Portfolio Management puzzles, solved step by step

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

Showing 11–20 of 37 · filtered from 100Clear filters

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
Try it first
Roughly what share of the 20-year gap opens in the final five years?
Show the worked solution
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
Try it first
Which is closest to the yield to maturity?
Show the worked solution
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?
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
Try it first
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?
040A bet pays even money and you win it 55% of the time. You can bet any fraction of your capital, as often as you like. What fraction maximises long-run growth, and what happens to growth if you stake double that?Probability and expected valueCoreHedge fundsQuantitative asset management
Try it first
What happens to long-run growth if you stake twice the growth-maximising fraction?
Show the worked solution
Stake 10% of capital each time; staking 20% drives long-run growth to about zero. For an even-money bet the growth-maximising, or Kelly, stake is p minus q, 0.55 minus 0.45, which is 10%. That earns about 0.50% a bet in log terms. At 20%, 0.55 x log 1.2 plus 0.45 x log 0.8 is about zero, so twice the stake gives no growth at all despite double the expected profit per bet.
Why does betting more of a winning edge ever make you poorer?
A shopkeeper who wins more often than she loses still goes under if one bad month can wipe out half the shop, because rebuilding from half takes a doubling. Wealth compounds, so what matters over many bets is the average log return, and a loss hurts the log more than an equal-sized gain helps it. With a small stake, the edge dominates. With a large stake, the losses' extra damage grows faster than the edge, and past a point it wins.
Long-run growth peaks at 0.50% a bet when 10% is staked and falls back to about zero at a 19.9% stake, so doubling the growth-maximising bet throws away the whole benefit of the edge. Half the Kelly stake keeps about three quarters of the peak growth.
How do you find the 10% and check the 20%?
Write growth per bet as g(f) = p log(1 + f) + q log(1 minus f), set its slope to zero, and the answer for even money is f = p minus q. The Kelly fraction for an even-money bet is simply the edge: 55% minus 45% is 10%. At 10%, g is 0.55 x 0.0953 minus 0.45 x 0.1054, about 0.50%. At 20% it is 0.55 x 0.1823 minus 0.45 x 0.2231, about -0.014%: essentially zero, and in fact a touch below it. Over 100 bets at the Kelly stake, capital grows by a factor of about 1.65 on the typical path; at 20% it goes nowhere.
The relationship
$g(f) = p\ln(1+f) + q\ln(1-f) \qquad f^{*} = p - q = 0.10$
f the fraction of capital staked on each bet
p, q the chances of winning and losing, 55% and 45%
g(f) the long-run growth rate per bet
f^{*} the growth-maximising stake
What it says in wordsLong-run growth is the probability-weighted log of each outcome, and it peaks when the stake equals the edge.
Practitioners usually bet a fraction of Kelly. Half Kelly, 5%, keeps about 75% of the peak growth with half the volatility, and it protects against the real problem: in markets you never know p exactly. If the true edge were 52.5% rather than 55%, the 10% stake would already sit near zero growth.
Where candidates lose it
Candidates reason from expected profit, which rises in a straight line with the stake, and conclude that more is always better with a positive edge. Others say bet everything, or bet 55%, confusing the win probability with the stake.
Frame it as compounding from the first sentence, give f equals p minus q, and then show the 20% case going to zero. That second number is usually why the question is asked.
What the interviewer asks next
What is the Kelly stake if the bet pays 2 to 1 and wins 40% of the time?
Why do most practitioners bet half Kelly or less?
How does uncertainty about the true win probability change the stake you choose?
042A fund category had 100 funds five years ago. Twenty of them were closed or merged away after averaging minus 4% a year; the 80 survivors averaged 11% a year. What was the true average return of the category, and how big is the survivorship bias in the survivors' figure?Performance measurementCoreFund selectionMutual funds
Try it first
What was the category's true average return?
Show the worked solution
About 8% a year, so the survivor average overstates the category by 3 points. Weight each group by its share of the 100 funds: 0.8 x 11% plus 0.2 x minus 4% is 8.8% minus 0.8%, or 8%. The survivors' 11% leaves out exactly the funds that did worst, so it describes the winners, not what the average investor in the category earned.
Why does a survivors-only average flatter the category?
Look at a school's alumni wall of fame and you would think every student became a success. The wall shows who is on it, not who was in the class. Funds that do badly are closed or merged into better ones, so a list of funds that exist today is a list of funds that did well enough to survive, and its average is biased upward. The bias is not random: the missing funds are exactly the ones with the worst numbers.
Of 100 funds, the 80 survivors averaged 11% while the 20 closed funds averaged minus 4%, so the full-category average was 8%. A table built only from survivors shows 11% and overstates what investors earned by 3 points a year.
How do you compute the true figure and the bias?
Take a weighted average by number of funds. The true category average is 80% of 11 plus 20% of minus 4, which is 8%, and the bias is the gap: 3 points a year. A quick way to see it: the closed funds were 15 points behind the survivors, and they were a fifth of the category, so they pull the average down by a fifth of 15, which is 3.
