Portfolio Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 31
- Topics
- 13
- Hard
- 30
009You buy a property for Rs 100 crore, collect Rs 7 crore of net rent at the end of each year for five years, and sell it at the end of year five for Rs 110 crore. What is the IRR?Real estate investmentReal assets
Try it first
Pick the closest IRR before you calculate.
Show the worked solution
About 8.7%. The IRR is the discount rate at which Rs 7 crore a year for five years plus Rs 110 crore at year five is worth exactly the Rs 100 crore paid. A quick estimate adds the 7% income yield to the annual growth in value, 100 to 110 over five years or 1.9% a year, which gives 8.9%. The exact answer is a little lower, 8.68%.
What is the IRR actually solving for?
Think of a savings account that pays you Rs 7 of interest each year on Rs 100 and then hands back Rs 110. You want the single interest rate that account must have been paying. The IRR is the one discount rate that makes the present value of everything you receive equal to what you paid. Try 9%: the rents are worth about 27.2 and the sale about 71.5, a total of 98.7, below 100, so 9% is too high. Try 8.5%: the total is about 100.7, so the answer sits between them, at 8.68%.
The property pays out 100 at year 0 and brings in 7 a year of rent plus 110 at the sale, and its IRR of 8.7% sits just under the rough sum of a 7.0% income yield and 1.9% a year of value growth. The relationshipr the internal rate of return 7 net rent each year, Rs crore 110 the sale price at year five, Rs crore What it says in wordsThe IRR is the rate that discounts the rents and the sale price back to the Rs 100 crore paid.Why is the rough split a little too high?
The rule of thumb, IRR roughly equals income yield plus growth, is exact only when the rent grows at the same rate as the value, so the yield stays at 7%. Here the rent is flat while the value rises, so by year five the rent is only 6.4% of the property's worth, and the average yield across the hold is below 7%. That is why the answer is 8.68% rather than 8.92%. In the room, give the rough split first to show the structure, then the exact figure. Say the limitation too: an IRR assumes the rents can be reinvested at the IRR itself, and it says nothing about how much money was at work.
Where candidates lose it
The common error is adding 7% of rent to 2% a year of price gain, calling it 9%, and stopping. That treats the gain as simple interest and ignores that it arrives only at the end.
The other way to lose the point is to try to solve the equation exactly out loud. Bracket it: 9% gives less than 100, 8.5% gives more, so the answer is about 8.7%.
What the interviewer asks next
- What is the IRR if the property sells for Rs 100 crore instead?
- What if the rent rises 2% a year in step with the value?
- How would 60% debt at 8% change the equity IRR?
022A private fund calls Rs 100 crore from an investor today and returns Rs 200 crore in five years, an IRR of about 14.9%. If the fund instead uses a credit line to delay the call by one year, at a borrowing cost of Rs 8 crore paid out of the final distribution, what happens to the IRR and to the multiple of money?Private markets
Try it first
With the credit line, what happens?
Show the worked solution
The IRR rises to about 17.7% while the multiple falls from 2.00x to 1.92x. With the line, the investor pays Rs 100 crore at year 1 instead of year 0 and receives Rs 192 crore at year 5 after the borrowing cost. That is 1.92 times the money over four years, 17.7% a year, against 2.00 times over five years, 14.9%. The investor ends with Rs 8 crore less, and the reported IRR looks better.
How can the return rise when the investor gets less money?
Imagine lending a friend money: getting back Rs 192 after four years can be a better annual rate than Rs 200 after five, even though Rs 200 is more money. IRR measures speed, not size, so anything that shortens the time the investor's money is out raises it, even at a cost. A subscription lineA short-term loan to a private fund, secured on investors' commitments, used to delay capital calls. does exactly that. The fund's deal is identical; only the investor's clock starts a year later.
Without the credit line the investor pays 100 at year 0 and gets 200 at year 5, an IRR of 14.9% and 2.00x; with it the investor pays 100 at year 1 and gets 192 at year 5, an IRR of 17.7% but only 1.92x. The relationship200, 192 the distribution at year 5 without and with the line, Rs crore 1/5, 1/4 one over the years the investor's money is at work What it says in wordsWith a single call and a single distribution, the IRR is the multiple raised to one over the years, minus one.Is the investor better or worse off?
