Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
Hard: 30

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016A value stock has a 5% dividend yield and 5% growth; a growth stock has a 1% yield and 9% growth. Both are priced at a 10% required return. If the required return rises to 10.5%, which price falls more, by how much, and what does that say about each stock's duration?Valuation riddlesHardBlackRockNew York · 2026
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Required return goes from 10% to 10.5%. How far does the growth stock fall?
Show the worked solution
The growth stock falls about 33% and the value stock about 9%. With a constant-growth model, price is dividend over required return minus growth. The value stock goes from 5 over 0.05, 100, to 5 over 0.055, 90.9. The growth stock goes from 1 over 0.01, 100, to 1 over 0.015, 66.7. Their implied durations are 20 and 100 years.
Why can a stock have a duration at all?
Think of two people selling their future income for cash today. One will earn a steady amount starting now; the other will earn little for years and a lot later. If the lender's interest rate rises, the second one's offer falls much more, because every rupee of it has to be discounted over a longer wait. A stock is a stream of future cash, and the further out the cash sits, the more its price moves when the discount rate moves: that sensitivity is its duration. In the constant-growth modela valuation that prices a share as the dividend expected next year divided by the required return minus a steady growth rate, price is D over (r minus g), and duration works out to 1 over (r minus g).
When the required return rises from 10% to 10.5%, the value stock falls from 100 to 90.9 while the growth stock falls from 100 to 66.7, because their implied durations are 20 and 100 years.
How do you work the two falls quickly?
Watch the gap, r minus g, not r. For the value stock the gap goes from 5 points to 5.5 points, ten per cent wider, so the price falls by 1 minus 5 over 5.5, which is 9.1%. For the growth stock the gap goes from 1 point to 1.5 points, fifty per cent wider, so the price falls by a third, 33.3%. The same half-point move is a small change to a wide gap and a large change to a narrow one.
The relationship
$P = \frac{D}{r-g},\qquad -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g}: \quad \frac{1}{0.05} = 20, \quad \frac{1}{0.01} = 100$
D next year's dividend per 100 of price: 5 for value, 1 for growth
r the required return, 10% rising to 10.5%
g the steady growth rate: 5% for value, 9% for growth
What it says in wordsPrice is the dividend over the gap between required return and growth, and the narrower the gap, the more a change in the required return moves the price.
The duration numbers also show their own limit. Twenty years times half a point predicts a 10% fall, close to the true 9.1%. A hundred years times half a point predicts 50%, well off the true 33.3%, because a straight-line estimate fails when the move is large next to the gap. And the model assumes growth runs forever at 9% against a 10% required return, which is fragile; real growth stocks are valued with a high-growth phase that fades, but the direction of the answer holds.
Where candidates lose it
The common slip is answering that both fall about the same because both are priced at 100 at the same required return. Equal prices hide very different timing of the cash, and timing is what rate sensitivity measures.
The second slip is applying duration in a straight line and saying the growth stock falls 50%. Mention duration, then show the exact repricing: when r minus g is only 1 point, the curve matters as much as the slope.
What the interviewer asks next
Why did long-duration growth stocks fall hardest when interest rates rose sharply?
What happens to the growth stock's price if expected growth falls from 9% to 8.5% at a 10% required return?
How would you estimate the duration of an index rather than one stock?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks MSE Forecasting equation
017Two funds' daily returns are negatively correlated within any given month, yet their yearly returns are positively correlated. How can both be true?Statistics, correlation and diversificationHardSquarepoint CapitalMontreal · 2024
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Which explanation fits?
Show the worked solution
A shared driver that moves slowly lifts or sinks both funds together across years, while short-term noise pushes them in opposite directions day to day. Within a month the slow driver barely changes, so daily correlation reflects only the opposing noise. Across years it dominates. With daily noise of 0.8% at -0.5 correlation and a shared yearly drift of 20% standard deviation, yearly correlation comes out at about +0.57.