The relationship
$\bar{R} = \frac{80 \times 11 + 20 \times (-4)}{100} = 8\% \qquad \text{bias} = 11 - 8 = 3 \text{ points}$
80, 20 the survivors and the closed funds
11, -4 each group's average annual return, per cent
What it says in wordsThe honest average counts every fund that existed at the start, including the ones that disappeared.
State the simplifications. This is an equal-weighted average across funds; an asset-weighted figure could differ, since closed funds are often small. The closed funds also ran for less than five years, so a precise study would compound each fund's returns over the time it existed. Neither changes the direction of the bias, and in real data sets the bias is larger over longer windows because more funds disappear.
Where candidates lose it
Most candidates either quote 11% or take the simple midpoint of 11 and minus 4, landing on 3.5% or 7.5%. The first ignores the dead funds; the second ignores that there are four survivors for each dead fund.
Weight by count, give 8% and 3 points, then say why it matters: a fund selector comparing a manager with a survivor-only peer average is holding the manager to a bar that no real investor earned.
What the interviewer asks next
How would you build a peer group for a manager review that avoids this bias?
If the closed funds were mostly small, how would an asset-weighted average differ?
Where else in investing does survivorship bias show up?
044A property earns Rs 8 crore a year of net operating income and is valued at an 8% cap rate. The owner borrowed 60% of that value. If cap rates fall to 7% with the income unchanged, what happens to the property's value, and to the owner's equity?Private and real asset mathsCoreReal estate investmentReal assets
Try it first
The value rises about 14%. How much does the owner's equity rise?
Show the worked solution
Value rises about 14.3%, from Rs 100 crore to Rs 114.3 crore, and the owner's equity rises about 35.7%. Rs 8 crore divided by 0.07 is Rs 114.3 crore. Debt stays at Rs 60 crore, so equity goes from Rs 40 crore to Rs 54.3 crore. At 60% debt the equity is 2.5 times as sensitive as the building, and the same leverage works in reverse if cap rates rise.
Why does a one-point fall in the cap rate lift value by 14%, not by one point?
A cap rate is a yield, and value is income divided by it, just as a bond's price rises when its yield falls. Going from 8% to 7% multiplies value by 8 over 7, because the same income is now capitalised at a lower yield: the building is worth 14.3 years of income instead of 12.5. A one-point move in a single-digit cap rate is a large move in value.
With income fixed at Rs 8 crore and debt fixed at Rs 60 crore, a fall in the cap rate from 8% to 7% lifts value 14.3% and equity 35.7%, while a rise to 9% cuts value 11.1% and equity 27.8%. The debt does not move, so the equity absorbs the whole change in value.
How does leverage turn 14% into 36%?
Buy a Rs 50 lakh flat with Rs 10 lakh of your own and a Rs 40 lakh loan. If the flat rises 10% to Rs 55 lakh, the loan is unchanged and your stake has risen from 10 to 15 lakh, or 50%. Debt is a fixed claim, so the whole change in asset value lands on the equity, and the equity's percentage move is the asset's move times value over equity. Here value over equity is 100 over 40, or 2.5, so 14.3% becomes 35.7%.
The relationship
$\Delta E\% = \Delta V\% \times \frac{V}{E} = 14.3\% \times \frac{100}{40} \approx 35.7\%$
V the starting property value, Rs 100 crore
E the starting equity, Rs 40 crore
\Delta V\% the percentage change in property value
What it says in wordsWith fixed debt, equity moves by the asset's percentage change scaled up by the ratio of value to equity.
Always give the reverse case, because the interviewer is testing whether you see both directions. If cap rates rise to 9%, value falls to Rs 88.9 crore, 11.1% down, and equity falls to Rs 28.9 crore, 27.8% down. The loan-to-value ratio climbs from 60% to about 68%, which may breach a lending covenant. This ignores interest, fees and any loan amortisation, which change the numbers but not the shape.
Where candidates lose it
The frequent error is saying the equity rises 14%, the same as the property, forgetting that the debt does not share in the gain. A second is treating the cap rate change as a one-point change in value.
Give the value change as 8 over 7, then the equity change through the ratio of value to equity, and finish with the downside at 9%. A candidate who volunteers the reverse case and the covenant risk sounds like someone who has owned a leveraged asset.
What the interviewer asks next
At what cap rate would the owner's equity be wiped out?
How would a 3% annual rise in NOI change the picture over five years?
Why do cap rates tend to move with long-term interest rates?
046A fixed deposit pays 8% a year. The depositor is taxed at 30% on the interest, and inflation runs at 6%. What is the real after-tax return?Compounding and fee dragCoreIndian wealth managementWealth management
Try it first
Before working it: is the depositor's purchasing power growing?
Show the worked solution
About -0.4% a year, a small real loss. Tax at 30% takes 2.4 points of the 8% interest, leaving 5.6%. Inflation at 6% then shrinks what that buys: 1.056 over 1.06 minus 1 is about -0.38%. The quick version, 5.6 minus 6, gives minus 0.4. Before tax the real return was about 1.9%, so the tax took more than all of it. Confirm current tax rates for any real case.
Why can a positive-looking rate lose purchasing power?