It depends on what the investor does with the Rs 100 crore during the extra year. If the idle money earns less than the Rs 8 crore the line costs, the investor is worse off even though the fund reports a higher IRR. That is why allocators look at the multiple and the IRR together, and increasingly ask for IRRs calculated both with and without the effect of credit lines. Say the limitation: this example uses one call and one distribution; real funds call and return money in many pieces, and the line's effect on IRR is largest in the early years of a fund.
Where candidates lose it
Candidates say both numbers fall, because the line costs money. They miss that IRR is time-weighted and rewards a later call.
The deeper trap is stopping at the arithmetic. The interviewer on a private markets desk wants to hear that a higher IRR here does not mean a better result for the investor, and that the multiple exposes it.
What the interviewer asks next
- What if the line delays the call by two years at a cost of Rs 16 crore?
- What return must the investor earn on the idle Rs 100 crore to break even?
- Why do some investors prefer to see a fund's multiple before its IRR?
031An office building has gross potential rent of Rs 10 crore a year. Vacancy runs at 8%, and operating costs are 30% of the rent actually collected. At an 8% cap rate, what is the building worth? Walk through it from gross potential rent.InvescoNew York · 2025
Try it first
Which number does the cap rate divide?
Show the worked solution
About Rs 80.5 crore. Vacancy of 8% takes Rs 10 crore of gross potential rent down to Rs 9.2 crore collected. Operating costs of 30% of that are Rs 2.76 crore, leaving net operating income of Rs 6.44 crore. Divide by the 8% cap rate: 6.44 over 0.08 is Rs 80.5 crore, or 12.5 times the income.
Why does the cap rate price net operating income and not the rent?
Think of buying a shop that a friend runs. You would not pay for the sales the shop could make if every shelf sold out; you pay for what is left after empty days and the electricity bill. A cap rate is net operating income divided by value, so the value is only ever as good as the income that actually reaches the owner after vacancy and running costs. Gross potential rent is the ceiling, not the income.
Rs 10 crore of gross potential rent loses Rs 0.8 crore to vacancy and Rs 2.76 crore to operating costs, leaving Rs 6.44 crore of net operating income. At an 8% cap rate that income is worth Rs 80.5 crore. What are the lines between gross rent and value, in order?
Say them as a ladder. Gross potential rent is what a full building earns at the rent roll. Take off vacancy and bad debt to get effective gross incomeThe rent a building actually collects after vacancy and unpaid rent, before any running costs.. Take off operating expenses, such as maintenance, insurance, property tax and management, to get net operating income. Every line above NOI moves the price by 12.5 times its size at an 8% cap rate, so a small leak near the top is a large leak in value. One extra point of vacancy costs Rs 0.1 crore of rent, Rs 0.07 crore of NOI, and Rs 0.875 crore of value.
The relationshipGPR gross potential rent, Rs 10 crore v vacancy rate, 8% o operating costs as a share of collected rent, 30% c the cap rate, 8% What it says in wordsValue is the income that survives vacancy and costs, divided by the yield buyers demand.For an exit value, the same arithmetic runs on the NOI expected in the sale year and an exit cap rate, which analysts often set a little above the entry cap rate to allow for an older building. Say which cap rate you are using and why, and say that capital spending such as a new roof sits below NOI and is not captured by this shortcut.
Where candidates lose it
The costly slip is dividing gross potential rent by the cap rate, which gives Rs 125 crore and overpays by Rs 44.5 crore. The other is applying the 30% cost ratio to gross rent rather than collected rent, which gives NOI of Rs 6.2 crore and a value Rs 3 crore too low.
Walk the ladder aloud, line by line, and name what each deduction is. Interest never appears: it depends on how the buyer finances the building, not on the building.
What the interviewer asks next
- The buyer expects NOI to grow 3% a year and plans to sell in five years at an 8.5% exit cap rate. What is the exit value?
- Why might a buyer's cap rate for this building differ from the seller's?
- What happens to value if operating costs rise to 35% of collected rent?
Asked at Invesco, Real Estate, New York, 2025 (Wall Street Oasis):
Walk me through how to get the exit value of a property from Gross Potential rent, using Cap rate.
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?Real 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 relationshipV 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?
054Without a calculator: what IRR turns money into 2.5 times in four years, and what IRR turns it into 3 times in five years?Neuberger BermanLondon · 2026
Try it first
Which pair is closest?
Show the worked solution
About 26% and about 25%. For 2.5 times in four years, take the square root of 2.5 twice: 1.58 and then 1.257, so 25.7% a year. For 3 times in five years, anchor on the fact that 2 times in three years is 26%; 3 times in five sits just below it at 24.6%. The bigger multiple over the longer hold is actually the lower annual return.
