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  1. 020A book holds Rs 60 crore of a stock with 30% volatility and Rs 40 crore of another with 20% volatility, and the two have a correlation of 0.5. What is the book's volatility in rupees, and what share of the risk comes from each position?Portfolio and risk mathsHardMan GroupBoston · 2022

    Try it first

    What share of the book's risk comes from the Rs 60 crore position?

    Show the worked solution

    The book's volatility is about Rs 23.1 crore a year, and the Rs 60 crore position carries about 74% of it on 60% of the capital. Stand-alone risks are Rs 18 crore and Rs 8 crore. Book variance is 18 squared plus 8 squared plus 2 x 0.5 x 18 x 8, which is 532, so volatility is Rs 23.07 crore. Each position's contribution is its covariance with the book over the book's volatility: Rs 17.17 crore and Rs 5.90 crore, which add back to the total.

    Why is risk not shared out like capital?

    Two friends share a taxi. One rides twice as far, straight through the traffic jam; the other gets off after a short hop. Splitting the fare by the number of bags each carries would be absurd. Risk belongs to a position in proportion to how much it moves and how much it moves with everything else, not to how much money sits in it. Here the first stock is larger, more volatile and positively correlated with the second, so it carries far more than its 60% of the capital.

    The bigger, more volatile name carries 74% of the risk on 60% of the capital18.0A alone60 x 30%+8.0B alone40 x 20%-2.93Diversifiedrho = 0.523.07BookRs crore60%40%Capital74.4%25.6%RiskPosition APosition B
    Stand-alone risks of Rs 18 crore and Rs 8 crore add to Rs 26 crore, diversification at a 0.5 correlation removes Rs 2.93 crore, and the book's Rs 23.07 crore of volatility splits 74.4% to the first position and 25.6% to the second, against a 60 to 40 split of capital.
    The relationship
    σbook2=a2+b2+2ρabRCA=a2+ρabσbookRCB=b2+ρabσbook\sigma_{book}^2 = a^2 + b^2 + 2\rho ab \qquad RC_A = \frac{a^2 + \rho ab}{\sigma_{book}} \qquad RC_B = \frac{b^2 + \rho ab}{\sigma_{book}}
    a, bstand-alone rupee volatilities: 60 x 30% = 18 and 40 x 20% = 8
    \rhothe correlation, 0.5
    RCa position's contribution to book volatility
    What it says in wordsEach position owns its own variance plus half the shared term, and dividing by the book's volatility turns that into rupees of risk.
    PositionCapital, Rs croreVolatilityStand-alone riskRisk contributionShare of risk
    A6030%18.017.1774.4%
    B4020%8.05.9025.6%
    Book10026.023.07100.0%
    Rs crore of annual volatility. The stand-alone risks add to Rs 26.0 crore, but the book's volatility is Rs 23.07 crore, of which position A contributes 74.4% and position B 25.6%.

    Why do the contributions add up exactly to the total?

    Split the variance. The cross term, 2 x 0.5 x 18 x 8 = 144, is shared equally, 72 to each position. So position A owns 324 + 72 = 396 of the 532 of variance and position B owns 64 + 72 = 136, and dividing each by the book's volatility of 23.07 gives rupee contributions that sum exactly to Rs 23.07 crore. The diversification benefit is the gap between the stand-alone total of Rs 26 crore and the book's Rs 23.07 crore.

    What does a risk manager do with the split?

    Cut where the risk is, not where the money is. Each rupee in position A carries 28.6 paise of marginal risk against 14.7 paise in position B, so trimming Rs 10 crore from A lowers book volatility by roughly Rs 2.9 crore. Recomputing exactly gives Rs 2.84 crore, close to the estimate. The limitation: the split is a snapshot at one correlation, and when correlations move, both the total and the split move with them.

    Where candidates lose it

    The quick answer shares risk like capital, 60 and 40, or like stand-alone risk, 18 and 8. The first ignores volatility and the second ignores correlation; neither sums to the book's actual Rs 23 crore of risk.

    The second slip is adding the stand-alone risks to get the book's risk, Rs 26 crore. Volatilities do not add unless the correlation is exactly 1; variances do, with the cross term included.

    What the interviewer asks next

    • If the correlation fell to zero, how would the risk split between the two positions?
    • How much of position B would you add to minimise the book's volatility, holding A fixed?
    • How do transaction costs change which position you trim first?

