Some Simple Stochastic Models for Analyzing Investment Guarantees p. 1/36

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1 Some Simple Stochastic Models for Analyzing Investment Guarantees Wai-Sum Chan Department of Statistics & Actuarial Science The University of Hong Kong Some Simple Stochastic Models for Analyzing Investment Guarantees p. 1/36

2 Introduction Sharp decline in stock markets around the world over the past few years Performance of World Stock Markets, Country Market Index Jan 2000 Dec 2002 Change(%) Australia All Ordinaries Stock Index % Canada TSE 300 Stock Index % France CAC 40 Stock Index % Hong Kong Hang Seng Stock Index % Japan Nikkei 225 Stock Index % United Kingdom FTSE 100 Stock Index % United States S&P 500 Stock Index % Some Simple Stochastic Models for Analyzing Investment Guarantees p. 2/36

3 Introduction Many actuaries are now being asked to employ stochastic models to measure solvency risk created by insurance products with equity-linked guarantees Some Simple Stochastic Models for Analyzing Investment Guarantees p. 3/36

4 Introduction Many actuaries are now being asked to employ stochastic models to measure solvency risk created by insurance products with equity-linked guarantees The March 2002 final report of the CIA Task Force on Segregated Fund Investment Guarantees (TFSFIG) provides useful guidance for appointed actuaries applying stochastic techniques to value segregated fund guarantees in a Canadian GAAP valuation environment. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 3/36

5 Introduction In the United States, the Life Capital Adequacy Subcommittee (LCAS) of the American Academy of Actuaries issued the C-3 Phase II Risk-Based Capital (RBC) proposal in December Some Simple Stochastic Models for Analyzing Investment Guarantees p. 4/36

6 Introduction In the United States, the Life Capital Adequacy Subcommittee (LCAS) of the American Academy of Actuaries issued the C-3 Phase II Risk-Based Capital (RBC) proposal in December In addition to the interest rate risk for interest-sensitive products, the C-3 Phase II report also addresses the equity risk exposure inherent in variable products with guarantees, such as (1) guaranteed minimum death benefits (GMDBs); (2) guaranteed minimum income benefits (GMIBs); and (3) guaranteed minimum accumulation benefits (GMABs). Some Simple Stochastic Models for Analyzing Investment Guarantees p. 4/36

7 Introduction In the United States, the Life Capital Adequacy Subcommittee (LCAS) of the American Academy of Actuaries issued the C-3 Phase II Risk-Based Capital (RBC) proposal in December In addition to the interest rate risk for interest-sensitive products, the C-3 Phase II report also addresses the equity risk exposure inherent in variable products with guarantees, such as (1) guaranteed minimum death benefits (GMDBs); (2) guaranteed minimum income benefits (GMIBs); and (3) guaranteed minimum accumulation benefits (GMABs). Stochastic scenario analysis is recommended to determine minimum capital requirements for these variable products. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 4/36

8 Introduction In Hong Kong, the Office of the Commissioner of Insurance issued the GN7: Guidance Note 7 on Reserving Standards for Investment Guarantees Class G insurance policies 99% level of confidence scenario testing based on a stochastic model Some Simple Stochastic Models for Analyzing Investment Guarantees p. 5/36

9 Introduction In Hong Kong, the Office of the Commissioner of Insurance issued the GN7: Guidance Note 7 on Reserving Standards for Investment Guarantees Class G insurance policies 99% level of confidence scenario testing based on a stochastic model ASHK issued AGN8: Process for determining liabilities under the Guidance Note 7 Some Simple Stochastic Models for Analyzing Investment Guarantees p. 5/36

10 Introduction There are a large number of potential stochastic models for equity returns. Regulators normally do not restrict the use of any model that reasonably fits the historical data. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 6/36

11 Introduction There are a large number of potential stochastic models for equity returns. Regulators normally do not restrict the use of any model that reasonably fits the historical data. Reasonable = pass a calibration test Some Simple Stochastic Models for Analyzing Investment Guarantees p. 6/36

12 Introduction There are a large number of potential stochastic models for equity returns. Regulators normally do not restrict the use of any model that reasonably fits the historical data. Reasonable = pass a calibration test The emphasis of the calibration process remains at the tails of the equity return distribution (percentiles) over different holding periods (1-5- and 10-year periods). Some Simple Stochastic Models for Analyzing Investment Guarantees p. 6/36