Why is the simple average so far off?
A savings account that pays 26% a year does not add 26 rupees every year to your 100. It adds 26 in year one, then 26% of 126 in year two, and so on, so by year four you have about 250. The IRR is the steady annual rate that compounds to the multiple, so it is always well below the total gain divided by the years. The simple average for 2.5 times in four years would be 150% over 4, or 37.5%, which overstates the true 25.7% by more than ten points.
The relationshipM the money multiple, cash back over cash in n the years held, with one cash flow in and one out What it says in wordsThe IRR is the nth root of the multiple, less one.How do you get there in your head?
Use one of two tricks. For four years, take the square root twice: the square root of 2.5 is about 1.58, and the square root of 1.58 is about 1.257. For odd holding periods, use logs: the natural log of 3 is about 1.10, divided by five is 0.22, and adding a little for compounding turns 22% into about 24.6%. Faster still is a small grid of anchors you know by heart, because interviewers ask the same handful of multiples and periods.
A grid of IRRs by multiple and holding period shows 2.5 times in four years at 25.7% and 3 times in five years at 24.6%, and a band of 2 times in three, 2.5 times in four and 3 times in five years all near 25%. Notice the diagonal band. Two times in three years, two and a half in four and three in five all land within a point of 25%. That one pattern lets you place almost any private equity outcome in a second. Say the limitation too: this shortcut assumes a single cash flow in and a single cash flow out. Real funds call and return money in stages, and the IRR then depends on the timing, not just the multiple.
Where candidates lose it
The trap is dividing the gain by the years, 150% over four and 200% over five, and answering 37.5% and 40%. It sounds confident and is wrong by more than ten points, and a private markets interviewer hears it as not understanding compounding.
The quieter miss is thinking 3 times must be the better deal because the multiple is bigger. The extra year costs more than the extra half turn of money earns.
What the interviewer asks next
- What multiple does a 20% IRR give over five years?
- A fund returns 2 times in three years and another 2.5 times in six. Which would you rather have, and what else would you ask?
- Why can two deals with the same multiple and the same holding period show different IRRs?
Asked at Neuberger Berman, Generalist, London, 2026 (Wall Street Oasis):
The associate interview was quite technical, covered 5-7 questions on valuation and understanding of returns in private equity
067A sponsor buys a business at 8 times EBITDA of Rs 100 crore, funded 50% with debt. EBITDA grows 8% a year, and Rs 40 crore of free cash flow repays debt every year. What exit multiple after five years gives the sponsor 2.5 times its money?Neuberger BermanNew York · 2022Neuberger BermanNew York · 2022
Try it first
Roughly what exit multiple does 2.5 times the money need?
Show the worked solution
About 8.2 times, barely above the 8 times paid. Entry value is Rs 800 crore, half debt, so equity is Rs 400 crore and the target is Rs 1,000 crore. Five years of Rs 40 crore repayments leave Rs 200 crore of debt, so the exit value must be Rs 1,200 crore. Year 5 EBITDA is 100 times 1.08 to the fifth, Rs 146.9 crore, and 1,200 over that is 8.17 times.
Why work backwards from the target?
Think of saving for a house: you start from the price you must reach and work out what monthly saving gets you there, instead of guessing savings and hoping. A paper LBO question that fixes the return is solved from the exit backwards, because the target return pins the equity value, and everything else follows from it. Equity in is half of Rs 800 crore, Rs 400 crore. 2.5 times that is Rs 1,000 crore of equity at exit.
The sponsor needs Rs 1,000 crore of equity and still owes Rs 200 crore of debt, so the business must sell for Rs 1,200 crore, which on year 5 EBITDA of Rs 146.9 crore is an exit multiple of 8.17 times. Which step do candidates drop?
The debt still owed. The equity holders only get what is left after the lenders are repaid, so the enterprise value at exit is the equity target plus the remaining debt: Rs 1,000 crore plus Rs 200 crore. Enterprise value belongs to lenders and owners together, so you always add the debt back before dividing by EBITDA. Dividing Rs 1,000 crore alone by Rs 146.9 crore gives 6.8 times and a wrong story about a deal that works even with a lower multiple.
