Hwai-Chung Ho Academia Sinica and National Taiwan University

A Stochastic Volatility Model for 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 )

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Outline Introduction LMSV models

Long-memory processesTaylor’s effectLong memory stochastic volatility models

VaR of integrated returns Equity-linked insurance policy with

maturity guaranteeConfidence intervals for VaR estimatesNumerical examples

Conclusions

Introduction

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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.

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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

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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

LMSV Models

Long-memory stochastic volatility models for asset returns

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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,

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Long-memory process

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series 1

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series squared_return

Autocorrelation FunctionShort-memory process

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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)

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Taylor’s effect

Autocorrelations of and are

positive for many lags whereas the return series

itself behaves almost like white noise.

tr 2 2, lnt tr r

ARCH Engle (1982)

GARCH Bollerslev (1986)

EGARCH Nelson (1991)

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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.

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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.

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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)

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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.

, exp 2t t t t tr u Z

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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)

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ACF of return series

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series return

S&P500 1977/1~2006/12

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series return

TSX 1977/1~2006/12

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S&P500 1977/1~2006/12

0 50 100 150 200

0.0

0.1

0.2

0.3

0.4

0.5

ˆ 0.41d

IGARCH

LMSV

GARCH/EGARCH

2ACF of ln tr

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TSX 1977/1~2006/12

0 50 100 150 200

0.0

0.1

0.2

0.3

0.4

0.5

ˆ 0.44d

IGARCH

LMSV

GARCH/EGARCH

2ACF of ln tr

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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.

VaR of Integrated Returns

Considering the maturity guarantee liability under long-duration equity-linked fund contracts

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Example

Use a single-premium equity-linked

insurance policy with guaranteed minimum

Maturity benefits (GMMBs) to illustrate the

calculation of VaR.

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Notation

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)

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Quantile Risk Measure The quantile of liability distribution is found

from

1TmTP G S e V

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A natural estimate for :

where

V

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Explicit expression for the true value of :

V

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Confidence Interval for VaR Risk Measure

exp 2 (2)t t tr Z u

When return is long memory stochastic volatility process:

(A)

When return is short memory stochastic volatility process:

Numerical Examples

(I) Simulation

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(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

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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

Conclusions

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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.

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THANK YOU!