Portfolio Management puzzles, solved step by step

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  1. 020A stock trades at 60 times earnings and pays no dividend, and its investors require 12% a year. If it should trade at 20 times earnings in ten years, what annual earnings growth does today's price require?Valuation riddlesHardFundamental asset managementAsset management

    Try it first

    Which growth rate does the 60x multiple imply?

    Show the worked solution

    About 25% a year for ten years. With no dividend, all of the 12% return must come from price, so the price in ten years must be 60 x 1.12 to the tenth, about 186 times today's earnings. If the stock then trades at 20 times, earnings must be 186 divided by 20, or 9.32 times today's. That is growth of 25.0% a year. At 15% growth the investor would earn only about 3% a year.

    How do you turn a multiple into a growth forecast?

    Think of paying 60 years of a shop's current profit for the shop. That only makes sense if the profit is going to be far bigger soon. A high multiple is a forecast you can read: work forward from the return investors want, and back from the multiple the stock should end at, and the growth in between is what the price assumes. Here the investor wants 12% a year with no dividend, so the price must grow 3.11 times in ten years, to about 186 times today's earnings. At a mature 20 times, earnings must reach 9.32 times today's level.

    What a 60x multiple needs: earnings growth of 25% a year for a decade2468100246810YearsEarnings, today = 1needed: 9.32at 15%: 4.05gapPrice in 10 years60 x 1.12^10 = 186.4At 20x, earningsmust be 9.32Growth needed25.0% a yearAt 15%: returnonly 3.0% a year
    To justify 60 times earnings today and 20 times in ten years at a 12% return, earnings must grow 25% a year to 9.32 times today's level, while a 15% path reaches only 4.05 and would leave the investor with about 3% a year.
    The relationship
    g=(PE0(1+r)10PE10)1/10−1=(60×3.10620)1/10−1≈25%g=\left(\frac{PE_0(1+r)^{10}}{PE_{10}}\right)^{1/10}-1=\left(\frac{60\times 3.106}{20}\right)^{1/10}-1\approx 25\%
    PE_0, PE_{10}the multiple today, 60, and in ten years, 20
    rthe required return, 12%, all from price since there is no dividend
    gthe annual earnings growth the price requires
    What it says in wordsThe earnings growth needed is the required price growth, adjusted for the multiple shrinking from 60 to 20.

    What do you do with the number once you have it?

    You ask how often a company sustains that growth for a decade, which is rarely, and you say so. The question the interviewer wants answered is not whether the company is good but whether the price has already paid for more than the company is likely to deliver. At a still strong 15% a year, earnings reach 4.05 times today's, the price at 20 times is about 81, and the investor earns about 3.0% a year rather than 12%. Say the limitations: the exit multiple of 20 is an assumption, and buybacks, dividends or a higher exit multiple would lower the growth needed.

    Where candidates lose it

    The common error is saying the stock needs to grow earnings at 12%, the required return. That ignores the multiple falling from 60 to 20, which on its own costs about 10% a year of price.

    The other trap is stopping at the arithmetic. On a fundamental desk the interviewer wants the judgement: 25% for a decade is a demanding assumption, and a reverse calculation like this is how you show a price is stretched without claiming to know the future.

    What the interviewer asks next

    • What growth is needed if the stock still trades at 40 times in ten years?
    • How does paying a 2% dividend change the answer?
    • What required return does today's price imply if earnings grow at 15%?
  2. 097A stock trades at 25 times trailing earnings, pays out 40% of earnings, earns a 20% return on equity, and has a 12% cost of equity. What growth rate does the price imply, and how does it compare with the growth its reinvestment could support?Valuation riddlesHardFundamental asset managementIndian equity research

    Try it first

    What perpetual growth does the 25x multiple imply?

    Show the worked solution

    The price implies about 10.2% growth for ever, below the 12% that reinvestment could support today. Setting 25 equal to 0.4 x (1 + g) / (0.12 - g) gives g of 2.6 / 25.4. Retaining 60% at a 20% return supports 12%, but 12% for ever equals the cost of equity and would make the stock infinitely valuable, so the market is pricing in returns fading to about 17%.

    What does it mean to work backwards from a multiple?

    If a flat rents for Rs 30,000 a month and sells for Rs 1.2 crore, the price tells you what buyers assume about future rents, whether or not anyone says it. A P/E is the same kind of compressed forecast: fix the payout and the cost of equity, and the multiple pins down the one perpetual growth rate the price assumes. Reverse engineering the price like this is safer than forecasting growth and seeing what multiple falls out.

    The relationship
    25=0.4 (1+g)0.12−g  ⇒  3−25g=0.4+0.4g  ⇒  g=2.625.4≈10.2%25 = \frac{0.4\,(1+g)}{0.12 - g} \;\Rightarrow\; 3 - 25g = 0.4 + 0.4g \;\Rightarrow\; g = \frac{2.6}{25.4} \approx 10.2\%
    0.4payout ratio
    0.12cost of equity
    gthe perpetual growth the price implies
    What it says in wordsThe multiple equals the payout grown one year, divided by the gap between the cost of equity and growth; solve for growth.
    Working backwards from 25x: the market assumes growth fades below 12%Implied by 25x P/E10.2%ROE 20% x retention 60%12.0%cost of equity 12%:growth here meansan infinite P/EPrice implies ROE fading to about 17.1% at the same 60% retentionJustified P/E by growth8%10.8x10%22.0x10.2%25.0x11%44.4x11.5%89.2xpayout 40%, cost of equity 12%
    The 25 times multiple implies 10.2% perpetual growth, below the 12% that 60% retention at a 20% return supports, and the justified P/E rises from 22 times at 10% growth to 89 times at 11.5%, which is why growth near the cost of equity cannot last for ever.

    Is the stock cheap because it can grow faster than the price assumes?

    Not so fast. Sustainable growth is return on equity times retention, 20% x 60% = 12%. But 12% growth for ever equals the cost of equity, which would make the stock worth an infinite multiple, so no price could reflect it; the gap tells you the market expects the 20% return on new investment to fade. At the same 60% retention, the price is consistent with a long-run return on equity of about 17.1%.

    That turns the question into a sharper one: will this business keep earning well above 17% on new money for a long time? That is a question about competition and moats, and it is the right place to spend the next five minutes of the interview. Say also that one-stage models are crude; a two-stage model with high growth fading to a lower rate is the usual next step.

    Where candidates lose it

    The common slip is declaring the stock undervalued because 12% beats 10.2%. Plugging 12% into the model gives division by zero, and a candidate who does not notice has not understood the formula.

    The second slip is using the earnings yield shortcut, 12% minus 4%, which ignores the payout ratio and lands near 8%. Set up the equation and solve it.

    What the interviewer asks next

    • What P/E would be justified if long-run growth were 9%?
    • Rebuild this with a two-stage model: 12% for five years, then 6%.
    • How would a lower payout ratio change the implied growth at the same 25x?