Portfolio Management puzzles, solved step by step

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

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011A fund compounds at 15% a year for ten years. What share of the total gain over the decade arrives in the last three years?Compounding and fee dragCoreAsset managementWealth management
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Guess before you calculate.
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About 45%. One rupee at 15% grows to 2.66 after seven years and 4.05 after ten. The total gain is 3.05, and the last three years add 4.05 minus 2.66, which is 1.39. That is 45.5% of the decade's gain in 30% of the time, because each year's 15% is earned on a larger base than the year before.
Why is the gain not spread evenly across the years?
Think of a snowball rolled down a long slope. In the first few metres it picks up a little snow; near the bottom each turn picks up much more, because the ball itself is bigger. Compounding earns the same rate on a growing base, so every year's rupee gain is 15% larger than the year before. Year 1 adds 0.15 on each rupee. Year 10 adds 0.53, three and a half times as much. The first three years together add only 0.52; the last three add 1.39.
At an assumed 15% a year, each rupee gains 0.15 in year 1 but 0.53 in year 10, so the last three years add 1.39 of the 3.05 total gain, 45.5% of it.
The relationship
$\frac{(1.15)^{10}-(1.15)^{7}}{(1.15)^{10}-1}=\frac{4.046-2.660}{3.046}\approx 0.455$
(1.15)^{10} the value of one rupee after ten years
(1.15)^{7} the value after seven years
(1.15)^{10}-1 the total gain over the decade
What it says in wordsThe last three years' share is the growth from year seven to year ten, divided by the growth over all ten.
What does this mean for an investor who leaves early?
Someone who exits after seven years has sat through 70% of the time but collected only 55% of the decade's gain. Because compounding back-loads the reward, leaving a long plan early costs far more than the fraction of time given up. The same arithmetic runs against the investor with fees: a charge taken every year compounds too, and its cost is also concentrated at the end. Say the limitation plainly: 15% is an assumed rate for the arithmetic, not a forecast, and real returns arrive unevenly, so the actual last three years could be the worst three.
Where candidates lose it
The trap is answering 30%, proportional to time, because the question sounds like a fraction of a decade. The interviewer is checking whether you picture compounding as a curve.
The other slip is dividing the last three years' gain by the final value, {P11_END:.2f}, rather than by the total gain, {P11_TOT:.2f}. Read the question again: it asks for a share of the gain, not of the ending pot.
What the interviewer asks next
At what rate would the last three years carry exactly half the gain?
A 1.5% annual fee is taken throughout. What share of the lost wealth falls in the last three years?
Why do long-horizon savers care more about the final years' return than the first years'?
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?
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?
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
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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?
057Fund A has an average annual return of 9% with 10% volatility. Fund B averages 11% with 30% volatility. Over the long run, which one compounds faster, and by roughly how much?Compounding and fee dragCoreMulti-assetWealth management
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Which fund grows a rupee more over twenty years?
Show the worked solution
Fund A, by about two points a year: roughly 8.5% against 6.5%. Long-run compound growth is approximately the average return less half the variance. For A that is 9% less half of 0.10 squared, 0.5 points. For B it is 11% less half of 0.30 squared, 4.5 points. Over twenty years a rupee in A grows to about Rs 5.11 and in B to about Rs 3.52.
Why is the average return not what you earn?
A shop that marks a price up 30% and then down 30% does not end where it started. Rs 100 becomes Rs 130 and then Rs 91, although the average move is zero. Returns compound by multiplying, so a swing up and an equal swing down always leave you behind, and the loss grows with the size of the swing. For moderate returns the loss is close to half the variance, half of volatility squared. That is why a fund's compound annual growth sits below its average annual return, and the gap widens sharply as volatility rises.
The relationship
$g \approx \mu - \tfrac{1}{2}\sigma^2 \qquad A: 9 - 0.5 = 8.5\% \qquad B: 11 - 4.5 = 6.5\%$
g long-run compound growth rate a year
mu the average annual return
sigma annual volatility, as a decimal
What it says in wordsCompound growth is roughly the average return less half the square of volatility.
Fund B has the higher average return but loses 4.5 points a year to volatility against half a point for fund A, so A compounds at about 8.5% and B at about 6.5%, and a rupee grows to Rs 5.11 in A against Rs 3.52 in B over twenty years.
How precise is the half-variance rule?
It is an approximation, good when returns and volatility are modest, and it overstates the drag when volatility is large. A more careful calculation that treats returns as lognormal puts A at 8.5% and B at 7.2%. The size of B's handicap moves with the method, but the ranking does not: the higher-volatility fund compounds more slowly. Say that in the room. The rule is a tool for spotting the direction and rough size, not a precise forecast.
The portfolio consequence is the reason the question exists. Two funds with the same average return are not equal if one is far more volatile, and a client who holds a single volatile fund lives with the compound number, not the average in the brochure. Combining volatile assets that do not move together lowers portfolio variance, which is one of the few ways to raise compound growth without raising the average return.
Where candidates lose it
The trap is picking B because 11% beats 9%. Candidates read the average return as the growth rate and treat volatility as a separate risk topic, when here it is a direct cost to the return itself.
The second loss is subtracting the volatility rather than half its square, which gives nonsense for B. Write the rule down, square first, halve second, and give both numbers.
What the interviewer asks next
At what volatility would fund B's compound growth fall to match fund A's?
Why do fund fact sheets that show the average annual return flatter volatile funds?
How does rebalancing between two volatile assets relate to this drag?
