Portfolio Management puzzles, solved step by step

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

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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
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Pick the closest one-year return.
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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
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What happens to the Sharpe ratio when you lever at the risk-free rate?
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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
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Before you calculate: roughly how likely is a losing ten-year average?
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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?
032A loan scoring model catches 90% of applicants who will go on to default, but it also flags 15% of good borrowers. If 4% of applicants default, what share of flagged applicants actually default?Statistics and forecastingCoreBlackRockWilmington · 2025
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Instinct first: what share of flagged applicants will default?
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Only 20%. Take 1,000 applicants. 40 will default and the model flags 36 of them. 960 are good and the model wrongly flags 15% of them, 144 people. The flagged pile holds 180, and 36 of those default: one in five. The model is good at catching defaulters, but defaulters are so rare that false alarms outnumber them four to one.
Why is 90% the wrong answer when the model catches 90% of defaulters?
Picture a smoke alarm that always sounds when there is a fire and also sounds now and then for burnt toast. Because toast burns far more often than houses do, most alarms in a year are toast. A flag's meaning depends on how common the thing it looks for is: when defaulters are 4% of applicants, even a modest false alarm rate on the other 96% produces more wrong flags than right ones. The 90% is the chance a defaulter gets flagged; the question asks the reverse, the chance a flag is a defaulter.
Of 1,000 applicants, 36 defaulters and 144 good borrowers are flagged, so only 36 of the 180 flagged applicants, 20%, actually default. The cleared pile is much cleaner: 4 defaulters in 820.
Why count people instead of using the formula?
Bayes' rule gives the same answer, but natural frequencies are faster to say and harder to get wrong under pressure. Turn every percentage into a count of people out of a round number, and the answer is simply the flagged defaulters over everyone flagged. The formula version is 0.04 x 0.9 over (0.04 x 0.9 plus 0.96 x 0.15), which is 0.036 over 0.18, or 20%. Offer it as the check after the counts.
The relationship
$P(D \mid F) = \frac{P(F \mid D)\,P(D)}{P(F \mid D)\,P(D) + P(F \mid G)\,P(G)} = \frac{0.9 \times 0.04}{0.9 \times 0.04 + 0.15 \times 0.96} = 0.20$
D the applicant will default
G the applicant is a good borrower
F the model flags the applicant
P(F|D) the catch rate, 90%
P(F|G) the false alarm rate, 15%
What it says in wordsThe chance a flag is real is the true flags divided by all flags, true and false.
Then say what a lender does with it. A flag at 20% is a reason for a closer look, not a rejection. The cleared pile, by contrast, holds only 4 defaulters in 820, about 0.5%, against 4% before the model, so the model is most useful for waving through the safe majority. Cutting the false alarm rate from 15% to 5% would lift the flagged default share to about 43%.
Where candidates lose it
Most candidates answer 90% or something close, swapping the chance of a flag given default for the chance of default given a flag. It is the same slip as reading a medical test's accuracy as the chance you are ill.
Say the base rate first, then count 1,000 people through the tree out loud. The interviewer mostly wants to hear that you know the base rate drives the answer.
What the interviewer asks next
What false alarm rate would make half of all flags real defaulters?
If the lender rejects every flagged applicant, how many good borrowers does it turn away per defaulter avoided?
How does the answer change for a riskier segment where 15% of applicants default?
Asked at BlackRock, Generalist, Wilmington, 2025 (Wall Street Oasis): Questions are pretty straightforward and test about statistics models about loan application and loan origination.
033An investor puts Rs 20,000 a month into the same equity scheme for 20 years. The portfolio earns 12% a year before costs. Through the regular plan the costs are 1.5% a year; through the direct plan they are 0.5%. How much less does the regular-plan investor end with, and what share of that gap opens in the last five years?Compounding and fee dragCoreIndian wealth managementMutual funds
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Roughly what share of the 20-year gap opens in the final five years?
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About Rs 19.5 lakh less, and about 62% of that gap opens in the last five years. Rs 48 lakh paid in grows to about Rs 1.73 crore at 11.5% net in the direct plan and Rs 1.54 crore at 10.5% in the regular plan. The gap is Rs 7.4 lakh after 15 years, then more than doubles in five. The fee is a percentage of a balance that is largest at the end.
