Mutual Fund Mastery puzzles, solved step by step
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006A broad equity index trades at 22 times earnings, an earnings yield of about 4.5%, while the 10-year government bond yields 7%. How much earnings growth does equity need just to match the bond, and which of the two is cheaper?PIMCOSan Diego · 2026
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Roughly what yearly earnings growth makes the index match the bond, before any extra reward for risk?
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About 2.5% a year of earnings growth just to tie, and about 5.5% if you demand an assumed 3 point premium for equity risk. Equity's expected return is roughly its earnings yield plus growth. At 22 times earnings the yield is 4.5%, 2.5 points below the bond. On starting yield the bond is cheaper; equity is cheaper only if you expect earnings growth comfortably above 5.5% a year.
How do you put a stock index and a bond on the same scale?
Think of two shops for sale. One pays its owner a fixed Rs 7 for every Rs 100 of price, forever. The other pays Rs 4.5 today, but its takings rise every year. Turn the P/E upside down to get an earnings yield, then add the growth the earnings will carry, and you have a number you can hold against the bond's yield. One over 22 is 4.55%, so the index starts 2.5 points behind the bond.
At 22 times earnings the index yields 4.5%, so it needs about 2.5% yearly earnings growth to match a 7% bond and about 5.5% once an assumed 3 point equity risk premium is added. The relationshipE/P the earnings yield, one over the P/E of 22 g the long-run growth rate of earnings 7.0% the government bond yield What it says in wordsEquity's rough expected return is what the earnings pay now plus how fast they grow, so the growth needed is the bond yield minus the earnings yield.So which is cheaper?
Answer with a condition, not a verdict. On starting yield the bond is cheaper, and equity is cheaper only if earnings can grow faster than about 5.5% a year for a long time. The 3 point premium is an assumption for this arithmetic; you should say you would set it from your own view of equity risk. Then compare the required growth with a sensible estimate of nominal earnings growth in that economy, and state which side of the line you think it falls.
Name the limitation that marks you out. The bond yield is nominal, while earnings rise with inflation, so comparing 4.5% directly with 7% mixes a real yield with a nominal one. This comparison, often called the Fed model, is a quick screen, not a valuation. It also ignores that some earnings are reinvested, and the payout and the return on that reinvestment both shape the growth you can expect.
Where candidates lose it
The common slip is comparing 4.5% with 7% and declaring bonds cheaper, full stop. That treats equity like a bond with a fixed coupon and throws away the growth that is the whole reason to own it.
The opposite slip is saying equity is cheaper because it grows, without putting a number on how much growth is already needed. The interviewer wants the 2.5% said out loud, the premium on top, and a view on whether that growth is achievable.
What the interviewer asks next
- If the bond yield falls to 6%, what P/E gives the same required growth?
- How would you adjust the comparison for inflation?
- Why might an index with a lower earnings yield still be priced fairly?
Asked at PIMCO, Debt Capital Markets, San Diego, 2026 (Wall Street Oasis):
Which is cheaper us bonds or us equities
043A project costs Rs 100 crore today and pays Rs 30 crore a year at the end of each of the next five years. What is its NPV at a 12% discount rate, and at what discount rate does the NPV fall to zero?VanguardMalvern · 2024
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Before you work it: roughly what is the NPV at 12%?
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An NPV of about Rs 8.1 crore at 12%, and the NPV reaches zero at about 15.2%, the IRR. Five payments of Rs 30 crore discounted at 12% are worth 30 times 3.605, about Rs 108.1 crore, against a cost of Rs 100 crore. Raising the rate shrinks the payments' present value, and at 15.2% they are worth exactly Rs 100 crore.
How do you value the five payments quickly?
A rupee promised next year is worth less than a rupee in your hand, the same way a friend's promise to repay in five years is worth less than cash today. Discount each payment back to today, and because the payments are equal you can use one annuity factor instead of five divisions. At 12% the five-year annuity factor is 3.605, so Rs 30 crore a year is worth about Rs 108.1 crore today. Subtract the Rs 100 crore cost: the NPV is about Rs 8.1 crore. Positive NPV means the project earns more than 12%.
