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  1. 030Two companies each have an enterprise value of 1,000, EBIT of 80 and a 25% tax rate. A has no debt. B has 500 of debt at 6% interest. Which trades on the lower P/E, and at what cost of debt would their P/Es match?Valuation riddlesHardBuy-side equity researchHedge fund long/short

    Try it first

    Same business, same enterprise value. What does B's debt do to its P/E?

    Show the worked solution

    B trades lower, at 13.3x against A's 16.7x, and they match when B's debt costs 8%. A earns 80 x 0.75 = 60 on equity of 1,000. B pays 30 of interest, earns 37.5 on equity of 500. The P/Es match when the after-tax cost of debt equals A's earnings yield of 6%: r x 0.75 = 6%, so r = 8%. Cheaper debt lowers B's P/E; dearer debt raises it.

    Why would the same business carry two different P/Es?

    Suppose a flat earns rent of Rs 6 for every Rs 100 of its price. Buy it with half cash and half a loan at 4.5% after tax, and your cash earns more than 6%, because the borrowed half costs less than it yields. P/E divides the equity's value by the equity's earnings, and debt changes both, by different amounts. Swapping equity for debt that costs less than the equity's earnings yield lowers the P/E; swapping for dearer debt raises it.

    A, no debtB, 500 of debt at 6%
    EBIT80.080.0
    Interest0.030.0
    Tax at 25%20.012.5
    Net income60.037.5
    Equity value1,000500
    P/E16.7x13.3x
    The same EBIT and the same enterprise value give two different P/Es once half of B is funded with debt.
    P/E of the levered company against what its debt costs0x10x20x30x2%4%6%8%10%B's pre-tax cost of debtA, no debt: 16.7x at any rateB: 26.7x at 11%B at 6%: 13.3xequal at 8%Why the lines cross at 8%A's earnings yield, 60 / 1,0006.0%B's debt after taxr x (1 - 25%)Equal when r x 0.75 = 6%r = 8%Cheaper debt: B's P/E below A'sDearer debt: B's P/E above A's
    B's P/E sits below A's 16.7x while its debt costs less than 8%, is 13.3x at 6%, and climbs above A once debt costs more than 8%, because leverage lowers P/E only while after-tax debt is cheaper than the earnings yield.

    Why is 8% the crossing point?

    Swapping 500 of equity for 500 of debt removes equity that was earning its share of A's 6% earnings yield, 30 of net income, and adds an after-tax interest cost of 500 x r x 0.75. If that cost is exactly 30, net income halves along with equity and the P/E does not move; 30 over 375 is 8% before tax. The general rule: leverage cuts P/E when the after-tax cost of debt is below the earnings yield, E/P. The limitation worth saying: B's lower P/E is not a sign it is cheaper. Its equity is riskier, so investors should demand a lower multiple for the same business.

    Where candidates lose it

    Candidates say B must trade on a higher P/E because leverage is risky, or on the same P/E because the business is identical. Both skip the arithmetic. Work the net income and the equity value and the answer drops out.

    The second loss is calling B cheaper because its P/E is lower. The interviewer wants to hear that a low P/E caused by leverage is a capital structure effect, which is why analysts compare levered companies on EV/EBIT instead.

    What the interviewer asks next

    • What are the two companies' EV/EBIT multiples, and why is that the fairer comparison?
    • If B's tax rate were zero, where would the crossing point be?
    • B is in a sector where peers carry no debt. How do you adjust its P/E before comparing?
  2. 031A company charges Rs 10 crore of depreciation in its books, but the tax rules let it claim Rs 25 crore of depreciation this year. The tax rate is 25%. What happens to the tax it pays, the tax it reports and its deferred tax liability?Three statement riddlesHardSell-side equity researchIndian brokerage research

    Try it first

    Compared with a year where both depreciation numbers were Rs 10 crore, what changes?

    Show the worked solution

    Cash tax falls by Rs 3.75 crore, reported tax does not change, and the deferred tax liability rises by Rs 3.75 crore. Tax is paid on taxable profit, which uses the Rs 25 crore tax depreciation. The P&L charges tax on book profit, which uses Rs 10 crore. The Rs 15 crore gap at 25% is Rs 3.75 crore of tax postponed, not saved: it comes back when tax depreciation later falls below book.

    Why can two depreciation numbers exist for the same machine?

