carrying 5% of earnings, so earnings fall from Rs 50 crore to Rs 47 crore while shares fall from 10 crore to 9 crore, and EPS rises from Rs 5.00 to Rs 5.22.
How do you get the exact number quickly?
Pick round figures and the answer falls out: Rs 1,000 crore of market value, Rs 50 crore of earnings and 10 crore shares. Earnings drop by 10% of the market value times 3%, which is Rs 3 crore, and shares drop by 10%, so EPS moves by 0.94 divided by 0.90. That is 1.0444, an accretion of 4.44%. The shortcut works for any size of company because only the ratios matter.
The relationship
$\frac{EPS_1}{EPS_0}=\frac{1-f\cdot PE\cdot y_c}{1-f}=\frac{1-0.1\times 20\times 0.03}{0.9}\approx 1.044$
f the fraction of shares bought back, 10%
PE the price to earnings multiple, 20
y_c the after-tax return on the cash spent, 3%
What it says in wordsEPS rises when the lost interest, as a share of earnings, is smaller than the share of the company bought back.
Does accretive mean the buyback was a good idea?
No, and a portfolio manager is expected to say so. Accretion is arithmetic about earnings per share; value is created only if the company paid less for its shares than they are worth. A company on a low P/E almost always gets an accretive buyback, even if the shares are overpriced, and one on a high P/E can create value with a dilutive buyback if the shares are cheap relative to future growth. Accretion also ignores risk: swapping cash for shares makes the remaining equity more levered.
Where candidates lose it
Candidates answer that a buyback always raises EPS because there are fewer shares. That forgets the lost income on the cash, and at a high enough P/E the buyback dilutes.
The second trap is stopping at accretive. On a buy-side desk the interviewer usually follows with whether it creates value, and the answer that separates candidates is that the two are different questions.
What the interviewer asks next
At what P/E does the same buyback become dilutive?
What if the buyback is funded with new debt at 8% before a 25% tax rate?
Why might a buyback that is accretive still destroy value for the remaining shareholders?
008A fund manager has a true information ratio of 0.5: genuine skill, with annual active returns averaging half their volatility. How many years of returns do you need before the track record is statistically significant at the 5% level?Performance measurementCoreFund selectionPerformance analysis
Try it first
Roughly how long must you wait?
Show the worked solution
About 15 years. The t-statistic of a track record is the information ratio times the square root of the number of years. Set 0.5 x root T equal to 1.96: the root of T is 3.92, so T is about 15.4 years. A manager twice as good, with an information ratio of 1.0, would still need nearly four years, and most careers and fund lives are shorter than this test demands.
Why does proof take so long even for a good manager?
Think of a cricketer whose true average is a little above the team's. In one season, luck swamps that small edge; only after many seasons does the average clearly separate. Skill adds up in proportion to time, but noise adds up in proportion to the square root of time, so the ratio between them grows only with the square root. An information ratio of 0.5 means one year's excess return is half a standard deviation of noise. It takes four years to reach a t-statistic of 1.0 and about 15.4 years to reach 1.96.
A manager with an information ratio of 0.5 sees the t-statistic of the track record reach the 1.96 significance line only after about 15.4 years, and even a manager with an information ratio of 1.0 needs about 3.8 years.
The relationship
$t=IR\sqrt{T}\ge 1.96 \;\Rightarrow\; T\ge\left(\frac{1.96}{0.5}\right)^2\approx 15.4$
IR the information ratio: mean active return over tracking error, per year
T years of track record
1.96 the two-sided 5% critical value
What it says in wordsThe years needed are the critical value divided by the information ratio, squared.
What should a fund selector do with that number?
Accept that statistics alone will not settle the question within a useful time. A fund selector who waits for significance hires managers after their best years are behind them; one who does not wait must lean on evidence other than the return series. That means the process, the people, turnover, whether the returns came from the bets the manager says they make, and how much of the record is explained by factors that could be bought cheaply. State the limitation: the calculation assumes independent years and a constant information ratio, and real skill decays as assets grow.
Where candidates lose it
The common answer is three to five years, because that is how long most reviews look back. The interviewer wants to see you compute rather than guess, and then notice how uncomfortable the answer is.
The other slip is squaring the wrong thing. Write t equals IR times root T, solve for root T first, then square.
What the interviewer asks next
Using monthly data, does the answer change?
How many years for a 90% confidence level instead?
If 1,000 managers have no skill, how many will look significant after 15 years?
009You buy a property for Rs 100 crore, collect Rs 7 crore of net rent at the end of each year for five years, and sell it at the end of year five for Rs 110 crore. What is the IRR?Private and real asset mathsCoreReal estate investmentReal assets