Imagine your rent rises 6% a year and your savings account adds 8% a year, but a third of that 8% goes to tax. You are left with 5.6% more money to meet 6% higher prices. Tax is charged on the nominal interest, including the part that only compensates for inflation, so it can take more than the whole real gain. The deposit looks like it earns 2 points above inflation; after tax it earns slightly less than inflation.
An 8% deposit loses 2.4 points to tax at 30%, and 6% inflation then takes the remaining 5.6 points and a little more, leaving a real after-tax return of about -0.38%. Before tax, the real return was 1.89%, so tax removed more than all of it.
How do you get the exact figure, and what does the tax really cost?
Divide growth factors rather than subtracting rates: the money grows by 1.056 while prices grow by 1.06, so real wealth changes by 1.056 over 1.06, a factor of 0.9962. Measured against the real return, the 30% tax rate works like a 120% tax, because it is levied on the inflation part of the interest too. Before tax, the real return was 1.08 over 1.06 minus 1, or 1.89%; after tax it is -0.38%. Rs 10 lakh on deposit buys about Rs 9.96 lakh of today's goods a year later.
The relationship
$r_{real} = \frac{1 + i(1-t)}{1 + \pi} - 1 = \frac{1.056}{1.06} - 1 \approx -0.38\%$
i the deposit rate, 8%
t the tax rate on interest, 30%
\pi inflation, 6%
What it says in wordsTake tax off the nominal interest first, then divide by the growth in prices.
Keep the numbers as illustrations. Tax slabs, deposit rates and inflation all change, and the treatment of interest depends on the depositor's total income, so confirm the current figures before advising anyone. The shape does not change: the higher inflation and the tax rate are, the more a taxed nominal return overstates what the saver keeps.
Where candidates lose it
The common answer is 2%, eight minus six, which forgets tax entirely. The next is to apply the tax to the real return, 30% of 2 points, and say 1.4%. Both miss that the tax is charged on the nominal interest.
Take the tax off first, then deflate. Give the exact figure and the one-line lesson: in a high-inflation, high-tax setting, a deposit can lose purchasing power while showing a positive rate on the statement.
What the interviewer asks next
What deposit rate would give a zero real after-tax return here?
How does the answer change for a depositor in a 10% tax slab?
Why might an investor still hold the deposit despite a negative real return?
048Estimate how much cash is withdrawn from ATMs in India on an average day.Market sizing and estimationCoreIndian asset managementAsset management
Try it first
Which approach gives the strongest estimate in the room?
Show the worked solution
Roughly Rs 6,000 crore a day, on stated assumptions. Assume about 2.5 lakh ATMs, each handling about 80 withdrawals a day: 2 crore withdrawals. Check from demand: 30 crore regular users withdrawing twice a month is 60 crore a month, also about 2 crore a day. At an average of Rs 3,000 a withdrawal, the total is about Rs 6,000 crore a day. Confirm against published payment statistics for a real figure.
Why build the count from the machines and from the people?
If you wanted to know how many cups of tea a railway station sells, you could count the stalls and ask how many cups each pours, or count the passengers and ask how many buy one. A supply-side estimate and a demand-side estimate rest on different assumptions, so when they land close together, each one checks the other. Here both roads meet at about 2 crore withdrawals a day, which is the number that matters; the rupee value is one multiplication away.
2.5 lakh ATMs at 80 withdrawals a day and 30 crore users at two withdrawals a month both give about 2 crore withdrawals a day. At Rs 3,000 each that is about Rs 6,000 crore a day, and the ticket size alone moves the answer between Rs 4,000 crore and Rs 8,000 crore.
Which assumptions need the most care?
Every number here is an assumption to be stated, not a fact to be quoted. The average ticket size is the loosest input: Rs 2,000 against Rs 4,000 moves the answer from Rs 4,000 crore to Rs 8,000 crore a day, a factor of two. Withdrawals per machine vary widely, too, between a busy city branch and a rural machine, so treat 80 as an average across very different sites. The user count is the easiest to reason about from the number of adults with bank cards who still rely on cash.
Say what would change the answer over time. Growth in digital payments lowers withdrawal counts; festivals, salary days and month ends raise them sharply, so an average day hides large swings. Converting to a year, about 365 times the daily figure, is a useful sanity check against any published annual total the interviewer might mention.
For an asset management desk, the point is less the number than the method: a stock-and-flow structure, two independent builds, and a clear statement of which input you trust least. That is the same discipline as sizing a company's addressable market before building a revenue forecast.
Where candidates lose it
Candidates often start from currency in circulation, which is a stock of notes rather than a daily flow, or from the whole population, which counts people who never use an ATM. Both produce a number without a way to check it.
Build it once from the machines, once from the users, and show they agree before multiplying by the ticket size. Then name the ticket size as the input you are least sure of.
What the interviewer asks next
How would the estimate change on the day after salaries are paid?
How would you estimate the share of ATM withdrawals that digital payments have replaced?
How would a bank use this number to plan cash replenishment?