The relationship30 the yearly payment, Rs crore 1.12 one plus the 12% discount rate 3.605 the five-year annuity factor at 12% What it says in wordsNPV is the present value of the payments less what you pay today.The project's NPV is Rs 50 crore with no discounting, about Rs 8.1 crore at 12%, and falls to zero at 15.2%, the IRR; any discount rate above that makes the project destroy value. How do you find the rate where NPV is zero without a calculator?
Bracket it. You need an annuity factor of 100 over 30, about 3.333. At 12% the factor is 3.605, too high, so the rate is above 12%. At 15% the factor is about 3.352, still a touch above 3.333; at 16% it is about 3.274, below. The IRRInternal rate of return: the discount rate at which the present value of a project inflows equals its cost, so the NPV is zero. is simply the discount rate at which the NPV curve crosses zero, here about 15.2%. In an interview, saying between 15% and 16%, closer to 15%, and showing the bracket, is a full answer.
State the limits of IRR alongside it. It assumes the payments can be reinvested at the IRR itself, it can mislead when comparing projects of very different size, and a project with cash flows that change sign more than once can have more than one IRR. NPV at the right cost of capital is the cleaner decision rule; IRR is the useful headline.
Where candidates lose it
The fast wrong answer is Rs 50 crore, five times 30 less 100, which ignores discounting entirely. Say the annuity factor out loud and the slip cannot happen.
The second loss is the IRR direction. Candidates who see a positive NPV at 12% sometimes guess the IRR is below 12%. A positive NPV at a rate means the project earns more than that rate, so the IRR is above it.
What the interviewer asks next
- The payments grow 5% a year instead of staying flat. Is the NPV higher or lower, and roughly by how much?
- A second project costs Rs 10 crore and has an IRR of 30%. Which would you take if you could take only one?
- Why does a higher discount rate hurt long-dated projects more than short-dated ones?
Asked at Vanguard, Mutual Funds, Malvern, 2024 (Wall Street Oasis):
A DCF walkthrough was asked for along with NPV with a whole question on CPV
061Three unnamed companies. A has a gross margin of 22% and 45 inventory days. B has a gross margin of 78% and no inventory. C has a gross margin of 35% and capital employed of three times sales. One is a grocery retailer, one a software firm and one a cement maker. Which is which?T. Rowe PriceBaltimore · 2022
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Which company is the cement maker?
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A is the grocery retailer, B is the software firm and C is the cement maker. A thin 22% gross margin with stock turning every 45 days is a reseller living on volume. A 78% gross margin with no inventory is a product that costs almost nothing to copy. Capital employed of three times sales means huge fixed plant, which among these three only cement needs.
What does each number tell you about how the business makes money?
Picture three shops on one street. The kirana store buys packets for Rs 78 and sells them for Rs 100 within weeks. The tutor sells the same recorded lesson a thousand times. The brick kiln on the edge of town cannot sell a single brick until it has built a furnace. Gross margin tells you how much of each sale the company keeps after paying for what it sells; inventory and capital tell you what it must hold to make that sale. Company A keeps 22 paise of each rupee and holds 45 days of stock, which is about 9.6% of a year's sales sitting on shelves: a reseller. Company B keeps 78 paise and holds no stock, so its costs sit below gross profit, in salaries: software.
Company A's 22% gross margin and 45 inventory days mark a grocery retailer, Company B's 78% gross margin with no stock marks a software firm, and Company C's capital employed of three times sales marks the cement maker. Why is the cement maker C and not A?
Cement is a commodity, so candidates often pin it to the lowest margin. But a cement maker does not resell anything; it turns limestone and fuel into a product, and its gross margin sits well above a pure reseller's. What gives it away is capital intensityHow much capital a business must tie up to produce a rupee of sales. Plant-heavy industries need several rupees; retailers and service firms need little.: capital employed of three times sales means Rs 300 of plant and working capital for every Rs 100 of sales. A grocer leases its stores and pays suppliers after it has sold the goods, so it might need only Rs 25 per Rs 100 of sales.