What kind of situation produces this?
Think of two ice cream stalls on the same beach. On any given day, a customer who buys from one does not buy from the other, so their daily sales move against each other. Across years, both do well in hot summers and badly in wet ones. Correlation is not one fixed number between two things; it depends on which driver dominates at the horizon you measure, and different drivers dominate at different horizons. For funds, the slow driver might be the economy's earnings cycle, which both portfolios share; the fast one might be money rotating between their two styles day to day.
In a simulated month the two funds move in opposite directions on 16 of 21 days, yet across years the shared drift adds 0.040 of covariance against 0.008 removed by the opposing noise, so yearly returns are positively correlated at about 0.57.
Can you show it with numbers?
Build each fund's yearly return from two parts. A shared drift, the same for both within a year but different from year to year, with a standard deviation of 20%. Plus daily noise of 0.8% a day for each fund, correlated at -0.5 between them, over 250 trading days. Within a month the drift is a constant, so it drops out of any correlation measured around the month's average, and the daily figure is the noise's -0.5. Across years, the drift contributes 0.2 squared, 0.040, to covariance, and the noise contributes -0.5 times 250 times 0.008 squared, -0.008, leaving +0.032.
The relationship
$\rho_{year} = \frac{\sigma_F^2 + \rho_d\, n\, \sigma_d^2}{\sigma_F^2 + n\,\sigma_d^2} = \frac{0.040 - 0.008}{0.040 + 0.016} = \frac{0.032}{0.056} \approx 0.57$
\sigma_F the standard deviation of the shared yearly drift, 20%
\rho_d the correlation of daily noise, -0.5
n trading days in a year, 250
\sigma_d each fund's daily noise, 0.8%
What it says in wordsYearly correlation is the shared drift's variance less the summed opposing noise, divided by each fund's total yearly variance.
Give the condition and a second mechanism. The sign flips only if the shared drift's variance is larger than the summed noise covariance; with a drift of 10% instead of 20%, covariance would be 0.010 minus 0.008 and correlation barely positive. A second route is timing: if one fund's holdings are priced with a lag, its daily moves look unrelated or even opposite to the other's, while over a year the lags wash out. Either way, the lesson for a fund analyst is that a diversification benefit measured on daily data may not exist at the horizon a client actually holds.
Where candidates lose it
The common failure is saying it is impossible, on the belief that correlation is a fixed property of two assets. The interviewer is checking whether you know correlation is horizon dependent and can name what drives each horizon.
The second failure is waving at small samples. Twelve monthly points are noisy, but noise is not an explanation; the strong answer builds a two-component model and states when the sign flips.
What the interviewer asks next
Using the same model, at what size of shared drift would yearly correlation be exactly zero?
Why might two funds that look like good diversifiers on daily data fail to diversify in a bear market?
How would stale prices in one fund distort its measured volatility as well as its correlation?
Asked at Squarepoint Capital, Hedge Fund, Montreal, 2024 (Wall Street Oasis): correlation can be negative intra-month but positive across a year, how?
021"Why should I buy your college, and how much would you sell it for?" Value a college with 8,000 students each paying Rs 2 lakh a year in fees, at a 35% operating margin.Estimation and market sizingHardWellington ManagementBoston · 2024
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What is the college's yearly operating profit?
Show the worked solution
Roughly Rs 560 to 784 crore, on Rs 56 crore of operating profit at assumed multiples of 10 to 14 times. Fees are 8,000 times Rs 2 lakh, Rs 160 crore; a 35% margin leaves Rs 56 crore. The case for buying is durability: a brand, accreditation and a campus that keep the seats full for decades. The multiple you ask for depends on how sure the buyer can be of that.
What is the interviewer really testing?
Think of selling the tea stall outside a busy railway station. The buyer is not paying for the kettle; he is paying for the queue that shows up every morning and the confidence it will keep showing up. Any institution can be valued as a stream of cash plus a judgement about how long and how reliably that stream lasts, and the question has two halves because those are the two halves of a valuation. "Why should I buy" asks for the durability story; "how much" asks for the numbers.