Why does a one-point fee gap cost so little early and so much late?
Think of a small leak in a water tank that is being filled from a tap. When the tank is nearly empty, the leak loses a trickle; when it is almost full, the same size of hole loses a lot more, because more water is pressing on it. A fund's costs are a percentage of the balance, and a monthly plan's balance is tiny in the early years and large in the last few, so most of the rupee cost falls at the end. On top of that, every rupee lost early would itself have compounded, so the gap grows faster than the balance.
The direct plan grows to Rs 1.73 crore and the regular plan to Rs 1.54 crore from the same Rs 48 lakh paid in. The gap is Rs 2.3 lakh after ten years, Rs 7.4 lakh after fifteen and Rs 19.5 lakh after twenty, so about 62% of it opens in the last five years.
How do you estimate the gap without a spreadsheet?
Use the future value of a monthly annuity at each net rate and subtract. The gap is about 13% of the regular plan's final value, which is far more than the one-point fee suggests, because the fee compounds for the whole life of every instalment. A quick mental check: the average rupee is invested for about ten years, and 1% a year for ten years is roughly 10% of the pot. The exact figure is a little higher because the early instalments sit in the fund for up to twenty years.
The relationship
$FV = P \sum_{k=1}^{240} (1+m)^{k}, \quad m = (1+r)^{1/12} - 1$
P the monthly instalment, Rs 20,000, invested at the start of each month
r the net annual return: 10.5% regular, 11.5% direct
m the equivalent monthly rate
What it says in wordsEach instalment compounds from the month it goes in to the end, and the net rate after costs decides how fast.
State the assumptions: a steady 12% gross return every year, costs that never change, and no tax. Real returns arrive unevenly, and expense ratios differ by scheme and change over time, so confirm the current figures for any real scheme. The limitation does not change the shape: whatever the numbers, the drag on a monthly plan is back-loaded.
Where candidates lose it
Candidates multiply 1% by 20 years and say the regular investor ends with about 20% less, or they say the cost is spread evenly across the years. Both miss that the fee is taken from a balance that is small early and large late.
The second loss is answering in percentages only. Put rupees on it: Rs 19.5 lakh is the number a client remembers, and most of it goes in the last five years, when the client is least likely to be watching the fee.
What the interviewer asks next
What would the gap be on a single lump sum of Rs 48 lakh invested on day one?
If the investor stops contributing after ten years but stays invested, how does the gap evolve?
Why might a client still choose a regular plan, and what should that service be worth?
034A five-year bond pays a 7% annual coupon and trades at 95. Without a calculator, estimate its yield to maturity.Bond mathsCoreFixed income
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Which is closest to the yield to maturity?
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About 8.2%. The bond pays 7 a year and also rises from 95 to 100 by maturity, which is about 1 point a year over five years. That is 8 a year of return on money that averages about 97.5 invested, and 8 over 97.5 is 8.21%. The exact yield with annual coupons is 8.26%, so the shortcut is within a few hundredths of a point.
Where does the return on a discount bond come from?
Buy a Rs 100 gift voucher for Rs 95 that also pays you Rs 7 each year until it can be cashed at full value in five years. You collect the Rs 7 every year, and you also pocket the Rs 5 discount at the end. A discount bond's yield is its coupon plus the discount it recovers as the price is pulled to par, spread across the years to maturity. Spread evenly, that is 1 point a year, so the bond earns roughly 8 a year.
The five-year 7% bond is priced at par when its yield is 7%, and at 95 its yield is 8.26%. The shortcut of coupon plus yearly pull to par, divided by the average price, gives 8.21%, while the current yield of 7.37% misses the pull to par entirely.
Why divide by the average price rather than 95?
The amount you have invested is not fixed at 95: in this rough picture the bond's value drifts up towards 100 over the five years, so the capital at work averages about 97.5. Dividing by the average price corrects most of the error from spreading the discount in a straight line. Dividing 8 by 95 instead gives 8.42%, too high. As a check, a bond priced at 8.21% comes out at 95.21, within a fraction of 95.
The relationship
$y \approx \frac{C + (F - P)/n}{(F + P)/2} = \frac{7 + 5/5}{97.5} = 8.21\%$
C the annual coupon, 7
F the face value repaid at maturity, 100
P today's price, 95
n years to maturity, 5
What it says in wordsYearly income plus the yearly share of the discount, divided by the average amount invested.
Know when the shortcut drifts. It is close for short bonds near par and gets worse for long maturities or deep discounts, where the true discounting curve bends away from a straight line. For a quick answer in the room, 8.2% with the one-line derivation is what the interviewer is after; then say that the exact figure is a touch higher.
Where candidates lose it
