Download Maksym Obrizan Lecture notes III

KYIV SCHOOL OF ECONOMICS
Financial Econometrics (2nd part):
Introduction to Financial Time Series
May 2011
Instructor: Maksym Obrizan
Lecture notes III
# 3. Value at Risk (VaR) is primarily concerned
with market risk
# 2. In this lecture:
Review the basic ideas behind Value at Risk
(VaR) calculations based on various time
series models:
1. RiskMetrics
2. Econometric models
3. Quantile models
4. Extreme value theory
Various types of risk in financial time series:
Credit risk, liquidity risk and market risk
# 4. One way to think about VaR is as of a
maximal loss associated with a rare (or
extraordinary) event
Value at Risk (VaR) –
Definitions of long and short financial positions:
A long financial position is –
Basic idea:
A short financial position is –
# 5. VaR under a probabilistic framework
The VaR of a long position over the time horizon
h with probability p is
# 7. The definition on slide #5 continues to apply
to a short position if one uses the distribution
of -∆Vt(h)
# 6. VaR defined on slide #5 typically assumes a
negative value when p is small
VaR is concerned with tail behavior of the CDF
Fh(x)
# 8. NOTES
# 9. For a known univariate CDF Fh(x) and
probability p one can simply use the pth
quantile
# 11. RiskMetricsTM
# 10. Calculation of VaR
# 12. In addition, RiskMetrics is built on an
IGARCH(1,1) process without a drift
Developed by J.P. Morgan
RiskMetrics assumes that
It can be shown that the conditional distribution
of rt[k] is
# 13. Thus, under this special IGARCH(1,1)
model the conditional variance of rt[k] is
proportional to the time horizon k
# 14. … and for a k-day horizon is
Thus, under RiskMetrics we have
VaR(k) = √k VaR
For the continuously compounded (i.e. log)
returns
# 15. Example
This rule is referred to the square root of time
rule in VaR calculation
# 16. Cont’d
# 17. The main advantage of RiskMetrics – it’s
simplicity
In addition, many stocks have non-zero means
of a return. For example,
# 19. VaR with multiple positions
Define ρij - the cross-correlation coefficient
between the two returns (i and j)
Then VaR can be generalized to m positions as
# 18. In this case, the distribution of k-period
return is
The 5% quantile used in k-period horizon VaR
calculation is then
# 20. NOTES
# 21. VaR based on a general time series model
Consider the log return of rt of a financial asset
# 22. The error term εt is often assumed to be
normal or a standardized Student-t
distribution
For a normal distribution obtain the 5% quantile
of a distribution for VaR calculations as
# 23. For a standardized Student-t distribution
the quantile is
Observe that if q is the pth quantile of a Student-t
distribution with v degrees of freedom then
Is the pth quantile of a standardized Student-t
distribution with v degrees of freedom
# 24. Thus, the 1-period horizon VaR at time t is
# 25. Example based on a standard normal εt
# 26. Cont’d
# 27. Example based on a standardized
Student-t εt
# 28. Cont’d
# 29. Quantile estimation –
# 30. Quantile and order statistics
This method makes no specific distributional
assumption
Use:
Empirical quantile directly
Quantile regression
# 31. Based on the asymptotic result one can
use r(h) to estimate the quantile xp where
h = np
For example, r(1) and r(n) are the sample min and
the sample max
# 32. Then the quantile xp can be estimated by
# 33. Check yourself
Daily log returns of Intel stock with 6,329
observations
# 34. NOTES
VaR of a long position of $10 mln?
# 35. Pros and Cons of Empirical Quantile:
“+”
“-”
Assumes that the distribution of return rt does not
change (i.e. loss cannot be greater than the
historical loss – not true!)
CONCLUSION:
# 36. Quantile regression
In practical applications, some explanatory
variables may be used to facilitate model
building
# 37. Quantile regression: choose β to minimize
# 38. Familiar estimator: Least Absolute Deviations
(LAD)
Minimizes the sum of absolute deviations
(OLS: sum of squared deviations)
Basic idea of quantile regression:
Quantile regression estimator is available in Stata
# 39. NOTES
# 40. NOTES