Portfolio Management puzzles, solved step by step

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  1. 058Every stock in a market has 35% volatility and every pair of stocks has a correlation of 0.25. What is the volatility of an equally weighted portfolio of 1 stock, of 10 stocks, and of infinitely many?Portfolio risk mathsCoreNorthern TrustChicago · 2025Northern TrustChicago · 2025

    Try it first

    With infinitely many stocks, where does portfolio volatility settle?

    Show the worked solution

    35% for one stock, about 20% for ten, and a floor of 17.5% for infinitely many. Portfolio variance is the stock variance times one over n, plus the correlation times what is left. With ten stocks that is 0.1225 times 0.325, a volatility of 20.0%. As n grows the one-over-n part vanishes and only the correlation term remains: 35% times the square root of 0.25, or 17.5%.

    Why does adding stocks lower risk at all?

    Ten shops in ten different towns do not all have a bad week at once; ten shops in one mall often do. Some of what moves a stock is its own news and some is the market everyone shares. Stock-specific shocks cancel out as you add names, because one company's bad quarter is offset by another's good one, but the shared market shock hits every name together and does not cancel. Correlation measures how much of each stock's movement is shared.

    The relationship
    σp2=σ2[1n+(1−1n)ρ]  →  σρ=0.35×0.5=17.5%\sigma_p^2 = \sigma^2\left[\frac{1}{n} + \left(1-\frac{1}{n}\right)\rho\right] \;\to\; \sigma\sqrt{\rho} = 0.35 \times 0.5 = 17.5\%
    sigmaeach stock's volatility, 35%
    rhothe correlation between any two stocks, 0.25
    nthe number of stocks, equally weighted
    What it says in wordsPortfolio variance is a shrinking stock-specific part plus a fixed shared part, and only the shared part survives as the portfolio grows.
    Adding stocks cuts risk fast, then hits a floor set by correlation0%10%20%30%40%1 stock: 35%10 stocks: 20.0%floor: 35% x square root of 0.25 = 17.5%shaded: stock-specific risk, removed by adding namesbelow the floor: market risk no number of stocks removes11020304050Number of stocks, equally weighted
    Portfolio volatility falls from 35% with one stock to 20.0% with ten and then flattens towards a floor of 17.5%, because adding names removes stock-specific risk but cannot remove the risk all the stocks share.

    How much of the benefit do the first ten stocks deliver?

    Most of it. Going from one stock to ten cuts volatility from 35% to 20.0%, about 86% of the whole distance to the floor. Going from ten to thirty takes it only to 18.4%. Diversification pays off quickly and then almost stops, and the level where it stops is set by correlation, not by the number of holdings. That is why a manager worried about risk gains more from adding assets that are less correlated than from adding a fortieth stock of the same kind.

    Say the limitation. Correlations are not fixed: in a sell-off they tend to rise together, which raises the floor exactly when diversification is needed. Real stocks also differ in volatility and correlation, so this uniform market is a teaching model; the shape of the curve survives, the exact numbers do not.

    Where candidates lose it

    The trap is saying diversification takes risk to zero, or reaching for the correlation without the square root and answering 8.75%. The floor is the square root of the shared variance, so it is volatility times the square root of the correlation.

    The second miss is getting 20% for ten stocks by guesswork and being unable to show it. Write the variance formula first and plug in: 0.1225 times 0.1 plus 0.9 times 0.25.

    What the interviewer asks next

    • What correlation would make a 10-stock portfolio half as risky as one stock?
    • Why do correlations tend to rise in a market sell-off, and what does that do to this floor?
    • How would you lower the floor itself rather than approach it?

    Asked at Northern Trust, Asset Management, Chicago, 2025 (Wall Street Oasis): First one was more technical and asked about my understanding of AM, portfolio diversification and strategy
    Asked at Northern Trust, Asset Management, Chicago, 2025 (Wall Street Oasis): Asked about my understanding of asset management, portfolio diversification and strategy

  2. 083Two trading desks each have a one-day 95% value at risk of Rs 10 lakh, and their daily P&Ls have a correlation of 0.3. Assuming normal returns, what is the combined value at risk, and how big is the diversification benefit?Portfolio risk mathsCoreBlackRockNew York · 2026

    Try it first

    Your first estimate of the combined value at risk?

    Show the worked solution

    About Rs 16.1 lakh, a diversification benefit of about Rs 3.9 lakh. Under normal returns value at risk is a fixed multiple of standard deviation, so it combines the same way: the square root of 10 squared plus 10 squared plus 2 x 0.3 x 10 x 10, which is the square root of 260. Adding the two desks' figures would overstate the risk by Rs 3.9 lakh.

    Why can you not just add the two numbers?

    Two friends each walk 10 minutes from a crossing, one north and one north-east. They do not end up 20 minutes apart; the angle between their paths matters. Under normal returns, a desk's value at risk is a fixed multiple of its standard deviation, and standard deviations combine like arrows: the angle between them is set by the correlation. Only at a correlation of 1 do the arrows point the same way and add to 20.

    Value at risk adds like arrows, not like numbersstraight sum: 20Desk A: Rs 10 lakhDesk B: Rs 10 lakhCombined: Rs 16.1 lakhcos = 0.3, 73 degreesCombined VaR, Rs lakhrho = 120.0rho = 0.316.1rho = 014.1Diversification benefit at 0.320.0 - 16.1 = Rs 3.9 lakh
    Placing the two Rs 10 lakh risks head to tail at the angle set by a 0.3 correlation gives a combined value at risk of Rs 16.1 lakh, against Rs 20 lakh if they moved together and Rs 14.1 lakh if they were uncorrelated.
    The relationship
    VaRA+B=102+102+2(0.3)(10)(10)=260≈16.1\text{VaR}_{A+B} = \sqrt{10^2 + 10^2 + 2(0.3)(10)(10)} = \sqrt{260} \approx 16.1
    10each desk's one-day 95% value at risk, Rs lakh
    0.3the correlation between the desks' daily P&Ls
    What it says in wordsSquare each desk's figure, add twice the correlation times their product, and take the square root.

    What assumption is doing the work, and when does it fail?

    The square-root rule holds only when returns are jointly normal, or close to it, so that value at risk is a clean multiple of standard deviation. With fat tails or options in the book, value at risk need not be subadditive, and the combined figure can even exceed the sum. Correlations also rise in a crisis, so the Rs 3.9 lakh benefit is thinnest on exactly the days it is needed. Firms often report both the diversified total and the sum of the parts for that reason.

    Where candidates lose it

    The fast wrong answer is Rs 20 lakh, which quietly assumes a correlation of 1. The interviewer then asks why banks bother measuring correlation at all, and the candidate has nowhere to go.

    The second loss is giving Rs 16.1 lakh without the normality condition. Say it: the square-root rule is a property of standard deviation, and value at risk inherits it only under normal returns.

    What the interviewer asks next

    • At what correlation is the combined value at risk exactly Rs 15 lakh?
    • How much does each desk contribute to the combined figure?
    • Why is expected shortfall preferred over value at risk for limits?

    Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Techincal and behavioral (VAR, market views, stock valuation) why blackrock, python experience?