Fees of Rs 160 crore less Rs 104 crore of costs leave Rs 56 crore of operating profit, which at assumed multiples of 10 to 14 times values the college at roughly Rs 560 to 784 crore.
How do you get from fees to a value?
Revenue is students times fees: 8,000 times Rs 2 lakh is Rs 160 crore a year. A 35% operating margin leaves Rs 56 crore, and the multiple you put on that profit is where the durability argument turns into a number. At 10 times it is Rs 560 crore, at 14 times Rs 784 crore. These multiples are assumptions for the exercise; say you would set them against what comparable education businesses have changed hands for, and against a cash flow valuation.
The relationship
$V = m \times (N \times F \times M) = m \times (8{,}000 \times 2 \text{ lakh} \times 0.35) = m \times 56 \text{ crore}$
N students, 8,000
F yearly fee per student, Rs 2 lakh
M the operating margin, 35%
m the multiple of operating profit, assumed 10 to 14
What it says in wordsProfit is students times fee times margin, and value is that profit times a multiple that reflects how durable it is.
Then sell the durability and test it. The selling points are a waiting list larger than the intake, accreditation that a new entrant would take years to earn, land owned rather than leased, and alumni who send their children. The tests are the risks: if enrolment falls 10% while costs stay fixed, operating profit drops from Rs 56 crore to about Rs 40 crore, because every rupee of lost fees falls straight to profit. A per-seat check helps too: Rs 560 crore over 8,000 seats is Rs 7 lakh a seat, three and a half years of fees.
Name one structural limit. Many colleges are run by trusts or societies that cannot distribute profit, so in practice a buyer may be acquiring a management contract, the land or a related company rather than the college itself; the cash a buyer can actually take out may be smaller than the operating profit. Asking who can receive the cash shows the interviewer you think like an owner.
Where candidates lose it
The common failure is answering only one half: a heartfelt speech about the college with no number, or a quick multiple with no reason a buyer should believe the profit lasts. The interviewer asked both questions on purpose.
The second failure is multiplying revenue instead of profit, or quoting a multiple as if it were a market fact. Build revenue, then profit, then state the multiple as your assumption and the range it gives.
What the interviewer asks next
What would a buyer pay if fees are capped by a regulator and costs rise 6% a year?
How would you value the college with a discounted cash flow instead of a multiple?
Which single number would you most want to verify before agreeing a price?
Asked at Wellington Management, Investment Research, Boston, 2024 (Wall Street Oasis): Why should I buy your College and how much would you sell it for?
039Give an equation for the surface area of an n by n by n Rubik's cube, counted in small unit squares. Then give an equation for how many of the small cubes show at least one face.Logic and numeracy brainteasersHardT. Rowe PriceBaltimore · 2020
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For a standard 3 by 3 by 3 cube, how many small cubes show at least one face?
Show the worked solution
The surface is 6n² unit squares, and n³ minus (n minus 2)³ cubes show at least one face. Six faces each carry n by n squares. For the cubes, count what is hidden: peeling one layer off every side leaves an (n minus 2) cube inside, so the visible ones are n³ minus (n minus 2)³, which expands to 6n² minus 12n plus 8. For a 3 by 3 by 3 cube that is 54 squares and 26 cubes.
Why are there two different counts here?
A house with a corner room has windows on two walls of that room, but it is still one room. Counting windows and counting rooms give different numbers. The surface counts squares, and a corner cube carries three squares and an edge cube two, so the square count is always larger than the number of cubes that show. Keeping the two counts apart is half the problem; candidates who blur them give 54 for both.
The surface is the easy part. Each of six faces is an n by n grid, so the area is 6n² unit squares: 54 for a standard cube. A quick check is to rebuild it from the cube types: 8 corners showing 3 squares, 12(n minus 2) edge cubes showing 2, and 6(n minus 2)² centre cubes showing 1. For n of 3 that is 24 plus 24 plus 6, which is 54 again.
On a 4 by 4 by 4 cube the skin carries 96 squares, but only 56 cubes show, because each corner cube shows three squares and each edge cube two; the dashed 2 by 2 by 2 core of 8 cubes is hidden, and 64 minus 8 is 56.
