Portfolio Management puzzles, solved step by step

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  1. 041Two stocks both have an 11% cost of equity. The value stock's dividends grow 3% a year forever and the growth stock's grow 8%. Using a constant-growth model, what is each stock's equity duration, and roughly how much does each fall if the cost of equity rises by 50 basis points?Valuation riddlesHardBlackRockNew York · 2026BlackRockNew York · 2026

    Try it first

    Which stock is more sensitive to a rise in the discount rate, and by roughly how much?

    Show the worked solution

    Durations of 12.5 and 33.3 years; the value stock falls about 6% and the growth stock about 14%. With price equal to D over (r minus g), duration is 1 over (r minus g): 1 over 0.08 and 1 over 0.03. A 50 basis point rise takes the spreads to 8.5 and 3.5 points, so prices fall by 1 minus 8/8.5, which is 5.9%, and 1 minus 3/3.5, which is 14.3%.

    Why does a stock have a duration at all?

    Think of two people valuing a lottery ticket: one pays out a small sum every year starting now, the other pays little now and a lot decades from now. If the interest rate rises, the second ticket loses far more value, because its money is further away and gets discounted for longer. An equity is a stream of future cash flows, so like a bond it has a duration: the weighted distance to its cash, and the growth stock's cash sits much further out. In the constant-growth model that distance comes out as 1 over (r minus g).

    Same cost of equity, very different sensitivity to itEquity duration, years = 1 / (r - g)12.5Valueg = 3%33.3Growthg = 8%Price change if r rises 0.5 point-5.9%Value-14.3%GrowthDashed: duration estimate, -6.25% and -16.7%
    At an 11% cost of equity, the value stock growing 3% has a duration of 12.5 years and loses 5.9% if the rate rises half a point, while the growth stock growing 8% has a duration of 33.3 years and loses 14.3%. The dashed lines show that the straight duration estimate overstates both falls.

    How do you get the price changes exactly, and why is the duration estimate too big?

    The price is proportional to 1 over (r minus g), so compare spreads before and after. For the value stock the spread moves from 8 to 8.5 points, a 5.9% price fall; for the growth stock it moves from 3 to 3.5, a 14.3% fall. Duration times the rate change gives 6.25% and 16.7%, a little larger, because price is a curved function of the rate: like a bond with convexity, the stock loses less than the straight-line estimate. The curvature matters more the longer the duration.

    The relationship
    P=D1r−gDeq=−1PdPdr=1r−gPnewPold=r−gr+Δr−gP = \frac{D_1}{r-g} \qquad D_{eq} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g} \qquad \frac{P_{new}}{P_{old}} = \frac{r-g}{r + \Delta r - g}
    D_1next year's dividend
    rthe cost of equity, 11%
    gthe perpetual growth rate, 3% or 8%
    D_{eq}equity duration, the percentage price change per point of rate change
    What it says in wordsIn a constant-growth model, sensitivity to the discount rate is one over the gap between the rate and the growth rate.

    This is the arithmetic behind a familiar market pattern: when real yields rise sharply, long-duration growth stocks usually fall more than value stocks. Say the limitation too. The model assumes growth is fixed while the rate moves; in practice rates often rise because growth is strong, which can offset part of the fall. And no real company grows at 8% forever, so the growth stock's duration is a stylised upper figure.

    Where candidates lose it

    Candidates often say both stocks move the same because they share a cost of equity, or they compute the percentage change of r, about 4.5%, and apply it to both prices. Neither uses the spread between r and g, which is the whole mechanism.

    State the formula, name the spread, and give the two durations first. Then offer the exact falls and say why they are smaller than the duration estimate.

    What the interviewer asks next

    • What happens to the growth stock's duration if its growth rate rises to 10%?
    • Why might a rate rise driven by stronger growth hurt growth stocks less than this model suggests?
    • How would you hedge the rate sensitivity of a growth-heavy portfolio?

    Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks
    Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? VaR, market views, stock valuation.

  2. 059Why should I buy your college, and how much would you sell it for? Suppose it earns an operating surplus of Rs 40 crore this year, the surplus grows 5% a year for the foreseeable future, and a buyer wants a 12% return.Valuation riddlesCoreWellington ManagementBoston · 2024

    Try it first

    What price does the growing surplus support, before land?

    Show the worked solution

    About Rs 600 crore for the operating business, before any value in the land. Next year's surplus is Rs 40 crore grown 5%, or Rs 42 crore. A surplus that grows forever at 5% and is valued at 12% is worth next year's amount divided by the 7-point gap: 42 over 0.07 is Rs 600 crore. The reason to buy is the durability of that surplus: steady demand for seats and fees that can rise with costs.

    What is the question really asking?

    It is a stock pitch in disguise. The interviewer wants two things in order: why this asset produces reliable cash, and what that cash is worth. Answer the why with the quality of the surplus, and the how much with a valuation you can do out loud. For a college the why is simple to say: students keep applying every year, fees are paid in advance, and a college with a good name can raise fees roughly in line with its costs. Those are the reasons the surplus can be treated as growing and durable.

    Why divide by 12% less 5%?

    Think of a rented flat whose rent rises every year. A buyer asking for a 12% return on a rent that grows 5% needs only 7% from the current rent; the other 5% arrives through growth. A cash flow growing at g forever, valued at a required return r, is worth next year's cash flow divided by r minus g. Here that is Rs 42 crore over 0.07, which is Rs 600 crore. Dividing this year's Rs 40 crore instead gives Rs 571 crore, a common small slip: the buyer receives next year's surplus, not this year's.