Try it first
Pick the closest IRR before you calculate.
Show the worked solution
About 8.7%. The IRR is the discount rate at which Rs 7 crore a year for five years plus Rs 110 crore at year five is worth exactly the Rs 100 crore paid. A quick estimate adds the 7% income yield to the annual growth in value, 100 to 110 over five years or 1.9% a year, which gives 8.9%. The exact answer is a little lower, 8.68%.
What is the IRR actually solving for?
Think of a savings account that pays you Rs 7 of interest each year on Rs 100 and then hands back Rs 110. You want the single interest rate that account must have been paying. The IRR is the one discount rate that makes the present value of everything you receive equal to what you paid. Try 9%: the rents are worth about 27.2 and the sale about 71.5, a total of 98.7, below 100, so 9% is too high. Try 8.5%: the total is about 100.7, so the answer sits between them, at 8.68%.
The property pays out 100 at year 0 and brings in 7 a year of rent plus 110 at the sale, and its IRR of 8.7% sits just under the rough sum of a 7.0% income yield and 1.9% a year of value growth.
The relationship
$100=\sum_{t=1}^{5}\frac{7}{(1+r)^t}+\frac{110}{(1+r)^5}\;\Rightarrow\; r\approx 8.7\%$
r the internal rate of return
7 net rent each year, Rs crore
110 the sale price at year five, Rs crore
What it says in wordsThe IRR is the rate that discounts the rents and the sale price back to the Rs 100 crore paid.
Why is the rough split a little too high?
The rule of thumb, IRR roughly equals income yield plus growth, is exact only when the rent grows at the same rate as the value, so the yield stays at 7%. Here the rent is flat while the value rises, so by year five the rent is only 6.4% of the property's worth, and the average yield across the hold is below 7%. That is why the answer is 8.68% rather than 8.92%. In the room, give the rough split first to show the structure, then the exact figure. Say the limitation too: an IRR assumes the rents can be reinvested at the IRR itself, and it says nothing about how much money was at work.
Where candidates lose it
The common error is adding 7% of rent to 2% a year of price gain, calling it 9%, and stopping. That treats the gain as simple interest and ignores that it arrives only at the end.
The other way to lose the point is to try to solve the equation exactly out loud. Bracket it: 9% gives less than 100, 8.5% gives more, so the answer is about 8.7%.
What the interviewer asks next
What is the IRR if the property sells for Rs 100 crore instead?
What if the rent rises 2% a year in step with the value?
How would 60% debt at 8% change the equity IRR?
010An ETF trades at 101.5 on the exchange while its indicative NAV is 100.0, and creating new units costs 0.3% of NAV. What does an authorised participant do, and what does it make per unit?Funds, ETFs and implementationCoreMutual fundsPortfolio implementation
Try it first
What does the authorised participant do?
Show the worked solution
It creates new units and sells them, keeping about 1.2 per unit, or 1.2% of NAV. The authorised participant buys the underlying basket at the NAV of 100, delivers it to the fund, receives new ETF units and sells them on the exchange at 101.5. Revenue of 101.5 less 100 for the basket and 0.3 of creation costs leaves 1.2. Its selling pushes the ETF price back toward NAV until the gap no longer covers the cost.
Why can anyone profit from the gap at all?
Picture a shop selling gift hampers for Rs 101.5 when the items inside cost Rs 100 at the market next door, and assembling a hamper costs Rs 0.3. Someone will buy the items, pack hampers and sell them until the hamper price falls. An ETF unit is a claim on a basket, and the creation and redemption window lets a large dealer convert one into the other, so the unit's price cannot drift far from the basket's value. The authorised participantA large dealer appointed by the fund to create and redeem ETF units in bulk directly with the fund. is that hamper maker.
The authorised participant buys the basket at 100, pays 0.3 in creation costs, receives new units and sells them at 101.5, keeping 1.2 per unit; the trade stops paying once the ETF price is back inside the band from 99.7 to 100.3.
Where does the arbitrage stop?
Every new unit sold adds supply on the exchange and every basket bought adds demand for the underlying shares, so the gap closes from both sides. The trade keeps paying until the premium falls to the creation cost of 0.3, so in a calm market the ETF price sits inside a band of roughly 99.7 to 100.3. Below 99.7 the trade reverses: buy cheap units, redeem them for the basket and sell the shares. The band is only as tight as the cost: an ETF holding illiquid bonds or foreign shares that trade in another time zone can show a wider gap for days, and that gap is a cost to whoever trades against it.
The relationship
$\text{gain}=P_{ETF}-NAV-c\cdot NAV=101.5-100-0.3=1.2$
P_{ETF} the exchange price of the ETF unit, 101.5
NAV the indicative net asset value per unit, 100
c the creation cost, 0.3% of NAV
What it says in wordsThe authorised participant keeps the premium over NAV less the cost of creating the unit.
