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Reserving for Investment Guarantees The Stochastic Modelling Approach Wai-Sum Chan, Ph.D., F.S.A., C.Stat. ASHK 6th Appointed Actuaries Symposium November 15, 2006 Reserving for Investment Guarantees p. 1/47 What is Stochastic Modelling?
Wai-Sum Chan, Ph.D., F.S.A., C.Stat. ASHK 6th Appointed Actuaries Symposium November 15, 2006
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SFs
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SFs
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SFs
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Concept of Arbitrage-Free — It is often referred to as no free lunch (NFL). NFL requires that on the basis of publicly available information, no investor can make a positive riskless return with no new investment.
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Concept of Arbitrage-Free — It is often referred to as no free lunch (NFL). NFL requires that on the basis of publicly available information, no investor can make a positive riskless return with no new investment. The important point to understand here is how the martingale property of a stochastic process relates to the idea of the absence of arbitrage.
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Informally, the Martingale Representation Theorem (an important theorem in Statistics) says that in equilibrium prices represented as the present (discounted) value
martingale under a given probability measure Q. This Q-measure is often called the risk neutral measure.
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Informally, the Martingale Representation Theorem (an important theorem in Statistics) says that in equilibrium prices represented as the present (discounted) value
martingale under a given probability measure Q. This Q-measure is often called the risk neutral measure. Why do I care about MRT? If our chosen stochastic models have the above characteristic, they are “guaranteed” arbitrage-free (or called no-arbitrage) models.
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Performance of World Stock Markets, 2000-2002 Country Market Index Jan 2000 Dec 2002 Change(%) Australia All Ordinaries Stock Index 3096 2976
Canada TSE 300 Stock Index 8481 6615
France CAC 40 Stock Index 5660 3064
Hong Kong Hang Seng Stock Index 15532 9321
Japan Nikkei 225 Stock Index 19540 8579
United Kingdom FTSE 100 Stock Index 6269 3940
United States S&P 500 Stock Index 1394 880
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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) report in June 2005.
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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) report in June 2005. 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).
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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) report in June 2005. 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).
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Reserving Standards for Investment Guarantees
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Reserving Standards for Investment Guarantees
liabilities under the Guidance Note 7
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⋆ Box-Jenkins ARIMA models ⋆ ARCH-type models
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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).
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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 Yt = log(Pt) − log(Pt−1) where Pt is the end-of-period stock price (or market index), with dividends reinvested.
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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 Yt = log(Pt) − log(Pt−1) where Pt is the end-of-period stock price (or market index), with dividends reinvested. The log-normal model assumes that log returns Yt are independently and identically distributed (IID) normal with mean µ and variance σ2.
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The independent lognormal (ILN) model is simple, scalable and tractable.
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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.
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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.
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The RSLN (Regime-Switching Log-Normal) model is defined as Yt = µSt + σSt εt where St = 1, 2, . . . , k denotes the unobservable state indicator which follows an ergodic k-state Markov process and εt is a standard normal random variable which is IID over time.
Pr{St+1 = j|St = i} = pij, 0 ≤ pij ≤ 1, &
k
pij = 1 for all i.
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The RSLN (Regime-Switching Log-Normal) model is defined as Yt = µSt + σSt εt where St = 1, 2, . . . , k denotes the unobservable state indicator which follows an ergodic k-state Markov process and εt is a standard normal random variable which is IID over time. The stochastic transition probabilities that determine the evolution in St (so that the states follow a homogenous Markov chain) is given by
Pr{St+1 = j|St = i} = pij, 0 ≤ pij ≤ 1, &
k
pij = 1 for all i.
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In most situations, k = 2 or 3 (that is two- or three-regime models) is sufficient for modelling monthly equity returns (Hardy, NAAJ, 2001)
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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
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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 The Canadian Institute of Actuaries (CIA) used a two-regime RSLN model for developing the calibration test.
