Private Equity puzzles, solved step by step

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  1. 076An investment is funded half with debt at 7% and half with equity. The asset earns 12% a year before financing, and tax is ignored. What return does the equity earn? What if the asset earns only 5%?Leverage and capital structureCoreHPS Investment Partnersnew york · 2024

    Try it first

    Before you calculate: with the asset at 12%, what does the equity earn?

    Show the worked solution

    Equity earns 17% when the asset earns 12%, and only 3% when it earns 5%. Per Rs 100 of asset, lenders take a fixed 3.5. At 12% equity keeps 8.5 on its 50; at 5% it keeps 1.5 on its 50. Leverage adds the spread between the asset return and the cost of debt, times debt over equity, in both directions.

    Why does borrowing push the equity return above the asset return?

    Think of buying a flat for Rs 100 lakh with Rs 50 lakh of your own money and a Rs 50 lakh loan at 7%. If the flat earns 12 lakh a year in rent and value, the bank takes 3.5 lakh and everything else is yours. Lenders get a fixed amount whatever the asset earns, so any return above the cost of debt on the borrowed half lands on the equity. The borrowed 50 earns 12% and costs 7%; that 5 point spread on 50 is 2.5, added to the 6 the equity's own half earns.

    Lenders take a fixed 3.5; equity keeps whatever is leftAsset earns 12%12Lenders: 7% x 50 = 3.5Equity keeps 12 - 3.5 = 8.58.5 on equity of 50= 17% return on equityUnlevered 12%, levered 17%: +5 pointsAsset earns 5%5Lenders: 7% x 50 = 3.5Equity keeps 5 - 3.5 = 1.51.5 on equity of 50= 3% return on equityUnlevered 5%, levered 3%: -2 pointsEach bar is Rs 100 of asset: Rs 50 of debt at 7%, Rs 50 of equity
    On Rs 100 of asset funded half with debt at 7%, lenders take 3.5 whatever happens: at a 12% asset return equity keeps 8.5 on 50 for 17%, and at 5% it keeps only 1.5 on 50 for 3%.
    The relationship
    rE=rA+(rA−rD)DE=12%+(12%−7%)×1=17%r_E = r_A + (r_A - r_D)\frac{D}{E} = 12\% + (12\% - 7\%)\times 1 = 17\%
    r_Ereturn on equity
    r_Areturn the asset earns before financing
    r_Dinterest rate on the debt
    D/Edebt over equity, here 50 over 50, which is 1
    What it says in wordsEquity earns the asset return plus the spread over the cost of debt, scaled up by how much debt there is for each rupee of equity.

    What happens when the asset earns less than the debt costs?

    The same formula runs backwards. At 5%, the spread is minus 2 points, and with one rupee of debt per rupee of equity the equity return falls to 5 minus 2, which is 3%. Leverage is not a return booster; it is a multiplier on the spread, and the spread can be negative. Below a 3.5% asset return equity earns nothing at all, and below zero it loses money faster than the asset does.

    In the room, give both numbers and then the rule in one sentence. A private credit interviewer is checking that you see the downside as clearly as the upside, because their job is to sit on the other side of that fixed 3.5.

    Where candidates lose it

    The fast wrong answer is 24%, doubling the asset return because equity is half the funding. That forgets the lenders are paid first. The other slip is subtracting the 7% rate from 12% and calling 5% the equity return, which treats the whole asset as if it were borrowed.

    Work in rupees on Rs 100 of asset: asset income, less interest, over equity. Then give the 5% case without being asked; that is the half of the answer that shows judgement.

    What the interviewer asks next

    • What equity return do you get at 70% debt and the same 12% asset return?
    • At what asset return does the equity earn exactly the same as the asset, and why?
    • Now add tax at 25%. What happens to the 17%?

    Asked at HPS Investment Partners, Financial Sponsors, new york, 2024 (Wall Street Oasis): I was asked to calculate the returns of an investment that consisted of 50% debt with an interest rate of 7%

  2. 077Estimate the annual revenue pool for school uniforms in India. Build it from enrolment, the share of students in schools that require uniforms, sets bought a year and the price of a set.Market sizing and estimationCoreAdvent InternationalBoston · 2022

    Try it first

    Which of the four inputs moves the answer the most across a sensible range?

    Show the worked solution

    About Rs 21,250 crore a year on these assumptions. Take roughly 25 crore school students, assume 85% attend schools that require uniforms, two sets a year and an average of Rs 500 a set: 25 x 0.85 x 2 x 500. Then say which input is weakest. Price is, because government and private sets differ several times over.

    How do you structure the estimate before any number?

    Say the chain out loud first: students, times the share who must wear a uniform, times sets a year, times price. It is how a shopkeeper would size the season: how many children in the area, how many of their schools insist on a uniform, how many sets each family buys, what they pay. The structure earns most of the marks, because the interviewer can disagree with a number and still follow your answer.

    Then put a number on each branch and label it an assumption. Enrolment is the one input with a public source, the government's school statistics; use about 25 crore and say you would confirm the current figure. The uniform share, sets a year and price are judgement calls: 85%, two sets and Rs 500 a set are reasonable starting points, not facts.

