Private Equity puzzles, solved step by step

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All topicsCredit and PIK maths8Returns maths10Mental paper LBOs8Operating levers and margin maths8Valuation riddles10Fund economics numeracy9Market sizing and estimation9Compounding and time value7Mental maths8Probability and expected value in deals8Leverage and capital structure9Logic and brainteasers6
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  1. 039A company sells two products with gross margins of 60% and 20%. Sales shift from 50/50 to 40/60, towards the low-margin product. What happens to the blended gross margin?Operating levers and margin mathsWarm upPortfolio operations team

    Try it first

    Where does the blended margin go?

    Show the worked solution

    The blended margin falls from 40% to 36%, with no change in either product's economics. Before, 0.5 x 60% + 0.5 x 20% = 40%. After, 0.4 x 60% + 0.6 x 20% = 24% + 12% = 36%. Moving 10 points of the mix from a 60% product to a 20% product costs 10% of the 40-point gap between them, which is 4 points of blended margin.

    How can margin fall when nothing got worse?

    A cafe earns 70% on coffee and 25% on sandwiches. If more customers come at lunch and order sandwiches, the cafe's overall margin drops even though neither the coffee nor the sandwich got less profitable. A blended margin is a sales-weighted average, so shifting weight towards the lower-margin product pulls the average down by itself. Each 1 point of mix moved costs 1% of the margin gap, here 0.4 points.

    Same products, same margins, a different mix: the blend falls 4 pointsA: 50 of sales60% margin, profit 30B: 50 of sales20% margin, profit 10Before: 50/50blended40%A: 40 of sales60% margin, profit 24B: 60 of sales20% margin, profit 12After: 40/60blended36%Revenue of 100 in both bars
    With product margins fixed at 60% and 20%, a 50/50 mix earns 30 plus 10 on revenue of 100, a 40% blend, while a 40/60 mix earns 24 plus 12, a 36% blend, so mix alone takes 4 points off the margin.
    The relationship
    m=wAmA+wBmB=0.4×60%+0.6×20%=36%m = w_A m_A + w_B m_B = 0.4 \times 60\% + 0.6 \times 20\% = 36\%
    w_A, w_Beach product's share of sales
    m_A, m_Beach product's gross margin
    What it says in wordsThe blended margin is each product's margin weighted by its share of sales.

    Why does a buyout team care, and what should it check next?

    When a target's margin falls, the first question is whether the products got worse or the mix moved. A mix-driven margin fall is a different problem from a pricing or cost problem, and it can come with rising profit. If the low-margin product is growing fast, total gross profit may still rise even as the percentage falls: a business at a 36% margin on revenue of 150 earns more than one at 40% on 100.

    Say the limitation. Gross margin ignores the overheads each product uses. A low-margin product that needs little selling effort can be more attractive after overheads than it looks here, so ask for contribution by product before judging the mix.

    Where candidates lose it

    The common loss is saying the margin stays at 40% because neither product changed. The interviewer is checking whether you know that a blend is a weighted average and moves with its weights.

    The second loss is reading a falling margin as bad news without asking about volume. Margin percentage and profit in rupees can move in opposite directions.

    What the interviewer asks next

    • What mix would bring the blended margin down to 30%?
    • If revenue grows from 100 to 130 with the new mix, does gross profit rise or fall?
    • How would you separate price, cost and mix effects in a margin bridge?
  2. 062A portfolio company has revenue of 730 a year. The operating team cuts days sales outstanding from 90 to 60. How much cash does that release?Operating levers and margin mathsWarm upPortfolio operations teamIndian mid-market PE

    Try it first

    How much cash comes out of receivables?

    Show the worked solution

    About 60, released once. Revenue of 730 is 2 a day. At 90 days, customers are holding 180 of unpaid invoices; at 60 days, 120. Collecting 30 days faster brings in the 60 difference as cash, one time. After that, receivables simply stay at the lower level, so the cash flow benefit does not repeat each year.

    What does a day of DSO actually hold?

    Picture a tailor who lets regular customers pay at the end of the month. On any given day, a month's worth of stitched clothes is out there unpaid, and that money is not in the tailor's drawer. Days sales outstanding counts how many days of sales are sitting with customers, so each day of DSO is one day of revenue held as receivables instead of cash. Here one day is 730 over 365, which is 2.

    Each day of DSO is a day of revenue sitting with customers instead of in the bankDSO 90Receivables 180DSO 60Receivables 120Cash released: 60Sales per day = 730 / 365 = 290 days x 2 = 180; 60 days x 2 = 120Released = 30 days x 2 = 60Once, not every yearYear 2 at DSO 60 releases nothing more.If revenue grows 20%, receivables riseto 144 and absorb 24 of cash.
    At 2 of sales a day, 90 days of receivables hold 180 and 60 days hold 120, so cutting DSO by 30 days releases 60 of cash, once, and later growth in revenue starts to absorb cash again.
    The relationship
    ΔCash=Revenue365×ΔDSO=730365×30=60\Delta\text{Cash} = \frac{\text{Revenue}}{365} \times \Delta\text{DSO} = \frac{730}{365} \times 30 = 60
    Revenue/365sales per day, here 2
    Delta DSOthe cut in days of receivables, 90 to 60
    What it says in wordsCash released equals one day of sales times the number of days cut.

    Why does a buyout fund care that it happens only once?

    Because a one-off release must not be valued like a recurring profit. The 60 can pay down debt or fund a dividend once, but it adds nothing to EBITDA and nothing to next year's cash flow. If a seller's numbers show a strong cash year driven by a receivables squeeze, a buyer strips it out before using that year to set the price. And as the company grows, receivables grow with it: 20% more revenue at 60 days lifts receivables to 144, absorbing 24 of cash.

    Where candidates lose it

    Two slips are common. The first is answering 30, the change in days, without converting days into money at 2 a day.

    The second is treating 60 as an annual saving and putting it into the free cash flow of every year. It is a one-time release from a lower balance; a fund that capitalises it as recurring overpays.

    What the interviewer asks next

    • Payables days go from 30 to 45 on cost of sales of 365. How much cash is released?
    • Why might cutting DSO cost the company revenue?
    • How would you spot a seller who squeezed receivables just before a sale?