A: grocer B: software C: cement Gross margin 22% 78% 35% Sales / capital employed 4.0x 2.5x 0.33x Illustrative EBIT margin 4% 25% 15% Return on capital = margin x turnover 16% 62.5% 5% Only C's turnover is given; the other turnovers and all three EBIT margins are illustrative, chosen to show how each model reaches its return on capital. The table shows why the pattern matters to an analyst, not just to a quiz. The grocer earns a thin margin many times over; software earns a fat margin on little capital; the cement maker needs a fair margin just to cover the plant it had to build, which is why cement returns swing so hard with prices and plant utilisation. The limit: companies define gross margin differently. Some put power and freight inside cost of sales, some below it, so check the definition before reading a margin across two companies.
Where candidates lose it
The fast wrong move is to sort by margin alone: lowest margin is the commodity, so cement is A. That ignores that a reseller's gross margin is thin by nature, because it buys finished goods.
The second miss is not using the capital line at all. Interviewers give three clues because each one rules out a different pairing; say which clue decides each match and the answer becomes an argument, not a guess.
What the interviewer asks next
- Add a fourth company: gross margin 60%, inventory days 200. What might it be?
- Which of the three would you expect to have negative working capital, and why?
- A cement maker's sales rise 10% with no new plant. What happens to its return on capital?
Asked at T. Rowe Price, Investments, Baltimore, 2022 (Wall Street Oasis):
Guess what type of company it is based on a few lines of income statement
078A commercial building has gross potential rent of Rs 12 crore a year. Vacancy runs at 8%, and operating costs are 20% of the rent actually collected. Buyers of similar buildings pay a cap rate of 8%. What is the building worth?InvescoNew York · 2025
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Which income figure do you divide by the cap rate?
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About Rs 110.4 crore. Start from Rs 12 crore of gross potential rent and take off 8% vacancy to reach Rs 11.04 crore collected. Take off operating costs of 20% of that, Rs 2.208 crore, to reach net operating income of Rs 8.832 crore. Divide by the 8% cap rate: Rs 8.832 crore over 0.08 is Rs 110.4 crore.
Why does a cap rate turn one year's income into a price?
Imagine a flat that brings you Rs 30,000 a month after society charges and the odd empty month, Rs 3.6 lakh a year. If similar flats sell for Rs 72 lakh, buyers are accepting a 5% yield, and you can price any flat on the street by dividing its net rent by 5%. A cap rate is that street yield for commercial property: value equals net operating income divided by the rate buyers accept. It works like a perpetuity with expected rent growth folded into the rate, which is why a lower cap rate means a higher price.
Rs 12 crore of rent on paper shrinks to Rs 11.04 crore collected after 8% vacancy and to Rs 8.832 crore of net operating income after running costs, which at an 8% cap rate is worth Rs 110.4 crore. Why does every cost line move the value so much?
Because each rupee of net income is multiplied by one over the cap rate, 12.5 times at 8%. A cost saving of Rs 10 lakh a year adds Rs 1.25 crore of value, and a rise in vacancy from 8% to 12% cuts value by about Rs 4.8 crore. That is why a buyer picks apart the rent roll and the service charge budget before arguing about the cap rate at all. The cap rate moves value too: the same income is worth Rs 117.8 crore at 7.5% and Rs 103.9 crore at 8.5%.
The relationshipGPR gross potential rent, if every unit were let all year v vacancy, 8% c operating costs as a share of collected rent, 20% cap the cap rate buyers accept, 8% What it says in wordsValue is the rent that is actually collected, less what it costs to run the building, divided by the market's yield.Say the limitation before the interviewer does. A single cap rate prices a stable, well-let building. A building with a large lease expiring next year, or one mid-refurbishment, needs its cash flows laid out year by year rather than one year's income capitalised. The cap rate method is a shortcut for the ordinary case, and the valuer's job is to spot when the case is not ordinary.