13 Stochastic Models for Equity Returns In this talk, we shall provide a brief review of some commonly used (in actuarial practice) stochastic investment return models: Independent Log-Normal models (ILN) Regime-Switching Log-Normal models (RSLN) Others: Independent Log-Stable models (ILS) Box-Jenkins ARIMA models ARCH-type models Some Simple Stochastic Models for Analyzing Investment Guarantees p. 7/36

14 Log-Normal Models The log-normal model has a long and illustrious history, and has become the workhorse of the financial asset pricing literature (Campbell et al., 1997, p.16). Some Simple Stochastic Models for Analyzing Investment Guarantees p. 8/36

15 Log-Normal Models The log-normal model has a long and illustrious history, and has become the workhorse of the financial asset pricing literature (Campbell et al., 1997, p.16). Let Y t = log(p t ) log(p t 1 ) where P t is the end-of-period stock price (or market index), with dividends reinvested. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 8/36

16 Log-Normal Models The log-normal model has a long and illustrious history, and has become the workhorse of the financial asset pricing literature (Campbell et al., 1997, p.16). Let Y t = log(p t ) log(p t 1 ) where P t is the end-of-period stock price (or market index), with dividends reinvested. The log-normal model assumes that log returns Y t are independently and identically distributed (IID) normal with mean µ and variance σ 2. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 8/36

17 Log-Normal Models The independent lognormal (ILN) model is simple, scalable and tractable. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 9/36

18 Log-Normal Models The independent lognormal (ILN) model is simple, scalable and tractable. But as attractive as the lognormal model is, it is not consistent with all properties of historical equity returns. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 9/36

19 Log-Normal Models The independent lognormal (ILN) model is simple, scalable and tractable. But as attractive as the lognormal model is, it is not consistent with all properties of historical equity returns. At short horizons, observed returns often have negative skewness and strong evidence of excess kurtosis with time varying volatility. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 9/36

20 RSLN Models The RSLN (Regime-Switching Log-Normal) model is defined as Y t = µ St + σ St ε t where S t = 1, 2,..., k denotes the unobservable state indicator which follows an ergodic k-state Markov process and ε t is a standard normal random variable which is IID over time. P r{s t+1 = j S t = i} = p ij, 0 p ij 1, & k p ij = 1 for all i. j=1 Some Simple Stochastic Models for Analyzing Investment Guarantees p. 10/36

21 RSLN Models The RSLN (Regime-Switching Log-Normal) model is defined as Y t = µ St + σ St ε t where S t = 1, 2,..., k denotes the unobservable state indicator which follows an ergodic k-state Markov process and ε t is a standard normal random variable which is IID over time. The stochastic transition probabilities that determine the evolution in S t (so that the states follow a homogenous Markov chain) is given by P r{s t+1 = j S t = i} = p ij, 0 p ij 1, & k p ij = 1 for all i. j=1 Some Simple Stochastic Models for Analyzing Investment Guarantees p. 10/36

22 RSLN Models In most situations, k = 2 or 3 (that is two- or three-regime models) is sufficient for modelling monthly equity returns (Hardy, NAAJ, 2001) Some Simple Stochastic Models for Analyzing Investment Guarantees p. 11/36

23 RSLN Models In most situations, k = 2 or 3 (that is two- or three-regime models) is sufficient for modelling monthly equity returns (Hardy, NAAJ, 2001) The use of RSLN processes for modelling maturity guarantees has been gaining popularity Some Simple Stochastic Models for Analyzing Investment Guarantees p. 11/36

24 RSLN Models In most situations, k = 2 or 3 (that is two- or three-regime models) is sufficient for modelling monthly equity returns (Hardy, NAAJ, 2001) The use of RSLN processes for modelling maturity guarantees has been gaining popularity Both the Canadian Institute of Actuaries (CIA) and the American Academy of Actuaries (AAA) used a two-regime RSLN model for developing the calibration test Some Simple Stochastic Models for Analyzing Investment Guarantees p. 11/36

25 RSLN Models A simple two-regime RSLN model: Y t = ε (1) t N(µ 1, σ 2 1), ε (2) with transition probability matrix P = t N(µ 2, σ 2 2), p 11 p 12, 0 < p ij < 1. (1) p 21 p 22 This implies that the vector of steady-state (ergodic) probabilities is π 1 = π 2 p 21 p 12 +p 21 p 12 p 12 +p 21. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 12/36

26 Log-Stable Models Stable distributions are a class of probability laws that are able to accommodate heavy tails and skewness, which are frequently seen in investment data Some Simple Stochastic Models for Analyzing Investment Guarantees p. 13/36

