Equity Research puzzles, solved step by step

Puzzles
100
Traced to a firm
19
Topics
11
Hard
30
Topic
All topicsProbability and brainteasers12Expected value and decisions8Market sizing and estimation12Returns and compounding9Valuation riddles12Three statement riddles10EPS and share count9Cost of capital and rates8Growth, mix and unit economics8Mental maths6Data and reasoning traps6
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–3 of 3 · filtered from 100Clear filters
  1. 008A company's unlevered cost of capital is 12% and it can borrow at 8%. Ignore taxes. What happens to its cost of equity and its WACC when it moves from no debt to debt equal to equity?Cost of capital and ratesHardBuy-side equity researchHedge fund long/short

    Try it first

    Debt at 8% replaces half the 12% capital. What is the new WACC?

    Show the worked solution

    The cost of equity rises from 12% to 16% and WACC stays at 12%. With debt equal to equity, shareholders carry the same business risk on half the capital, so their required return rises by the spread between the unlevered rate and the debt rate, 4 points, times debt over equity. Half at 16% and half at 8% is still 12%. Without taxes, cheap debt only moves risk around.

    Why can cheap debt not lower the cost of capital on its own?

    Think of two friends buying a food truck. If one lends at a fixed rate and the other takes whatever is left after paying the loan, the truck's takings are no less risky; the owner just now carries all of the ups and downs on a smaller stake. Borrowing does not change the business, so it cannot change the total return the business must earn for all its funders; it only shifts risk from lenders to shareholders.

    No taxes: borrowing raises the cost of equity by exactly enough to keep WACC flat0%4%8%12%16%20%00.511.52Debt to equity16%10%Cost of equity 20%WACC stays 12%Cost of debt 8%Wrong: hold equity at 12% and WACC seems to fall,to 10% at 1x and 9.3% at 2x
    Without taxes the cost of equity rises in a straight line from 12% at no debt to 20% at debt twice equity, while WACC stays flat at 12%; the dashed red line is the mistaken WACC that holds equity at 12% and falls to 10% at debt equal to equity.

    How do you get the 16%?

    The Modigliani and MillerThe 1958 result, from Franco Modigliani and Merton Miller, that in a world without taxes or distress costs the value of a firm does not depend on how it is financed. relation gives it directly. The cost of equity is the unlevered rate plus the gap between the unlevered rate and the debt rate, scaled by debt over equity. Here that is 12% plus (12% minus 8%) times 1, which is 16%. Check with WACC: half at 16% plus half at 8% is 12%, exactly the unlevered rate.

    The relationship
    rE=rU+(rU−rD)DE=12%+4%×1=16%r_E = r_U + (r_U - r_D)\frac{D}{E} = 12\% + 4\% \times 1 = 16\%
    r_Uthe unlevered cost of capital, the return the business itself must earn
    r_Dthe cost of debt
    D/Edebt over equity at market values
    What it says in wordsShareholders demand the business's return plus a premium for each rupee of debt standing ahead of them.

    Now add back what the puzzle removed. With tax, interest is deductible, so debt does lower WACC a little; at high debt, the cost of debt itself rises and distress costs appear. That is why an analyst who sees WACC fall sharply as a model adds debt should check whether the cost of equity was left unchanged. In practice this shows up as re-levering beta: the equity beta must rise when leverage rises.

    Where candidates lose it

    The trap is averaging 12% and 8% and announcing a WACC of 10%. It holds the cost of equity fixed while the equity becomes riskier, and it quietly creates value from nothing.

    The second loss is getting 12% and not being able to say why. The one line to say is that financing slices the same cash flows differently; it does not change them.

    What the interviewer asks next

    • Add a 25% tax rate. What is the WACC at debt equal to equity now?
    • If the debt cost rises to 10% at this leverage, what happens to the cost of equity?
    • How do you re-lever a peer's beta for a company with more debt?
  2. 033You have a US dollar cost of equity of 9% for a company. Expected inflation is 5% in India and 2.5% in the US. What is the equivalent rupee cost of equity for discounting rupee cash flows?Cost of capital and ratesHardBuy-side equity researchLong-only asset management

    Try it first

    Which conversion keeps the valuation the same in both currencies?

    Show the worked solution

    About 11.7%. Scale the dollar rate by relative inflation: (1 + 9%) x (1 + 5%) / (1 + 2.5%) = 1.1166, so the rupee cost of equity is 11.66%. Adding the 2.5 point gap gives 11.5%, close but slightly low. The conversion keeps the value the same in both currencies, because rupee cash flows grow faster by exactly the same factor.

    Why must the rate change when the business has not?

    Think of a salary quoted two ways. A Rs 10 lakh salary rising with 5% inflation and the same salary quoted in dollars rising with 2.5% inflation are one job. If you discount the faster-rising rupee salary at the slower dollar rate, you make the same job look more valuable. A discount rate carries the inflation of its currency, so moving between currencies means moving the rate by the inflation gap, in the same way the cash flows move.

    The relationship
    1+kRs=(1+k$)×1+πIndia1+πUS=1.09×1.051.0251 + k_{Rs} = (1 + k_{\$}) \times \frac{1 + \pi_{India}}{1 + \pi_{US}} = 1.09 \times \frac{1.05}{1.025}
    k_$the dollar cost of equity, 9%
    k_Rsthe rupee cost of equity
    piexpected inflation in each country, 5% and 2.5%
    What it says in wordsOne plus the rupee rate equals one plus the dollar rate, scaled up by the ratio of the two inflation factors.
    From a dollar discount rate to a rupee one, through the inflation gap9.00%US dollarcost of equity+2.50Inflation gap5.0% less 2.5%+0.16Compoundingterm11.66%Rupeecost of equityThe exact conversion(1 + 9%) x 1.05 / 1.025= 1.09 x 1.0244= 1.116611.66%Adding the gap alone gives11.50%, close but slightly lowSame value in either currency
    A 9.00% dollar rate plus the 2.5 point inflation gap reaches 11.50%, and the compounding term adds 0.16 points more, so the rupee cost of equity is 11.66% rather than a spread added by feel.

