Portfolio Management puzzles, solved step by step
- Puzzles
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- Topics
- 13
- Hard
- 30
086Two private deals each return 2.0 times the money invested, in a single payout at the end. One pays out after three years, the other after seven. What IRR does each earn?InvescoNew York · 2025
Try it first
Roughly, what IRR does the seven-year deal earn?
Show the worked solution
About 26% a year for the three-year deal and 10.4% for the seven-year deal. The IRR of a single payout is the multiple to the power one over the years, less one: 2 to the one third and 2 to the one seventh. Same money back, very different speed. The multiple says how much, the IRR says how fast, and a deal needs both.
Why does the same multiple give such different returns?
Two friends each double their savings; one takes three years, the other seven. Nobody would call them equally good investors. A multiple of money ignores time entirely, while IRR is a rate per year, so the same 2.0x falls from 26% to 10.4% as the wait stretches from three years to seven. The curve is steep early: each extra year cuts the IRR most when the holding period is short.
A fixed 2.0 times multiple earns about 26% a year if it arrives in three years and only 10.4% if it takes seven; to match the three-year IRR, the seven-year deal would need about 5.0 times the money. The relationshipM multiple of money, 2.0 T years until the single payout What it says in wordsFor one cash flow in and one out, the IRR is the yearly growth rate that turns the investment into the multiple.So which number should an investor care about?
Both, because each can be gamed alone. A short, quick flip can post a high IRR on a small absolute gain, while a long hold can post a big multiple at a return below the cost of capital. Limited partners look at IRR for speed and at the multiple, often called the equity multiple or MOIC, for how much wealth was actually created. To match 26% over seven years, the second deal would need about 5.0 times the money, not 2.0.
One caution for the room: this clean formula holds only with one outflow and one inflow. With interim distributions, the IRR assumes those cash flows are reinvested at the IRR itself, which is rarely true, so a quoted IRR on a deal with early payouts flatters it.
Where candidates lose it
The common slip is linear: 100% gain over seven years is about 14% a year. That forgets compounding and overstates the return by more than three points.
The second loss is treating a higher IRR as automatically better. Say that a 26% IRR on a small cheque held briefly can create less wealth than 10% on a large one held for years.
What the interviewer asks next
- What multiple does a 20% IRR produce over five years?
- A deal returns 1.5x in one year. Is that better than 2.0x in three?
- Why do sponsors sometimes use a credit line to delay capital calls, and what does it do to IRR?
Asked at Invesco, Real Estate, New York, 2025 (Wall Street Oasis):
Lots of basic questions asked about IRR, EM, Cap Rates etc
087Estimate how many new passenger cars are sold in India in a year.AllianceBernsteinNew York · 2021
Try it first
Where should the estimate start?
Show the worked solution
Roughly 40 lakh new cars a year, built from stated assumptions. About 30 crore households, 8% owning a car, gives about 2.4 crore cars. Ownership rising half a point a year adds 15 lakh first-time buyers; a 12-year life means 20 lakh replacements; fleets and second cars add about 5 lakh. Then compare with published industry sales data.
How do you turn a stock of cars into yearly sales?
A school's intake each year is not its total strength; it is the students leaving that year plus any growth in the roll. New car sales work the same way: they are the growth in the number of cars owned plus the cars being replaced, so the answer needs a stock and two flows. Say this structure first; the interviewer is scoring the tree more than the final number.
About 30 crore households with 8% owning a car gives a stock of about 2.4 crore cars, and new sales come from 15 lakh first-time buyers, 20 lakh replacements and about 5 lakh fleet and second cars, roughly 40 lakh a year. Which assumption would you defend first, and which would you flag?
Take the population as roughly 140 crore and confirm the current figure; at about 4.7 people per household that is 30 crore households. The 8% ownership share is the number to flag, because ownership is concentrated in cities and in the top income bands. Replacement is the largest flow, so the assumed 12-year life matters most: at 10 years replacements rise to 24 lakh, at 15 they fall to 16.
