What is the theoretical relationship between N-day VaR and 1-day VaR under the square root of time rule?

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BlackTom AI · Accepted Answer

Context: The question asks about the theoretical relationship between N-day VaR and 1-day VaR under the square root of time rule. The given option states: 'N-day VaR = 1-day VaR × √N.'

First, consider the core idea of the square root of time rule (SRT) in risk measurement. SRT posits that the risk (as measured by VaR) scales with the square root of the time horizon when returns are jointly normally distributed and time-homogeneous, and when there is little serial correlation in returns. In mathematical terms, VaR over a longer horizon N days can be approximated by scaling the one-day VaR by √N, assuming the incremental risks over each day are independent and identically distributed.

Applying this to the given option:
- The expression N-day VaR = 1-day VaR × √N mirrors the SRT formula, implying that moving from 1 day to N days increases VaR by the square root of N. This aligns with the standard result under the stated assumptions.

Now, what would make this statement sound incorrect or incomplete? If the assumptions break down—such as presence of serial autocorrelation in returns, non-stationarity, heavy tails, or nonlinear risk factors—then the square root of time rule may not hold exactly, and the relation could be biased or require alternative scaling (e.g., using a time-varying volatility model or CVaR adjustments).

However, given the conventional theoretical framework for SRT and VaR, the proposed relationship expressed in the option is the classic and widely cited form: scaling 1-day VaR by √N to obtain N-day VaR, under the usual IID and normality assumptions.

In summary, the statement captures the fundamental SRT concept: risk grows with the square root of the holding period, so extending from 1 day to N days involves multiplying the one-day VaR by √N, provided the underlying assumptions hold true.

BU.231.710.51.FA25 PoP Quiz - 1 (Week 4)

What is the theoretical relationship between N-day VaR and 1-day VaR under the square root of time rule?

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