Why count the hidden cubes instead of the visible ones?
The hidden cubes form one clean block, (n minus 2) on every side, so counting them and subtracting from n³ avoids every double count. Counting the visible cubes directly means adding corners, edges and faces while remembering that each edge already lost its corners. Both routes give the same answer, and the second makes a good check.
The relationship
$V(n) = n^3 - (n-2)^3 = 6n^2 - 12n + 8 \qquad V(3) = 27 - 1 = 26$
n small cubes along one edge
(n-2)^3 the hidden core after peeling one layer from every side
V(n) cubes showing at least one face
What it says in wordsVisible cubes are all the cubes minus the core you cannot see; expanded, it is the surface area less a correction for corners and edges.
The expanded form is worth a sentence because it links the two answers. 6n² is the square count; the minus 12n plus 8 removes the extra squares that edge and corner cubes contribute. Say the edge case too: the formula needs n of at least 2. For n of 1 it gives 2, but a single cube is one cube.
Where candidates lose it
The trap is giving 6n² as the answer to both questions. The interviewer is checking whether you notice that one cube can show up to three squares. Name the two counts before you write anything.
The second loss is trying to count visible cubes from the surface and getting tangled in double counts at the edges. Go to the hidden core first, then offer the corner, edge and face breakdown as the check.
What the interviewer asks next
How many small cubes show exactly two faces, as a formula in n?
For which n is the hidden core larger than the visible shell?
How would the counts change for a 4 by 5 by 6 box?
Asked at T. Rowe Price, Equities, Baltimore, 2020 (Wall Street Oasis): Give an equation that yields the surface area of an n by n by n Rubic's cube based on number of blocks per side
052A company trades at 10 times EBITDA of Rs 200 crore. It has net debt of Rs 500 crore, depreciation of Rs 40 crore, interest of Rs 50 crore and a 25% tax rate. What P/E is the market paying?Valuation riddlesHardHoulihan LokeyChicago · 2026
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Before you work it: which is closest to the P/E?
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About 18.2x. Enterprise value is 10 x 200, Rs 2,000 crore. Take off Rs 500 crore of net debt and shareholders own Rs 1,500 crore. Earnings for shareholders are EBITDA of 200 less depreciation of 40 and interest of 50, which is 110 before tax and 82.5 after 25% tax. Rs 1,500 crore over Rs 82.5 crore is 18.2x.
Why can you not read the P/E straight off the EV multiple?
Think of a house worth Rs 1 crore with a Rs 40 lakh home loan. The owner's stake is Rs 60 lakh, and the rent left for the owner is the rent after the loan interest is paid. An EV multiple compares the whole business with profit before lenders and the tax office are paid; a P/E compares the shareholders' slice with the profit left for them. So you must take both the value and the earnings down to the shareholder line before you divide. Converting one side and not the other is the usual slip.
Enterprise value of Rs 2,000 crore less Rs 500 crore of net debt leaves Rs 1,500 crore of equity, and EBITDA of Rs 200 crore less depreciation, interest and tax leaves Rs 82.5 crore of net profit, so the P/E is 18.2x.
Why does the P/E come out higher than the EV multiple here?
Both sides shrink on the way down, but not by the same share. Value falls from 2,000 to 1,500, a quarter lost to lenders. Earnings fall from 200 to 82.5, well over half lost to depreciation, interest and tax, so the denominator shrinks faster and the multiple rises. Heavy depreciation, heavy interest or a high tax rate all push the P/E above the EV/EBITDA; a debt-free, low capex business has the two much closer together.
The relationship
$\text{P/E} = \frac{10 \times 200 - 500}{(200 - 40 - 50)(1 - 0.25)} = \frac{1{,}500}{82.5} \approx 18.2$
10 x 200 enterprise value, EV/EBITDA times EBITDA
500 net debt, the lenders' claim
200 - 40 - 50 profit before tax after depreciation and interest
1 - 0.25 the share of pre-tax profit kept after 25% tax
What it says in wordsTake the lenders out of the value and the lenders, the asset wear and the tax out of the earnings, then divide.
A quick check catches the common half-conversion. If you remember the tax but forget the interest, earnings are 160 x 0.75, which is 120; divide the full EV of 2,000 by that and you get 16.7x, a number that mixes the whole business with a shareholders' profit line. State the assumption that net debt is the only claim between EV and equity: minorities or preference capital would sit there too.
Where candidates lose it
