Investment Banking puzzles, solved step by step

Puzzles
100
Traced to a firm
36
Topics
12
Hard
29
Topic
All topicsProbability7Mental maths and counting11Growth and compounding9Valuation riddles11Logic and brainteasers11Rates, risk and options9Estimation and market sizing7Accounting riddles9Expected value and games7Enterprise value and dilution6Deal maths6DCF and cost of capital7
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 31–32 of 32 · filtered from 100Clear filters
  1. 097A stock trades at 20x next year's earnings, pays out 60% of earnings as dividends, and its dividends are expected to grow at 7% a year forever. What cost of equity is the market implying?Valuation riddlesCoreElite boutique IBBulge bracket IB

    Try it first

    What return is a buyer at this price implicitly expecting?

    Show the worked solution

    About 10%. The Gordon growth model says price equals next year's dividend over (cost of equity minus growth). Rearranged, cost of equity equals dividend yield plus growth. The dividend yield is payout over P/E, 0.6 / 20, which is 3%. Adding 7% growth gives 10%. If the 20x were on trailing earnings, the answer would be slightly higher, about 10.2%.

    How does a price turn into a return?

    If a flat costs Rs 1 crore and rents for Rs 3 lakh a year, rising 7% a year, a buyer is earning 3% in rent now and 7% a year from the rent growing, about 10% in total. A buyer of a growing income stream earns its current yield plus its growth, so a price plus a growth assumption tells you the return the buyer must be expecting. For a share, the income is the dividend.

    Rearrange the Gordon model and a price becomes a returnP/E 20earnings yield 1/20= 5%x payout 60%dividend yield 0.6 / 20= 3%+ growth 7%cost of equity= 10%Return thebuyer expects3% paid as dividends7% from the dividend growing10% cost of equity implied by the priceIf the 20x is on trailing earnings, next year's dividend is 7% bigger: 3% x 1.07 + 7% = 10.21%.Growth is the soft input: at 6% growth the same price implies 9%.
    A P/E of 20 and a 60% payout give a dividend yield of 3%, and adding 7% expected dividend growth implies that buyers at this price expect a 10% return on equity.

    Why is the dividend yield payout over P/E?

    The relationship
    P=D1ke−g  ⇒  ke=D1P+g=0.6×E20×E+0.07=3%+7%=10%P = \frac{D_1}{k_e - g} \;\Rightarrow\; k_e = \frac{D_1}{P} + g = \frac{0.6 \times E}{20 \times E} + 0.07 = 3\% + 7\% = 10\%
    Pshare price, 20 times next year's earnings E
    D_1next year's dividend, 60% of E
    k_ecost of equity implied by the price
    gperpetual dividend growth, 7%
    What it says in wordsThe return implied by a price is next year's dividend yield plus the dividend's growth rate.

    Dividend over price is (payout x earnings) over (P/E x earnings), and earnings cancel, leaving 0.6 / 20 = 3%. You never need the share price or the earnings per share, only the two ratios. That is the trick the interviewer is looking for: spotting that the units cancel.

    How much should you trust the 10%?

    Only as far as the growth input. The dividend yield is observable; the 7% perpetual growth is an opinion. Drop it to 6% and the same price implies 9%. Check consistency too: growth of 7% with 40% of earnings retained implies a return on retained equity of about 17.5%, since growth equals retention times return on equity. If the company cannot earn that, the 7% is too high.

    Where candidates lose it

    The fast wrong answer is 5%, the earnings yield, which treats all earnings as paid out and ignores growth. A close second is answering 7%, quoting the growth rate as if it were the return.

    The quieter slip is mixing trailing and forward. Say which earnings the 20x is on; if trailing, grow the dividend one year before dividing.

    What the interviewer asks next

    • What P/E would the stock trade on if the cost of equity were 12% with the same payout and growth?
    • Check the 7% growth against retention and return on equity: is it consistent?
    • Why might the implied cost of equity differ from a CAPM estimate for the same company?
  2. 099A company with a 10% WACC and an equity beta of 1.0 is considering an all-equity project in a riskier business, where pure-play peers have an asset beta of 1.5. The risk-free rate is 7% and the equity risk premium is 5%. What discount rate should the project use?DCF and cost of capitalCoreBulge bracket IBMiddle market IB

    Try it first

    Which rate belongs in the project's DCF?

    Show the worked solution

    14.5%, from the project's own beta, not the company's 10% WACC. The project is all-equity, so its discount rate is its cost of equity at the peers' asset beta: 7% + 1.5 x 5% = 14.5%. Using 10% would accept projects that cannot earn their own risk. A project returning 12% looks like a winner at 10% but destroys value at 14.5%.

    Why is the company's own WACC the wrong rate?

    A bank with a low cost of funds still charges a risky borrower a high interest rate, because the rate should reflect the loan, not the bank. Projects work the same way. The discount rate is the return investors demand for the risk of these particular cash flows, so it belongs to the project, not to the company that happens to own it. The company's 10% WACC is the right rate for a project as risky as the company's average business, and this one is riskier.

    Where does the risk come from? Pure-play peers, companies that do only this riskier business, have an asset betaThe beta of a business with its debt stripped out: how much its operating cash flows move with the market, regardless of how it is financed. of 1.5. Because the project is all-equity, its equity beta equals that asset beta. Plug it into CAPM: 7% + 1.5 x 5% = 14.5%. For comparison, the parent's own cost of equity at a beta of 1.0 is 12%, and its WACC is lower still because it blends in cheaper debt.

    Discount the project at its own risk, not its owner'sCompany WACCthe wrong hurdle10.0%Company cost of equity7% + 1.0 x 5%12.0%Project rate, asset beta 1.57% + 1.5 x 5%14.5%project earns 12%At 10%: 12 / 0.10 = 120NPV +20: looks like a winnerWrong rate, wrong decisionAt 14.5%: 12 / 0.145 = 82.8NPV -17.2: destroys valueCost 100, pays 12 a year forever
    The project's own rate is 14.5% from an asset beta of 1.5, against the company's 10% WACC, so a project earning 12% shows an NPV of +20 at the wrong rate and -17.2 at the right one.

    What does the wrong rate cost in practice?

    The relationship
    rproject=rf+βasset×ERP=7%+1.5×5%=14.5%r_{\text{project}} = r_f + \beta_{\text{asset}} \times \text{ERP} = 7\% + 1.5 \times 5\% = 14.5\%
    r_frisk-free rate, 7%
    beta_assetthe pure-play peers' asset beta, 1.5
    ERPequity risk premium, 5%
    What it says in wordsAn all-equity project's rate is the risk-free rate plus its own beta times the market premium.

    Take a project costing 100 that pays 12 a year forever, an IRR of 12%. At 10% it is worth 120, an NPV of +20. At 14.5% it is worth 82.8, an NPV of -17.2. A company that discounts everything at its own WACC systematically accepts risky projects that destroy value and rejects safe ones that would create it. The limitation: peer betas are estimated with error, so use several peers and a median, and if the project were part-funded with debt you would relever the asset beta for that financing.

    Where candidates lose it

    The standard slip is answering 10% because the question opens with the company's WACC. That number is there as bait: the interviewer wants to hear that the discount rate follows the risk of the cash flows.

    The second loss is relevering the 1.5 with the parent's debt when the question says the project is all-equity. Read the financing assumption before adjusting the beta.

    What the interviewer asks next

    • If the project were funded 30% with debt at 9% pre-tax and a 25% tax rate, what rate would you use?
    • How would you find the asset beta if the peers carry debt?
    • Why might a conglomerate's divisions argue for a single company-wide hurdle rate, and what goes wrong?