A Rs 500 crore equity fund has daily volatility of 1.1%. Compute the one-day 99% VaR, the ten-day VaR by the square-root-of-time rule and the monthly VaR, then explain how daily autocorrelation of 0.2 breaks the scaling.

One-day 99% VaR is about Rs 12.8 crore, ten-day Rs 40.5 crore and 21-day Rs 58.6 crore by square-root scaling. That assumes independent days. With autocorrelation of 0.2, each day's loss partly repeats the next day, multi-day variance gains covariance terms, and ten-day VaR rises to about Rs 48.5 crore, monthly to about Rs 71.1 crore, roughly a fifth higher.

Start from one day. The 99% point of a normal distribution is 2.33 standard deviations, so one-day VaR is 2.33 times 1.1% times Rs 500 crore, about Rs 12.8 crore. If days are independent, variance adds across days, so volatility and VaR grow with the square root of the horizon: ten days is about 3.16 times one day, Rs 40.5 crore, and 21 days about 4.58 times, Rs 58.6 crore. Many risk reports have built ten-day figures this way for years; if you are matching a regulatory definition, confirm the current rule for scaling and for the confidence level.

The square-root rule adds only the daily variances. When returns are autocorrelatedCorrelated with their own past values, so that a return today carries information about the return tomorrow., the variance of a sum also includes every pair of days, weighted by their correlation. With a correlation of 0.2 between neighbouring days, and 0.04 two days apart, a ten-day variance is about 14.4 days' worth instead of 10, so ten-day VaR is about 20% higher, Rs 48.5 crore. Over 21 days the uplift is about 21%, Rs 71.1 crore. A tap that drips steadily and a tap whose drips come in bursts fill the same bucket on average per drop, but the bursty one gives a much more uneven bucket over an hour.

Report the monthly number from monthly behaviour, not from daily behaviour scaled up. The cleanest fix is to estimate the longer-horizon volatility directly, from overlapping multi-day returns or from monthly returns, which captures the autocorrelation automatically. The autocorrelation here also has a cause worth naming: thinly traded stocks whose prices update late. Their daily volatility looks low and their risk shows up over weeks, so the one-day figure itself understates the fund's true risk. Negative autocorrelation, common in some mean-reverting books, does the opposite, and square-root scaling then overstates the risk.

Add the other two limits briefly. Daily returns have fatter tails than the normal, so the 2.33 multiplier understates the one-day number too, and volatility clusters, so a calm month's VaR is not a good guide to the next turbulent one.