Portfolio Management puzzles, solved step by step

Puzzles
100
Traced to a firm
31
Topics
13
Hard
30
Topic
All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 11–12 of 12 · filtered from 100Clear filters
  1. 096A manager holds the index but overweights stock A by 5 percentage points and underweights stock B by 5 points. A has 30% volatility, B has 25%, and they correlate at 0.6. What tracking error does this pair of bets create?Portfolio risk mathsHardMSCIMonterrey · 2013

    Try it first

    Before working it: is the tracking error above or below the 1.5% that the A bet alone would create?

    Show the worked solution

    A tracking error of about 1.25% a year. Tracking error is the volatility of the active weights. A's bet contributes (5% x 30%) squared, 2.25; B's contributes (5% x 25%) squared, 1.5625; and because the bets are opposite on correlated stocks, the covariance term is minus 2.25. The sum is 1.5625, whose square root is 1.25%.

    Why does adding a second bet reduce the risk?

    Buying an umbrella and selling a raincoat leaves you with little net exposure to rain, because both move with the weather. Overweighting A and underweighting B, when the two stocks tend to move together, is partly a hedge: when both rise, the gain on A is partly offset by the shortfall on B. Tracking error measures the risk left after that offset.

    Two opposite bets on correlated stocks partly cancelActive weightsStock A, vol 30%+5 pointsStock B, vol 25%-5 pointscorrelation 0.6one bet up, one down:the covariance termturns negativeVariance terms, squared per cent+2.25A's own+1.56B's own-2.25A with B1.56TotalTE = 1.25%Tracking errorrho 0.6, opposite bets1.25%rho 0, opposite bets1.95%rho 0.6, same direction2.46%
    The variance of the active bets is A's own term 2.25 plus B's 1.5625 minus a covariance term of 2.25, which leaves 1.5625 and a tracking error of 1.25%, against 1.95% if the stocks were unrelated and 2.46% if both bets pointed the same way.
    The relationship
    TE2=wA2σA2+wB2σB2+2 wAwB ρ σAσB=2.25+1.5625−2.25TE^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2\,w_A w_B\,\rho\,\sigma_A\sigma_B = 2.25 + 1.5625 - 2.25
    w_A, w_Bactive weights, +5% and -5%
    sigma_A, sigma_Bvolatilities, 30% and 25%
    rhocorrelation, 0.6
    What it says in wordsTracking error is the portfolio volatility formula applied to the active weights instead of the holdings.

    What is the neat coincidence, and what does it hide?

    Here the covariance term exactly cancels A's own variance, so the answer equals B's bet alone, 5% x 25% = 1.25%. That is a coincidence of these numbers, not a rule: the covariance term is 2 x 0.6 x 30 x 25, which happens to equal 30 squared. Change the correlation to 0.5 and the answer moves. The general lesson holds, though: tracking error depends on the size of the bets and on how much they cancel.

    Say the limits. Correlations are estimated and unstable, so a pair that looks like a hedge in calm markets can decouple when a stock-specific event hits. Real tracking error also includes every other small active weight, and many small bets can add up to more than one large one.

    Where candidates lose it

    The common slip is adding the two bets' risks, 1.5% plus 1.25%, as if they were independent and in the same direction. That ignores both the correlation and the opposite signs of the weights.

    The quieter slip is getting the sign of the covariance term wrong. The weights have opposite signs, so the term is negative; say that out loud before plugging in numbers.

    What the interviewer asks next

    • At what correlation would the tracking error be zero?
    • What tracking error would a 2% overweight in a stock with 40% volatility add on its own?
    • How would you decompose a portfolio's tracking error into contributions from each bet?

    Asked at MSCI, Financial Tools, Monterrey, 2013 (Wall Street Oasis): What's the tracking error formula?

  2. 099Project A costs Rs 100 crore and returns Rs 130 crore after one year. Project B costs Rs 100 crore and returns Rs 20 crore a year for ten years. You can do only one. Which ranks higher on IRR, which on NPV at a 10% cost of capital, and which would you take?Private and real asset mathsHardPIMCOMunich · 2024

    Try it first

    At a 10% cost of capital, which project creates more value?

    Show the worked solution

    A wins on IRR, 30% against about 15%; B wins on NPV at 10%, Rs 22.9 crore against Rs 18.2 crore. Take B. IRR measures a rate, NPV measures rupees of value at your actual cost of capital. The two disagree because A returns its money in one year and B keeps earning for ten. Above about 11%, the ranking flips.

    How can one project have the higher return and the other create more value?

    Would you rather earn 30% on Rs 100 for a day, or 15% on Rs 100 for ten years? The first is a better rate; the second makes you more money. IRR is a rate per year and says nothing about how long the money earns it, while NPV counts the rupees created at your cost of capital, so when projects differ in timing the two can rank them in opposite order.

    IRR ranks A first; at a 10% cost of capital, NPV ranks B first5%10%15%20%25%30%35%-40501000cost of capital 10%B: NPV 22.9 at 10%A: NPV 18.2 at 10%lines cross at 11.2%IRR B15.1%IRR A 30%B: 20 a year, 10 yearsA: 130 in one yearDiscount rateRs croreIRRA 30.0 / B 15.1NPV at 10%A 18.2 / B 22.9NPV measuresvalue created
    Project B's NPV falls steeply with the discount rate from Rs 100 crore at 0% and crosses zero at its 15.1% IRR, while A's starts at Rs 30 crore and crosses at 30%; the lines meet at about 11.2%, so at a 10% cost of capital B is worth Rs 22.9 crore to A's Rs 18.2 crore.
    The relationship
    NPVA=1301.10−100=18.2,NPVB=20×1−1.10−100.10−100=22.9NPV_A = \frac{130}{1.10} - 100 = 18.2, \qquad NPV_B = 20 \times \frac{1 - 1.10^{-10}}{0.10} - 100 = 22.9
    130A's single payout after one year, Rs crore
    20B's yearly payout for ten years, Rs crore
    10%the cost of capital
    What it says in wordsDiscount each project's cash at the cost of capital and subtract what it costs.

    When would A be the right choice after all?

    Two cases. If the cost of capital is above about 11.2%, the crossover rate, A's NPV is the higher one; and if A's Rs 130 crore could be put straight into another project earning well above 10%, the pair of projects might beat B. Both are really statements about the reinvestment rate. IRR implicitly assumes A's proceeds are reinvested at 30%; NPV assumes the cost of capital, which is usually the more honest assumption.

    In real estate and private markets this is the everyday tension between a quick flip and a long hold. Give both numbers, name the crossover, and say that for mutually exclusive projects NPV decides.

    Where candidates lose it

    The common slip is picking A because 30% beats 15%. With mutually exclusive projects of different lengths, IRR ranks rates, not value, and the interviewer asks this to see if you know the difference.

    The second loss is stopping at take B without saying when that changes. Name the crossover rate and the reinvestment assumption; that is the part that sounds like an investor.

    What the interviewer asks next

    • At what cost of capital are you indifferent between the two?
    • What if B's cash flows ran for twenty years instead of ten?
    • Why might a fund manager paid on IRR prefer A anyway?

    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