    Asked at Man Group, Investment Management, Boston, 2022 (Wall Street Oasis): How do you understand portfolio risk and transaction cost?

  2. 095Daily returns have an autocorrelation of 0.1 at a one-day lag, decaying geometrically: 0.01 at two days, 0.001 at three, and so on. By how much is the true volatility over a 21-day month higher than daily volatility times the square root of 21?Portfolio and risk mathsHardAQR Capital ManagementGreenwich · 2022

    Try it first

    How much higher is the true monthly volatility?

    Show the worked solution

    About 10% higher. The variance of a sum of 21 daily returns is 21 daily variances plus twice every covariance between pairs of days. With autocorrelation 0.1 at lag one, 0.01 at lag two and so on, the covariance terms add about 4.42 daily variances, lifting the total from 21 to about 25.4, a 21% rise. Volatility, the square root, rises by about 10%.

    Why does a small autocorrelation matter over a month?

    Think of a queue where each person who joins makes it slightly more likely that the next person joins too. Each nudge is small, but over a month of days they add up to longer queues than pure chance would give. The variance of a sum counts every pair of days, and there are 20 neighbouring pairs in a 21-day month, each counted twice, so a lag-one correlation of 0.1 alone adds 4 daily variances to the 21.

    Twenty neighbouring pairs of days lift a month's variance by 21%Variance of the month, in daily variancesSquare-root rule2121.00With autocorrelation21+4.025.42lag 2: +0.38lags 3 to 20: +0.04Volatility of the month, in daily volatilitiesDaily x root 214.58True, root of 25.425.04about+10% volatility
    The square-root rule counts 21 daily variances for a month, but one-day lags add 4.0, two-day lags 0.38 and longer lags 0.04, for 25.42, so monthly volatility is 5.04 daily volatilities rather than 4.58, about 10% higher.

    How do you do the sum?

    Write the monthly variance as the daily variance times 21, plus twice the sum over every lag. Lag one appears in 20 pairs with correlation 0.1, lag two in 19 pairs with 0.01, lag three in 18 with 0.001: twice (2.0 + 0.19 + 0.018 and so on) is about 4.42. The total is 25.42 daily variances instead of 21, a ratio of 1.210, and the volatility ratio is its square root, 1.100. Over a long horizon the ratio tends to (1 + 0.1)/(1 - 0.1) = 1.222, so a month is already close to the limit.

    The relationship
    Var⁡(∑t=1nrt)=σ2[n+2∑k=1n−1(n−k)ρk]=σ2 [21+4.42]\operatorname{Var}\Big(\sum_{t=1}^{n} r_t\Big) = \sigma^2\Big[n + 2\sum_{k=1}^{n-1}(n-k)\rho_k\Big] = \sigma^2\,[21 + 4.42]
    sigma^2the variance of one day's return
    ndays in the month, 21
    rho_kthe autocorrelation at a lag of k days, 0.1 to the power k
    n - khow many pairs of days in the month are k days apart
    What it says in wordsA month's variance is the sum of the daily variances plus twice every covariance between pairs of days, weighted by how many such pairs the month holds.

    What does this mean for risk and performance numbers?

    Scaling daily volatility by the square root of time silently assumes zero autocorrelation. Positive autocorrelation, which shows up in trend-following returns and in portfolios of thinly traded assets priced from stale quotes, makes the square-root rule understate longer-horizon risk. Negative autocorrelation, typical of mean-reverting strategies, does the reverse: at minus 0.1 the monthly variance is only 0.83 times the rule's. The same effect is why correlations between two assets measured on daily returns can differ from those measured on monthly returns: prices that react to the same news on different days look less related day by day than month by month.

    Where candidates lose it

    Candidates either dismiss 0.1 as too small to matter or report 21% as the answer. The first ignores that the correlation enters twenty times over; the second forgets that volatility is the square root of variance.

    The other loss is refusing to estimate without doing every lag. The lag-one term alone gives 21 + 4.0, close to the full 25.42; say that the higher lags add about 0.42 and that you checked.

    What the interviewer asks next

    • What is the ratio if the lag-one autocorrelation is minus 0.1 instead?
    • A fund holds thinly traded assets priced from stale quotes. Which way does its reported volatility err?
    • Why can the correlation between two assets look higher on monthly returns than on daily returns?

    Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): describe what covariance means what would be the difference between the correlation of daily vs monthly returns of a given year