Where candidates lose it
The common slip is dividing gross potential rent by the cap rate and quoting Rs 150 crore. Gross potential rent is a ceiling that assumes every square foot is let and nothing costs anything to run; no market cap rate was ever quoted on it.
The second slip is taking the 20% cost ratio on gross rent instead of collected rent. That gives costs of Rs 2.4 crore and a value of Rs 108.0 crore. Read which base a percentage is quoted on before you multiply.
What the interviewer asks next
- The buyer funds 60% of the price with debt at 9%. Does that change the value of the building?
- What cap rate would make the building worth Rs 120 crore?
- Why do cap rates tend to rise when interest rates rise?
Asked at Invesco, Real Estate, New York, 2025 (Wall Street Oasis):
Walk me through how to get the exit value of a property from Gross Potential rent, using Cap rate.
100A central bank cuts rates by 50 basis points. For a stock valued as next year's earnings divided by (r minus g), with the required return r falling from 12% to 11.5% and growth g at 7%, what happens to the fair P/E? And why might the market fall anyway?PIMCOLos Angeles · 2024
Try it first
What happens to the fair P/E if growth stays at 7%?
Show the worked solution
The fair P/E rises from 20x to about 22.2x, about 11%, if growth holds. P/E equals 1 over (r minus g): 1/0.05 is 20x and 1/0.045 is 22.2x. But a cut often comes because the economy is weakening. If growth expectations fall to 6.5%, the gap is back to 5 points and the P/E is unchanged; at 6%, the P/E falls to 18.2x. The market may also have priced the cut in already.
Why does a small rate cut move the P/E so much?
Think of a ladder leaning against a wall: move its foot a few centimetres when it is nearly upright and the top barely moves; move it when the ladder is almost flat and the top drops a long way. The P/E depends on the gap between the required return and growth, and when that gap is small, a half-point change in it moves the P/E by a large fraction. Here the gap goes from 5 points to 4.5, a tenth smaller, and the fair P/E rises by about 11%.
The relationshipP/E the price over next year's earnings r the return investors require, falling from 12% to 11.5% g the long-run growth rate of earnings, 7% What it says in wordsThe fair multiple is one over the gap between the required return and growth, so narrowing the gap lifts the multiple.With growth held at 7%, the cut lifts the fair P/E from 20.0x to 22.2x, but if the cut comes with growth expectations falling to 6.5% the P/E stays at 20.0x, and at 6% it drops to 18.2x, 9% below where it began. So why might the market fall on a rate cut?
Three reasons, all visible in the formula. First, growth: central banks usually cut because they see the economy slowing, and if the market lowers its growth estimate by as much as the cut lowers r, the gap and the P/E do not change at all; lower it by more and the P/E falls, to 18.2x at 6% growth. Second, expectations: if a 50 basis point cut was fully expected, it was already in the price, and only the surprise moves the market; a cut smaller than hoped can disappoint. Third, earnings themselves: E in the ratio is next year's earnings, and a weakening economy can lower that number even as the multiple rises.
Say the simplification. Writing P/E as 1 over (r minus g) assumes all earnings are paid out and growth is steady forever; with partial payout, the numerator is the payout ratio instead of one. The direction of every effect survives, but the sizes are illustrative. The general point holds firmly: a rate cut raises fair value only if what it says about growth does not undo it.
Where candidates lose it
The common slip is assuming the P/E rises by about the size of the cut, half a percent. The P/E moves on the gap between r and g, and a half-point change in a five-point gap shifts the P/E by about 11%.
The other loss is answering only the first half. The question is designed to test whether you know why markets can fall on good-sounding news: say that a cut carries information about growth, and that expected cuts are already in the price.
What the interviewer asks next
- What growth rate would leave the fair P/E exactly unchanged after the cut?
- Why does the same cut matter more for a high-growth stock than for a low-growth one?
- How might a rate cut affect a bank's earnings differently from a consumer company's?
Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis):
had only one 'technical' question about the effect of a rate cut on equity prices
