Historical VaR: Relative Returns for Assets and Rate Changes for Yields

# Historic Value at Risk - Ratios vs. Differences


# Historic Value at Risk - Ratios vs. Differences












Quick Summary on Historic VaR Let $S_0,...,S_n$ be the daily values of some stock (where $S_0$ is the current value). Then for $i=1,\ldots,n$ we let $$\hat r_i:=S_{i-1}/S_i \quad \text{and}\quad \hat S_i := S_0\cdot \hat r_i$$ Now we can estimate e.g. the 95% 1-day-VaR by looking at the $(0.95n)$-th smallest number amongst all the scenarios $\hat S_1,...,\hat S_n$ and then subtracting $S_0$.

The above approach works fine when we look at stocks since $S_i>0$ for all $i$. But what if we consider an interest rate that is potentially close to zero, and even worse, may go negative (which is the case for the German short-term treasury bills at the moment), then we have that the $\hat r_i$ become very large and potentially negative, which renders the scenarios $\hat S_i$ completely useless.

Questions One solution might be to look at the differences, i.e. $\overline r_i := S_{i-1}-S_i$ and $\overline S_i = S_0+\overline r_i$. But this completely ignores the order of magnitude of an asset. So my questions are:

1) Does anybody have an idea as to what approach is usually used in practice

2) Are differences a sensible approach at all?

3) Are there potentially other methods to avoid this problem?

## Answer by Richi Wa (score 4, accepted)

https://quant.stackexchange.com/a/14103

As a short summary and adaption of the question: You better redefine $\hat{r}_i= \frac{S_{i-1}}{S_1}-1$ and $\hat{S}_i = (1+\hat{r}_i)S_0$.

The above definition of $\hat{S}_i$ yields a sample of potential values for $S$ for the future day. This approach is usually applied in historical simulation. The aim here is to use information of the past about the distribution of invariant quantities. The stock price itself is not invariant but returns can be assumed to be invariant. It then makes sense to form a sample using these. The good thing is if we assume that $\hat{r}_i$ is e.g. normally distributed then $\hat{S}_i$ is so too. One could reformulate the set-up to a log-normal world.

For interest rates we rather think of an additive evolution. They change in bps and market participants add bps to the yield curve when they think about future developments. In my experience this is what is done in the historical simulation of interest rates. You form samples of the form $$ \hat{r}_i = r_0 + (r_{i-1}-r_i). $$ Doing this we again use invariant quantities of the past and apply it to the present situation. This is often done fo the whole interest rate curve and then used to price bonds or derivatives.

The term invariant in this context is borrowed from Attilio Meucci.

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/14116

It might help to think of the two as special cases of $$S_{i+1}-S_i = \sigma (c+S_i)^\beta \epsilon$$ which looks like a Constant Elasticity of Variance extension. Taking squares of both sides and then logs will (nearly) linearise it, allowing you to carry some basic estimation using OLS.

The parameter $c$ will control the lower bound and can impose some volatility at zero.

$\beta=0$ corresponds to absolute shocks, while $\beta=1$ corresponds to relative shocks. But there are other possibilities, for example 1/2 like CIR. Values bigger than one would indicate a very wild behaviour (similar to strictly local martingales).