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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?

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I'm not sure I follow: the definition of $\hat{S}_i$ does not not seem logical to me with regard to the definition of $\hat{r}_i$. What is $\hat{r}_i$ exactly, and why would it become large or negative (it can't be large and negative). Can you clarify? –  Bob Jansen Jul 20 at 13:31
    
I apologise for not being precise enough. The $\hat r_i$ represent the (past) return of the asset/stock over one day. Assuming that the past is a good indicator as to what the future value of the stock might be, we may see $\hat S_1,...,\hat S_n$ as tomorrow's potential values of the stock. So these values give us a distribution of the stock's value (or a histrogram if you will). Now, if e.g. $S_1,S_2$ are very small (but still different), then the ratio $\hat r_2$ becomes very unpredictable and potentially very large (as we divide by a small number). –  Tom Jul 20 at 14:57
    
To add a bit more: If e.g. $S_1=103,S_2=110$ then the return is $\hat r_2 = 0.936$ (i.e. a decrease of around 6.4% over one day). This is a reasonable number. However, if $S_1=0.013,S_2=0.005$ then the return is $\hat r_2 = 2.6$, i.e. an increase of around 160% over one day. And if all $\hat r_i$ have this unpredictable order of magtnitude the $\hat S_i$ are just not reasonable any more. –  Tom Jul 20 at 15:03

2 Answers 2

up vote 2 down vote accepted

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.

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Thank you for this. I don't know if this is too much to ask but do you happen to know any literature where these things are explained more rigorously? Because it seems to me that this subtlety may be quite important in certain scenarios. –  Tom Jul 21 at 19:34
    
I don't really know one single book where historical simulation is explained in detail. I have seen the above approach being applied successfully in practice. Meucci wrote the book "Risk and Asset Allocation" published at Springer. A lot of ressources can also be found at his web page symmys.com. I hope that helps. –  Richard Jul 22 at 6:30
    
You might find the paper Neither 'Normal' nor 'Lognormal': Modeling Interest Rates Across All Regimes interesting. –  Richard Jul 22 at 6:38

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).

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