Portfolio Management puzzles, solved step by step

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

Showing 1–10 of 100

001A fund's returns have an R squared of 0.81 against its benchmark index. The fund's volatility is 20% a year and the index's is 18%. What are the correlation, the beta and the fund's residual volatility?Statistics and forecastingWarm upPerformance analysisAsset management

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Before you work it: how much of the fund's 20% volatility does the index fail to explain?

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Correlation 0.9, beta 1.0 and residual volatility of about 8.7%. Correlation is the square root of R squared, so 0.9. Beta is correlation times the ratio of volatilities, 0.9 x 20 / 18, which is exactly 1.0. The unexplained 19% of the fund's variance of 400 is 76, and the square root of 76 is 8.7%: the fund's own risk, on top of what the index explains.

Why does 81% explained still leave so much unexplained?

Think of a household's monthly spending. If rent explains most of how the bill moves, the groceries, travel and surprises that make up the rest can still swing it by a lot. The fund is the same. R squared splits variance, and variance is volatility squared, so a small share of variance becomes a much larger share once you take the square root back. The fund's variance is 20 squared, 400. The index explains 81% of it, which is 324. The other 76 belongs to the fund alone, and the square root of 76 is 8.72%.

The index explains 324 of the fund's variance of 400 and leaves 76 unexplained; in volatility terms that is 18 points from the index and 8.7 points of the fund's own, which combine to 20 only because volatilities add in squares.

How do you get the correlation and the beta from R squared?

In a regression on a single index, R squared is simply the correlation squared, so the correlation is the square root of 0.81, which is 0.9. Beta is the correlation scaled by how volatile the fund is relative to the index: 0.9 times 20 over 18 is exactly 1.0. So the fund moves one for one with the index on average, and carries about 8.7 points of volatility the index does not explain. Mention the sign: the root could be minus 0.9, but a long-only equity fund with a positive slope takes the positive root.

The relationship

\rho=\sqrt{R^2}=0.9 \qquad \beta=\rho\,\frac{\sigma_f}{\sigma_i}=0.9\times\frac{20}{18}=1.0 \qquad \sigma_\varepsilon=\sigma_f\sqrt{1-R^2}=20\sqrt{0.19}\approx 8.7\%

R^2	the share of the fund's variance the index explains, 0.81
\rho	the correlation between fund and index returns
\sigma_f, \sigma_i	the volatilities of the fund, 20%, and the index, 18%
\sigma_\varepsilon	the residual volatility, the part of the fund's risk the index does not explain

What it says in wordsCorrelation is the root of R squared, beta rescales it by the volatility ratio, and the residual volatility is the fund's volatility times the root of the unexplained share.

What does the residual number tell a portfolio manager?

With a beta of 1.0, the residual volatility is the fund's tracking errorThe volatility of the difference between a fund's return and its benchmark's return. against the index. An R squared of 0.81 sounds index-like, but 8.7 points of tracking error is a genuinely active book: almost half as volatile as the market itself. Say the limitation too. The split assumes the relationship is linear and stable over the sample; a fund whose beta drifted during the period shows a lower R squared for reasons that have nothing to do with stock picking.

Where candidates lose it

The common slip is treating R squared as a share of volatility and answering 19% of 20%, which is 3.8%. The interviewer is checking whether you know that variances add and volatilities do not, which is the same fact that sits under every portfolio risk calculation.

The second slip is computing beta as 0.9 and stopping, forgetting that beta needs the volatility ratio. Say the three formulas in order and the numbers follow.

What the interviewer asks next

If the fund's beta were 1.2 with the same volatilities, what R squared would that imply?
How would you tell whether the 8.7 points are skill or just unintended sector bets?
Why might R squared against a style index be much higher than against the broad market?