The common wrong answers are 7%, which confuses coupon with yield, and 7.37%, the current yield, which forgets that the bond will be repaid at 100. A few candidates add the whole 5-point discount to a single year and say 12%.
Say the two sources of return aloud, coupon and pull to par, then give the formula. The direction check helps too: the bond trades below par, so its yield must be above the coupon.
What the interviewer asks next
The same bond trades at 105. Estimate its yield.
Why does the shortcut get worse for a 20-year bond at 80?
If yields rise one point from here, roughly what happens to the price?
035Estimate how many 5G smartphones are sold in India in a year.Market sizing and estimationCoreAllianceBernsteinNew York · 2022
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Which assumption moves the answer most?
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Roughly 12 crore 5G phones a year, on stated assumptions. Take about 60 crore smartphone users out of 140 crore people. If they replace their phone every 4 years, that is 15 crore replacements, plus about 1 crore first-time buyers, so 16 crore phones. If three quarters of new phones sold are 5G, the answer is about 12 crore. A household check, one phone every two years across 30 crore households, gives 15 crore, close enough.
Why start from users and replacement, not from the population?
Think of how many school shoes a town buys in a year. The number of children matters, but the answer is set by how often each child outgrows a pair. In a market where most people already own the product, yearly sales are the user base divided by the replacement cycle, plus a smaller flow of first-time buyers. Population only tells you the ceiling; the cycle turns the stock of users into a yearly flow.
Sixty crore smartphone users replacing every four years, plus one crore first-time buyers, gives sixteen crore phones a year and about 12 crore 5G phones at a 75% share. Moving only the replacement cycle between three and five years swings the answer from 15.75 crore to 9.75 crore.
Which assumptions should you say out loud, and how do you check them?
Every number here is an assumption to be stated, not a fact to be quoted: the user base, the cycle, the first-time flow and the 5G share. The replacement cycle deserves the most care because a one-year change in it swings the answer by between a fifth and a third; the population barely matters by comparison. A three-year cycle gives 15.75 crore and a five-year cycle 9.75 crore. The 5G share is the other moving part, because it changes fast from one year to the next; ask which year the interviewer means.
Then check from a different direction. With about 30 crore households, one new phone per household every two years gives 15 crore phones, close to the 16 crore from the user build. Two methods that land near each other are more convincing than one precise-looking number. For a real estimate, replace every assumption with published industry shipment data and confirm the current figures.
If the interviewer wants value rather than units, multiply by an assumed average selling price and say that 5G phones are skewed to the middle and upper price bands, so the price assumption needs as much care as the cycle.
Where candidates lose it
Candidates often start with the population and multiply by a smartphone share, then forget to turn a stock of owners into a yearly flow, and announce 60 crore phones sold a year. Others spend the whole time debating the population figure, which is the least uncertain input.
Draw the tree first, say that sales are mostly replacements, and spend your care on the cycle and the 5G share. Close with a sanity check from households or another angle.
What the interviewer asks next
How would you turn this into a market size in rupees?
How does the answer change if the replacement cycle lengthens because phones last longer?
What would you look at to check the 5G share for a given year?
Asked at AllianceBernstein, Equity Research, New York, 2022 (Wall Street Oasis): Estimate the market size of 5G smartphone sales in 2022.
036Twenty fund managers have no skill at all: each has a 50% chance of beating the benchmark in any year, independently. What is the chance that at least one of them beats the benchmark five years running?Behavioural and decision trapsCoreFund selectionMulti-manager allocation
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Instinct first: how likely is at least one five-year streak among 20 unskilled managers?
Show the worked solution
About 47%, close to a coin toss. One manager beats the benchmark five years running with probability one half to the fifth, or 1 in 32. The chance that none of the 20 does it is 31/32 to the 20th power, about 53%. So the chance that at least one does is about 47%. In a crowd of unskilled managers, a five-year streak is almost as likely as not.
Why is a five-year streak weak evidence of skill?
Put 20 people in a room and ask each to call five coin tosses. Somebody calling all five correctly would look gifted, but with that many people, somebody usually does. The chance of a streak for one named person is small, but the chance that someone in a crowd produces one is large, and fund selection looks at the crowd. An investor who screens a category for managers with five straight winning years is sampling exactly the lucky tail of this distribution.
