Portfolio Management puzzles, solved step by step

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  1. 021Analyst A has an information coefficient of 0.10 on 25 independent bets a year. Analyst B has an information coefficient of 0.03 on 400 independent bets a year. Whose information ratio is higher?Performance measurementHardQuantitative asset managementSystematic investing

    Try it first

    Which analyst has the higher information ratio?

    Show the worked solution

    Analyst B, with 0.6 against 0.5. By the fundamental law of active management, the information ratio is roughly the information coefficient times the square root of the number of independent bets. A gets 0.10 times the root of 25, which is 5, for 0.5. B gets 0.03 times the root of 400, which is 20, for 0.6. Breadth beats accuracy here, but only if B's bets really are independent.

    How can a less accurate analyst add more value?

    Think of two shopkeepers. One makes a large margin on a few sales a day; the other makes a thin margin on hundreds. The second can earn more, and with less swing day to day, because many small edges average into a steadier total. Active return grows with the number of bets while its noise grows only with the square root, so the ratio of the two rises with the square root of breadth. This is Grinold's fundamental lawRichard Grinold's result that a strategy's information ratio is approximately its information coefficient times the square root of its breadth. of active management. B's skill is under a third of A's, but B's root breadth is four times A's.

    Information ratio = skill x square root of independent betsAnalyst A: accurate, few betsSkill (IC)0.10Root of betsroot 25 = 5Information ratio0.5bars scaled to the larger of the twoAnalyst B: thin edge, many betsSkill (IC)0.03Root of betsroot 400 = 20Information ratio0.6bars scaled to the larger of the twoB's edge rests on breadth. If B's 400 bets move together like 100 independent ones:0.03 x root 100 = 0.3, and A's 0.5 is ahead again.
    Analyst A's skill of 0.10 on a root breadth of 5 gives an information ratio of 0.5, while analyst B's skill of 0.03 on a root breadth of 20 gives 0.6, unless B's bets are correlated enough to count as only 100, which cuts B to 0.3.
    The relationship
    IR≈IC×BRA:0.1025=0.5B:0.03400=0.6IR\approx IC\times\sqrt{BR} \qquad A: 0.10\sqrt{25}=0.5 \qquad B: 0.03\sqrt{400}=0.6
    ICthe information coefficient: the correlation between forecasts and outcomes
    BRbreadth: the number of independent bets a year
    IRthe information ratio: active return per unit of tracking error
    What it says in wordsSkill times the square root of independent bets gives the information ratio.

    Where does the argument break?

    At the word independent. Four hundred bets that all lean on the same factor, say cheap stocks, behave like far fewer independent bets, and breadth has to be counted in independent bets, not trades. If B's 400 positions carry the information of only 100 independent ones, B's ratio is 0.03 times 10, which is 0.3, and A is ahead again. The law also ignores costs: 400 bets a year means more turnover, and a thin edge of 0.03 is the first thing trading costs eat. A portfolio manager hiring between the two would ask B how correlated the signals are before believing the 0.6.

    Where candidates lose it

    The instinctive answer is A, because a coefficient of 0.10 sounds far better than 0.03. The interviewer is testing whether you know breadth enters the formula at all.

    The second trap is taking B's 400 at face value. The follow-up is almost always what if the bets are correlated, and the answer is that breadth shrinks and B's advantage can vanish.

    What the interviewer asks next

    • How many independent bets would A need to match B?
    • How would you estimate the effective number of independent bets in a portfolio?
    • Why do trading costs hit analyst B harder than analyst A?
  2. 098A portfolio holds 60% equities returning 10% and 40% bonds returning 6%. Its benchmark holds 50% equities returning 12% and 50% bonds returning 5%. Split the 0.1 point shortfall into allocation, selection and interaction effects.Performance measurementHardPerformance analysisMulti-asset

    Try it first

    Was the decision to overweight equities a good one?

    Show the worked solution

    Allocation added 0.70 points, selection cost 0.50 and interaction cost 0.30, netting to the -0.1 shortfall. Allocation prices the weight bets at benchmark returns: +10% x 12 and -10% x 5. Selection prices the return gaps at benchmark weights: 50% x (10 - 12) and 50% x (6 - 5). Interaction is the weight bets times the return gaps.

    Why split the result at all, if it only lagged by 0.1?

    A team can lose a match by one run after a brilliant bowling spell and a dismal batting collapse; the scoreline hides both. Attribution separates the decision of how much to hold in each asset class from the decision of what to hold within it, so each can be judged on its own. Here the allocation call was right and the equity picking was poor, which is a very different story from a flat 0.1 point miss.

    The allocation call added value; the stock and bond picking gave it back8.50Benchmark+0.70Allocation-0.50Selection-0.30Interaction8.40Portfolio7.5%8.0%8.5%9.0%9.5%axis starts at 7.5%Net-0.10points0.7 - 0.5 - 0.3
    From the benchmark's 8.5%, the equity overweight adds 0.70 points, weak equity selection takes 0.50 and the interaction takes 0.30, arriving at the portfolio's 8.4%.
    SegmentAllocationSelectionInteractionTotal
    Equities+1.20-1.00-0.20-0.00
    Bonds-0.50+0.50-0.10-0.10
    Total+0.70-0.50-0.30-0.10
    Equities carry the story: the overweight earned +1.20 while picking cost -1.00 and the interaction -0.20; bonds lost -0.50 on the underweight but picked well.

    What is the interaction term, and why does it bite here?

    Interaction is the weight bet times the return gap. The manager overweighted equities by 10 points and underperformed within them by 2 points, so the extra equity weight magnified the poor picking by 0.2; underweighting bonds, where the picking was good, gave up another 0.1. Some firms fold interaction into selection, since the weights are the manager's choice; say which convention you are using.

    One more check an interviewer may ask for. The Brinson-Fachler version measures allocation against the total benchmark return, (weight bet) x (segment return minus 8.5). It gives the same total, 0.70, here, because the weight bets sum to zero, but it splits the credit between segments differently.

    The relationship
    A=∑(wp−wb)Rb,S=∑wb(Rp−Rb),I=∑(wp−wb)(Rp−Rb)A = \sum (w_p - w_b) R_b, \quad S = \sum w_b (R_p - R_b), \quad I = \sum (w_p - w_b)(R_p - R_b)
    w_p, w_bportfolio and benchmark weights
    R_p, R_bportfolio and benchmark returns within each segment
    What it says in wordsAllocation prices the weight bets, selection prices the return gaps, interaction prices the two together.

    Where candidates lose it

    The common slip is judging the allocation by the portfolio's own equity return, 10%, instead of the benchmark's 12%. That mixes the picking into the allocation and understates a good decision.

    The second slip is losing track of signs on the underweight. Ten fewer points of bonds at 5% is minus 0.5, and in selection the bonds' 1 point outperformance at 50% weight is plus 0.5. Write the table before speaking.

    What the interviewer asks next

    • If the manager had held benchmark weights, what would the portfolio have returned?
    • Why do some firms fold interaction into selection?
    • How does attribution work for a portfolio that holds a segment the benchmark does not?