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A simple two-regime RSLN model: Yt = ε(1)
t
∼ N(µ1, σ2
1),
ε(2)
t
∼ N(µ2, σ2
2),
with transition probability matrix P = p11 p12 p21 p22 , 0 < pij < 1.
(1)
This implies that the vector of steady-state (ergodic) probabilities is π1 π2 =
p21 p12+p21 p12 p12+p21
.
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The American Academic of Actuaries (AAA, 2005) in its C3 Phase II document proposes the SLVM model and used it for developing the US calibration test.
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The American Academic of Actuaries (AAA, 2005) in its C3 Phase II document proposes the SLVM model and used it for developing the US calibration test. Yt = µt/12 + (σt/ √ 12)zy,t, where µt = A + Bσt + Cσ2
t ,
ln(σt) = νt = (1 − φ)νt−1 + φ ln(τ) + σνzν,t where (zy,t, zν,t) have a standard bivariate normal distribution with fixed correlation ρ.
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The American Academic of Actuaries (AAA, 2005) in its C3 Phase II document proposes the SLVM model and used it for developing the US calibration test. Yt = µt/12 + (σt/ √ 12)zy,t, where µt = A + Bσt + Cσ2
t ,
ln(σt) = νt = (1 − φ)νt−1 + φ ln(τ) + σνzν,t where (zy,t, zν,t) have a standard bivariate normal distribution with fixed correlation ρ. AAA provided some prerecorded scenarios for “plug-and-play” use by actuaries.
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ARMA (autoregressive moving average) or called Box-Jenkins models
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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.
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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)Yt = θ(L)at, where L is the backshift operator such that LsYt = Yt−s, φ(L) = 1 − φ1L − . . . − φpLp, θ(L) − θ1L − . . . − θqLq.
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Robert Engle, who proposed this class of models in 1982, won the 2003 Nobel Prize in Economic Sciences
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Robert Engle, who proposed this class of models in 1982, won the 2003 Nobel Prize in Economic Sciences Autoregressive conditional heteroscedastic (ARCH) models
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Robert Engle, who proposed this class of models in 1982, won the 2003 Nobel Prize in Economic Sciences Autoregressive conditional heteroscedastic (ARCH) models
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
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Let at = (Yt − µ) be the mean-corrected log return.
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Let at = (Yt − 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: εt = at
and εt in the above equation has a marginal t distribution with mean zero, unit variance and degrees
following representation ht = ω + β ht−1 + α a2
t−1.
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be fitted using maximum likelihood estimation (MLE) or a similar statistical procedure
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The MLE of the stochastic models discussed in this talk can be obtained by:
– EXCEL
– EXCEL, freeware from SoA
– MATLAB, Many others
– SAS & Many others
– GAUSS, Many others
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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 Am = Pm P0
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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 Am = Pm P0 Three different holding periods: 1 year, 5 years and 10 years
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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 Am = Pm P0 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
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In the United States, the AAA uses an approach similar to the CIA, but SLVM, instead of RSLN model, is fitted to monthly total return data (Benchmark series: S&P 500 total returns), January 1945 to October 2002
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In the United States, the AAA uses an approach similar to the CIA, but SLVM, instead of RSLN model, is fitted to monthly total return data (Benchmark series: S&P 500 total returns), January 1945 to October 2002 Four different holding periods: 1 year, 5,10 and 20 years
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In the United States, the AAA uses an approach similar to the CIA, but SLVM, instead of RSLN model, is fitted to monthly total return data (Benchmark series: S&P 500 total returns), January 1945 to October 2002 Four different holding periods: 1 year, 5,10 and 20 years A set of 22 accumulated wealth calibration points that must be materially met by the equity model used to determine capital requirements
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Calibration Standard for Total Return Wealth Ratios Percentile 1 Year 5 Years 10 Years 20 Years 2.5% 0.78 0.72 0.79 n/a 5.0% 0.84 0.81 0.94 1.51 10.0% 0.90 0.94 1.16 2.10 90.0% 1.28 2.17 3.63 9.02 95.0 % 1.35 2.45 4.36 11.70 97.5% 1.42 2.72 5.12 n/a
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Note: Canadian Calibration Points are based on TSE 300 Total Return Accumulation Factors
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No official calibration points have been set for the Hong Kong market
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No official calibration points have been set for the Hong Kong market Hang Seng total return historical experience as a proxy for returns on a broadly diversified Hong Kong equity fund
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No official calibration points have been set for the Hong Kong market Hang Seng total return historical experience as a proxy for returns on a broadly diversified Hong Kong equity fund But Heng Seng total return data are only officially available from HSI Services Ltd starting from 1991.