    Think of a salaried employee who is allowed to claim a deduction early in the year for an expense she will actually use over three years. Her tax bill falls now, but she cannot claim it again later. Tax rules often allow depreciation faster than the books show it, so the same asset produces a larger deduction early and a smaller one later. The total over the asset's life is the same; only the timing differs. Confirm the current depreciation rates in the tax rules before using real numbers.

    Same year, two depreciation charges, two tax numbers, Rs croreIn the books: what the P&L showsEBITDA110Book depreciation(10)Profit before tax100Tax expense at 25%25.00In the tax return: what is paidEBITDA110Tax depreciation(25)Taxable profit85Tax paid at 25%21.25Tax charged in the P&L25.00Tax actually paid21.25Gap 3.75: added to the deferred tax liability on the balance sheet
    The books charge tax of Rs 25.00 crore on profit of Rs 100 crore while the tax return pays Rs 21.25 crore on Rs 85 crore, and the Rs 3.75 crore gap is parked in the deferred tax liability until the timing difference reverses.

    Where does each number land in the three statements?

    The income statement shows a tax expense of Rs 25.00 crore, split into current tax of Rs 21.25 crore and deferred tax of Rs 3.75 crore, so net income is Rs 75 crore either way. The cash flow statement adds back the Rs 3.75 crore of deferred tax as a non-cash charge, so operating cash flow is Rs 3.75 crore higher than it would be if both depreciation numbers matched. On the balance sheet the deferred tax liability rises by Rs 3.75 crore, matched by the extra cash.

    YearBook depreciationTax depreciationLiability movementLiability at year end
    11025+3.753.75
    21015+1.255.00
    31010+0.005.00
    4100-2.502.50
    5100-2.500.00
    Over a five year asset life both methods deduct Rs 50 crore, so the liability builds early and unwinds to zero by the end.

    The analyst's point: a growing company that keeps buying assets keeps adding new early-year gaps, so its deferred tax liability can grow for years and behave almost like permanent free funding. When capex slows, the reversals arrive and cash tax rises above reported tax. That is worth one sentence in the room.

    Where candidates lose it

    The common loss is saying depreciation is non-cash, so nothing happens to cash. The tax saved by depreciation is cash, and here the tax return claims more of it than the books.

    The second loss is lowering the reported tax charge. The P&L follows book profit; the difference is recorded as deferred tax, not as lower expense. Say that the saving is a postponement, and show when it reverses.

    What the interviewer asks next

    • What happens in year four, when tax depreciation is zero and book depreciation is still Rs 10 crore?
    • Why might an analyst treat a steadily growing deferred tax liability as closer to equity than to debt?
    • Give an example of a timing difference that creates a deferred tax asset instead.
  3. 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?
  4. 041A company has an enterprise value of 5,000 and net cash of 500. It has 100 shares in issue and 10 options with a strike price of 40. What is the value per share using the treasury stock method?Valuation riddlesHardSell-side equity researchBuy-side equity research

    Try it first

    Where does the answer land?

    Show the worked solution

    About 53.64 per share. Equity value is 5,000 plus 500 of net cash, 5,500. If the price is P, the options add 10 shares and their 400 of exercise cash buys back 400 / P shares, so P = 5,500 / (110 - 400/P). Solving gives P = 5,900 / 110 = 53.636. Iterating from the naive 55 gets there in three rounds.

    Why can you not just divide by a share count?

    Think of splitting a restaurant bill where one late guest pays a fixed Rs 40 whatever the bill, and the rest is shared. How much the others pay depends on the bill, and the bill depends on who is sharing it. Under the treasury stock methodA way to count option dilution: assume in-the-money options are exercised and the exercise cash is used to buy back shares at the current price., how many net new shares the options create depends on the share price, and the share price depends on how many shares there are. The dilution and the answer have to be found together.

    Price sets dilution, dilution sets price: the loop settles at one numberShare price P= 5,500 / diluted sharesDiluted shares= 100 + 10 - 400 / PP sets how manyshares the 400buys back53.554.054.555.0fixed point 53.636step 055.00 naivestep 1step 1: 53.540step 2step 2: 53.643step 3step 3: 53.636step 4Each step: price to diluted shares to a new price
    Starting from the naive 55, each round of price to diluted shares to new price moves closer to 53.636, the one price at which the dilution and the value per share agree; when dilution depends on price, set up one equation and solve it.

    How do you solve it in one line instead of looping?