002A trading book's one-day 99% value at risk is Rs 2 crore. What are its 10-day and its one-month (21 trading day) value at risk under the usual scaling rule, and when does that rule fail?Portfolio risk mathsWarm upAQR Capital ManagementGreenwich · 2022
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Pick the 10-day value at risk before you calculate.
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About Rs 6.3 crore over 10 days and Rs 9.2 crore over 21 days. With independent daily returns, variance adds across days, so volatility and value at risk scale with the square root of time: 2 x the root of 10 and 2 x the root of 21. The rule fails when returns trend or mean revert, when tails are fat, and when the book changes during the period.
Why the square root of time and not time itself?
Think of a person taking random steps left or right. After a hundred steps they are rarely a hundred steps from the start, because the left steps cancel the right ones; the typical distance is about ten, the square root of a hundred. Daily returns behave the same way when each day is independent. Variances add across independent days, so the spread of a ten-day return is the daily spread times the square root of ten, not times ten. Value at risk at a fixed confidence level is a multiple of that spread, so it scales the same way: Rs 2 crore becomes Rs 6.32 crore over ten days and Rs 9.17 crore over 21.
Scaled by the square root of time, a one-day value at risk of Rs 2 crore becomes about Rs 6.3 crore at ten days and Rs 9.2 crore at 21 days, far below the Rs 20 crore and Rs 42 crore that scaling by the number of days would give.
The relationship
$\text{VaR}_T=\text{VaR}_1\sqrt{T} \qquad 2\sqrt{10}\approx 6.32 \qquad 2\sqrt{21}\approx 9.17$
\text{VaR}_1 the one-day value at risk, Rs 2 crore
T the holding period in trading days
What it says in wordsMultiply the one-day figure by the square root of the number of days, which is valid only for independent, identically spread daily returns and an unchanged book.
When does the rule give the wrong answer, and in which direction?
The rule rests on three assumptions, and each one breaks in real markets. If returns trend, so a bad day tends to follow a bad day, the true ten-day loss is larger than Rs 6.3 crore; if they mean revert, it is smaller. Fat tails make the 99% point further out than a normal curve suggests, and the ratio between the tail and the spread need not hold across horizons. And a book is not frozen: over a month a desk cuts losing positions, which the scaling ignores. Say which way each one pushes the number and the interviewer knows you understand the rule rather than having memorised it.
One more thing worth saying: the scaling also assumes the expected daily return is zero. Over a day that is harmless. Over a year, drift matters and a simple square root rule starts to overstate the loss for a portfolio with a positive expected return.
Where candidates lose it
The fast wrong answer is Rs 20 crore, which treats ten independent days as ten worst days in a row. Candidates who know the square root rule sometimes lose the point anyway by stating it without its assumptions.
The follow-up is almost always when it fails. Have the three failures ready, trending returns, fat tails and a changing book, and say in which direction each pushes the number.
What the interviewer asks next
If daily returns have a positive autocorrelation of 0.2, is the true 10-day value at risk above or below Rs 6.3 crore?
Why do regulators ask for a 10-day horizon rather than one day?
What is the annual value at risk under the same rule, using 250 trading days?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): Specific statistics questions on financial concepts. daily vs monthly return, VAR, more that i don't remember
003A car leaves town A for town B, 100 miles away, at 50 miles an hour. At the same moment a bird leaves B at 100 miles an hour, flies to meet the car, turns back to B, then turns again toward the car, and keeps shuttling until the car reaches B. How far does the bird fly?Logic brainteasersWarm upBlackRockNew York · 2025
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Answer inside ten seconds.
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200 miles. The car needs 100 miles at 50 miles an hour, which is two hours. The bird flies without stopping for those two hours at 100 miles an hour, so it covers 200 miles, however many times it turns. Summing the legs gives the same answer: 66.7 there and back, then 22.2 there and back, each pair a third of the one before, which adds to 200.
Why is summing the legs the slow way?
