Mutual Fund Mastery puzzles, solved step by step

Puzzles
100
Traced to a firm
29
Topics
13
Hard
30
Topic
All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 033A lender's shares trade at 3 times book value and it earns a 15% return on equity. What P/E is that? If its ROE falls to 12% and the price-to-book stays at 3 times, what happens to the P/E?Valuation riddlesCoreEquity research at AMCsGlobal asset managers

    Try it first

    Before you work it: what P/E does 3 times book and a 15% ROE imply?

    Show the worked solution

    A P/E of 20x, rising to 25x if ROE falls to 12% while the P/B stays at 3. Earnings are ROE times book value, so P/E equals P/B divided by ROE: 3 over 0.15 is 20. At 12% ROE the same P/B gives 3 over 0.12, or 25. The shares look no dearer on book, but each rupee of earnings now costs 25% more.

    How do P/B and ROE give you the P/E?

    Suppose a shop is sold for three times the money the owner has put into it, and the shop earns 15% a year on that money. A buyer paying 300 for every 100 put in gets 15 of profit a year for her 300, which is 20 years of profit. ROE is the bridge between book value and earnings, so P/E is simply P/B divided by ROE. Price sits on top of both ratios, and it cancels.

    The relationship
    PE=P/BE/B=P/BROE=3.00.15=20×3.00.12=25×\frac{P}{E} = \frac{P/B}{E/B} = \frac{P/B}{ROE} = \frac{3.0}{0.15} = 20\times \qquad \frac{3.0}{0.12} = 25\times
    P/Bprice to book value per share, 3.0
    ROEearnings divided by book value, 15% then 12%
    P/Eprice to earnings per share
    What it says in wordsDivide price-to-book by the return on equity and you have price-to-earnings.
    Same price, same book, lower ROE: the P/E risesROE 15%, P/B stays 3.0xRs 300PriceRs 100BookRs 15Earningsearnings drawnat 4x scaleP/E = 3.0 / 15% = 20xROE falls to 12%, P/B stays 3.0xRs 300PriceRs 100BookRs 12Earningsearnings drawnat 4x scaleP/E = 3.0 / 12% = 25x
    With book value of Rs 100 a share and a price of Rs 300, a 15% ROE gives earnings of Rs 15 and a P/E of 20x. If ROE falls to 12% and the P/B stays at 3, earnings drop to Rs 12 and the same price becomes a P/E of 25x.

    What does an unchanged P/B after a fall in ROE tell you?

    If returns fall and the price-to-book does not, the shares have become more expensive per rupee of profit, even though nothing on the book-value screen moved. For a lender, book value is the natural anchor, and many investors screen on P/B alone. This is the case where that screen misleads: the same 3.0x now buys 12% returns instead of 15%. To hold a 20x P/E at 12% ROE, the P/B would have to fall to 2.4x.

    There are fair reasons the market might hold the P/B: it may expect ROE to recover, or the fall may be a one-off provision. Say that, then say the test: if ROE stays at 12%, a P/B of 3.0 is paying for returns that are no longer there. The ratios here are illustrations; the relationship holds for any company whose book value is meaningful.

    Where candidates lose it

    The common slip is multiplying instead of dividing: 3 times 0.15 gives 0.45, which some candidates then misread as 4.5x. Earnings are smaller than book here, so the P/E has to be larger than the P/B, not smaller. A quick sense check catches it.

    The second trap is saying the P/E falls when ROE falls, because lower returns sound cheaper. With the price and book unchanged, lower earnings mean a higher P/E.

    What the interviewer asks next

    • What P/B would keep the P/E at 20x if ROE is 12%?
    • If the cost of equity is 12%, what does a 12% ROE suggest about a fair P/B for a lender that does not grow?
    • Why do analysts value lenders on P/B more often than on P/E?