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No official calibration points have been set for the Hong Kong market Hang Seng total return historical experience as a proxy for returns on a broadly diversified Hong Kong equity fund But Heng Seng total return data are only officially available from HSI Services Ltd starting from 1991. Hence, we look at monthly Hang Seng Index (HSI) with dividend reinvested series, from June 1973 to September 2003 (n = 363), as an example. The data were downloaded from the DataStream Database (CUHK).
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0.0 0.1 0.2 0.3 0.4 0.5 0.6 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 Year Total Return
0.00 0.05 0.10 0.15 0.20 0.25 Volatility Total Return Volatility
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AAA C-3 Phase I — 95th percentile standard
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AAA C-3 Phase I — 95th percentile standard AAA C-3 Phase II — RBC requirment would be based
www.actuary.org/pdf/life/phase2.asp
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AAA C-3 Phase I — 95th percentile standard AAA C-3 Phase II — RBC requirment would be based
www.actuary.org/pdf/life/phase2.asp
The AAA proposed standard would be based on the average required surplus for the worst α (%) of the scenarios (i.e., CTE 100 − α(%)). For examples, CTE(90) for TAR; and CTE(75) VA reserve.
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AAA C-3 Phase I — 95th percentile standard AAA C-3 Phase II — RBC requirment would be based
www.actuary.org/pdf/life/phase2.asp
The AAA proposed standard would be based on the average required surplus for the worst α (%) of the scenarios (i.e., CTE 100 − α(%)). For examples, CTE(90) for TAR; and CTE(75) VA reserve. What is CTE? How to compute CTE for stochastic model generated scenarios?
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1 2 3 4 5 6
0.1 0.2 0.3 Logarithm of Monthly Total Returns f(x)
CTE
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The P-measure type of models for equity return discussed in Part I of this talk can also be used for fitting interest rate variables.
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The P-measure type of models for equity return discussed in Part I of this talk can also be used for fitting interest rate variables. However, market practioners often prefer no-arbitrage types of interest-rate models. Because there are huge volume of trading bonds, bond options and bond futures of different maturities
want to price these contracts in a way which is consistent with the closest-matching traded derivatives.
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The P-measure type of models for equity return discussed in Part I of this talk can also be used for fitting interest rate variables. However, market practioners often prefer no-arbitrage types of interest-rate models. Because there are huge volume of trading bonds, bond options and bond futures of different maturities
want to price these contracts in a way which is consistent with the closest-matching traded derivatives. There are many many classes of stochastic interest-rate models. A good reference book for actuaries is: Interest Rate Models: An
Introduction by Andrew J.G. Cairns, PhD, FIA, FFA, Princeton
University Press (2004).
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The short rate, usually written rt is the (annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t. Specifying the current short rate does not specify the entire yield curve. However no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of rt as a stochastic process under a risk-neutral measure Q then it characterises the whole term structure.
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The short rate, usually written rt is the (annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t. Specifying the current short rate does not specify the entire yield curve. However no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of rt as a stochastic process under a risk-neutral measure Q then it characterises the whole term structure. There are many short rate models, for examples:
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The model, proposed by Ho and Lee (1986), is the first no-arbitrage interest rate model. The continuous-time form of the model is given by dr(t) = µ(t)dt + σ(t)dz(t) where r(t) is the short rate at time t, µ(t) is the drift term, σ(t) is the instantaneous standard deviation (volatility) of the short rate and dz(t) is a Brownian Motion (or called Winener Process).