The portfolio management point: this machinery is why an ETF can trade close to its holdings without the fund itself selling anything. Mutual fund investors transact at NAV once a day; ETF investors transact at a market price that stays near NAV only because this arbitrage is open.
Where candidates lose it
Candidates reach for the wrong direction, buying units and redeeming them, because redeem sounds like cashing in a profit. Name the cheap side and the dear side first, and the direction follows.
The quieter miss is forgetting the cost. A premium of 1.5 is not the profit; 1.2 is, and a premium of 0.2 is not an opportunity at all.
What the interviewer asks next
The ETF trades at 99.0. Walk through the trade.
Why do bond ETFs sometimes trade at large discounts in a stressed market?
Who bears the cost when an ETF persistently trades at a premium?
011A fund compounds at 15% a year for ten years. What share of the total gain over the decade arrives in the last three years?Compounding and fee dragCoreAsset managementWealth management
Try it first
Guess before you calculate.
Show the worked solution
About 45%. One rupee at 15% grows to 2.66 after seven years and 4.05 after ten. The total gain is 3.05, and the last three years add 4.05 minus 2.66, which is 1.39. That is 45.5% of the decade's gain in 30% of the time, because each year's 15% is earned on a larger base than the year before.
Why is the gain not spread evenly across the years?
Think of a snowball rolled down a long slope. In the first few metres it picks up a little snow; near the bottom each turn picks up much more, because the ball itself is bigger. Compounding earns the same rate on a growing base, so every year's rupee gain is 15% larger than the year before. Year 1 adds 0.15 on each rupee. Year 10 adds 0.53, three and a half times as much. The first three years together add only 0.52; the last three add 1.39.
At an assumed 15% a year, each rupee gains 0.15 in year 1 but 0.53 in year 10, so the last three years add 1.39 of the 3.05 total gain, 45.5% of it.
The relationship
$\frac{(1.15)^{10}-(1.15)^{7}}{(1.15)^{10}-1}=\frac{4.046-2.660}{3.046}\approx 0.455$
(1.15)^{10} the value of one rupee after ten years
(1.15)^{7} the value after seven years
(1.15)^{10}-1 the total gain over the decade
What it says in wordsThe last three years' share is the growth from year seven to year ten, divided by the growth over all ten.
What does this mean for an investor who leaves early?
Someone who exits after seven years has sat through 70% of the time but collected only 55% of the decade's gain. Because compounding back-loads the reward, leaving a long plan early costs far more than the fraction of time given up. The same arithmetic runs against the investor with fees: a charge taken every year compounds too, and its cost is also concentrated at the end. Say the limitation plainly: 15% is an assumed rate for the arithmetic, not a forecast, and real returns arrive unevenly, so the actual last three years could be the worst three.
Where candidates lose it
The trap is answering 30%, proportional to time, because the question sounds like a fraction of a decade. The interviewer is checking whether you picture compounding as a curve.
The other slip is dividing the last three years' gain by the final value, {P11_END:.2f}, rather than by the total gain, {P11_TOT:.2f}. Read the question again: it asks for a share of the gain, not of the ending pot.
What the interviewer asks next
At what rate would the last three years carry exactly half the gain?
A 1.5% annual fee is taken throughout. What share of the lost wealth falls in the last three years?
Why do long-horizon savers care more about the final years' return than the first years'?
014You backtest 20 independent trading strategies, none of which has any real edge, and test each one at the 5% significance level. What is the chance at least one looks significant, and what per-test threshold would hold that overall false alarm rate at 5%?Statistics and forecastingCoreQuantitative researchSystematic investing
Try it first
Chance that at least one of the 20 worthless strategies passes?
Show the worked solution
About 64%, and a per-test threshold of about 0.25%. Each worthless strategy passes by luck 5% of the time, so all 20 fail with chance 0.95 to the 20th, 35.8%, and at least one passes 64.2% of the time. To hold the overall rate at 5%, test each at 5% divided by 20, which is 0.25%; the exact version, 1 minus 0.95 to the power of one twentieth, is 0.256%.
Why does testing more ideas throw up a false winner?
Ask a room of 20 people to each flip a coin five times, and there is a fair chance someone gets five heads. Nobody in the room has a lucky hand; there were simply enough tries. A 5% test lets one worthless idea in twenty through by chance, so a researcher who tests twenty ideas should expect about one false winner, not be impressed by it. The expected number of false positives here is 20 x 0.05, exactly 1, and the chance of at least one is 64.2%.
Testing each worthless strategy at 5%, the chance that at least one looks significant reaches 64.2% at 20 strategies, while testing each at 0.25% holds it near 4.9%.
The relationship
$P(\ge 1 \text{ false}) = 1-(0.95)^{20}\approx 0.642 \qquad \alpha_{each}=\frac{0.05}{20}=0.25\%$
0.95 the chance a worthless strategy fails a 5% test
20 the number of independent strategies tested
\alpha_{each} the per-test threshold that caps the overall false alarm rate near 5%
