Project 2 – CV-VaRAnnamária LakiSupervisor: Áron Varga

Summary of 1st semester ( Project 1 )In the previous semester, our project was about the square root of time scaling used for 1-day VaR to estimate n-day VaR. (Where VaR is a certain quantile of the PnL distribution.) The reason for this is that we don’t have enough avaliable data to compute Regulatory Capital, where 10-day VaR is required. This scaling factor is . Why ? Because

if I.I.D Assuming, that the PnL distribution is I.I.D: Weekly: Bi-weekly (Regulatory): Monthly:

Value at Risk (VaR) is a widely used risk measure of the risk of loss on a specific portfolio of financial exposures. For a given portfolio, time horizon, and probability p, the p VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is p. This assumes mark-to-market pricing, and no trading in the portfolio.

Profit and Loss summarizes the revenues, costs and expenses incurred during a specific period of time, usually a fiscal quarter or year.

Computing: Portfolio (eg. S&P500) Initial capital Time-horizon Confidence level (95%, 99%)The question arises, how big of a mistake do we make if we use the square-root of time scaling. The regulators would like to see that the firms are on the conservative side (i.e. make sure to set aside higher capital than needed)

Conclusion: Nowadays is conservative estimation but it depends on market conditions. for weekly VaR is clearly worse than for bi-weekly VaR.

Further Conclusion: Underlying assumption if I.I.D does not hold. It would be wiser to incorporate this in our model. When volatility increases we will underestimate our risk.

Conditional Volatility VaR ( CV-VaR )Starting point: We saw that the I.I.D assumption does not hold.

CV-VaR: We assume that PnL distribution depends on time through volatility

Where ℱ is distribution that „exists in the background”. So, we use time dependent volatility to get better estimated risk.

Remark: Autocorrelation is ignored for the moment.

Computing CV-VaRWe need: Portfolio ( S&P500 ) ( simplified compared to project 1 ) Confidence-level Time-horizon

Volatility: Volatility is a statistical measure of the dispersion of returns for a given security or market index. Volatility can either be measured by using the standard deviation or variance between returns from that same security or market index. Commonly, the higher the volatility, the riskier the security.

Step 1.: The portfolio’s Value ( from Marks)Step 2.: PnLsStep 3.: Computing 30-day ’s (StDEV) of the PnL..

Step 4.: After these steps we can compute the CV-VaRs with different ( mostly used 99% and 95% ) confidence-levels.

CV- where T: current day. Eg.

:

Representing in ExcelProblem: What can we use to fill empty cells for the starting period? Solution: constant back-filling.

Representing in Matlab

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sigma

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F

The PnL and F distributionAs we see, PnL distribution is not heteroskedastic, so we should try modelling volatility.

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VaRCV VaR

As we can see, CV-VaR reacts to changing market environments faster.

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1-day VaR

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n-day VaR

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Ratios exceeding ( Confidence-level. 95% ): Holding period

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n-day VaR / 1-day VaRn-day CV VaR / 1-day CV VaR3.1623

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n-day VaR / 1-day VaRn-day CV VaR / 1-day CV VaR2.2361

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n-day VaR / 1-day VaRn-day CV VaR / 1-day CV VaR3.1623

Conclusion

CV-VaR is an improvement over VaR in terms of measuring risk. ( Regulators are happy )

I.I.D is still not achieved, It is possible to improve estimation. However, the plot of F doesn’t

look bad. Next idea is to work on autocorrelation.