We saw in Project I. that I.I.D does not hold for our modell. The idea try CV-VaR, where the PnL distribution depends on the time through volatility. The results give us a better estimation over VaR. However, I.I.D still not achived. Modell coded in Matlab.

1. Project 2 – CV-VaR
Annamária Laki
Supervisor: Áron Varga

2. 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 𝑛. Why 𝑛? Because
if 𝑋1, … , 𝑋 𝑛 I.I.D → 𝐷( 𝑋𝑖) = 𝑛 𝐷𝑋1
Assuming, that the PnL distribution is I.I.D:
 Weekly: 5
 Bi-weekly (Regulatory): 10
 Monthly: 20
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.

3. 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 10 is conservative
estimation but it depends on market
conditions. 5 for weekly VaR is clearly
worse than 10 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.

4. 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
𝑃𝑛𝐿 ~ 𝜎𝑡 ∙ ℱ
 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-VaR
We 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.

5. Step 1.: The portfolio’s Value ( from Marks)
Step 2.: PnLs
Step 3.: Computing 30-day 𝜎𝑡’s (StDEV) of the PnL..
𝑃𝑛𝐿 ~ 𝜎𝑡 ∙ ℱ → ℱ~
𝑃𝑛𝐿
𝜎𝑡
Step 4.: After these steps we can compute the CV-VaRs with different ( mostly
used 99% and 95% ) confidence-levels.
CV-𝑉𝑎𝑅% = 𝜎 𝑇 ∙ ℱ%
−1
where T: current day.
 Eg.
CV−𝑉𝑎𝑅99% = 𝜎 𝑇 ∙ ℱ−1
99%
ℱ−1
99% : Percentile(PnL, 0.01)

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

7. Representing in Matlab
0 2000 4000 6000 8000 10000 12000 14000 16000
0
1000
2000
3000
S&P 500
0 2000 4000 6000 8000 10000 12000 14000 16000
-200
0
200
PnL
0 2000 4000 6000 8000 10000 12000 14000 16000
0
20
40
60
sigma
0 2000 4000 6000 8000 10000 12000 14000 16000
-10
0
10
F

8. The PnL and F distribution
As we see, PnL distribution is not heteroskedastic, so we should try modelling volatility.
0 2000 4000 6000 8000 10000 12000 14000 16000
-150
-100
-50
0
50
100
150
PnL
F

9. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
S&P 500
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
50
100
150
VaR
CV VaR
As we can see, CV-VaR reacts to changing market environments
faster.

10. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
50
100
150
1-day VaR
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
100
200
300
n-day VaR
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
1.5
2
2.5
3
n-day CV VaR / 1-day CV VaR

11. Ratios exceeding n ( Confidence-level. 95% ):
Holding period
Number of days
for volatility
5 10
30
40
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
0.5
1
1.5
2
2.5
3
3.5
4
n-day VaR / 1-day VaR
n-day CV VaR / 1-day CV VaR
2.2361
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
1
2
3
4
5
6 n-day VaR / 1-day VaR
n-day CV VaR / 1-day CV VaR
3.1623
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
0.5
1
1.5
2
2.5
3
3.5
4
n-day VaR / 1-day VaR
n-day CV VaR / 1-day CV VaR
2.2361
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
1
2
3
4
5
6 n-day VaR / 1-day VaR
n-day CV VaR / 1-day CV VaR
3.1623

12. 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 𝜎𝑡 estimation. However, the plot of F doesn’t look bad.
 Next idea is to work on autocorrelation.