27 Log-Stable Models Stable distributions are a class of probability laws that are able to accommodate heavy tails and skewness, which are frequently seen in investment data A random variable Y t is S0(α, β, γ, δ), if its characteristic function takes the form: Ψ(t) = E[exp(itY )] [ ( exp ( γ α t α 1 + iβsign(t) = ( [ ( exp γ t 1 + iβsign(t) 2 π tan πα 2 )( )] ) γt 1 α 1 + iδt, α 1, )( )] ) ln(γ t ) + iδt, α = 1. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 13/36

28 Log-Stable Models Stable distributions are a class of probability laws that are able to accommodate heavy tails and skewness, which are frequently seen in investment data A random variable Y t is S0(α, β, γ, δ), if its characteristic function takes the form: Ψ(t) = E[exp(itY )] [ ( exp ( γ α t α 1 + iβsign(t) = ( [ ( exp γ t 1 + iβsign(t) 2 π tan πα 2 )( )] ) γt 1 α 1 + iδt, α 1, )( )] ) ln(γ t ) + iδt, α = 1. It should be noted that Gaussian distributions are special cases of stable laws with α = 2 and β = 0; more precisely, N(µ, σ 2 ) = S0(2, 0, σ/ 2, µ). Some Simple Stochastic Models for Analyzing Investment Guarantees p. 13/36

29 ARMA Models ARMA (autoregressive moving average) or called Box-Jenkins models Some Simple Stochastic Models for Analyzing Investment Guarantees p. 14/36

30 ARMA Models ARMA (autoregressive moving average) or called Box-Jenkins models Wilkie s model (1995, BAJ); Chan s model (2002, BAJ); and Frees (1997, NAAJ) models are this types of models. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 14/36

31 ARMA Models ARMA (autoregressive moving average) or called Box-Jenkins models Wilkie s model (1995, BAJ); Chan s model (2002, BAJ); and Frees (1997, NAAJ) models are this types of models. An ARMA(p, q) model: φ(l)y t = θ(l)a t, where L is the backshift operator such that L s Y t = Y t s, φ(l) = 1 φ 1 L... φ p L p, θ(l) θ 1 L... θ q L q. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 14/36

32 ARCH Models Robert Engle, who proposed this class of models in 1982, won the 2003 Nobel Prize in Economic Sciences See the webpage Some Simple Stochastic Models for Analyzing Investment Guarantees p. 15/36

33 ARCH Models Robert Engle, who proposed this class of models in 1982, won the 2003 Nobel Prize in Economic Sciences See the webpage Autoregressive conditional heteroscedastic (ARCH) models Engle (1982): ARCH Bollerslev (1986): GARCH Some Simple Stochastic Models for Analyzing Investment Guarantees p. 15/36

34 ARCH Models Robert Engle, who proposed this class of models in 1982, won the 2003 Nobel Prize in Economic Sciences See the webpage Autoregressive conditional heteroscedastic (ARCH) models Engle (1982): ARCH Bollerslev (1986): GARCH Useful because these models are able to capture empirical regularities of asset returns such as thick tails of unconditional distributions, volatility clustering and negative correlation between lagged returns and conditional variance Some Simple Stochastic Models for Analyzing Investment Guarantees p. 15/36

35 ARCH Models Let a t = (Y t µ) be the mean-corrected log return. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 16/36

36 ARCH Models Let a t = (Y t µ) be the mean-corrected log return. In this talk, we consider a GARCH(1,1) process and its innovation following a Student t distribution: / ε t = a t ht, and ε t in the above equation has a marginal t distribution with mean zero, unit variance and degrees of freedom ν, and the conditional variance has the following representation h t = ω + β h t 1 + α a 2 t 1. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 16/36

37 Maximum Likelihood Estimation The final report of the CIA Task Force on Segregated Fund Investment Guarantees states, in its Appendix D, that the model should be fitted using maximum likelihood estimation (MLE) or a similar statistical procedure Some Simple Stochastic Models for Analyzing Investment Guarantees p. 17/36

38 Maximum Likelihood Estimation The MLE of the stochastic models discussed in this talk can be obtained by: Independent Log-Normal models (ILN) EXCEL Regime-Switching Log-Normal models (RSLN) EXCEL, freeware from SoA Independent Log-Stable models (ILS) VB, freeware from Nolan Box-Jenkins ARMA models SAS & Many others ARCH-type models GAUSS & Many others Some Simple Stochastic Models for Analyzing Investment Guarantees p. 18/36

39 Calibration Tests Even though there is no mandatory class of stochastic models for fitting the baseline data, it is recommended that the final model be calibrated to some specified tail distribution percentiles of the accumulation factors A m = P m P 0 Some Simple Stochastic Models for Analyzing Investment Guarantees p. 19/36

