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. 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