The relationship2.5 x 400 the equity the sponsor needs back, Rs crore 400 - 5 x 40 debt still owed after five annual repayments 100 x 1.08^5 year 5 EBITDA, Rs crore What it says in wordsThe exit multiple is the equity target plus remaining debt, divided by exit EBITDA.Of the Rs 600 crore gain, EBITDA growth valued at the entry multiple supplies Rs 375 crore and debt paydown Rs 200 crore, so the multiple only needs to add Rs 25 crore, a rise from 8.0 to 8.17 times. Then give the view. 2.5 times in five years is an IRR of about 20%, and this deal gets there almost entirely from growth and paydown. That is the comfortable kind of LBO: the return does not depend on a buyer paying more than the sponsor did. Say the simplifications as well: free cash flow is held flat at Rs 40 crore although EBITDA grows, and fees, interest on the debt and taxes are folded into that figure.
Where candidates lose it
The trap is forgetting the Rs 200 crore of debt still outstanding and dividing the equity target by EBITDA, which gives 6.8 times and makes the deal look safer than it is. The exit value must cover the lenders before the sponsor sees a rupee.
The second loss is stopping at the number. The interviewer wants to hear that 8.2 times against 8 times paid means the return is driven by growth and paydown, not by hoping for multiple expansion.
What the interviewer asks next
- What IRR does the deal earn if the exit multiple falls to 7 times?
- How does the answer change if the sponsor uses 60% debt and repays the same Rs 40 crore a year?
- Why might free cash flow for debt repayment not stay flat at Rs 40 crore as EBITDA grows?
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis):
The most difficult was the more advanced industry-specific technicals and paperback LBOs
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis):Interviews 4-5 were very technical again and also included multiple paperback LBOs and other, more advanced technicals.
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
099Project A costs Rs 100 crore and returns Rs 130 crore after one year. Project B costs Rs 100 crore and returns Rs 20 crore a year for ten years. You can do only one. Which ranks higher on IRR, which on NPV at a 10% cost of capital, and which would you take?PIMCOMunich · 2024
Try it first
At a 10% cost of capital, which project creates more value?
Show the worked solution
A wins on IRR, 30% against about 15%; B wins on NPV at 10%, Rs 22.9 crore against Rs 18.2 crore. Take B. IRR measures a rate, NPV measures rupees of value at your actual cost of capital. The two disagree because A returns its money in one year and B keeps earning for ten. Above about 11%, the ranking flips.
How can one project have the higher return and the other create more value?
Would you rather earn 30% on Rs 100 for a day, or 15% on Rs 100 for ten years? The first is a better rate; the second makes you more money. IRR is a rate per year and says nothing about how long the money earns it, while NPV counts the rupees created at your cost of capital, so when projects differ in timing the two can rank them in opposite order.
Project B's NPV falls steeply with the discount rate from Rs 100 crore at 0% and crosses zero at its 15.1% IRR, while A's starts at Rs 30 crore and crosses at 30%; the lines meet at about 11.2%, so at a 10% cost of capital B is worth Rs 22.9 crore to A's Rs 18.2 crore. The relationship130 A's single payout after one year, Rs crore 20 B's yearly payout for ten years, Rs crore 10% the cost of capital What it says in wordsDiscount each project's cash at the cost of capital and subtract what it costs.When would A be the right choice after all?
Two cases. If the cost of capital is above about 11.2%, the crossover rate, A's NPV is the higher one; and if A's Rs 130 crore could be put straight into another project earning well above 10%, the pair of projects might beat B. Both are really statements about the reinvestment rate. IRR implicitly assumes A's proceeds are reinvested at 30%; NPV assumes the cost of capital, which is usually the more honest assumption.
In real estate and private markets this is the everyday tension between a quick flip and a long hold. Give both numbers, name the crossover, and say that for mutually exclusive projects NPV decides.
Where candidates lose it
The common slip is picking A because 30% beats 15%. With mutually exclusive projects of different lengths, IRR ranks rates, not value, and the interviewer asks this to see if you know the difference.
The second loss is stopping at take B without saying when that changes. Name the crossover rate and the reinvestment assumption; that is the part that sounds like an investor.
What the interviewer asks next
- At what cost of capital are you indifferent between the two?
- What if B's cash flows ran for twenty years instead of ten?
- Why might a fund manager paid on IRR prefer A anyway?
Asked at PIMCO, Real Estate, Munich, 2024 (Wall Street Oasis):
just asked a bunch of questions on recent real estate news, as well as a couple of technicals including IRR, NPV
