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# Homoscedasticity vs heteroscedasticity

#### Maxim Rastorguev

##### Member
Dear all, I now try to calculate the factor VAR for my fixed income portfolio. The factor VAR assumes that each and every asset in the portfolio has an exposure on a set of the same factors. It’s greatest advantage is no need to calculate too many volatilities and correlations ( I have some 70 bonds in the portfolio so you can imagine the size of variance-covariance matrix). All you need is a restricted set of vols and corrs for the factors you choose. One another advantage IMHO for a bond portfolio is that these factors are quite simple and observable: base rates and credit spreads are good candidates. On the other hand historical var can be hard to calculate because too many bonds are new or illiquid or have strange history or whatever. So my approach is to regress yield of a bond or a set of similar bonds to base rates and spreads to move from yields of bonds to factors sets. Sounds simple and logical. But when doing so I faced with heteroscedasticity, which is as we know a norm in reality (while homoscedasticity is a more ideal world). As a result I just can not regress them. I am sure that the base rates and credit spreads are the leading factors which can explain 90% of volatility of each bond in the portfolio. But this heteroscedasticity is something I just don’t know how to handle. Can anyone advice?

#### ami44

##### Active Member
Subscriber
1. Heteroscedasticity is not that bad. The least square estimator will not be the best estimator anymore, but it‘s still consistent and unbiased. So first order solution is to ignore the problem

2. There are Methods to deal with heteroscedasticity https://en.wikipedia.org/wiki/Weighted_least_squares

3. If I understand correctly you want to calculate the portfolio VaR by Variance-Covariance method. You are using two risk factors, interest and credit spread and now you need the sensitivity of your bond prices to these risk factors. Since you know the sensitivity of the prices to the bond yield, you are trying to find the dependency between yield and the risk factors for each bond. Is that correct?

My first order approximation would be, that yield is 100% correlated to both risk factors. I suspect that deviations from this assumption will not be linear. If you want to be more precise, you probably need to restrict your regression to a small interval around the current value. But I feel by neglecting term structures you make a bigger error, than by assuming 100% correlation of yield and interest rates.

I have never done this though. I would be interested to here how it worked out at the end.

#### Maxim Rastorguev

##### Member
Thank you ami44, Yes you are absolutely correct in 3. If Ya stands for bond’s A yield to maturity , Br base rate (risk free), Cr credit spread, then what I want is to express Ya as Ya=(alpha)*Br + (beta)*Cr + epsilon. Where the sum ((alpha)*Br + (beta)*Cr) stands for systematic part of the whole exposure and epsilon is unsystematic one. Accordingly I get alpha and beta as exposures to the risk factors from regression. And this epsilon doesn’t seem to have constant variance and this is where my heteroscedasticity really plays out. It is not constant so you can calculate the variance for a certain historical window but it is problematic to expect it will be more or less the same in future, so you can underscore or overscore the volatility.
As to concrete exposures they are not 100% , which seems natural as corporates bonds often lag what government curve does and afterwards they can overshot because of stop loss selling by many trades etc.

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