    Write P x (110 - 400/P) = 5,500. The P cancels in the second term, leaving 110P - 400 = 5,500, so P = 5,900 / 110 = 53.636. When the options are in the money, the consistent price is simply equity value plus exercise cash, divided by all shares including the options. The check: at 53.636, the 400 of cash buys back 7.458 shares, so 2.542 net new shares take the count to 102.542, and 5,500 over that is 53.636.

    StepPrice inDiluted sharesPrice out
    055.000102.72753.540
    153.540102.52953.643
    253.643102.54353.636
    353.636102.54253.636
    Each round overshoots and cuts the gap to the answer to about a fourteenth; a spreadsheet with iterative calculation switched on does exactly this.

    Say the limitation. The method ignores the time value of options and assumes exercise today. A model that values the options properly would subtract their value from equity instead, and the answer would come out slightly lower.

    Where candidates lose it

    Candidates usually give 55, forgetting the options, or 50, adding the new shares but forgetting the exercise cash. Both are one-step answers to a problem that loops.

    The other loss is announcing that the model is circular and stopping. Show the one-line algebra, give the number, and check it by running one round of the loop out loud.

    What the interviewer asks next

    • What if the strike were 60? Do the options dilute at all?
    • How would you treat convertible bonds in the same valuation?
    • Why do some analysts use fully diluted shares on all options, regardless of strike?
  5. 051You have n cars, each with a full tank that lasts exactly 1,000 miles, and fuel can be passed from one car to another in the middle of the journey. What is the farthest one car can get? Give the answer for three cars, and say what happens as n grows without limit.Probability and brainteasersHardMillennium ManagementLondon · 2024

    Try it first

    With three cars, how far can the last car get?

    Show the worked solution

    With three cars the last car reaches about 1,833 miles; with n cars it reaches 1,000 x (1 + 1/2 + ... + 1/n). The cars drive together until they have burned one tankful between them, then one car refills the rest and stops. That happens after 1,000/n miles, then 1,000/(n - 1), down to the last car's 1,000. The sum has no ceiling, but it grows only like the logarithm of n.

    Why does a car drop out after exactly 1,000/n miles?

    Picture a group of hikers sharing water on a long desert walk. Once the group has drunk one person's worth between them, that person can pour what is left into the others' bottles, fill every one back up, and wait by the path. Carrying that hiker any further only costs water. The cars work the same way. After d miles each car has burned d miles of fuel and has room for d more. One car can refill the other n - 1 cars exactly when (n - 1) x d equals what it has left, 1,000 - d, which solves to d = 1,000/n.

    Each car that drops out hands over one tank; each stretch is shorter than the last3 cars333 miles2 cars500 miles1 car, its own full tank1,000 miles03338331,833 miles0 to 333: three cars burn 3 x 333 = 1,000 miles of fuel. Car 3 tops up cars 1 and 2, then stops.333 to 833: two cars burn 2 x 500 = 1,000 miles of fuel. Car 2 tops up car 1, then stops.833 to 1,833: car 1 drives the whole of its own tank, 1,000 miles.Total distance by number of cars: no ceiling, but it grows like 1,000 x ln n1 car1,0002 cars1,5003 cars1,83310 cars2,929100 cars5,187
    Three cars drive 333 miles together, two cars drive the next 500 and the last car drives its own 1,000, for 1,833 miles in all; each stretch burns one full tank, and ten cars reach only 2,929 miles.

    Why do three cars buy 1,833 miles and not 3,000?

    Follow the road left to right. Three cars run 333 miles together and burn 1,000 miles of fuel between them, one full tank. Car 3 has 667 miles of fuel left and hands 333 to each of the other two, which fills them. Two cars then run 500 miles, burning another tankful, and car 2 refills car 1. Car 1 runs its own 1,000. Every stretch burns exactly one tank, but each extra car buys a shorter stretch, because its tank has to move more cars.

    The relationship
    D(n)=1000(1+12+13+⋯+1n)≈1000 (ln⁡n+0.577)D(n) = 1000\left(1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots + \tfrac{1}{n}\right) \approx 1000\,(\ln n + 0.577)
    D(n)the farthest one car gets with n cars, in miles
    1/kthe stretch driven while k cars are still on the road, in tankfuls
    0.577the Euler-Mascheroni constant, the gap between the harmonic sum and ln n
    What it says in wordsAdd one over k for every car count from n down to one; the total grows like the natural logarithm of n.