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

C	the annual coupon, 7
F	the face value repaid at maturity, 100
P	today's price, 95
n	years to maturity, 5

0.5^5	one manager's chance of five straight wins, 1 in 32
20	the number of independent managers

S_0	today's rate, 84 rupees per dollar
\pi_{IN}	Indian inflation, 5%
\pi_{US}	US inflation, 2%

Try it first

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

How do you estimate the gap without a spreadsheet?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the return on a discount bond come from?

Why divide by the average price rather than 95?

Where candidates lose it

What the interviewer asks next

Try it first

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

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

Where candidates lose it

What the interviewer asks next

Try it first

Why should higher inflation weaken a currency?

What does this mean for an investor holding assets abroad?

Where candidates lose it

What the interviewer asks next

Try it first

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

Why does it matter most in strong markets?

Where candidates lose it

What the interviewer asks next

Try it first

Why does betting more of a winning edge ever make you poorer?

How do you find the 10% and check the 20%?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a survivors-only average flatter the category?

How do you compute the true figure and the bias?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a one-point fall in the cap rate lift value by 14%, not by one point?

How does leverage turn 14% into 36%?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a positive-looking rate lose purchasing power?

How do you get the exact figure, and what does the tax really cost?

Where candidates lose it

What the interviewer asks next

Try it first

Why build the count from the machines and from the people?

Which assumptions need the most care?

Where candidates lose it

What the interviewer asks next