When Square Root of Time VaR Scaling Breaks Down

# How accurate is the square root of time rule for VaR for a portfolio containing several different types of instruments


# How accurate is the square root of time rule for VaR for a portfolio containing several different types of instruments












Assuming that your value at risk model is based on normality assumptions, e.g. using a Delta-Gamma normal model does the approximation hold perfectly for a portfolio of stocks and options? What about other instruments?

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/53590

Effectively, I sense two questions here, 1) around the validity of the $\sqrt{T}$-assumption in the scaling of the risk horizon ; and 2) the quality of the $ \Delta$-$\Gamma$-approximation in relationship to that.

## Re question 1.

The square root of time scaling results from the i.i.d. assumption of the underlying random variable. If your r.v. (which might be the portfolio PnL) is truly independent in time and identical across time points, then

$$ \sigma^2\left(Z_k\right)\equiv \sigma^2\left(\sum_{1}^{k}x_i\right)=k\sigma_x^2, $$ i.e. the volatility scales with $\sqrt{k}$. Again: Effectively, this result only holds for i.i.d. variables. The graph below depicts a quick example using a normal distribution (green) and a student-t-distributed example (red). Each dot is a 99% quantile based on a simulation with 5000 scenarios and the number of components that were added on the x axis. As an example, the top right dot (red) is the 99% quantile of the sum of 1000 student-t-distributed r.v.s with degrees of freedom equal to 3. In both cases, the $\sqrt{t}$-rule is able to (perfectly) match the 99%-quantile at each time horizon.

## Re question 2:

Since your risk factors, i.e. the driving force behind your modelled pnl, may well be i.i.d. (at least we may assume that), we need to make sure that the subsequent aggregation of "small" local / non-local shocks at time 0 (Delta-Gamma-Approach) add up to the same thing under a 'larger' move.

### Linear example:

For a linear product, this holds with equality, i.e. for $f=a+bx$,

$$ f(x_0+ x)-f(x_0)=bx $$

which is linear. So if $x$ is i.i.d. in this case, the $\sqrt{T}$-rule will hold for $f(x)$ as well.

### Non-linear example (Delta gamma)

Let's see how the non-linear contributions might add up for $f(x)=a+bx+cx^2$:

$$ f(x_0 + x)-f(x)=(b+2cx_0)x+cx^2 $$

As you can see from this very simple example, $$f(x_0+\sum_i x_i)-f(x_0)\neq \sum_if(x_0+ x_i)-f(x_0)$$

and hence we cannot simply 'pass' the risk factor (and its i.i.d.) properties through.

Find below another very simply example around that. Assume that we have again an i.i.d. risk factor, say this is a stock return. Then, the full reval pnl of the stock is $$f(x)=S(e^{x}-1) \approx S(x+0.5x^2)$$

A plot of this simple ansatz (below) shows that for small time horizons, the linear component of the pricing function (the "$bx$" term) dominates, and so will the $\sqrt{t}$ property of the i.i.d. risk factor. At a certain point, the higher order ($x^2$ and higher) contributions have an impact and will dominate the function. At this point, the square-root-of-time rule is not applicable any more in its standard form.

## Practical summary

In practice, I often encounter that the rule is applied in practice. For short time horizons, 'nice' risk factors (sufficiently i.i.d) and "simple" products, its application seems reasonable. On longer time scales ($>$ 10 business days), we usually see that it cannot be applied anymore due to, for example, autocorrelation or non-linear payoffs.