Monte Carlo VaR

[Si100]

Hi,

I am a novice at VaR and I'm currently trying to work out Monte Carlo VaR but am having zero luck. My task is as follows:


Estimate the 1-day 95% VaR and the 1-day 99% VaR for an equity portfolio using Monte Carlo simulation with Student t marginal distributions, a Normal copula and 10,000 trials.

I have a portfolio with three variables which includes data with two stock market index prices and an exchange rate. I have so far calculated the 1) Multivariate Standard Normal simulations 2) Normal Copula simulations 3) Dependant Student t Returns.

I am stuck on the "simulated returns with required means and volitilities" as I have literally no idea how to work out the 1)Expected Excess Return and 2) Risk Factor Senstivities for the three variables. I'v found methods to work out 1) such as Jenson's Alpha, actual returns-CAPM and the GMB formulas, but im not sure what to do? Because I'm dealing with stock markets (ie FTSE and CAC) and an exchange rate (euro/pound sterling) I can't work out which formula to use.

If you can steer me in the right direction with working out the expected excess return and risk factor sensitivies for my three variables I would be extremely grateful.

Thank you

 

[Aleksander Hansen]

Which question are you looking at?

 

[Si100]

I was under the impression that I needed to work out the expected excess returns and risk factor sensitivities for each variable in order to calculate simulated returns with required means and volitilities. So I would like to know how to work out the first 2 in order to work out the simulated returns with required means and volitilities if possible.

 

[Si100]

If it helps I'v taken the skeleton framework from the Alexander spreedsheet IV.4 provided with his book, I would post the file but it doesn't seem to be working for some reason.

 

[Aleksander Hansen]

the risk factor sensitivities are just the betas from a regression on the risk factors (which you have to make sure are mapped risk factors and not just variables).
You say you have the dependent t-returns, does that include the d.o.f., which you would need to include in addition to your mean and variance.

 

[Si100]

Ok I think I'v got a method to work out the excess returns. What I'v done is take the growth rate in the 2 stock indexs and exchange rate over the two year daily data. I'v then downloaded 3-month treasury bill data for the US for the same time period and worked out the difference between the growth in the variables against the risk-free rate.

Again thank you for your quick reply, that makes sense about the betas for risk factor sensitivities, so in terms of regression, what would I regress my three variables against? Would I regress each individual 3 against the portfolio value, portfolio growth or average returns of the portfolio? I'm also not sure what you mean by "make sure are mapped risk factors and not just variables", how do I do this. If you could explain in the most basic format I would appricate it as I only came into contact with VaR as a concept at the end of last year so I am struggling with a few concepts/terminology.

And yes the dependent t-returns does include the individual degrees of freedom for each variable, which I have calculated already.

Thanks again

 

[Aleksander Hansen]

All three (so multivariate) in terms of log-returns against the log-return of your portfolio, assuming the log-returns are the invariants (which may or may not be the case but I suspect it is, especially for exchange rates).

VaR mapping is hard to explain in just a few sentences so I would encourage you to do a quick Google search. Basically if you have, say stocks, bonds and currency, then log-returns of stocks would be one risk factor and log-returns of currency would be another, and the changes in bond yields would be another.

One question though, what decomposition did you use on the correlation matrix to produce the correlated returns?

 

[Si100]

I have logged the returns of each risk factor and the portfolio and have regressed each risk factor return against the portfolio returns. Then I have assumed that each beta coefficient is equal to the 3 risk factor sensitivities.
Could you possibly verify that this is correct and what you meant and if you agree with the way I found the expected excess returns found in a previous post (ie downloading the treasury bill data)?

I have to conduct 10,000 trials overall, does this mean I have to simulate 10,000 observations for each risk factor (ie drag each risk factor column down 10,000 times)?

I used the Cholesky Matrix Decomposition to produce the correlated returns

Thanks again

 

[Aleksander Hansen]

1. You simulate the correlated array of risk factors together, not separately. So 10,000 paths for your correlated risk factors.
2. It would appear as if what you have done is correct, although it's a little hard for me to tell without looking at it in detail
3. You should calculate the expected excess return using OIS (Overnight Indexed Swap) data, which is what the banks use as their risk-free rate. Treasury bill data is not a good proxy for the risk-free rate.
4. When you used the Cholesky decomposition did it work without any adjustments?

 

[novice]

Hello, I have been given a task to perform Monte-Carlo simulation using cholesky decomposition matrix on a series of assets. However, I have no idea how to go about conducting this process. Do I need to first transform my data to a normal series? If so how do I go about doing this.
Thanks for your help.

 

[chiyui]

Hello, I have been given a task to perform Monte-Carlo simulation using cholesky decomposition matrix on a series of assets. However, I have no idea how to go about conducting this process. Do I need to first transform my data to a normal series? If so how do I go about doing this.
Thanks for your help.

Try to see the attachment. It's a zipped powerpoint file showing the procedure of using cholesky factors to generate 2 or more correlated random numbers.
Ask me if you have any difficulties of understanding the powerpoint slides.

 

[bPan]

Hi Alexander / Chiyui,

I want to compute Marginal VaRs by Monte Carlo way. How correct am I to say
Marginal VaR = (Portfolio VaR/Portfolio Value)*Beta
=(Portfolio VaR/Portfolio Value)*(Covariance (Asset, Portfolio)/Portfolio variance)?

This method is as mentioned in Jorion's Chapter 7. So for MC Marginal VaR, can I just replace Parametric VaR in the example with MC VaR value?

Thanks

 

[chiyui]

I haven't seen (or I forgot, haha shrug) a term of "Marginal VaR" before.
Where did you see this term?
And where did you see the formula of Marginal VaR = (Portfolio VaR/Portfolio Value)*Beta? How does it come out?
And which Jorion's book are you referring to?

 

[bPan]

Take a look at this video by David Harper...

I'm wondering if the same concept can be extended to Monte Carlo (MC).