    Four numbers multiply to the pool; one of them carries most of the doubtStudents enrolled25 crorex 85%In uniform schools21.25 crorex 2 setsSets bought a year42.5 crorex Rs 500Revenue poolRs 21,250 crSwing each assumption across a sensible range, holding the othersShare in uniform schools 75% to 95%18,75023,750Sets a year 1.5 to 2.515,93826,562Price a set Rs 300 to Rs 80012,75034,000base case Rs 21,250 crorePrice is the widest bar because the government and private school mix sets the average.
    Twenty five crore students, 85% in uniform schools, two sets a year and Rs 500 a set give a pool of about Rs 21,250 crore; swinging the price from Rs 300 to Rs 800 moves that from Rs 12,750 crore to Rs 34,000 crore, the widest range of the four inputs.

    Which branch do you name as the biggest uncertainty, and why?

    Swing each input across a range you could defend, holding the others. The uniform share from 75% to 95% moves the pool by about Rs 5,000 crore. Sets a year from 1.5 to 2.5 moves it by about Rs 10,625 crore. Price from Rs 300 to Rs 800 moves it by Rs 21,250 crore, so price is where you would spend your first hour of real work. The reason is the mix: many state schools supply uniforms cheaply in bulk, while private schools sell branded sets at a premium.

    A strong close splits the pool into those two segments and says which one a buyout fund would care about: the private school segment, where the price is set by the school and the supplier relationship is sticky.

    Where candidates lose it

    Candidates rush to a single number and stop. The interviewer then asks which assumption they are least sure of, and there is no answer, because the chain was never tested.

    The second miss is presenting the enrolment figure as a precise fact from memory. Round it, call it an assumption to be confirmed, and move on to the inputs that actually decide the answer.

    What the interviewer asks next

    • Split the pool into government and private schools. Which is the better business for a supplier?
    • How would you check the Rs 500 a set assumption in a week?
    • What share of this pool could one regional manufacturer realistically win?

    Asked at Advent International, Private Equity, Boston, 2022 (Wall Street Oasis): First two were behavioral/personality interviews. Last one was a market sizing test.

  3. 079How many coffee shops can a metro of 2 crore people support? Build it from people, cups drunk a week, the share bought outside the home and the cups a shop sells a day.Market sizing and estimationCoreVista Equity PartnersAustin · 2023

    Try it first

    Which branch of the chain would you flag as the weakest?

    Show the worked solution

    About 2,000 shops on these assumptions. Of 2 crore people, assume 30% drink coffee, a cup a day each: 4.2 crore cups a week. If 10% are bought outside the home, that is 42 lakh a week, 6 lakh a day. At 300 cups a shop a day the metro supports about 2,000. The share bought outside is the branch to test.

    What is the chain, and why build it before any number?

    Shops exist to serve demand, so go from people to cups to shops. Picture a single street: how many of the people passing drink coffee, how often, and how many of those cups are bought rather than made at home. A sizing answer is a chain of stated assumptions, and the interviewer is testing the chain, not the final figure. Say the chain first, then fill it.

    A chain of stated assumptions from people to shopsPeople in the metro2 crorex 30% drink coffeeCoffee drinkers60 lakhx 7 cups a week eachCups drunk a week4.2 crorex 10% bought outside homeCups bought outside a week42 lakh/ 7 daysCups bought outside a day6 lakh/ 300 cups a shop a dayShops supportedabout 2,000What the interviewer testsWhich branch is weakest?Share bought outside home:5% gives 1,000 shops10% gives 2,000 shops15% gives 3,000 shopsOne branch triples the answer.
    Two crore people, 30% of whom drink seven cups a week, with 10% of those cups bought outside the home, give 6 lakh cups a day and about 2,000 shops at 300 cups each; moving the share bought outside from 5% to 15% moves the answer from 1,000 to 3,000 shops.

    How do you sanity check the supply side?

    Three hundred cups a day is the number to defend. A shop open twelve hours selling 25 cups an hour reaches it; a quiet neighbourhood outlet may sell half that and a busy office-district one double. Check the answer from the other end: about one shop for every 10,000 people, which you can compare with what you see on a busy street. If your answer implied a shop for every 500 people, a branch is wrong, and you would say which.

    A growth equity interviewer often follows with how many of those shops a single chain could own, which turns the sizing into a market share question. Have the number of shops in your answer split into chains and independents if you can, even roughly.

    Where candidates lose it

    The common loss is jumping to an answer from a vague sense of how many cafes a city has. Even a good guess scores poorly, because there is nothing for the interviewer to probe.

    The second is treating every coffee drinker as a cafe customer. Most coffee is made at home or in offices, and missing the share bought outside inflates the answer by ten times or more.

    What the interviewer asks next

    • How does the answer change if you include tea, and should you?
    • What would make a shop sell 600 cups a day rather than 300?
    • How would you estimate the share of cups bought outside the home without a survey?

    Asked at Vista Equity Partners, Healthcare, Austin, 2023 (Wall Street Oasis): market sizing - how many coffee shops in US