Twenty unskilled managers thin to an expected 0.625 unbeaten after five years, yet the chance that at least one of them completes the streak is 47%. With 100 managers it rises to 96%, so a streak in a large category says very little on its own.
How do you compute at least one without adding up cases?
Use the complement. The chance that at least one of many independent trials succeeds is one minus the chance that every trial fails. Each manager fails the streak with probability 31/32. All 20 fail with probability (31/32) to the 20th, which is about 0.53. One minus that is 0.47. The expected number of streaks is 20 over 32, or 0.625, which is a separate and also useful number: less than one streak on average, yet close to even odds of seeing at least one.
The relationship
$P(\text{at least one}) = 1 - \left(1 - 0.5^{5}\right)^{20} = 1 - \left(\tfrac{31}{32}\right)^{20} \approx 0.47$
0.5^5 one manager's chance of five straight wins, 1 in 32
20 the number of independent managers
What it says in wordsFind the chance every manager misses the streak, then take it away from one.
The limitation runs in two directions. Real managers are not independent, since many hold similar stocks, which makes streaks cluster and the calculation rougher. And some managers do have skill; the point is only that a streak on its own cannot tell the skilled from the lucky. That takes longer records, consistent process and a view on why the edge should persist.
Where candidates lose it
The common wrong answer is one in 32, about 3%, which answers the question for one named manager rather than for any of twenty. The next most common is adding 20 times 1/32 to get 62.5%, which counts the cases where two managers both succeed twice.
Say the complement method aloud and give 47%. Then draw the lesson for manager selection in one sentence: a screen for long winning streaks mostly selects luck.
What the interviewer asks next
How many unskilled managers would you need for a 90% chance of at least one ten-year streak?
If one manager in the group truly beats the benchmark 60% of the time, how likely is that manager to show a five-year streak?
How would you design a fund selection test that is not fooled by this effect?
037Inflation runs at 5% a year in India and 2% in the US, and the exchange rate is 84 rupees to the dollar. Under relative purchasing power parity, where should the rate be in ten years?Currency and global returnsCoreGlobal investingMacro
Try it first
Pick the closest answer before you calculate.
Show the worked solution
About 112 rupees to the dollar. Relative PPP says the higher-inflation currency loses value at the inflation gap. Each year the rate rises by 1.05 over 1.02, about 2.94%. Over ten years that compounds to a factor of 1.336, and 84 times that is about 112.2. The simple 3% shortcut gives 112.9, close enough in the room.
Why should higher inflation weaken a currency?
If a cup of tea costs Rs 20 this year and Rs 21 next year while the same cup abroad goes from 1 dollar to 1.02, the rupee has lost more buying power than the dollar. If money is to buy roughly the same basket in both countries over time, the exchange rate has to move by the gap between the two inflation rates. That is relative purchasing power parity: it does not say what the rate should be today, only how it should drift.
Relative PPP takes the rate from 84 to about 112.2 over ten years by compounding a drift of about 2.94% a year. Actual rates can sit well away from that path for years, so the line describes a long-run tendency, not a forecast.
What does this mean for an investor holding assets abroad?
For a rupee-based investor, a US asset earns its local return plus the rupee's depreciation. A US bond yielding 3 points less than an Indian bond is not worse on this reasoning, because the expected currency drift gives back roughly the same gap. Measured the other way, the rupee loses about 25% of its dollar value over the ten years, which is what a dollar-based investor in Indian assets must earn back through higher local returns.
The relationship
$S_{10} = S_0 \left(\frac{1 + \pi_{IN}}{1 + \pi_{US}}\right)^{10} = 84 \times \left(\frac{1.05}{1.02}\right)^{10} \approx 112.2$
S_0 today's rate, 84 rupees per dollar
\pi_{IN} Indian inflation, 5%
\pi_{US} US inflation, 2%
What it says in wordsThe rate rises each year by the ratio of the two price levels' growth.
Say the limitation plainly. PPP is a weak guide over one or two years: capital flows, interest rate moves and risk appetite can push the rate far from this path and hold it there. Over a decade the evidence for the drift is better, but it is a tendency, not a rule. The inflation rates themselves are assumptions; for a real view, confirm current figures and consider what each central bank is targeting.
Where candidates lose it
Candidates get the direction right and then apply the 3% gap once, answering 86.5 or 87. Others compound India's 5% alone and land near 137, forgetting that the dollar is losing value too.
Say the drift is the ratio of the two inflation rates, compound it ten times, and offer the simple 3% version as a check. Then add one sentence on why the short-run rate need not follow.
What the interviewer asks next
