Portfolio Management puzzles, solved step by step

Puzzles
100
Traced to a firm
31
Topics
13
Hard
30
Topic
All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–2 of 2 · filtered from 100Clear filters
  1. 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?Private and real asset mathsHardNeuberger 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.

    Paper LBO worked backwards: from the target return to the one number that must hold1,000equity needed2.5 x 400+ 200debt still owed400 - 5 x 401,200exit EVRs croreDivide by year 5 EBITDA100 x 1.08 to the 5th = 146.91,200 / 146.9Exit multiple = 8.17xEntry was 8.0x: the plan needs almostno multiple expansion to reach 2.5x
    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 relationship
    mexit=2.5×400+(400−5×40)100×1.085=1,200146.9=8.17×m_{exit} = \frac{2.5 \times 400 + (400 - 5 \times 40)}{100 \times 1.08^5} = \frac{1{,}200}{146.9} = 8.17\times
    2.5 x 400the equity the sponsor needs back, Rs crore
    400 - 5 x 40debt still owed after five annual repayments
    100 x 1.08^5year 5 EBITDA, Rs crore
    What it says in wordsThe exit multiple is the equity target plus remaining debt, divided by exit EBITDA.
    Where the Rs 600 crore of gain comes from, Rs croreEquity in400EBITDA growth+375Debt paydown+200Multiple 8.0x to 8.17x+25Equity out1,000 = 2.5x
    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.

  2. 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?Private and real asset mathsHardPIMCOMunich · 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.

    IRR ranks A first; at a 10% cost of capital, NPV ranks B first5%10%15%20%25%30%35%-40501000cost of capital 10%B: NPV 22.9 at 10%A: NPV 18.2 at 10%lines cross at 11.2%IRR B15.1%IRR A 30%B: 20 a year, 10 yearsA: 130 in one yearDiscount rateRs croreIRRA 30.0 / B 15.1NPV at 10%A 18.2 / B 22.9NPV measuresvalue created
    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 relationship
    NPVA=1301.10−100=18.2,NPVB=20×1−1.10−100.10−100=22.9NPV_A = \frac{130}{1.10} - 100 = 18.2, \qquad NPV_B = 20 \times \frac{1 - 1.10^{-10}}{0.10} - 100 = 22.9
    130A's single payout after one year, Rs crore
    20B'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