Forecasting Volatility of Hang Seng Index and its Application on Reserving for Investment Guarantees. Herbert Tak-wah Chan Derrick Wing-hong Fung

Forecasting Volatility of Hang Seng Index and its Application on Reserving for Investment Guarantees Herbert Tak-wah Chan Derrick Wing-hong Fung

This presentation represents the view of the presenters and does not represent our employer.

Objective This presentation aims to explain the 2006 ASHK Annual Best Paper Forecasting Volatility of Hang Seng Index and its Application on Reserving for Investment Guarantees jointly written by Herbert Chan and Derrick Fung, which can be downloaded at http://www.actuaries.org.hk/doc/paper06_herbertchan&derrickfung.pdf

The paper The paper can be generally divided into 2 parts (1) Forecast Hang Seng Index (HSI) daily volatility based on historical HSI data and validation of these forecasted daily volatilities (2) Discuss the application of forecasted volatility on reserving for investment guarantees

Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees

HSI Data Daily return of HSI price Historical volatility Realized volatility Implied volatility

Daily Return We define daily return R n at day n as logarithmic return: R n = 100 (lnp n -lnp n-1 ) where P n is the closing price of HSI at day n Why not R n = 100 (P n /P n-1-1)?

Daily Return lnp n -lnp n-1 = ln(p n /P n-1 ) = ln(1+r n ) = R n R n2 /2 + R n3 /3 R n4 /4 + (Taylor series) ~ R n for small values of R n

Daily Return Logarithmic return is commonly used in financial literature because of its additive nature Cumulative Return at day n = 100 (lnp n lnp 0 ) = R n + R n-1 + + R 1

Historical Volatility Historical volatility at day n is defined as the variance of daily returns in the preceding 30 transaction days

Historical Volatility Historical volatility at day n: 1 29 n m = n 29 ( R m R where R is the mean of return in the preceding 30 transaction days A total of 1235 historical volatilities are calculated 15 February 1999 to 20 April 2004 2 )

Realized Volatility Realized Volatility is the variance of 5-minute returns within a day Record 5-minute returns of HSI R n,d = 100 (lnp n,d lnp n,d-1 ) where P n,d is the asset price at trading day n, at the 5-minute mark d.

Realized Volatility Realized volatility at day n is defined as: where day n ~ σ R n 2 n 54 1 = 52 d = 2 ( ) 2 R R 55 n, d n is the mean of 5-minute return at

Realized Volatility A total of 1235 realized volatilities are calculated 15 February 1999 to 20 April 2004

Implied Volatility Using Black-Scholes model, implied volatility is calculated from options whose underlying asset is HSI It is calculated on a daily basis and is obtained from Hong Kong Exchange and Clearing Limited 1235 implied volatilities are obtained 15 February 1999 to 20 April 2004

HSI Data 1235 historical volatilities, realized volatilities and implied volatilities are obtained from 15 February 1999 to 20 April 2004

GARCH model Financial studies show that stable periods and volatile periods tend to be protracted, resulting in clusters GARCH model can capture these volatility clusters

GARCH model Generalized autoregressive conditional heteroskedasticity (GARCH) model was developed by Bollerslev (1986) R σ n 2 n = μ σ ++ = ω n α ε R n 2 n 1 + βσ 2 n 1 ε n ~ NID(0,1)

GARCH model GARCH model with historical volatility, realized volatility or implied volatility ~ NID(0,1) ε n 2 1 2 1 2 1 2 + + + = + = n n n n n n n s R R γ βσ α ω σ ε σ μ

Estimation Simple GARCH Observations:1235 Standard Approx Parameter DF Estimate Error t Value Pr > t μ ω α β 1 0.0425 0.0440 0.97 0.3335 1 0.0353 0.0146 2.43 0.0153 1 0.0422 0.009499 4.44 <.0001 1 0.9460 0.0124 76.35 <.0001

Fitting When HSI data are fitted into the GARCH model, empirical results show satisfactory model fitting performance.