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The model, proposed by Ho and Lee (1986), is the first no-arbitrage interest rate model. The continuous-time form of the model is given by dr(t) = µ(t)dt + σ(t)dz(t) where r(t) is the short rate at time t, µ(t) is the drift term, σ(t) is the instantaneous standard deviation (volatility) of the short rate and dz(t) is a Brownian Motion (or called Winener Process). In practice, we have to transform the model into discrete form: ∆r(t) = µ(t)∆t + σ(t)∆z(t)
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Details of practical implementation of HL model can be found in many textbooks, it is very simple to program.
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Details of practical implementation of HL model can be found in many textbooks, it is very simple to program. Shortcomings: not mean reversion; could produce negative interest rates.
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The model, proposed by Hull and White (1986), is a no-arbitrage interest rate model with mean reversion. The continuous-time form of the model is given by dr(t) = α θ(t) α − r(t)
where r(t) is the short rate at time t, µ(t) is the drift term, σ(t) is the instantaneous standard deviation (volatility) of the short rate and dz(t) is a Brownian Motion (or called Winener Process).
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The model, proposed by Hull and White (1986), is a no-arbitrage interest rate model with mean reversion. The continuous-time form of the model is given by dr(t) = α θ(t) α − r(t)
where r(t) is the short rate at time t, µ(t) is the drift term, σ(t) is the instantaneous standard deviation (volatility) of the short rate and dz(t) is a Brownian Motion (or called Winener Process). Under the HW model, the short rate at time t reverts to the mean θ(t)/α.
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Ho-Lee Model Hull-White Model Inputs Term Structure of Interest rates Term Structure of Interest rates Term Structure of Volatilites Term Structure of Volatilites Rate of Mean Reversion, α Mean reversion No Yes Distribution of r(t) Gaussian Gaussian
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An ASHK Interest Rate Committee Project.
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An ASHK Interest Rate Committee Project. EFBNs’ transcation data downloaded from the Hong Kong Monetary Authority’s Web
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An ASHK Interest Rate Committee Project. EFBNs’ transcation data downloaded from the Hong Kong Monetary Authority’s Web Bootstrapping method and spline smooth
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An ASHK Interest Rate Committee Project. EFBNs’ transcation data downloaded from the Hong Kong Monetary Authority’s Web Bootstrapping method and spline smooth The following graph shows the Hong Kong weekly Term Structure of Interest Rates
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Take care of correlations among different asset classes.
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Take care of correlations among different asset classes. Classical Wilkie’s (1995, BAJ) Model (inflation rates, share returns, long-term interest rates) for UK data
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Take care of correlations among different asset classes. Classical Wilkie’s (1995, BAJ) Model (inflation rates, share returns, long-term interest rates) for UK data Advanced Chan’s (2002, BAJ) mutiple model for the UK data
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Take care of correlations among different asset classes. Classical Wilkie’s (1995, BAJ) Model (inflation rates, share returns, long-term interest rates) for UK data Advanced Chan’s (2002, BAJ) mutiple model for the UK data Classical Frees’ (1997, NAAJ) model for the US data
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Take care of correlations among different asset classes. Classical Wilkie’s (1995, BAJ) Model (inflation rates, share returns, long-term interest rates) for UK data Advanced Chan’s (2002, BAJ) mutiple model for the UK data Classical Frees’ (1997, NAAJ) model for the US data Advanced Chan’s (2004, NAAJ) non-linear threshold-type models
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Returns,” North American Actuarial Journal, 2005, 9(4), 83-94.
for Actuarial Use,” North American Actuarial Journal, 2004, 8 (4), 37-61.
Approach,” British Actuarial Journal, 2002, Vol. 8, 545-591.
Actuarial Journal, 1998, 637-652.
Series Approach,” Australian Actuarial Journal, 1998, Vol.2, 127-141.
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