What it says in wordsThe chance of at least one false winner is one minus the chance that every test correctly fails; dividing the level by the number of tests caps it.
What does a quant desk actually do about it?
Dividing the threshold by the number of tests is called the Bonferroni correctionA rule that divides the significance level by the number of tests run, so the chance of any false positive across all of them stays near the original level.. The real discipline is counting every test you ran, including the ones you dropped quietly, because the correction is only as honest as that count. A researcher who tried 200 variants and reports the best 20 has a far bigger multiple testing problem than the 20 suggest. Desks also hold out data the research never touched and demand a reason for the edge before the backtest. Say the limitation: the correction assumes independent tests, and for correlated strategies it is too strict, which costs real ideas.
Where candidates lose it
The fast wrong answers are 5%, which ignores that there are 20 tests, and 100%, which adds the chances. Say the complement and the answer arrives in one line.
The second trap is naming the fix without the cost. A tighter threshold throws away some genuine strategies too, and an interviewer on a systematic desk expects you to say that trade-off out loud.
What the interviewer asks next
If the 20 strategies are highly correlated, is the true chance of a false winner higher or lower than 64%?
Of 1,000 strategies, how many worthless ones pass at 5%?
Why is out-of-sample testing a better defence than a stricter threshold?
019The three-year government yield is 7.0% and the two-year yield is 6.6%. You buy a three-year bond with a 7% coupon at par. If the yield curve does not move over the next year, roughly what does the bond return?Bond mathsCoreFixed income
Try it first
Pick the closest one-year return.
Show the worked solution
About 7.7%. You collect the 7 coupon, and a year later the bond is a two-year bond. If the curve has not moved, it is priced at the two-year yield of 6.6%, which makes it worth 100.73. Coupon 7.00 plus a price gain of 0.73 is 7.73%. A shortcut gives the same: 7% plus the two-year duration of about 1.82 times the 0.4 point fall in yield.
Why does an unchanged curve still move the bond's price?
Think of walking down a gentle slope while standing still relative to the hillside: the ground stays put, but you end up lower because you moved along it. A bond ages along the curve; if the curve slopes up and stays still, the bond's yield falls as its maturity shortens, and its price rises. Today it is a three-year bond at 7.0%. In a year it will be a two-year bond, and two-year bonds yield 6.6%. That 0.4 point fall in yield is the roll downThe price gain a bond earns as it ages into a shorter maturity with a lower yield on an upward sloping curve..
The three-year bond bought at 7.0% becomes a two-year bond priced at 6.6% a year later, so an unchanged curve returns the 7.00 coupon plus a 0.73 price gain, 7.73% against 7.00 on a flat curve.
How do you check it without a calculator?
Use duration. A two-year bond with a 7% coupon has a modified duration of about 1.82, so a 0.4 point fall in yield lifts its price by about 1.82 x 0.4, which is 0.73. Add the 7 coupon and the one-year return is about 7.73%, within a hair of the exact 7.73%. Say the limitation: the answer depends entirely on the curve staying put. If two-year yields rise to 7.0% by next year, the roll down disappears and the return is the coupon alone.
The relationship
$P_1=\frac{7}{1.066}+\frac{107}{1.066^2}\approx 100.73 \qquad r=\frac{7+(100.73-100)}{100}\approx 7.73\%$
P_1 the bond's price in a year, as a two-year bond at a 6.6% yield
7 the annual coupon
100 the price paid today, at par
What it says in wordsThe one-year return is the coupon plus the price gain from repricing at the lower two-year yield, over the price paid.
Where candidates lose it
The common answer is 7%, the yield at purchase, which is right only if the curve is flat. The interviewer asked about an unchanged curve precisely to see whether you notice that the bond moves along it.
The overshoot is adding the whole slope of the curve or forgetting that the price gain depends on duration. The gain is duration times the yield change, not the yield change alone.
What the interviewer asks next
What would the return be if the curve were inverted, with the two-year at 7.4%?
Which point on this curve gives the most roll down per unit of duration?
How much must the two-year yield rise over the year to wipe out the roll down?
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.
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
$E[r_L]=r_f+L(E[r]-r_f)=6+1.5(4)=12\% \qquad \sigma_L=L\sigma=18\% \qquad S=\frac{6}{18}=\frac{4}{12}\approx 0.33$
L the leverage, 1.5 times
r_f the risk-free rate, 6%, also the borrowing rate here
S the 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%?
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.
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
$z = \frac{0 - \mu}{\sigma / \sqrt{n}} = \frac{-10}{15/\sqrt{10}} = -2.11 \qquad P(\bar{R} < 0) = \Phi(-2.11) \approx 1.75\%$
\mu the expected annual return, 10%
\sigma the annual volatility, 15%
n the number of independent years averaged, 10
\Phi the 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?