Picture a dog on a walk that runs ahead to the gate and back to you, over and over, until you reach the gate. Nobody counts the dog's sprints; you just ask how long the walk took and how fast the dog runs. When something moves at a constant speed for a known time, distance is speed times time, whatever path it traces. The bird's zig-zag looks like the hard part of the question. It is a distraction. The only thing that matters is when the flying stops, and that is when the car arrives.
The car takes two hours to cover 100 miles at 50 miles an hour, and the bird zig-zags between B and the car for exactly those two hours, so at 100 miles an hour it flies 200 miles; the shrinking legs of 66.7, 66.7, 22.2, 22.2 and so on add to the same total.
How do you check 200 the long way, in case the interviewer asks?
The bird and the car close at 150 miles an hour, so they first meet after 100 / 150 of an hour, 40 minutes, when the car is 33.3 miles from A and the bird has flown 66.7 miles. The bird flies 66.7 miles back to B. By then the car is 66.7 miles along, 33.3 miles from B, and the same geometry repeats on a gap a third as large. Each round trip is a third of the one before, so the legs form a geometric series: 133.3 times one over one minus a third, which is 200.
The relationship
$d=v_{bird}\times\frac{D}{v_{car}}=100\times\frac{100}{50}=200 \qquad \text{check: } 133.3\times\frac{1}{1-\tfrac{1}{3}}=200$
D the distance from A to B, 100 miles
v_{car}, v_{bird} the speeds, 50 and 100 miles an hour
What it says in wordsThe bird's distance is its speed times the car's travel time; the geometric sum of its legs confirms it.
Why ask this on a quantitative research desk? Because the same move, stepping back from the path to the total, is how you price anything path-dependent in your head: ask what is conserved or fixed before tracing every step.
Where candidates lose it
Candidates start computing the first meeting point, then the second, and lose the room in arithmetic. The interviewer is watching for the moment you ask how long the bird flies; some interviewers stop you once you begin summing legs.
The second trap is saying infinite because the bird turns infinitely often. A sum of infinitely many shrinking terms can be finite, and here it is.
What the interviewer asks next
What if the bird flew at 150 miles an hour?
If both towns sent a car toward each other at 50 miles an hour, how far does the bird fly?
How many times does the bird touch the car?
Asked at BlackRock, Quantitative Research, New York, 2025 (Wall Street Oasis): A car starts at point A going 50 miles an hour towards point B, and a bird starts at point B going towards point A at 100 miles per hour
004A portfolio has a 25% chance of a down year, and each year is independent of the others. What is the chance of at least one down year over ten years?Behavioural and decision trapsWarm upWealth managementRetirement and pensions
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Quick instinct: how likely is at least one down year in ten?
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About 94%. The chance of avoiding a down year every single year is 0.75 multiplied by itself ten times, which is 5.6%. At least one down year is everything else: 1 minus 0.056, or 94.4%. Over a decade a falling year is close to certain, so a plan that treats one as a surprise is a plan built on the wrong base case.
Why work through the chance of it never happening?
Ask a commuter how likely they are to miss at least one train in a year of mornings, and the honest answer is: nearly certain, even if they miss one in a hundred. "At least one" questions have many routes to yes and a single route to no. Counting the one way it never happens and subtracting from 1 is always faster than counting every way it can happen. Here the only route to no down year is ten good years in a row, each with chance 0.75, so 5.6% of decades are clean and 94.4% are not.
With a one in four chance of a down year, the chance of at least one down year passes 50% by the third year and reaches 94.4% by the tenth, because the chance of dodging every one shrinks to 5.6%.
The relationship
$P(\text{at least one})=1-(1-p)^n=1-0.75^{10}\approx 0.944$
p the chance of a down year, 25%
n the number of years, 10
What it says in wordsThe chance of at least one down year is one minus the chance that every year is up.
Why does a wealth desk ask a probability question like this?
Because clients judge a portfolio year by year and plan over decades. A 25% chance in any one year sounds like a risk you might avoid; over ten years a down year is close to certain. The point of the number is to change the conversation from whether a down year comes to what the plan does when it does. Say the limitation as well: independence is an assumption. Real market years are not coin flips, and a regime of bad years clusters, which changes the count of down years without changing the lesson.