Forecasting HSI Volatilities After fitting HSI data from 1999 to 2004, one day ahead volatilities are forecasted by the GARCH model.

Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 1235 data

Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 1 to 735 736

Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 2 to 736 737

Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 500 to 1234 1235

Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 736 to 1235

Forecasted Volatility Validation Compare actual volatility with forecasted volatility from day 736 to day 1235? However, actual volatility is not observable!

Value at Risk approach Since volatilities are not observable, we use the value at risk approach to validate the accuracy of forecasted volatilities In the GARCH model, return is assumed to be normally distributed with mean and variance 2 σ n μ n

Value at Risk approach μ 1. 645 n σ n μn

Value at Risk approach Now we have 500 forecasted daily volatilities. If we assume forecasted daily volatility and estimated mean at day n are accurate there should be around 25, i.e. 5% of 500, of them falling into the colored area In other words, if we observe around 25 falling into the colored area, we can conclude that those 500 forecasted daily volatilities and estimated means are accurate.

Value at Risk approach The new problem is: are we safe to conclude that forecasted daily volatilities and estimated means are accurate if we observe 24 falling into the colored area? What if 23, 26 or 27 falling into the colored area? What is our tolerance limit?

Binomial Test H 0 :forecasted daily volatility and estimated mean are accurate H 1 :forecasted daily volatility and estimated mean are not accurate Rejection Rule: With n=500, p=0.05, np(1-p)=23.75, α =5%, we reject H 0 if test statistic< np 1.96 np(1 p) =15.4 Or test statistic> np + 1.96 np (1 p) =34.6 Conclusion: If the number of returns falling into the colored area is between 16 and 34, we fail to reject H 0 at 5% significance level

Binomial Test Results GARCH Simple With historical volatility With realized volatility With implied volatility No. of returns falling into the colored area 16 17 18 18 p-value 0.0648 0.1007 0.1509 0.1509

Validation Result Based on value at risk approach and results of binomial test, it is found that the forecasted volatilities from the GARCH model are accurate.

Application of Forecasted Volatilities Reserving for investment guarantees Asset management Option pricing Calculation of VaR for asset portfolios

Reserving for investment guarantees For illustrative purposes, the remaining slides demonstrate a very simple simulation method of reserving for investment guarantees Conditional tail expectation (CTE), lapse rates, management expenses, mortality, withdrawal, future contributions, interest rate, etc., are not discussed in the simple simulation method

Reserving for investment guarantees Actuaries shall pay due regard to GN7 and AGN8 issued by the OCI and ASHK respectively for reserving principles, modeling process, calibration standard and modeling constraints

HSI price at worst case scenario Now we know the daily return follows properties in GARCH model and forecasted volatilities are accurate, we can simulate the most adverse HSI price by using random generator.

We now forecast most adverse HSI price 100 days later (1) 100 random numbers between (0,1) are generated by a random generator and considered to be F(x) of returns from day 1 to 100 (2) Mean of return μ n for day 1 to 100 is assumed to be the same and estimated by the GARCH model by fitting historical data into the model (3) Daily volatility for day 1 is forecasted by the GARCH model by fitting historical data into the model

We now forecast most adverse HSI price 100 days later (4) As we have the cumulative distribution function, estimated mean and forecasted daily volatility for day 1, we can determine daily return for day 1, hence the HSI price at day 1 (5) Daily volatility for day 2 is forecasted by the GARCH model by fitting historical data and simulated data for day 1 into the model (6) HSI price at day 2 is simulated (7) Similarly, HSI price at day 100 is simulated

We now forecast most adverse HSI price 100 days later (8) Repeat step 1 to 7 for 2000 times, we get 2000 simulated HSI price at day 100 (9) The most adverse HSI price at day 100 with 99% level of confidence is the 21 st simulated price in ascending order among those 2000 simulated prices

Reserving for Investment Guarantees A simple provision (ignoring mortality, lapse, expense, interest rate, etc.) for investment guarantees wholly invested in HSI is: Reserve = Guaranteed benefit most adverse HSI price at 99% confidence interval

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