    What happens as n goes to infinity?

    The harmonic series never converges, so enough cars can reach any distance at all. But the growth is logarithmic: ten cars reach about 2,929 miles and a hundred cars about 5,187, so every tenfold increase in cars adds only about 2,300 miles. When asked for the asymptotic answer, say both halves: unbounded, and slow. State the assumptions too: fuel moves between cars without loss, and a car that drops out is simply left behind. If the helper cars had to drive home, the answer would shrink sharply.

    Where candidates lose it

    The two fast wrong answers are 1,000 miles, because fuel cannot be created, and 3,000 miles, because three tanks were bought. Both miss that every car on the road burns fuel at the same rate, so the fleet spends most of its fuel carrying itself forward.

    The second loss comes at the asymptotic part. Candidates who say the distance levels off, or that it grows in proportion to n, lose the point. Say unbounded, growing like 1,000 times ln n, and give the ten and hundred car figures to show you can use the result.

    What the interviewer asks next

    • How many cars do you need to cover 3,000 miles? (11)
    • What changes if every helper car must keep enough fuel to drive back to the start?
    • Where else does the harmonic series turn up, in probability or in markets?

    Asked at Millennium Management, Investments, London, 2024 (Wall Street Oasis): What is the maximum distance you can get with the cars if you can transfer petrol in the middle of the journey?

  6. 052A game multiplies your stake by 1.5 when a fair coin lands heads and by 0.6 when it lands tails, and you must stake everything you have on every flip. The expected return per flip is plus 5%. After 100 flips, what does a typical player hold?Expected value and decisionsHardHedge fund long/shortLong-only asset management

    Try it first

    Before working it: after 100 flips the typical player holds...

    Show the worked solution

    About 0.5% of the starting stake. One head and one tail together multiply wealth by 1.5 x 0.6 = 0.90, so the typical growth factor per flip is the square root of 0.90, or 0.949, a loss of about 5.1% a flip. The median path has 50 heads and 50 tails and ends at 0.9 to the power 50, about 0.005. The average ends near 131.5x, carried by rare lucky paths.

    Why does a plus 5% game shrink the typical player?

    Think of a shop that raises a price 50% one month and cuts it 40% the next. The two changes average plus 5%, yet an item tagged Rs 100 ends at Rs 90. Wealth compounds by multiplying, not by adding. So the rate that decides where one player ends up is the geometric mean of the multipliers, not their arithmetic average. Here the geometric mean is the square root of 1.5 x 0.6, which is 0.949 a flip, and 0.949 applied a hundred times is a very small number.

    The relationship
    g=1.5×0.6=0.949g100=0.950≈0.005g = \sqrt{1.5 \times 0.6} = 0.949 \qquad g^{100} = 0.9^{50} \approx 0.005
    gthe typical growth factor per flip, the geometric mean of the two multipliers
    1.5, 0.6the multipliers on heads and on tails
    0.9^50fifty head and tail pairs, the median outcome after 100 flips
    What it says in wordsThe typical player's wealth grows at the geometric mean of the multipliers, and here that mean is below one.
    Log scale: the average rises, the typical player sinks0.00001x0.0001x0.001x0.01x0.1x1x, start10x100x1,000xAverage of all players: 131.5xTypical player: 0.9^50 = 0.005x0255075100FlipsOnly 13.6% of players finish above 1x
    On a log scale the average of all players climbs in a straight line to about 131.5 times the stake, while the typical player falls to about 0.5% of it; and only 13.6% of all players finish above where they started.

    If the typical player loses, where does the plus 5% average come from?

    From a very small number of paths with far more heads than tails. You need at least 56 heads in 100 flips just to finish ahead, and only about 13.6% of players get there. The average is pulled up by the few players who land 70 or more heads and finish hundreds of thousands of times richer, while most players finish near zero. A player with exactly 70 heads ends at about 468,733 times the stake. The mean is a true number, but almost no individual player experiences it.

    What does this have to do with running money?

    A portfolio compounds exactly like the game. Volatility pulls the growth rate below the average return by roughly half the variance, so a strategy with a positive expected return can still shrink a typical account if it is run at too much size. Here the average return is 5% with a swing of 45% either way; half of 45% squared is about 10%, which is why the typical path loses about 5% a flip. The fix is sizing, not the odds: staking a quarter of wealth each flip, the Kelly fractionThe share of wealth to stake on each bet that maximises the long-run growth rate of wealth. here, lifts the typical player to about 1.86x after 100 flips.

