Portfolio Management puzzles, solved step by step

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  1. 007A company trades at 20 times earnings. It uses cash that was earning 3% after tax to buy back 10% of its shares at the market price. Is the buyback accretive to earnings per share, and by how much?Valuation riddlesCoreFundamental asset managementAsset management

    Try it first

    Which comparison decides whether the buyback lifts earnings per share?

    Show the worked solution

    Yes, it is accretive, by about 4.4%. Take Rs 50 crore of earnings on 10 crore shares at Rs 100, so EPS is Rs 5.00. The buyback spends Rs 100 crore, which was earning Rs 3 crore, so earnings fall to Rs 47 crore while the share count falls to 9 crore. EPS becomes Rs 5.22. It is accretive because the 5% earnings yield beats the 3% return on cash.

    What is the company actually swapping?

    Suppose you hold a fixed deposit paying 3% after tax and use it to buy out a partner's share of a shop that earns 5% on its price. Your income goes up, because you replaced a 3% asset with a 5% one. A buyback is the same trade. The company gives up the after-tax return on its cash and gets back a slice of its own earnings, priced at the earnings yield, which is one over the P/E. At 20 times earnings that yield is 5%, so the swap raises earnings per share. At 33.3 times earnings the yield would be 3% and the buyback would leave EPS unchanged.

    Accretive because the shares earn more than the cash they replaceCash earns, after tax3.0% (what you give up)Shares earn: 1 / P/E of 205.0% (what you buy)Break-even P/E = 1 / 3% = 33.3x. Below it the buyback adds to EPS.Rs 5.00Before: 50 / 10Rs 5.22After: 47 / 9+4.4% EPSAccretion is arithmetic, not value:value is created only if the shareswere bought below what they areworth, whatever EPS does
    The buyback gives up cash earning 3% after tax and buys shares carrying 5% of earnings, so earnings fall from Rs 50 crore to Rs 47 crore while shares fall from 10 crore to 9 crore, and EPS rises from Rs 5.00 to Rs 5.22.

    How do you get the exact number quickly?

    Pick round figures and the answer falls out: Rs 1,000 crore of market value, Rs 50 crore of earnings and 10 crore shares. Earnings drop by 10% of the market value times 3%, which is Rs 3 crore, and shares drop by 10%, so EPS moves by 0.94 divided by 0.90. That is 1.0444, an accretion of 4.44%. The shortcut works for any size of company because only the ratios matter.

    The relationship
    EPS1EPS0=1−f⋅PE⋅yc1−f=1−0.1×20×0.030.9≈1.044\frac{EPS_1}{EPS_0}=\frac{1-f\cdot PE\cdot y_c}{1-f}=\frac{1-0.1\times 20\times 0.03}{0.9}\approx 1.044
    fthe fraction of shares bought back, 10%
    PEthe price to earnings multiple, 20
    y_cthe after-tax return on the cash spent, 3%
    What it says in wordsEPS rises when the lost interest, as a share of earnings, is smaller than the share of the company bought back.

    Does accretive mean the buyback was a good idea?

    No, and a portfolio manager is expected to say so. Accretion is arithmetic about earnings per share; value is created only if the company paid less for its shares than they are worth. A company on a low P/E almost always gets an accretive buyback, even if the shares are overpriced, and one on a high P/E can create value with a dilutive buyback if the shares are cheap relative to future growth. Accretion also ignores risk: swapping cash for shares makes the remaining equity more levered.

    Where candidates lose it

    Candidates answer that a buyback always raises EPS because there are fewer shares. That forgets the lost income on the cash, and at a high enough P/E the buyback dilutes.

    The second trap is stopping at accretive. On a buy-side desk the interviewer usually follows with whether it creates value, and the answer that separates candidates is that the two are different questions.

    What the interviewer asks next

    • At what P/E does the same buyback become dilutive?
    • What if the buyback is funded with new debt at 8% before a 25% tax rate?
    • Why might a buyback that is accretive still destroy value for the remaining shareholders?
  2. 084A bank earns a 16% return on equity, grows at 8% a year and has a 13% cost of equity. What price to book is justified? What happens if its return on equity falls to 13%?Valuation riddlesCoreFundamental asset managementIndian equity research

    Try it first

    At a 13% return on equity, what is the justified price to book?

    Show the worked solution

    1.6 times book at a 16% return on equity, falling to 1.0 times at 13%. The justified multiple is the return on equity less growth, over the cost of equity less growth: 8 over 5 at 16%, and 5 over 5 at 13%. When a bank earns exactly its cost of equity, a rupee of book is worth a rupee, and growth adds nothing.

    Why does a bank earning its cost of equity trade at book?

    Suppose a friend offers to take Rs 1 lakh and pay you exactly the return you could get anywhere else with the same risk. The deal is worth Rs 1 lakh; no more, however large your friend makes it. A bank reinvesting at a return equal to its cost of equity creates no value on the money it keeps, so its book is worth exactly book. Only the spread between the two rates earns a premium.

    A bank is worth its book only when it earns its cost of equity9%11%13%15%17%19%21%0.5x1.0x1.5x2.0x2.5xROE = cost of equity: 1.0xROE 16%: 1.6xworth less than bookvalue createdReturn on equityP/B = (ROE - g)/ (r - g)16%: 8 / 5 = 1.6x13%: 5 / 5 = 1.0xg = 8%, r = 13%held fixed
    With growth at 8% and a 13% cost of equity, the justified price to book is 1.0 times where return on equity equals 13% and rises to 1.6 times at 16%; below 13% the line falls under book, because growth then destroys value.

    Where does the formula come from?

    Start from the dividend discount model. Next year's dividend is book times return on equity times the share paid out. To grow at 8% with a 16% return, the bank must retain half its earnings, since growth is return on equity times retention; that leaves a payout of 1 minus g over ROE. Substituting gives price over book equal to (ROE minus g) over (r minus g). At 16%, retention is 50%, payout 50%, and the multiple 1.6 times.

    The relationship
    PB=ROE−gr−g=0.16−0.080.13−0.08=1.6\frac{P}{B} = \frac{ROE - g}{r - g} = \frac{0.16 - 0.08}{0.13 - 0.08} = 1.6
    ROEreturn on equity, 16%
    glong-run growth, 8%
    rcost of equity, 13%
    What it says in wordsThe premium to book is the return spread over the cost spread, both measured from the growth rate.

    The limit is that every input is a long-run steady state. A bank with a cyclical credit cost earns 20% in good years and 8% in bad ones, and the formula wants the through-cycle figure. Say so, then say which input you would argue about first.

    Where candidates lose it

    The common slip is scaling: 13 is less than 16, so the multiple falls in proportion to about 1.3 times. The relationship is not proportional; it is anchored at 1.0 times where the two rates meet.

    The deeper miss is not seeing that growth is only good when return on equity beats the cost of equity. Below 13%, faster growth lowers the multiple, and saying that out loud is what the question is testing.

    What the interviewer asks next

    • The bank earns 11% on equity. What happens to the multiple if growth rises from 8% to 10%?
    • What return on equity does a price to book of 2.0 times imply at the same growth and cost of equity?
    • Why do lenders with similar returns on equity trade at very different multiples?