If Indian one-year rates are 7% and US rates 4.5%, what forward rate does interest parity imply for one year out?
Why can a currency stay far from PPP for several years?
How should a rupee-based investor think about hedging a ten-year US equity holding?
038An equity fund keeps 8% of its assets in cash earning 6.5% a year, while its equities return 14%. How much return does the cash cost the fund, and when is that cost largest?Funds, ETFs and implementationCorePortfolio implementationMutual funds
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Roughly how much does the 8% cash holding cost the fund this year?
Show the worked solution
About 0.6 points a year here. Cash drag is the cash weight times the gap between what equities and cash earn: 0.08 x (14 minus 6.5) is 0.6. The fund returns 13.4% instead of 14%. The drag is largest in strong markets and turns into a cushion when equities fall: in a year equities lose 10%, the same cash saves about 1.3 points.
Why is the cost the gap, not the whole equity return?
Suppose you keep a little of your salary in a savings account instead of a higher-paying deposit. You have not lost the whole deposit rate on that money, only the difference between the two rates. Cash in an equity fund still earns something, so the return given up is the cash weight times the gap between the equity return and the cash return. Here that is 8% of 7.5 points, which is 0.6 of a point for the whole fund.
Holding 8% in cash takes the fund from 14.0% to 13.4%, a drag of 0.6 points. Because the drag is 8% of the equity-cash gap, it grows in strong markets, is zero when equities earn exactly 6.5%, and becomes a cushion of 1.32 points when equities fall 10%.
Why does it matter most in strong markets?
The drag scales with the equity return. In a 30% year the same 8% of cash costs about 1.9 points, in a 14% year 0.6, and in a year equities lose 10% it adds about 1.32 points. That is why cash drag shows up most in the years a fund is judged hardest: a fund that trails its benchmark in a strong rally often simply held more cash than the index, which holds none.
The relationship
$\text{drag} = w_c\,(R_e - R_c) = 0.08 \times (14 - 6.5) = 0.6$
w_c the cash weight, 8%
R_e the equity return, 14%
R_c the return on cash, 6.5%
What it says in wordsThe fund gives up the cash weight times whatever equities earned above cash.
Funds hold cash for real reasons: redemptions, new money not yet invested, and dry powder for opportunities. The implementation answer is not zero cash but equitising it, for example by holding index futures against the cash balance so that the fund keeps market exposure while holding liquidity. Say that as the practical close, and note that futures carry their own costs and margin needs.
Where candidates lose it
The common slip is multiplying the cash weight by the whole equity return and saying 1.1 points, which assumes cash earns nothing. The second is treating cash drag as a fixed annual fee rather than a bet against the market that pays off only in down years.
Give the formula, the 0.6, and then the shape: bigger in rallies, negative in falls. That second half is what separates an implementation answer from arithmetic.
What the interviewer asks next
How would you equitise the cash, and what does that cost?
A manager says the cash is a deliberate market call. How would you judge it over five years?
If the fund's benchmark returns 14% and the fund's stocks return 14.5%, does the fund beat the benchmark after the cash drag?

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

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

\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

D	the applicant will default
G	the applicant is a good borrower
F	the model flags the applicant
P(F\|D)	the catch rate, 90%
P(F\|G)	the false alarm rate, 15%

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

How do you check it without a calculator?

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Why does leverage leave the Sharpe ratio alone?

When does leverage lower the Sharpe ratio?

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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?

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Why is 90% the wrong answer when the model catches 90% of defaulters?

Why count people instead of using the formula?

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Why does a one-point fee gap cost so little early and so much late?

How do you estimate the gap without a spreadsheet?

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Where does the return on a discount bond come from?

Why divide by the average price rather than 95?

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Why start from users and replacement, not from the population?

Which assumptions should you say out loud, and how do you check them?

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Why is a five-year streak weak evidence of skill?

How do you compute at least one without adding up cases?

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Why should higher inflation weaken a currency?

What does this mean for an investor holding assets abroad?

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Why is the cost the gap, not the whole equity return?

Why does it matter most in strong markets?

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