    Where candidates lose it

    The trap is answering with the expected value, 1.05 to the power 100, about 131.5 times the stake. That is the average across every possible player and the right answer to a different question. The interviewer asked what a typical player holds, which is the median.

    The second loss is calling the game bad. The odds are good; the sizing is bad. Say that staking a fraction of wealth each flip turns the same odds into a growing account, and you have shown why the question is asked on an investing desk.

    What the interviewer asks next

    • What fraction of your wealth should you stake each flip to maximise long-run growth?
    • How many heads out of 100 do you need to finish ahead?
    • Would you play this game once for your whole savings? Would you play it 100 times with a quarter each time?
  7. 053Estimate the annual market in India, in Rs crore, for metformin, the usual first-line tablet for type 2 diabetes. Build it from the adult population, prevalence, the diagnosis rate, the treatment rate, the share of treated patients on this molecule and the daily cost of therapy.Market sizing and estimationHardSell-side equity researchResearch KPO and GCC

    Try it first

    Once the chain is built, which input moves the answer the most?

    Show the worked solution

    About Rs 1,560 crore a year, on stated assumptions. Take 95 crore adults and 10% prevalence for 9.5 crore people with diabetes. Half are diagnosed, 4.75 crore; half of those take regular tablets, 2.38 crore; 60% of them are on this molecule, 1.43 crore patients. At Rs 3 a day for 365 days each patient spends Rs 1,095 a year, which gives about Rs 1,560 crore.

    How do you structure a market size before you pick any number?

    Sizing a drug market is like working out how many raincoats a town buys: not everyone gets caught in the rain, not everyone who gets wet buys a coat, and those who buy choose among brands. Write the chain first and say it out loud: people, times the share with the disease, times the share who know they have it, times the share treated, times the share on this molecule, times the annual cost. Stating the chain before any number shows the interviewer the logic, and lets them correct one input without the estimate collapsing.

    Each step after prevalence keeps only about half the poolStart: 95 crore adults x 10% prevalence9.5 crore have diabetesx 50% are diagnosed4.75 crore diagnoseddrop outx 50% take regular tablets2.38 crore on regular tabletsdrop outx 60% are on this molecule1.43 crore on this molecule1.43 crore patients x Rs 3 a day x 365 daysabout Rs 1,560 crore a year
    From 9.5 crore people with diabetes, half are diagnosed, half of those take regular tablets and 60% of those are on this molecule, leaving 1.43 crore patients and a market of about Rs 1,560 crore at Rs 3 a day.
    StepAssumptionPool or value
    Adultsassumed95 crore
    With diabetes10% prevalence9.5 crore
    Diagnosed50%4.75 crore
    On regular tablets50% of diagnosed2.38 crore
    On this molecule60% of treated1.43 crore
    Cost per patientRs 3 a day x 365Rs 1,095 a year
    Marketpatients x annual costRs 1,560 crore
    Each row multiplies the one above it, and the market is 1.43 crore patients times Rs 1,095 a year.

    Which assumption deserves the most care?

    The one you are least sure of, because in a multiplicative chain every input moves the answer in the same proportion. Raising the diagnosis rate from 50% to 60% lifts the market by 20%, exactly as much as raising prevalence from 10% to 12%. Candidates spend their effort on prevalence because it is the headline statistic and wave the diagnosis rate through, when it is often the less certain number. Every figure here is an assumption for the exercise; check each against a published national survey before using it for anything.

    How do you sanity check the answer?

    Cross-check from the other end. Taking India's population as roughly 140 crore, also an assumption, Rs 1,560 crore works out to about Rs 11 per person per year, which is plausible for one cheap, widely used tablet. Then name what the estimate leaves out: combination tablets that contain the molecule, patients who take it irregularly, and the price gap between branded and generic packs. Each moves the number, and saying so is worth more than an extra decimal.

    Where candidates lose it

    Candidates lose this by starting with a number instead of a chain. They say ten crore diabetics and then improvise, and when the interviewer questions one step there is no structure to adjust. Write the chain first, then fill it in.

    The second loss is treating every person with diabetes as a patient on the drug. Skipping the diagnosis and treatment steps gives about Rs 6,242 crore, four times the answer, and the gap between having a disease and being treated for it is the point of the question.