The fast wrong answer is 10x, treating the two multiples as the same thing with a different name. The second wrong answer converts only one side: equity value over EBITDA, or EV over net profit, each of which pairs a value with an earnings line that belongs to someone else.
Say the matching rule before you calculate: enterprise value goes with profit before interest, equity value goes with profit after interest and tax. Then the arithmetic takes thirty seconds.
What the interviewer asks next
The company repays Rs 200 crore of debt from cash. What happens to the P/E if the EV multiple stays at 10x?
Which is the better multiple for comparing two companies with very different debt levels, and why?
Interest falls to zero and the tax rate rises to 30%. What is the P/E now?
Asked at Houlihan Lokey, Private Funds Advisory, Chicago, 2026 (Wall Street Oasis): Valuation ratio questions like if EV/EBITDA is 10x, what is
068Option A turns Rs 1 lakh into Rs 3 lakh in 12 years. Option B turns Rs 1 lakh into Rs 1.5 lakh in 4 years. Which has the higher IRR, which has the higher NPV at a 6% discount rate, and why do the two measures disagree?Performance measurement and returnsHardPIMCOMunich · 2024
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Which statement is right?
Show the worked solution
B has the higher IRR, 10.7% against 9.6%, but A has the higher NPV at 6%, Rs 0.49 lakh against Rs 0.19 lakh. IRR measures speed of growth; NPV measures rupees of value at your cost of money. B grows faster but stops after four years, while A keeps compounding above 6% for twelve. The two rankings swap at 9.05%: below that A wins on NPV, above it B does.
What does each measure reward?
Compare a short, well-paid contract with a long, steady job. The contract pays more per month, but it ends quickly and you must find something else. IRR ranks by speed: the rate at which the money grows. NPV ranks by size: how many rupees of value the option creates once everything is discounted at your own cost of money. A's IRR is 3 to the power 1/12, less one, 9.59%. B's is 1.5 to the power 1/4, less one, 10.67%. B is faster.
At a 6% discount rate, A's Rs 3 lakh in year 12 is worth 3 / 1.06 to the 12th, Rs 1.49 lakh today, an NPV of Rs 0.49 lakh. B's Rs 1.5 lakh in year 4 is worth Rs 1.19 lakh, an NPV of Rs 0.19 lakh. A creates more than twice the value, because it earns well above 6% for three times as long.
Option A's NPV starts higher and falls faster as the discount rate rises, crossing B's at 9.05%, so A creates more value at a 6% cost of money while B has the higher IRR, 10.7% against 9.6%.
Where do the rankings swap, and what does that rate mean?
The NPV curves cross where both options are worth the same: 3 / (1 + r) to the 12th = 1.5 / (1 + r) to the 4th, so (1 + r) to the 8th = 2 and r = 9.05%. That crossover has a plain meaning: B returns Rs 1.5 lakh in year 4, and to match A it would need to double over the next eight years, which takes 9.05% a year. If you can reinvest B's proceeds above that, B is better; below it, A is. IRR silently assumes you can reinvest at B's own 10.7%, which is why it favours the short option.
The relationship
$\frac{3}{(1+r)^{12}} = \frac{1.5}{(1+r)^4} \;\Rightarrow\; (1+r)^8 = 2 \;\Rightarrow\; r \approx 9.05\%$
3 option A's payout in year 12, Rs lakh
1.5 option B's payout in year 4, Rs lakh
r the discount rate at which the two options are worth the same
What it says in wordsThe crossover rate is the return B's proceeds must earn afterwards to catch up with A.
For a fund desk this is the reinvestment question in plain clothes. A client choosing between a short product with a high stated yield and a long one with a lower yield is really asking what the short money will earn when it comes back. The limit: NPV needs a discount rate, and a different rate can flip the answer; at 10%, A's NPV turns negative, about Rs 4,411 below zero, while B's stays positive at about Rs 2,452, so B wins.
Where candidates lose it
The trap is answering that the higher IRR must also have the higher NPV. That holds only when the options have the same size and the same timing; here the timing differs by eight years.
The second miss is picking a winner without naming the discount rate. Say that the answer depends on the cost of money, find the crossover rate, and explain it as the reinvestment rate B must beat.
What the interviewer asks next
B can be repeated three times in a row at the same terms. Does that change the comparison?
Why do private funds report IRR rather than NPV to their investors?
At what discount rate does option A's NPV become zero?
Asked at PIMCO, Real Estate, Munich, 2024 (Wall Street Oasis): just asked a bunch of questions on recent real estate news, as well as a couple of technicals including IRR, NPV