Where candidates lose it
The trap is adding: 25% times ten is 250%, which is obviously wrong, and candidates who spot that often retreat to 25%, which is just as wrong. The interviewer wants the complement said out loud.
The quieter trap is getting 94% and stopping. On a wealth or pensions desk, the follow-up is what that means for a client, so have one sentence ready on setting expectations before the first bad year arrives.
What the interviewer asks next
What is the chance of at least two down years in ten?
How many years before a down year is more likely than not?
If down years cluster, does the chance of at least one go up or down?
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?
006What are the Macaulay duration and the modified duration of a three-year bond paying a 6% annual coupon and priced at par?Bond mathsCorePIMCOLos Angeles · 2024
Try it first
Before you calculate: where does the Macaulay duration sit?
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Macaulay duration about 2.83 years and modified duration about 2.67. At par the yield equals the 6% coupon, so the cash flows discount to 5.66, 5.34 and 89.00, which add to 100. Weight each year by its share of the price: 1 x 0.0566 + 2 x 0.0534 + 3 x 0.8900 gives 2.83. Divide by 1.06 for modified duration: a one point rise in yield cuts the price by roughly 2.67%.
What is duration, if not the time to maturity?
Picture a seesaw with three children sitting at the one, two and three metre marks. If the child at three metres is much heavier, the pivot that balances the seesaw sits close to three, not at the middle. Macaulay duration is that pivot: the average time you wait for your money, with each payment weighted by its present value. Here the three weights are the discounted coupons of 5.66 and 5.34 and the discounted final payment of 89.00. The last payment is so heavy that the balance point, 2.83 years, sits only two months short of maturity.
The bond's discounted cash flows of 5.66, 5.34 and 89.00 sit at years 1, 2 and 3 and balance at 2.83 years, which is the Macaulay duration; dividing by 1.06 gives a modified duration of 2.67.
Year Cash flow Present value at 6% Share of price Year x share
1 6 5.66 0.0566 0.0566
2 6 5.34 0.0534 0.1068
3 106 89.00 0.8900 2.6700
Total 118 100.00 1.0000 2.8334
Weighting each payment date by its share of the price gives a Macaulay duration of 2.8334 years.
Why divide by 1.06 to get modified duration?
Macaulay duration is a time. Modified duration is a price sensitivity: the percentage change in price for a one point change in yield. With annual compounding the two differ by a factor of one plus the yield, so 2.833 divided by 1.06 is 2.673. Say what it means: if yields rise from 6% to 7%, the bond loses about 2.67% of its price, a little less in reality because the price-yield curve bends. That bend is convexity, and it is the natural next question.
The relationship
$D_{mac}=\sum_t t\,\frac{PV_t}{P}=\frac{1(5.66)+2(5.34)+3(89.00)}{100}\approx 2.83 \qquad D_{mod}=\frac{D_{mac}}{1+y}\approx 2.67$
PV_t the present value of the payment at year t
P the bond's price, 100 at par
y the yield, 6%
What it says in wordsMacaulay duration is the value-weighted average payment date; modified duration divides it by one plus the yield to turn it into a price sensitivity.
Where candidates lose it
The fast wrong answer is three years, which is true only of a zero coupon bond. The second is averaging the dates without weighting them, which gives two. The interviewer wants the words present value weighted before any number.
Candidates also mix up the two durations. Say which is a time and which is a sensitivity, and use modified duration for any question about how much the price moves.
What the interviewer asks next
What happens to the duration if the coupon rises to 10% and the bond still trades at par?
Estimate the price if yields jump to 7%, then say whether the true price is higher or lower.
What is the duration of a three-year zero coupon bond?
Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis): Lots of random bond math questions -- duration of this bond with x coupon sold at par
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?

\text{VaR}_1	the one-day value at risk, Rs 2 crore
T	the holding period in trading days

D	the distance from A to B, 100 miles
v_{car}, v_{bird}	the speeds, 50 and 100 miles an hour

p	the chance of a down year, 25%
n	the number of years, 10

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

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