Assumption Base If wrong Effect on the total Households owning a car 8% 10% Replacement rises to 25 lakh Yearly rise in ownership 0.5 point 0.3 point First-time buyers fall to 9 lakh Life of a car 12 years 10 years Replacement rises to 24 lakh Fleet and second cars 15% on top 25% on top Adds about 3.5 lakh The total is most sensitive to the replacement branch, because it is the largest flow and rests on two soft assumptions, ownership and car life. Finish by naming what you would check. Industry bodies publish domestic sales every month, and an interviewer at a fund expects you to know the order of magnitude. If your estimate is far off, say which branch you would revisit rather than adjusting the total to fit.
Where candidates lose it
The common loss is starting from population times the share who want a car. That sizes the stock of potential owners, not the yearly flow, and the answer comes out ten times too big.
The second loss is mixing units: crore households and lakh sales in the same sentence without saying so. Label the stock in crore and the flows in lakh, and read the tree back once before giving the total.
What the interviewer asks next
- How would the answer change if used cars absorbed most first-time buyers?
- Size the market for two-wheelers the same way.
- Which listed-sector metric would you track to see whether your replacement assumption holds?
Asked at AllianceBernstein, Investment Banking, New York, 2021 (Wall Street Oasis):
Case study market sizing question
088An n by n by n cube, like a Rubik's cube, is built from small cubes. As a formula in n, how many of the small cubes show at least one face on the outside?T. Rowe PriceBaltimore · 2020
Try it first
For n = 3, a standard Rubik's cube, how many small cubes show a face?
Show the worked solution
n cubed minus (n minus 2) cubed, which expands to 6n squared minus 12n plus 8. Everything not on the surface forms a hidden cube with one layer peeled from each side, so its side is n minus 2. For n = 3 that is 27 minus 1, or 26; for n = 10 it is 488. If the interviewer means the coloured squares instead, the answer is 6n squared.
Why count what you cannot see?
To count the tiles round the edge of a courtyard, it is easier to measure the whole yard and subtract the inner lawn than to walk the border without counting corners twice. The cubes not on the surface form one clean block of side n minus 2, so subtracting it from n cubed avoids every double-counting trap at the edges and corners.
In a 3 by 3 by 3 cube only the centre cube is hidden, so 26 of the 27 show a face; the same subtraction of a hidden core of side n minus 2 gives 8, 26, 56, 98 and 488 for n equal to 2, 3, 4, 5 and 10. The relationshipn^3 all the small cubes (n-2)^3 the hidden core after peeling one layer from every side What it says in wordsSurface cubes are all cubes less the core you cannot see.How do you check the formula a second way?
Count by type. There are always 8 corner cubes, 12 edges each carrying n minus 2 cubes between the corners, and 6 faces each with an (n minus 2) by (n minus 2) centre. That is 8 plus 12(n minus 2) plus 6(n minus 2) squared, which expands to 6n squared minus 12n plus 8. Two routes agreeing is the check the interviewer wants to hear.
Two edge cases show care. The formula needs n of at least 2; a single cube, n = 1, is all surface, and the formula would wrongly give 2. And the question is sometimes asked as surface area: the number of coloured squares is 6n squared, 54 for a standard cube. Ask which is meant before answering.
Where candidates lose it
The fast wrong answer is 6n squared, which counts the stickers, not the cubes: every edge cube is counted twice and every corner three times. On a standard cube that gives 54 against the true 26.
The second loss is giving the formula with no check. Say the corner, edge and face split once; it takes ten seconds and proves the algebra.
What the interviewer asks next
- How many small cubes show exactly two faces, as a formula in n?
- For what n are more than half the cubes hidden?
- Now the cube is n by n by m. Generalise the count.
Asked at T. Rowe Price, Equities, Baltimore, 2020 (Wall Street Oasis):
Give an equation that yields the surface area of an n by n by n Rubic's cube based on number of blocks per side.