086A client needs her money in exactly three years. Why does a bond fund with a duration of about three years protect her whether interest rates rise or fall? Show it with Rs 1 crore and yields jumping from 7% to 8% on the first day.Bond maths and durationHardPIMCOLondon · 2022
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Yields jump to 8% on day one and the fund's value falls. Where does she stand at year three?
Show the worked solution
Because a fall in price and a rise in reinvestment income cancel at a horizon equal to the duration. If yields jump to 8%, Rs 1 crore at a three-year duration drops about Rs 2.74 lakh on day one, then compounds at 8% instead of 7%. By year three it is worth Rs 122.53 lakh against Rs 122.50 lakh had nothing moved. If yields fall to 6%, the gain today offsets the lower reinvestment rate in the same way.
Why do rising rates both hurt and help a bond investor?
Think of a tenant who has locked in a lease. If market rents rise, the lease itself is worth less to sell on, because a new buyer could get a better deal elsewhere, but any new space the tenant takes from now on costs more too. A bond investor faces the mirror image. When yields rise, bonds already held fall in price, but every coupon and every maturing bond is reinvested at the new, higher yield. One effect is immediate and the other builds over time. Which one wins depends only on how long you hold.
Make it concrete with a fund that has a duration of exactly three years. Picture its Rs 100 lakh as two holdings of Rs 50 lakh each at 7%: paper that matures in one year, and paper that matures in five. The value-weighted average of one and five is three, so the fund's Macaulay durationThe average time until a bond portfolio pays back its cash, with each payment weighted by its share of present value. is 3.0 years. Left alone at 7%, both halves grow to Rs 61.25 lakh by year three, Rs 122.50 lakh in all.
A jump to 8% knocks Rs 2.74 lakh off the fund on day one, but reinvesting at 8% wins it back by about year 3.0, so at her three-year date she holds Rs 122.53 lakh against a plan of Rs 122.50 lakh; a fall to 6% gives the mirror image.
What happens to each holding when yields jump to 8%?
The one-year paper matures at Rs 53.50 lakh and is rolled for two more years at 8%, reaching Rs 62.40 lakh, Rs 1.15 lakh more than the plan. The five-year paper still has two years to run at year three and must be sold at an 8% yield, fetching Rs 60.12 lakh, Rs 1.13 lakh less than the plan. One leg carries reinvestment risk and the other carries price risk, and a duration equal to the horizon sets them against each other in equal size. Held alone, each leg would fail: all short paper with yields falling to 6% gives only Rs 119.10 lakh, and all five-year paper with yields at 8% gives only Rs 120.25 lakh.
The relationship
$V_3 = P(y)\,(1+y)^3 \qquad P(8\%) = \frac{53.50}{1.08} + \frac{70.13}{1.08^5} = 97.26 \;\Rightarrow\; 97.26 \times 1.08^3 = 122.53$
P(y) the fund's value today at yield y, Rs lakh
53.50, 70.13 the amounts the one-year and five-year paper pay at maturity
V_3 the value at year three if all cash is reinvested at y
What it says in wordsWhatever the new yield, the fund's value at year three is today's repriced value grown at that yield, and at a horizon equal to the duration the two changes offset.
Where does this protection stop working?
In three places, and naming them is what the interviewer is after. Her horizon shrinks by exactly a year every year, but the fund's duration does not keep pace on its own: maturing paper has to be reinvested and coupons and rate moves shift the average, so the fund has to be rebalanced to stay matched. The offset assumes all yields move together; if short yields rise while long yields fall, the two legs no longer cancel. And the match only protects someone whose date is fixed: an open-ended fund whose manager keeps duration at three years forever is matched to nobody's date in particular. A product that runs down towards a fixed maturity close to her date does the matching more naturally. The small surplus either way, about Rs 0.02 lakh, comes from convexity and is a bonus, not the point.
Where candidates lose it
The common answer is that a shorter fund is safer, so she should use a liquid fund. That removes price risk but leaves her fully exposed to falling rates: if yields drop to 6% and stay there, three years of rolling short paper leaves her at Rs 119.10 lakh, about Rs 3.4 lakh short of plan.
The other loss is saying duration is a measure of price sensitivity and stopping there. This question is about the second meaning of duration, a time horizon at which price risk and reinvestment risk balance. Say both meanings, then show the cancellation with one number.