    How do you prove the value is the same in both currencies?

    Take a cash flow worth 100 in today's money, received in year five. In dollars it grows at 2.5% inflation and is discounted at 9%: 1.025 to the fifth over 1.09 to the fifth gives a discount factor of 0.7353. In rupees it grows at 5% and is discounted at 11.66%: the factor is 0.7353. The two factors match to the fourth decimal in every year, which is the proof that the conversion is right.

    Say the limitation. This converts the currency, nothing else. If the dollar rate was built for a US listed peer, it may not carry any premium for country risk, and whether to add one is a separate judgement you should state and defend, not a number to fold silently into the conversion.

    Where candidates lose it

    The common loss is keeping 9% for rupee cash flows, which values Indian inflation at a US discount rate and inflates the answer. The second is adding a few points for India by feel, which mixes two questions, currency and country risk, into one unexplained number.

    Give the exact formula, the number, and the 11.5% approximation, then say the value check in one sentence.

    What the interviewer asks next

    • If the rupee is expected to depreciate 3% a year against the dollar, what does that imply about the inflation gap?
    • How would you convert a dollar risk-free rate to a rupee one?
    • When would you add a country risk premium, and where in the build would it go?
  3. 058A stock is priced as a growing perpetuity: next year's dividend grows at 6% a year forever and investors require 12%. If the cost of equity rises to 13%, with growth unchanged, by how much does the fair price fall?Cost of capital and ratesHardLong-only asset managementBuy-side equity research

    Try it first

    Roughly how far does the fair price fall?

    Show the worked solution

    About 14%. In a growing perpetuity the price is next year's dividend divided by (r - g). The gap goes from 12% - 6% = 6 points to 13% - 6% = 7 points, so the price is multiplied by 6/7, a fall of 14.3%. With a Rs 6 dividend the price goes from Rs 100 to Rs 85.7. A stock growing only 2%, priced at the same Rs 100, falls just 9.1%.

    Why does one percentage point matter so much?

    Think of a shop you could buy whose rent rises every year. What you would pay depends on the return you want minus the rate at which the rent grows: the faster the rent grows, the less of your required return has to come from today's income. Price depends on the gap r minus g, not on r itself, so when growth is high and the gap is narrow, a one point move in r is a large change in the gap. Here the gap goes from 6 points to 7, a rise of a sixth, so the price falls to six sevenths.

    The relationship
    P=D1r−gP13%P12%=0.12−0.060.13−0.06=67=0.857P = \frac{D_1}{r-g} \qquad \frac{P_{13\%}}{P_{12\%}} = \frac{0.12-0.06}{0.13-0.06} = \frac{6}{7} = 0.857
    Pthe fair price today
    D_1next year's dividend
    rthe cost of equity
    gthe growth rate of the dividend, forever
    What it says in wordsThe new price over the old is the old gap over the new gap, because the dividend does not change.
    Price depends on the gap r - g, so fast growers move most when r moves010020030012%13%Grows 6%, Rs 6 dividendGrows 2%8%10%12%14%16%Cost of equity, rFair price, RsPrice fall when r goes 12% to 13%Grows 6%: gap 6 points to 7-14.3%Grows 2%: gap 10 points to 11-9.1%Rs 100 to Rs 85.7 andRs 100 to Rs 90.9: same start,different sensitivity
    Both stocks are worth Rs 100 at a 12% cost of equity, but when it rises to 13% the stock growing 6% falls 14.3% while the stock growing 2% falls only 9.1%, because the fast grower's gap r - g is narrower.

    Which stocks are most exposed to a rise in rates?

    Compare a slow grower priced the same way. A stock with a Rs 10 dividend growing at 2% is also worth Rs 100 at 12%; at 13% it is worth Rs 90.9, a fall of 9.1%. The more growth is built into a price, the further in the future its cash arrives, and the more the price moves when the discount rate moves. The durationThe sensitivity of a price to its discount rate, roughly the percentage price change for a one point change in the rate. of a growing perpetuity is about 1/(r - g): 16.7 for the fast grower and 10 for the slow one. The duration shortcut predicts a 16.7% fall for the fast grower; the actual 14.3% is smaller because the curve bends.

    What are the limits of this answer?

    The perpetuity assumes 6% growth forever and a discount rate that shifts cleanly by one point with nothing else changing. Real companies do not grow steadily forever, and a rise in rates often arrives with a change in growth expectations too. Treat 14.3% as the sensitivity of the valuation to its discount rate, not as a forecast of the share price. The ranking survives the simplification: high-growth, long-dated cash flows are the most rate sensitive.

    Where candidates lose it

    The quick wrong answer is about 8%, because 13 is 8.3% more than 12. The price does not depend on the discount rate alone but on its gap over growth, and that gap rose by a sixth.

    The second loss is stopping at the number. Add that a slow grower priced the same would fall only about 9%, and you have explained in one sentence why high-growth stocks sell off hardest when rates rise.

    What the interviewer asks next

    • What growth rate would make a one point rise in r cut the price by a third?
    • The cost of equity rises to 13% but growth expectations also rise to 6.5%. What happens to the price?
    • Why is the duration shortcut less accurate for large moves in the rate?