    The relationship
    V=S1r−g=40×1.050.12−0.05=420.07=600V = \frac{S_1}{r - g} = \frac{40 \times 1.05}{0.12 - 0.05} = \frac{42}{0.07} = 600
    S_1next year's surplus, Rs crore
    rthe buyer's required return
    gthe permanent growth rate of the surplus
    What it says in wordsA growing perpetuity is worth next year's payment divided by the gap between the required return and the growth rate.
    Value of a growing surplus: the gap between return and growth does the work05001,0001,50012%: Rs 600 crore11%: Rs 700 crore13%: Rs 525 croreat 8%: Rs 1,400 croreone point either side of 12%moves value by Rs 100 and Rs 75 crore8%10%12%14%16%Buyer's required return, growth fixed at 5%Rs crore
    With growth fixed at 5%, the college is worth Rs 600 crore at a 12% required return, but Rs 700 crore at 11% and Rs 525 crore at 13%, because value depends on the gap between return and growth and that gap is small.

    Now say what the number is sensitive to. One point on the required return moves the value by Rs 100 crore up or Rs 75 crore down, because a 7-point gap becoming 6 or 8 is a large change in proportion. The land and buildings may be worth more than the operating surplus, so a seller would also ask what the campus fetches as property. And check the structure before promising anyone the surplus: many colleges, in India among other places, are run by trusts or societies, and whether an owner can take surplus out at all is a legal question to confirm.

    Where candidates lose it

    The trap is diving into a formula without answering why. The question starts with why should I buy, and a candidate who opens with a number has skipped the half the interviewer cares about most.

    The arithmetic trap is dividing Rs 40 crore by 12%, which treats a growing surplus as flat and values it at Rs 333 crore. Name the growth, use next year's surplus, and divide by the gap.

    What the interviewer asks next

    • What growth rate is the seller implicitly assuming if he asks Rs 800 crore?
    • How would you value the land separately, and when would it exceed the value of the operating business?
    • What would make you use a higher required return for this college than for a listed education company?

    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?

  3. 072An equity index trades at 22 times forward earnings, pays out 40% of earnings as dividends, and its long-run earnings growth is expected to be 10% a year. The 10-year government bond yields 7%. Which asset is cheaper?Valuation riddlesHardPIMCOSan Diego · 2026

    Try it first

    The earnings yield is 4.5% and the bond yields 7%. What does that comparison tell you?

    Show the worked solution

    On expected return, equities offer about 11.8% against the bond's 7%, a premium of about 4.8 points. The earnings yield, 1 over 22 or 4.5%, looks worse than 7%, but it ignores growth. The dividend yield is 40% of 4.5%, or 1.82%, and adding 10% growth gives about 11.8%. Whether equities are cheaper depends on whether a 4.8-point premium pays enough for equity risk.

    Why is 4.5% against 7% the wrong comparison?

    Compare a fixed-rent lease with a shop whose profits grow each year. The lease might pay more in year one, but the shop's income keeps rising. A bond's yield is everything it will ever pay, while an earnings yield is only the starting point of a stream expected to grow, so the two cannot be compared directly. The earnings yield of 4.5% sits 2.5 points below the bond, and that gap says almost nothing about which is cheaper.

    Compare expected returns, not an earnings yield with a bond yieldNaive: earnings yield vs bond yield0%4%8%12%4.5%E/P = 1/227.0%10-yr bondequities look 2.5 points worseLike with like: expected returns0%4%8%12%1.8+10 growth11.8%equities7.0%10-yr bondequity premium about 4.8 points
    The earnings yield of 4.5% looks worse than the bond's 7%, but the index's expected return, a 1.8% dividend yield plus 10% growth, is about 11.8%, a premium of about 4.8 points over the bond.

    How do you put them on the same footing?

    Estimate the equity's expected return the way you would a bond's. For a stock or an index, that is roughly the dividend yield plus long-run growth, the logic of the Gordon growth model. The dividend yield is the payout ratio times the earnings yield, and the growth rate does the rest. Here 40% of 4.55% is 1.82%, and 10% growth takes the total to 11.82%. Against 7% on the bond, equities offer 4.82 extra points a year.

    The relationship
    E[R]≈DP+g=0.40×122+10%=1.82%+10%=11.82%E[R] \approx \frac{D}{P} + g = 0.40 \times \frac{1}{22} + 10\% = 1.82\% + 10\% = 11.82\%
    D/Pthe dividend yield, the payout ratio times earnings over price
    glong-run growth in earnings and dividends, 10%
    What it says in wordsAn equity's expected return is roughly its dividend yield plus the rate at which its dividends grow.

    Then test the growth number, because the answer rests on it. Growing earnings 10% while paying out 40% means reinvesting 60% at a return on equity of about 16.7%, which is demanding for a whole market. If growth were 8%, the premium would shrink to about 2.8 points. A view on which asset is cheaper is really a view on whether that premium, after testing growth, pays enough for the extra risk of equities. That is the judgement to state, with the numbers that drive it.

    Where candidates lose it

    The trap is comparing the earnings yield with the bond yield and declaring bonds cheaper. That comparison ignores growth and treats a rising income stream as if it were fixed.

    The second loss is taking the 10% growth at face value. Check it against the payout ratio: growth needs reinvestment, and the implied return on equity tells you whether the number is plausible.

    What the interviewer asks next

    • What equity risk premium would you need to call equities and bonds fairly valued here?
    • How does inflation change the comparison between an earnings yield and a nominal bond yield?
    • What growth rate makes the index's expected return exactly equal to 7%?

    Asked at PIMCO, Debt Capital Markets, San Diego, 2026 (Wall Street Oasis): Which is cheaper us bonds or us equities