40 Calibration Tests Even though there is no mandatory class of stochastic models for fitting the baseline data, it is recommended that the final model be calibrated to some specified tail distribution percentiles of the accumulation factors A m = P m P 0 Three different holding periods: 1 year, 5 years and 10 years Some Simple Stochastic Models for Analyzing Investment Guarantees p. 19/36

41 Calibration Tests Even though there is no mandatory class of stochastic models for fitting the baseline data, it is recommended that the final model be calibrated to some specified tail distribution percentiles of the accumulation factors A m = P m P 0 Three different holding periods: 1 year, 5 years and 10 years The CIA report (TFSFIG, 2002) recommends a set of model calibration points based on TSE 300 monthly total return series (the benchmark series), from January 1957 to December Some Simple Stochastic Models for Analyzing Investment Guarantees p. 19/36

42 Calibration Tests In the United States, the AAA uses an approach similar to the CIA, where a regime-switching log-normal model is fitted to monthly total return data (Benchmark series: S&P 500 total returns), January 1945 to October 2002 Some Simple Stochastic Models for Analyzing Investment Guarantees p. 20/36

43 Calibration Tests In the United States, the AAA uses an approach similar to the CIA, where a regime-switching log-normal model is fitted to monthly total return data (Benchmark series: S&P 500 total returns), January 1945 to October 2002 Three different holding periods: 1 year, 5 years and 10 years Some Simple Stochastic Models for Analyzing Investment Guarantees p. 20/36

44 Calibration Tests In the United States, the AAA uses an approach similar to the CIA, where a regime-switching log-normal model is fitted to monthly total return data (Benchmark series: S&P 500 total returns), January 1945 to October 2002 Three different holding periods: 1 year, 5 years and 10 years A set of 30 accumulated wealth calibration points that must be materially met by the equity model used to determine capital requirements Some Simple Stochastic Models for Analyzing Investment Guarantees p. 20/36

45 AAA Calibration Points Some Simple Stochastic Models for Analyzing Investment Guarantees p. 21/36

46 CIA Calibration Points Note: Canadian Calibration Points are based on TSE 300 Total Return Accumulation Factors Some Simple Stochastic Models for Analyzing Investment Guarantees p. 22/36

47 The Hong Kong Market In the second part of this talk, we shall attempt to develop a set of calibration points for the Hong Kong market Some Simple Stochastic Models for Analyzing Investment Guarantees p. 23/36

48 The Hong Kong Market In the second part of this talk, we shall attempt to develop a set of calibration points for the Hong Kong market Hang Seng total return historical experience as a proxy for returns on a broadly diversified Hong Kong equity fund Some Simple Stochastic Models for Analyzing Investment Guarantees p. 23/36

49 The Hong Kong Market In the second part of this talk, we shall attempt to develop a set of calibration points for the Hong Kong market Hang Seng total return historical experience as a proxy for returns on a broadly diversified Hong Kong equity fund Hence, monthly Hang Seng Index (HSI) total return series, from June 1973 to September 2003 (n = 363), is chosen as the benchmark series in this study Some Simple Stochastic Models for Analyzing Investment Guarantees p. 23/36

50 HSI Total Return Total Return Volatility Total Return Volatility Year Some Simple Stochastic Models for Analyzing Investment Guarantees p. 24/36

51 Fitted Models Maximum Likelihood Estimation (MLE) Some Simple Stochastic Models for Analyzing Investment Guarantees p. 25/36

52 Fitted Models Maximum Likelihood Estimation (MLE) ILN µ σ Some Simple Stochastic Models for Analyzing Investment Guarantees p. 25/36

53 Fitted Models Maximum Likelihood Estimation (MLE) ILN RSLN µ σ Regime 1 Regime 2 µ 1 = µ 2 = σ 1 = σ 2 = p 11 = p 22 = p 12 = p 21 = π 1 = 74% π 2 = 26% Some Simple Stochastic Models for Analyzing Investment Guarantees p. 25/36

54 Fitted Models Other models Some Simple Stochastic Models for Analyzing Investment Guarantees p. 26/36

55 Fitted Models Other models ILS α β γ δ Some Simple Stochastic Models for Analyzing Investment Guarantees p. 26/36

56 Fitted Models Other models ILS α β γ δ GARCH(1,1) µ ω α β ν 3.96 Some Simple Stochastic Models for Analyzing Investment Guarantees p. 26/36

57 The Final Model Both the CIA and AAA employed a two-regime RSLN model for their baseline series Some Simple Stochastic Models for Analyzing Investment Guarantees p. 27/36

