Portfolio Management puzzles, solved step by step

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  1. 002A trading book's one-day 99% value at risk is Rs 2 crore. What are its 10-day and its one-month (21 trading day) value at risk under the usual scaling rule, and when does that rule fail?Portfolio risk mathsWarm upAQR Capital ManagementGreenwich · 2022

    Try it first

    Pick the 10-day value at risk before you calculate.

    Show the worked solution

    About Rs 6.3 crore over 10 days and Rs 9.2 crore over 21 days. With independent daily returns, variance adds across days, so volatility and value at risk scale with the square root of time: 2 x the root of 10 and 2 x the root of 21. The rule fails when returns trend or mean revert, when tails are fat, and when the book changes during the period.

    Why the square root of time and not time itself?

    Think of a person taking random steps left or right. After a hundred steps they are rarely a hundred steps from the start, because the left steps cancel the right ones; the typical distance is about ten, the square root of a hundred. Daily returns behave the same way when each day is independent. Variances add across independent days, so the spread of a ten-day return is the daily spread times the square root of ten, not times ten. Value at risk at a fixed confidence level is a multiple of that spread, so it scales the same way: Rs 2 crore becomes Rs 6.32 crore over ten days and Rs 9.17 crore over 21.

    Value at risk grows with the square root of time, not with time369121510152125Holding period, trading daysVaR, Rs crorescaling by days: 20 at 10 days (wrong)1 day: 2.010 days: 6.321 days: 9.2Holds only if daily returns areindependent and the book is nottraded down in between
    Scaled by the square root of time, a one-day value at risk of Rs 2 crore becomes about Rs 6.3 crore at ten days and Rs 9.2 crore at 21 days, far below the Rs 20 crore and Rs 42 crore that scaling by the number of days would give.
    The relationship
    VaRT=VaR1T210≈6.32221≈9.17\text{VaR}_T=\text{VaR}_1\sqrt{T} \qquad 2\sqrt{10}\approx 6.32 \qquad 2\sqrt{21}\approx 9.17
    \text{VaR}_1the one-day value at risk, Rs 2 crore
    Tthe holding period in trading days
    What it says in wordsMultiply the one-day figure by the square root of the number of days, which is valid only for independent, identically spread daily returns and an unchanged book.

    When does the rule give the wrong answer, and in which direction?

    The rule rests on three assumptions, and each one breaks in real markets. If returns trend, so a bad day tends to follow a bad day, the true ten-day loss is larger than Rs 6.3 crore; if they mean revert, it is smaller. Fat tails make the 99% point further out than a normal curve suggests, and the ratio between the tail and the spread need not hold across horizons. And a book is not frozen: over a month a desk cuts losing positions, which the scaling ignores. Say which way each one pushes the number and the interviewer knows you understand the rule rather than having memorised it.

    One more thing worth saying: the scaling also assumes the expected daily return is zero. Over a day that is harmless. Over a year, drift matters and a simple square root rule starts to overstate the loss for a portfolio with a positive expected return.

    Where candidates lose it

    The fast wrong answer is Rs 20 crore, which treats ten independent days as ten worst days in a row. Candidates who know the square root rule sometimes lose the point anyway by stating it without its assumptions.

    The follow-up is almost always when it fails. Have the three failures ready, trending returns, fat tails and a changing book, and say in which direction each pushes the number.

    What the interviewer asks next

    • If daily returns have a positive autocorrelation of 0.2, is the true 10-day value at risk above or below Rs 6.3 crore?
    • Why do regulators ask for a 10-day horizon rather than one day?
    • What is the annual value at risk under the same rule, using 250 trading days?

    Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): Specific statistics questions on financial concepts. daily vs monthly return, VAR, more that i don't remember

  2. 015A Rs 50 crore equity portfolio has a beta of 1.2 to the index. How much index futures notional must you sell to bring the portfolio's beta down to 0.5?Portfolio risk mathsWarm upPortfolio implementationHedge funds

    Try it first

    How much notional do you sell?

    Show the worked solution

    Sell Rs 35 crore of index futures notional. At a beta of 1.2 the portfolio moves like Rs 60 crore of the index. At the target of 0.5 it should move like Rs 25 crore. The difference, (1.2 minus 0.5) x Rs 50 crore, is Rs 35 crore, assuming the futures move one for one with the index. At an assumed Rs 10 lakh a contract, that is about 350 contracts.

    Why is the portfolio's market exposure not simply Rs 50 crore?

    Think of a car that goes 1.2 km for every km a reference car goes. Holding Rs 50 crore of it is like holding Rs 60 crore of the reference. Beta converts a portfolio's value into index-equivalent exposure, so a Rs 50 crore book at beta 1.2 carries Rs 60 crore of market risk. Once you see the exposure in index rupees, the hedge is a subtraction: you want Rs 25 crore left, so you take away Rs 35 crore by selling futures, which carry a beta of one to the index.

    Hedge the change in beta, not the whole portfolioPortfolio value50its value in rupeesMarket exposure now6050 x 1.2Sell index futures-35(1.2 - 0.5) x 50Exposure left2550 x 0.5 = the target betaAt an assumed Rs 10 lakh a contract, Rs 35 crore is about 350 contracts.
    The Rs 50 crore portfolio at beta 1.2 carries Rs 60 crore of market exposure; selling Rs 35 crore of index futures leaves Rs 25 crore, which is a beta of 0.5 on the portfolio.
    The relationship
    N=(βtarget−βnow)×V=(0.5−1.2)×50=−35N=(\beta_{target}-\beta_{now})\times V=(0.5-1.2)\times 50=-35
    Nfutures notional to trade, Rs crore; negative means sell
    \beta_{now}, \beta_{target}the current beta 1.2 and the target 0.5
    Vthe portfolio's value, Rs 50 crore
    What it says in wordsThe futures notional equals the change in beta times the portfolio value, with a minus sign meaning a sale.

    What does the hedge not do?

    It removes market risk, not stock risk. After the hedge the portfolio still carries every stock-specific bet it had; only its sensitivity to the index has been cut. That is often the point: a manager who likes the stocks but not the market can keep the stock picks and trim the market bet. Say the limitations: beta is estimated from history and drifts, so the hedge is right only on average; futures need margin and must be rolled at expiry, and the futures price can move slightly differently from the index, which is called basis risk.

    Where candidates lose it

    The common answers are Rs 50 crore, hedging the whole value, and Rs 60 crore, hedging the whole beta-weighted value. Both take the beta to zero, not to 0.5. Hedge the change in beta.

    Candidates also forget the direction. Lowering beta means selling futures; raising it means buying them. Say the sign with the number.

    What the interviewer asks next

    • How much would you trade to raise the beta to 1.5 instead?
    • If the portfolio falls 10% in value, is the hedge still right?
    • Why might the hedged portfolio still lose money in a market fall?