    What the interviewer asks next

    • How does the market change if a national screening drive lifts diagnosis to 70%?
    • How would you size the market for a newer, far more expensive class of diabetes drug?
    • What would you check to test the Rs 3 a day assumption?
  8. 055A holding company has a market value of Rs 10,000 crore. It owns 50% of a listed subsidiary whose market value is Rs 16,000 crore, and it also runs its own business, which earns Rs 300 crore a year. What multiple is the market paying for that own business?Valuation riddlesHardSell-side equity researchIndian brokerage research

    Try it first

    What earnings multiple is the market paying for the holding company's own business?

    Show the worked solution

    About 6.7x, on an implied stub value of Rs 2,000 crore. The 50% stake is worth half of Rs 16,000 crore, Rs 8,000 crore, at the subsidiary's own market price. Take that out of the holding company's Rs 10,000 crore and the market is paying Rs 2,000 crore for everything else. Against Rs 300 crore of earnings that is 6.7 times, assuming the holding company carries no debt or cash of its own.

    How do you find the price of a business that has no price of its own?

    A thali costs Rs 300 and includes a sweet the same restaurant sells alone for Rs 80. The rest of the meal is costing you Rs 220. When a company's value contains something with its own visible price, subtract that price to see what the market is paying for the rest. Analysts call what is left the stubThe value the market implicitly assigns to a holding company's own business after subtracting the market value of its listed stakes.. Here the visible item is the listed stake, worth Rs 8,000 crore at the subsidiary's share price.

    Subtract what has a visible price to see the price of what does not, Rs croreOther holders' 50%Holding co's 50%8,000Listed subsidiaryworth 16,000Stake at market8,000Stub 2,000Holding companyworth 10,000Stub = 10,000 - 8,000 = 2,000Own business earns 3002,000 / 300 = 6.7xWith a 20% holding discountStake counted at 6,400Stub 3,6003,600 / 300 = 12.0xWrong: 10,000 / 300 = 33xcharges the business for the stake
    Half of a Rs 16,000 crore subsidiary is Rs 8,000 crore, which leaves only Rs 2,000 crore of the holding company's Rs 10,000 crore for its own business, so the market pays 6.7x that business's Rs 300 crore of earnings.
    The relationship
    stub=10,000−0.5×16,000=2,0002,000300=6.7×\text{stub} = 10{,}000 - 0.5 \times 16{,}000 = 2{,}000 \qquad \frac{2{,}000}{300} = 6.7\times
    10,000the holding company's market value, Rs crore
    0.5 x 16,000its stake in the listed subsidiary at market value
    300the own business's annual earnings, Rs crore
    What it says in wordsThe stub is the holding company's value less the market value of its stake, and the multiple is the stub over the own business's earnings.

    Why might 6.7x not be the whole story?

    Holding companies usually trade below the value of what they own. If the market applies a 20% holding-company discount to the stake, it is valuing the stake at Rs 6,400 crore, so the stub rises to Rs 3,600 crore and the implied multiple to 12x. The discount reflects tax on any eventual sale of the stake, dividends that may never reach the holding company's own shareholders, and the cost of running the holding company. Give both numbers and say which assumption produces each.

    What would you check before calling the stub cheap?

    Three things. Whether the holding company carries debt, which the stub has to absorb; whether the Rs 300 crore of earnings is recurring or flattered by one-off items; and whether the stake is ever likely to be sold or distributed. A discount that never closes is not a mispricing, so a low stub multiple is a question to investigate, not a conclusion.

    Where candidates lose it

    The trap is dividing the whole Rs 10,000 crore by Rs 300 crore and quoting 33x, which charges the operating business for a stake it does not contain. The interviewer made the stake most of the value precisely so that mistake would be large.

    The second loss is stopping at 6.7x without mentioning the holding-company discount. Give 6.7x on market value, then 12x with a 20% discount, and say the real answer depends on why the discount exists.

    What the interviewer asks next

    • The subsidiary falls 25% and the holding company's price does not move. What is the stub multiple now? (13.3x)
    • Why do holding-company discounts persist for years?
    • What pair of positions would isolate the stub, and what risks would remain?
  9. 056A company spends Rs 40 crore on developing software this year. Instead of expensing it, the company capitalises the full amount and amortises it over four years. Compared with expensing, what changes in year one for EBITDA, EBIT, operating cash flow and investing cash flow? Ignore tax for the first pass, then add it.Three statement riddlesHardSell-side equity researchBuy-side equity research

    Try it first

    Compared with expensing, what happens to operating cash flow in year one?