What the interviewer asks next
After one year, what has happened to the fund's duration and to her horizon, and what should change?
Short yields rise 1% and long yields fall 1% on the same day. Does the protection hold?
Why does the matched fund end slightly above plan whichever way yields move?
Asked at PIMCO, Sales, London, 2022 (Wall Street Oasis): Typically the product interview was toughest with questions regarding applications of duration
087The street values a refining company at 6 times its current EBITDA of Rs 1,500 crore. You think mid-cycle EBITDA, at normal refining margins, is Rs 1,050 crore. What multiple of mid-cycle EBITDA is the street really paying, and what does that say about your variant view?Valuation riddlesHardFranklin TempletonSan Mateo · 2024
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What multiple of mid-cycle EBITDA does the street's price imply?
Show the worked solution
About 8.6 times mid-cycle EBITDA, so the stock is cheap only if today's margins last. Six times Rs 1,500 crore is an enterprise value of Rs 9,000 crore. On your normal earnings of Rs 1,050 crore that is 8.6x. If 6x is a fair multiple of normal earnings, the business is worth about Rs 6,300 crore, 30% less. Your variant view is really a view on earnings, not on the multiple.
Why can a low multiple be expensive?
A cricketer who scores 600 runs in a dream season is not a 600-run player; his normal year is nearer 400. Pay for 600 every year and you have overpaid, however modest the fee looks against this season's scorecard. A multiple is a price divided by one year's earnings, so when that year is a cyclical peak, the multiple looks cheap precisely because the earnings are temporarily high. Refiners are a textbook case: their profit swings with the gap between crude oil costs and product prices, which moves in cycles.
The street's Rs 9,000 crore is 6.0 times peak EBITDA of Rs 1,500 crore but 8.6 times mid-cycle EBITDA of Rs 1,050 crore, and at 6 times normal earnings the business would be worth Rs 6,300 crore, 30% less.
How do you turn the gap into a variant view?
Hold the price still and change only the earnings. The street pays Rs 9,000 crore of enterprise value. Divided by your mid-cycle Rs 1,050 crore, that is 8.6 times normal earnings, a premium multiple hiding behind a discount one. Suppose refiners have usually traded near 6 times normal earnings. Then the price you can defend is 6 x 1,050, or Rs 6,300 crore, and the street is implicitly assuming that today's Rs 1,500 crore is the new normal. Your disagreement is a 30% lower view of sustainable EBITDA.
The relationship
$\frac{EV}{EBITDA_{mid}} = \frac{6 \times 1{,}500}{1{,}050} = \frac{9{,}000}{1{,}050} = 8.6\text{x}$
EV enterprise value the street's price implies, Rs crore
1,500 current EBITDA, Rs crore
EBITDA_{mid} your estimate of EBITDA at normal margins, Rs crore
What it says in wordsKeep the price the street pays, swap in normal earnings, and see what multiple you are really being asked to pay.
Why does the equity fall further than the enterprise value?
Because debt does not shrink when margins do. With an assumed Rs 3,000 crore of net debt, the street's equity value is Rs 6,000 crore; at Rs 6,300 crore of enterprise value it is Rs 3,300 crore. A 30% fall in the value of the business becomes a 45% fall for shareholders, because all of the loss lands on the equity slice. Say the weak point of your own view too: mid-cycle is an estimate of an average over a cycle whose length and depth nobody knows, and if capacity is genuinely scarce for years, today's margins may be closer to normal than you think. The defence is to show the history of margins and say what would make you wrong.
Where candidates lose it
Candidates argue about the multiple: refiners deserve 5x, or 7x, or the sector average. That misses the point of the question, which is that the street and you can agree on 6x and still disagree by Rs 2,700 crore, because you are applying it to different earnings.
The second loss is calling the stock cheap at 6x without asking which year's EBITDA sits underneath. For a cyclical business, always ask whether the denominator is a peak, a trough or a normal year before reading the multiple at all.
What the interviewer asks next
At the bottom of the cycle EBITDA is Rs 600 crore and the stock trades at 15x. Is it expensive?
How would you estimate mid-cycle EBITDA for a refiner, and what history would you need?
What would make you abandon the variant view?
Asked at Franklin Templeton, Oil & Gas, San Mateo, 2024 (Wall Street Oasis): Why did I have a variant view of the multiple I applied to a refiner company relative to street expectations.