070A portfolio has fallen 40% from its peak. What annual return does it need to get back to that peak within three years?Compounding and fee dragCoreWealth managementRisk management
Try it first
Pick the annual return needed before you calculate.
Show the worked solution
About 18.6% a year. After a 40% fall, 100 has become 60, and getting back to 100 needs a gain of 40 on 60, which is 66.7%. Spread over three years with compounding, the annual return is the cube root of 100 over 60, about 1.186, so 18.6% a year. Dividing the loss or the gain by three gives the wrong answer both ways.
Why does a 40% loss need more than 40% to repair?
A tree that loses 40% of its branches does not regrow them at the rate it lost them; it regrows from what is left. A loss is measured against the old, larger value, but the recovery is measured against the new, smaller one, so the gain needed is always bigger than the loss. Here 100 falls to 60. Getting back to 100 needs 40 on a base of 60, which is 66.7%. The same effect grows quickly: a 60% loss needs 150%.
The gain needed to recover rises faster than the loss, from 25% after a 20% fall to 66.7% after a 40% fall and 150% after a 60% fall, and repairing the 40% fall in three years takes about 18.6% a year.
The relationship
$r = \left(\frac{1}{1-0.40}\right)^{1/3} - 1 = 1.667^{1/3} - 1 = 18.6\%$
0.40 the drawdown, as a decimal
1/(1-0.40) the multiple needed to get back to the peak, 1.667
1/3 spread over three years with compounding
What it says in wordsFind the multiple needed to recover, then take the root for the number of years.
What should you say to a client after the number?
That 18.6% a year for three years is a demanding return for most portfolios, and chasing it by taking more risk can deepen the hole. The honest conversation is about the time it takes to recover, not about finding a return high enough to recover quickly. Allow five years and the requirement falls to about 10.8% a year. The deeper lesson for portfolio construction is that avoiding large drawdowns is worth more than it looks, because the repair costs rise so steeply.
Where candidates lose it
The trap is answering 13.3%, the 40% loss divided by three years. It makes two mistakes at once: it measures the gain on the wrong base and it ignores compounding.
The half-right answer, 22.2%, gets the 66.7% right and then divides by three. Take the cube root, and say both steps out loud so the interviewer hears you knew each one.
What the interviewer asks next
What loss would need exactly 100% to recover, and why?
If the portfolio earns 10% a year from here, how long does recovery take?
Why do risk managers set drawdown limits well before the loss gets large?
090Two retirees each start with Rs 1 crore and withdraw Rs 10 lakh at the end of every year. One earns minus 20%, plus 10% and plus 30% over three years; the other earns the same returns in reverse order. Who ends with more, and by how much?Compounding and fee dragCoreRetirement and pensionsWealth management
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Who ends with more?
Show the worked solution
The retiree with the good year first ends with Rs 87.6 lakh, against Rs 77.1 lakh, a gap of Rs 10.5 lakh. Without withdrawals both would end at Rs 114.4 lakh, because the order of multiplication does not matter. Withdrawals break that: in the bad-first order, the money taken out early misses the strong returns that come later.
Why does the order suddenly matter?
A shop owner who must pay a fixed rent every month is hurt more by a bad first quarter than a bad last one: an early slump forces them to dip into stock just when prices are low. Without withdrawals, 0.8 x 1.1 x 1.3 gives the same answer in any order; with fixed withdrawals, each rupee taken out misses every return after it, so losses early in retirement do lasting damage.
Starting from Rs 1 crore with Rs 10 lakh withdrawn each year, the bad-year-first path ends at Rs 77.1 lakh and the good-year-first path at Rs 87.6 lakh, although both would reach Rs 114.4 lakh with no withdrawals.
Where exactly does the Rs 10.5 lakh go?
Think of each withdrawal as money that would otherwise have kept compounding. The ending balance equals the no-withdrawal result less each withdrawal grown at the returns it missed. In the bad-first order the first Rs 10 lakh misses a 10% and a 30% year, so it costs 14.3 lakh of ending wealth; in the good-first order it misses a 10% gain and a 20% fall, so it costs only 8.8.
Withdrawal Bad year first: cost at the end Good year first: cost at the end
End of year 1 14.3 8.8
End of year 2 13.0 8.0
End of year 3 10.0 10.0
Total taken from the no-withdrawal Rs 114.4 lakh 37.3 26.8
Each Rs 10 lakh withdrawal costs the ending balance its own value grown at the returns it missed, which totals Rs 37.3 lakh in the bad-first order and Rs 26.8 lakh in the good-first order.
This is sequence-of-returns risk, and it is why retirement planners hold a cash or short-bond buffer for the first few years of withdrawals, or vary withdrawals with markets. The same logic runs in reverse for someone still saving: early losses hurt a saver less, because little money is invested yet.
Where candidates lose it
The trap is answering that order cannot matter because multiplication commutes. That is true for a lump sum left alone and exactly wrong once money moves.
The second loss is getting the two balances without explaining why. Give the one-line mechanism: each withdrawal misses the returns that follow it.
What the interviewer asks next
What happens to the gap if the withdrawal doubles to Rs 20 lakh?
Does the order matter for someone making equal yearly contributions instead?
How would you protect a new retiree against a bad first year?

(1.15)^{10}	the value of one rupee after ten years
(1.15)^{7}	the value after seven years
(1.15)^{10}-1	the total gain over the decade

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

i	the deposit rate, 8%
t	the tax rate on interest, 30%
\pi	inflation, 6%

g	long-run compound growth rate a year
mu	the average annual return
sigma	annual volatility, as a decimal

Try it first

Why is the gain not spread evenly across the years?

What does this mean for an investor who leaves early?

Where candidates lose it

What the interviewer asks next

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

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 is the average return not what you earn?

How precise is the half-variance rule?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a 40% loss need more than 40% to repair?

What should you say to a client after the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the order suddenly matter?

Where exactly does the Rs 10.5 lakh go?

Where candidates lose it

What the interviewer asks next