    Show the worked solution

    EBITDA rises Rs 40 crore, EBIT rises Rs 30 crore, operating cash flow rises Rs 40 crore and investing cash flow falls Rs 40 crore; total cash does not change. Capitalising takes the spend off the income statement and puts Rs 10 crore of amortisation below EBITDA instead. On the cash flow statement the same Rs 40 crore moves from the operating section to the investing section. The company is no richer; it only looks richer on the two most quoted lines.

    What does capitalising actually move?

    A family that buys a Rs 40,000 laptop to last four years of college can tell itself it spent Rs 40,000 this month, or Rs 10,000 a year for four years. The bank balance is the same either way. Capitalising a cost is the second story: the money leaves in year one, but the income statement recognises it a quarter at a time. The spend becomes an asset on the balance sheet, and the cash paid for it is reported under investing activities instead of operating ones.

    Same Rs 40 crore spend, year one, Rs crore: the lines change, the cash does notExpensedEBITDAspend -40110Amortisation0EBIT110Operating cash flowspend -40110Investing cash flow0Net cash110Capitalised over 4 yearsEBITDA150Amortisation-10EBIT140Operating cash flow150Investing cash flow-40Net cash110Operating cash flow +40, investing cash flow -40, net cash 110 both ways
    Expensed, the Rs 40 crore spend cuts EBITDA and operating cash flow to Rs 110 crore; capitalised, both stay at Rs 150 crore while investing cash flow shows minus Rs 40 crore, so net cash is Rs 110 crore either way.

    Which lines improve, and by how much?

    Take a company with EBITDA of Rs 150 crore before this spend and nothing else going on. Expensed, EBITDA, EBIT and operating cash flow are all Rs 110 crore. Capitalised, EBITDA is Rs 150 crore, amortisation takes Rs 10 crore, EBIT is Rs 140 crore, operating cash flow is Rs 150 crore and investing cash flow is minus Rs 40 crore. EBITDA and operating cash flow each gain the full Rs 40 crore, EBIT gains only Rs 30 crore, and net cash is Rs 110 crore both ways.

    Year one, Rs croreExpensedCapitalisedChange
    EBITDA110150+40
    Amortisation0-10-10
    EBIT110140+30
    Operating cash flow110150+40
    Investing cash flow0-40-40
    Net cash1101100
    Capitalising lifts EBITDA and operating cash flow by the full spend and EBIT by the spend less one year of amortisation, while net cash is unchanged.

    What happens once tax is added?

    It depends on the tax rules, which vary by country and should be confirmed for the case at hand. If the tax authority allows the deduction on the spend whatever the accounts say, cash tax is the same both ways, and book profit after a 25% tax is Rs 22.5 crore higher, with a Rs 7.5 crore deferred tax liability. If tax instead follows the books, capitalising raises year-one cash tax by Rs 7.5 crore, so the choice that flatters profit actually costs cash.

    Why does an analyst care?

    Because the two lines that gain are the two most used in valuation and quality screens: EV/EBITDA and cash conversion. A company that capitalises heavily looks cheaper on EV/EBITDA and converts more of its profit into operating cash than a peer that expenses the same spend. The fair comparison is free cash flow after all capital spending, including capitalised development, which is Rs 110 crore in both cases. If the spend recurs every year, amortisation builds to Rs 40 crore by year four and EBIT converges, but EBITDA stays Rs 40 crore higher for good.

    Where candidates lose it

    The usual slip is saying operating cash flow does not change because cash is cash. Total cash does not change; the split between sections does, and the split is what most people quote when they talk about cash conversion.

    The second slip is giving EBIT the full Rs 40 crore uplift. Rs 10 crore of amortisation sits above EBIT, so EBIT gains Rs 30 crore in year one, and that gap closes to nothing once a steady spend has built up four years of amortisation.

    What the interviewer asks next

    • The company spends Rs 40 crore every year. What do EBITDA and EBIT look like in year four?
    • How would you adjust two peers, one capitalising and one expensing, so they compare fairly on EV/EBITDA?
    • What happens on each statement if the capitalised project is abandoned in year two?
  10. 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?