58 The Final Model Both the CIA and AAA employed a two-regime RSLN model for their baseline series The HSI total returns are reasonably fiited by a two-regime RSLN process Some Simple Stochastic Models for Analyzing Investment Guarantees p. 27/36

59 The Final Model Both the CIA and AAA employed a two-regime RSLN model for their baseline series The HSI total returns are reasonably fiited by a two-regime RSLN process Interpretation of the RSLN2 model: Regime 1 Usual, Stable Normal-Volatility State Regime 2 Occasional, Unstable High-Volatility State Some Simple Stochastic Models for Analyzing Investment Guarantees p. 27/36

60 The Final Model Density Function for Monthly Return, Hang Seng Index f(x) Logarithm of Monthly Total Returns Some Simple Stochastic Models for Analyzing Investment Guarantees p. 28/36

61 HK Calibration Points Calibration points can be developed based on the final fitted RSLN model Calibration Points Quantile (%) 1-Year 5-Year 10-Year Some Simple Stochastic Models for Analyzing Investment Guarantees p. 29/36

62 CTE AAA C-3 Phase I 95th percentile standard Some Simple Stochastic Models for Analyzing Investment Guarantees p. 30/36

63 CTE AAA C-3 Phase I 95th percentile standard AAA C-3 Phase II proposed RBC requirment would be based on a modified Conditional Tail Expectation (CTE) measure. The AAA report can be found at: Some Simple Stochastic Models for Analyzing Investment Guarantees p. 30/36

64 CTE AAA C-3 Phase I 95th percentile standard AAA C-3 Phase II proposed RBC requirment would be based on a modified Conditional Tail Expectation (CTE) measure. The AAA report can be found at: The AAA proposed standard would be based on the average required surplus for the worst 10% of the scenarios (CTE 90) Some Simple Stochastic Models for Analyzing Investment Guarantees p. 30/36

65 CTE AAA C-3 Phase I 95th percentile standard AAA C-3 Phase II proposed RBC requirment would be based on a modified Conditional Tail Expectation (CTE) measure. The AAA report can be found at: The AAA proposed standard would be based on the average required surplus for the worst 10% of the scenarios (CTE 90) What is CTE? How to compute CTE for RSLN generated scenarios? Some Simple Stochastic Models for Analyzing Investment Guarantees p. 30/36

66 CTE f(x) 3 CTE Logarithm of Monthly Total Returns Some Simple Stochastic Models for Analyzing Investment Guarantees p. 31/36

67 Applications Risk Measures for maturity guarantee liability under equity-linked contracts: ξ the probability that the guarantee will not be exercised Q α quantile risk measure (value-at-risk) of the liability CTE(α) = E[X X > Q α ] Some Simple Stochastic Models for Analyzing Investment Guarantees p. 32/36

68 Applications Risk Measures for maturity guarantee liability under equity-linked contracts: ξ the probability that the guarantee will not be exercised Q α quantile risk measure (value-at-risk) of the liability CTE(α) = E[X X > Q α ] Given a stochastic equity model and details of the contracts, these measures can be computed analytically or by simulations. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 32/36

69 An Example We adopt a simple example from Hardy (2001, NAAJ): A segregated fund contract that matures in 10 years; the guarantee is equal to the fund market value at the start of the projection. Management fees of 0.25% per month, compounded continuously, are deducted. Lapses and deaths are ignored for simplicity. Some Simple Stochastic Models for Analyzing Investment Guarantees p. 33/36

70 An Example ξ, Q α and CTE(α) are computed using fitted RSLN models for the U.S. (S&P500), Canadian (TSE300) and Hong Kong (HSI) markets. Markets HSI TSE300 S&P500 ξ Q Q Q CTE(0.900) CTE(0.950) CTE(0.975) Note: Q α and CTE(α) are expressed as percentage of the fund value Some Simple Stochastic Models for Analyzing Investment Guarantees p. 34/36

71 A Scenario Generator Some Simple Stochastic Models for Analyzing Investment Guarantees p. 35/36

72 End of Presentation Thank You! Some Simple Stochastic Models for Analyzing Investment Guarantees p. 36/36

Temporal Aggregation of Equity Return Time-Series Models

Temporal Aggregation of Equity Return Time-Series Models 1 Chan, W.S., 2 L.X. Zhang, and 3 S.H. Cheung 1 Department of Finance, The Chinese University of Hong Kong, Hong Kong, PR China. E-Mail: waisum@baf.msmail.cuhk.edu.hk