Portfolio Management puzzles, solved step by step

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  1. 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?Private and real asset mathsCoreReal 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%.

    Cash flows in and out, and the IRR split into income and growth-100Y0+7Y1+7Y2+7Y3+7Y4+117Y57 rent+ 110 saleRough split of the returnincome 7.0%+1.9%growth: 100 to 110 over 5 yearsRough sum: 8.9%Exact IRR8.7%Slightly below the rough sum: flat rentis a falling yield on a rising value.
    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 relationship
    100=∑t=157(1+r)t+110(1+r)5  ⇒  r≈8.7%100=\sum_{t=1}^{5}\frac{7}{(1+r)^t}+\frac{110}{(1+r)^5}\;\Rightarrow\; r\approx 8.7\%
    rthe internal rate of return
    7net rent each year, Rs crore
    110the 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?
  2. 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.

    The debt stays at Rs 60 crore, so every move in value lands on the equityDebt 60Equity 28.9Value 88.9Cap rate 9%value -11.1%equity -27.8%Debt 60Equity 40.0Value 100.0Cap rate 8%Debt 60Equity 54.3Value 114.3Cap rate 7%value +14.3%equity +35.7%NOI fixed at Rs 8 crore; debt fixed at 60% of the Rs 100 crore starting 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
    ΔE%=ΔV%×VE=14.3%×10040≈35.7%\Delta E\% = \Delta V\% \times \frac{V}{E} = 14.3\% \times \frac{100}{40} \approx 35.7\%
    Vthe starting property value, Rs 100 crore
    Ethe 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?
  3. 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?Private and real asset mathsCoreInvescoNew 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.

    The same 2.0x, earned over different lengths of time1234567891010%20%30%40%3 years: 26.0% a year7 years: 10.4% a yearYears to get 2.0x backBoth deals2.0x moneyTo match 26%over 7 years5.0xis needed,not 2.0x
    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 relationship
    IRR=M1/T−1:21/3−1=26.0%,21/7−1=10.4%\text{IRR} = M^{1/T} - 1: \quad 2^{1/3} - 1 = 26.0\%, \quad 2^{1/7} - 1 = 10.4\%
    Mmultiple of money, 2.0
    Tyears 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