Hwai-Chung Ho Academia Sinica and National Taiwan University

1. A Stochastic Volatility Modelfor Reserving Long-Duration Equity-linked Insurance: Long-Memory v.s. Short-Memory Hwai-Chung Ho Academia Sinica and National Taiwan University December 30, 2008 (Joint work with Fang-I Liu and Sharon S.Yang )

2. Outline Introduction LMSV models Long-memory processes Taylor’s effect Long memory stochastic volatility models VaR of integrated returns Equity-linkedinsurance policy with maturity guarantee Confidence intervals for VaR estimates Numerical examples Conclusions 2

3. Introduction

4. Motivation • 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 models to measure the potential risk for equity-linked life insurance guarantee. • As the duration of life insurance designed is very long, the long-term nature of the asset model should be taken into account.

5. Literature Review Asset models used in actuarial practice: • Hardy (2003) • Regime switching lognormal model • Hardy, Freeland and Till (2006) • ARCH, GARCH, stochastic log-volatility model

6. Purpose of this Research • Propose an asset model with LMSV for valuing long-term insurance policies • Derive analytic solutions to VaR for long-term returns • Derive the confidence interval of VaR for equity-linked life insurance with maturity guarantee • Numerical illustration

7. LMSV Models Long-memory stochastic volatility models for asset returns

8. Long memory process • Long memory 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. • Synonyms: Long-range dependence , persistent memory, strong dependence, Hurst effect, 1/f phenomenon,

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

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

11. Taylor’s effect Autocorrelations of and are positive for many lags whereas the return series itself behaves almost like white noise.

12. ARCH • Engle (1982) • GARCH • Bollerslev (1986) • EGARCH • Nelson (1991)

13. Long-memory phenomenon in asset volatility • Ding, Granger, and Engle (1993) • Autocorrelation function of the squared or absolute-valued series of speculative returns often decays at a slowly hyperbolic rate, while the return series itself shows almost no serial correlation.

15. 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 both the squared and absolute-valued daily returns.

16. In addition to index returns, the phenomenon of long memory in stochastic volatility is also observed in • individual stock return • Ray and Tsay (2000) • minute-by-minute stock returns • Ding and Granger (1996) • foreign exchange rate returns • Bollerslev and Wright (2000)

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

18. Empirical Evidence of Long Memory in Stock Volatility • Data • S&P500 daily log returns • TSX daily log returns • 1977/1~2006/12 • Model fitting • GARCH (1,1),EGARCH (2,1) and IGARCH (1,1) • LMSV • Estimation of long-memory parameter d in FARIMA(0,d,0) • GPH estimator -Geweke and Porter-Hudak (1983)

19. ACF of return series S&P500 1977/1~2006/12 TSX 1977/1~2006/12

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

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

22. Validation of LMSV model • ACF of fitted short-memory GARCH and EGARCH models decays too rapidly and that of long-memory IGARCH model seems too persistent. Neither is suitable to model these data. • LMSV model is able to reproduce closely the empirical autocorrelation structure of the conditional volatilities and thus replicates the behaviors of index returns well.

23. VaR of Integrated Returns Considering the maturity guarantee liability under long-duration equity-linked fund contracts

24. Example Use a single-premium equity-linked insurance policy with guaranteed minimum Maturity benefits (GMMBs) to illustrate the calculation of VaR.

25. Notation

26. Policy setting • Single Premium P = S0 • Payoffs at Maturity date = Max [ G, F(T)] • F(T) = Account value at maturity date • F(T)=P．(ST ／S0 )．exp(-Tm) = ST exp(-Tm)

27. Quantile Risk Measure • The quantile of liability distribution is found from

28. A natural estimate for : where

29. Explicit expression for the true value of :

30. (A) Confidence Interval for VaR Risk Measure When return is long memory stochastic volatility process:

31. When return is short memory stochastic volatility process:

32. Numerical Examples(I) Simulation

33. (II)Real Data G=100,S0=100, management fees=0.022% per day 25-year single-premium equity-linked(S&P500 Jan. 1981- Dec. 2006) insurance policy with maturity guarantee 33

34. G=100,S0=100, management fees=0.022% per day • 30-year single-premium equity-linked(S&P500 Jan. 1977- Dec. 2006)insurance policy with maturity guarantee

35. Conclusions

36. 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. 36

37. THANK YOU!