5/6	the chance a single roll is not a six
35/36	the chance a roll of two dice is not a double six

f	the fraction of shares bought back, 10%
PE	the price to earnings multiple, 20
y_c	the after-tax return on the cash spent, 3%

IR	the information ratio: mean active return over tracking error, per year
T	years of track record
1.96	the two-sided 5% critical value

r	the internal rate of return
7	net rent each year, Rs crore
110	the sale price at year five, Rs crore

Try it first

Why does multiplying the trials by six not keep the odds the same?

How many rolls would the double six bet need to be favourable?

Where candidates lose it

What the interviewer asks next

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What is the company actually swapping?

How do you get the exact number quickly?

Does accretive mean the buyback was a good idea?

Where candidates lose it

What the interviewer asks next

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Why does proof take so long even for a good manager?

What should a fund selector do with that number?

Where candidates lose it

What the interviewer asks next

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What is the IRR actually solving for?

Why is the rough split a little too high?

Where candidates lose it

What the interviewer asks next

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Why can anyone profit from the gap at all?

Where does the arbitrage stop?

Where candidates lose it

What the interviewer asks next

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Why is the gain not spread evenly across the years?

What does this mean for an investor who leaves early?

Where candidates lose it

What the interviewer asks next

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Why does testing more ideas throw up a false winner?

What does a quant desk actually do about it?

Where candidates lose it

What the interviewer asks next

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Why does an unchanged curve still move the bond's price?

How do you check it without a calculator?

Where candidates lose it

What the interviewer asks next

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Why does leverage leave the Sharpe ratio alone?

When does leverage lower the Sharpe ratio?

Where candidates lose it

What the interviewer asks next

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

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

Where candidates lose it

What the interviewer asks next