The Effect of Long-memory Stochastic Volatility on the Anaysis of Confidence Interval for VaR Risk Measure

1. The Effect of Long-memory Stochastic Volatility on the Anaysis of Confidence Interval for VaR Risk Measure Hwai-Chung Ho Institute of Statistical Science, Academia Sinica Sharon S. Yang Department of Business Mathematics, Soochow University Fang-I Liu Ph.D candidate, Department of Finance, National Taiwan University

2. Outline Introduction Long memory process Literature Review Long memory stochastic volatility model Modeling volatility of stock returns Confidence interval for VaR risk measure Conclusion 2

3. Introduction • Risk management for investment guarantee has become a critical topic in the insurance industry. • The regulator has required the actuary to use the stochastic asset liability modeling to measure the potential risk for equity-linked life insurance guarantee. • The asset model that can project future investment return shall be used. • The duration of life insurance is very long. Thus, the long-term nature of the asset model shall be considered.

4. Literature Review • Asset Models used in actuarial practice • Wilkie (1995) • Hardy (2003) • Regime switching lognormal model • Hardy, Freeland and Till (2007) • GARCH, ARCH, Stochastic log-volatility Model

5. Literature Review-Con’t • Risk Measure • Wirch and Hardy (1999) and Down and Blake(2006) • Dowd and Cairns (2005)

6. Purpose of this Research • To propose an asset model with LMSV for valuing long-term insurance policies. • Validation of LMSV using daily data on equity indexes. • Derive analytic solution VaR risk measure. • Derivate the Confidence Interval of VaR for Equity-linked Life Insurance with Maturity Guarantee. • Numerical illustration

7. An introduction tolong memory process

8. Long memory process • Long-range dependence in a stationary time series occurs if its autocovariance function can be represented as for 0<d<1/2. • The covariances of a long-memory process tend to zero like a power function and decay so slowly that their sums diverge. On the contrary, short-memory processes are usually characterized by rapidly decaying, summable covariances.

9. Autocorrelation Function Long-memory process Short-memory process

10. Fractional ARIMA process • It is a natural extension of the classic ARIMA modes and usually denoted as FARIMA (p,d,q). • Note that FARIMA has long-range dependence if and only if 0<d<1/2. • FARIMA (0,d,0)

11. Existence of long-memory phenomenon in asset volatility • Ding, Granger, and Engle (1993) • Autocorrelation function of the square or absolute-valued series of an high frequency asset return often decays at a slowly hyperbolic rate, even though the return series has no serial correlation.

12. Existence of long-memory phenomenon in asset volatility • Lobato and Savin (1998) • Lobato and Savin examine the S&P 500 index series for the period of July 1962 to December 1994 and find that strong evidence of persistent correlation exists in squared daily returns.

13. Existence of long-memory phenomenon in asset volatility • Except for index returns, the phenomenon of long memory in stochastic volatility is apparent to individual stock return (Ray and Tsay (2000)) and even stronger in some high-frequent data such as minute-by-minute stock returns (Ding and Granger (1996)) and foreign exchange rate returns (Bollerslev and Wright (2000)).

14. Long memory stochastic volatility model (LMSV) • Breidt, Crato and De Lima (1998) • The LMSV model is constructed by incorporating a FIRIMA process in a standard stochastic volatility scheme, which is defined by where • σ>0 • {Zt} is a FIRIMA process • {Zt} is independent of {ut} • {ut} is a sequence of independent and identical random variables with mean zero and variance one

15. LMSV • An appealing property of LMSV model is that it is easy to analyze after taking logarithm of the squared series where is a i.i.d. sequence. • It is evident that the autocovariances of log squared series are identical to those of FIRIMA process except at lag zero.

16. Empirical Evidence of Long Memory in Stock Volatility • Data • S&P500 daily log returns • TSX daily log returns • 1977/1~2006/12 • Estimation of long-memory parameter d • GPH estimator • Geweke and Porter-Hudak (1983)

17. Validation of LMSV model

18. IGARCH LMSV GARCH/EGARCH S&P500 1977/1~2006/12

19. IGARCH LMSV GARCH/EGARCH TSX 1977/1~2006/12

20. Validation of LMSV model • It is obviously that the ACF of fitted short-memory GARCH and EGARCH models decays rapidly but that of long-memory IGARCH model seems too persistent to model these data. • Only LMSV model is able to reproduce closely the empirical autocorrelation structure of the conditional volatilities and thus replicates the behaviors of index returns well.

21. Application for LMSV model Modeling the maturity guarantee liability under equity-linked fund contracts

22. Example • We use a single-premium equity-linkedinsurance policy with maturity guarantee to illustrate the calculation of VaR. • Policy setting • Single Premium=S0 • Payoffs at Maturity date =Max [ G, F(T)] • F(T) denotes the account value at Maturity date • F(T)=ST

23. Notations

24. Quantile Risk Measure • The quantile of liability distribution is found from • Explicit expression for : • Consistent estimates for : where

25. Confidence Interval for VaR Risk Measure (A) (B)

26. Numerical Example • G=100,S0=100 • management fees=0.022% per day • 30-year single-premium equity-linkedinsurance policy with maturity guarantee

27. Numerical Example • G=100,S0=100 • management fees=0.022% per day • 20-year single-premium equity-linkedinsurance policy with maturity guarantee

28. Numerical Example • G=100,S0=100 • management fees=0.022% per day • 10-year single-premium equity-linkedinsurance policy with maturity guarantee

29. Conclusions The numerical results show that the LMSV effect makes the VaR estimate more uncertain and results in a wider confidence interval. Therefore, when using VaR risk measure for risk management, ignoring the effect of long-memory in volatility may underestimate the variation of VaR estimate. 29