D	next year's dividend per 100 of price: 5 for value, 1 for growth
r	the required return, 10% rising to 10.5%
g	the steady growth rate: 5% for value, 9% for growth

\sigma_F	the standard deviation of the shared yearly drift, 20%
\rho_d	the correlation of daily noise, -0.5
n	trading days in a year, 250
\sigma_d	each fund's daily noise, 0.8%

N	students, 8,000
F	yearly fee per student, Rs 2 lakh
M	the operating margin, 35%
m	the multiple of operating profit, assumed 10 to 14

n	small cubes along one edge
(n-2)^3	the hidden core after peeling one layer from every side
V(n)	cubes showing at least one face

Try it first

Why can a stock have a duration at all?

How do you work the two falls quickly?

Where candidates lose it

What the interviewer asks next

Try it first

What kind of situation produces this?

Can you show it with numbers?

Where candidates lose it

What the interviewer asks next

Try it first

What is the interviewer really testing?

How do you get from fees to a value?

Where candidates lose it

What the interviewer asks next

Try it first

Why are there two different counts here?

Why count the hidden cubes instead of the visible ones?

Where candidates lose it

What the interviewer asks next

Try it first

Why can you not read the P/E straight off the EV multiple?

Why does the P/E come out higher than the EV multiple here?

Where candidates lose it

What the interviewer asks next

Try it first

What does each measure reward?

Where do the rankings swap, and what does that rate mean?

Where candidates lose it

What the interviewer asks next

Try it first

Why do rising rates both hurt and help a bond investor?

What happens to each holding when yields jump to 8%?

Where does this protection stop working?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a low multiple be expensive?

How do you turn the gap into a variant view?

Why does the equity fall further than the enterprise value?

Where candidates lose it

What the interviewer asks next