Historical Simulation for interest rate products

Discussion on Value-at-Risk Models
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Joined: Wed Jan 27, 2010 1:40 pm

Historical Simulation for interest rate products

Postby safia » Mon Feb 01, 2010 2:48 pm


I have a question regarding calculating VaR for interest rate products using Historical Simulation. I have looked in several books and apparently there are a flora of different models. Let's take a simpel zero coupon bond with one year to maturity. That is Present Value ( PV) = N*exp(-r*1) where N is the nominal and r is the 1 year interest rate.

* For example you can view the interest rate (r) as the risk factors and calculate log returns (ln(r_t/r_(t-1)) = relchange. And then when you want to calculate the different scenarios (hypothetical new values of the bond) you take PV_new = N*exp(-r*(1+relchange)).

* But... I have also seen some examples where you instead of taking log returns just take the absolute changes (r_t - r_(t-1)) = change. And then calc the scenarios as: PV_new = N*exp(-(r+change)).

* Furthermore I have seen examples where you view the absolute changes of the discount factors (exp(-r*T) - exp(r_(t-1)*T) = discountchange and calc new scenarios as
PV_new = exp(-r*1) + discountchange

I have tested all the different methods and they give quite different results, especially if you want to calculate VaR on a historical period (lets say in the 1990's) but with todays bond. So that you have r_0 today but look at historical changes of r_0 back in 1992. Back then, the interest levels were so much higher today so if you take aboslute changes of the interest rates you can easily end up with negative interest rates (which of course doesn't matter if your portfolio only exists of one long bond).

So I guess my question is really what do you think of all these different models. Which one should you use and why? Maybe one should use different methods depending on if you are in "present time" or if you look at "1992"?

Best regards, Safia

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Joined: Sun Sep 28, 2008 10:30 pm

Re: Historical Simulation for interest rate products

Postby coalexander » Mon Feb 01, 2010 3:14 pm

Hi Safia

One should consider whether you need to be internally consistent with an interest rate option pricing model. The price of a tradable asset must be modelled by a scale-invariant (i.e. geometric) process, otherwise the relative price of an option to the underlying asset would not be invarinat under a numeriare change (see my work with Leonardo Nogueira, downloadable from my publications pages). Hence the bond price must be geometric, i.e. the interest rate must be arithmetic, so you should use absolute changes in interest rates. However, this is the risk-neutral world, and since you are posting on vol IV forum I assume you are in the real world.

This is a question that has interested me in the past, and which I have described in the attached. I conclude that it depends on the time-period, as you surmized. The choice between relative or absolute changes depends on which are the more stable. In countries with very high interest rates, or when interest rates have been trending during the sample period, relative changes tend to be more stable than absolute changes.

Best Wishes, Carol
Last edited by coalexander on Thu Jan 01, 1970 1:00 am, edited 0 times in total.
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Re: Historical Simulation for interest rate products

Postby safia » Mon Feb 01, 2010 3:42 pm

Thank you so much for your quick answear!This will really help me.

BR, Safia

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