損失分配法下作業風險值快速蒙地卡羅法的設計

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科技部補助專題研究計畫成果報告

期末報告

損失分配法下作業風險值快速蒙地卡羅法的設計

計畫類別：個別型計畫計畫編號： MOST 102-2410-H-004-062- 執行期間： 102 年 08 月 01 日至 104 年 01 月 31 日執行單位：國立政治大學風險管理與保險學系計畫主持人：謝明華計畫參與人員：碩士班研究生-兼任助理人員：鄭韶昉碩士班研究生-兼任助理人員：陳怡如碩士班研究生-兼任助理人員：陳安柔碩士班研究生-兼任助理人員：邱齡儀碩士班研究生-兼任助理人員：陳思雅報告附件：出席國際會議研究心得報告及發表論文處理方式： 1.公開資訊：本計畫可公開查詢 2.「本研究」是否已有嚴重損及公共利益之發現：否 3.「本報告」是否建議提供政府單位施政參考：否

中華民國 104 年 04 月 27 日

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中文摘要：近年來，作業風險的量化已經成為金融機構監理的一個重要議題。例如，保險監理的 Solvency II 與銀行監理的巴塞爾協定都要求保險公司與銀行需要計提作業風險資本。在巴塞爾協定的進階測量方法 (Advanced Measurement

Approaches) 下，金融機構有自由去選擇使用的隨機模型。損失分配法 (Loss distribution approach) 是一個符合這個目的的標準隨機模型。在損失分配法下，事業單位與損失形態的組合組成一個矩陣；而矩陣中的每一個元素有自己的損失分配。這些損失分配的相關性通常是透過 copulas 來做連結。金融監理上對作業風險資本計提的需求, 通常是需要金融機構計算一年內，在九十九點九的信賴度下，作業風險可能帶來的最大損失。在這樣的高標準要求下，傳統的蒙地卡羅法無法提供一個準確的估計值。因此，本計畫的主要目的是設計一個有效率的蒙地卡羅演算法，以達成快速且正確計算作業風險值的目標。中文關鍵詞：作業風險, 金融機構監理, 進階測量方法, 損失分配法, 蒙地卡羅法

英文摘要： In recent years, quantification of operational risk becomes an important issue for regulation in

financial industry. For example, Solvency II for insurers and Basel Accord for banks are required insurance companies and banks to allocate capital for operation risk. Loss distribution approach (LDA), also known as Actuarial Model, is a standard and popular method of Advanced Measurement Approaches (AMA). LDA concerns the measurement of risk for random losses generated from a huge m by d matrix whose element corresponds to a complex combination of business line and event type. Two major challenges of LDA are data collection and usage, and computation issue of getting an accurate risk measure for

operational risk. Chapelle et al. (2008), Dahen and Dionne (2010), and many other research discuss data collection and usage, but much less attention has been paid to the computational efficiency issue. The risk measure used for regulatory capital purposes reflects a holding period of one-year and a

confidence level of 99.9% (The Basel Accord). It is almost infeasible to get an accurate estimate of such

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(Asmussen and Glynn 2007). Therefore, in this project, we propose an efficient Monte Carlo

simulation algorithm for computing such risk measure. 英文關鍵詞： operational risk, advanced measurement approaches,

loss distribution approach, Monte Carlo simulation, variance reduction

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An efficient Monte Carlo method

for estimation of operational VaR and ES

under loss distribution approach

ABSTRACT: In recent years, quantification of operational risk becomes an important issue for regulation in financial industry. For example, Solvency II for insurers and Basel Accord for banks are required insurance companies and banks to allocate capital for operation risk. Loss distribution approach (LDA), also known as Actuarial Model, is a standard and popular method of Advanced Measurement Approaches (AMA). LDA concerns the measurement of risk for random losses generated from a huge m by d matrix whose element corresponds to a complex combination of business line and event type. Two major challenges of LDA are data collection and usage, and

computation issue of getting an accurate risk measure for operational risk. Chapelle et al. (2008), Dahen and Dionne (2010), and many other research discuss data collection and usage, but much less attention has been paid to the computational efficiency issue. The risk measure used for regulatory capital purposes reflects a holding period of one-year and a confidence level of 99.9% (The Basel Accord). It is almost

infeasible to get an accurate estimate of such risk measure if naïve Monte Carlo approach is used (Asmussen and Glynn 2007). Therefore, in this paper, we propose an efficient Monte Carlo simulation algorithm for computing such risk measure.

KEY WORDS: operational risk, advanced measurement approaches, loss distribution approach, Monte Carlo simulation, variance reduction

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Introduction

The definition of operational risk in BIS (2004) is “the risks of losses resulting from inadequate or failed internal processes, people and systems or from external events. This definition includes legal risks, but excludes strategic and reputational risks.” In recent years, quantification of operational risks becomes an important issue for regulations in financial industry. For example, insurance companies and banks must follow Solvency II and Basel Accord respectively when allocating operation risk capital. There are two basic methods proposed by Basel II (i.e. Basic Indicator Approach (BIA), and Standardized Approach (SA)) to define the operational risk capital of a bank as a fraction of its gross income. In (BIS 2006), the Advanced

Measurement Approach (AMA), on the other hand, allows banks to develop their own models for assessing the regulatory capital which covers their yearly operational risk exposure within a confidence interval of 99.9% (henceforth, this exposure is called operational value at risk, or OpVaR). To sum up, there are three different methods for calculating operational risk charges (BIA, SA, AMA). Levels of model sophistication and risk sensitivity are increased correspondingly. For further detailed descriptions of these three methods in BIS (2004) are as follows:

1. BIA - Banks using the Basic Indicator Approach must hold capital for operational risk equal to the average over the previous three years of a fixed percentage (denoted alpha) of positive annual gross income.

2. SA - The capital charge for each business line is calculated by multiplying gross income by a factor (denoted beta) assigned to that business line. Beta serves as a proxy for the industry-wide relationship between the operational risk loss experience for a given business line and the aggregate level of gross income for that business line. It should be noted that in the Standardised Approach gross income is measured for each business line, not the whole institution.

3. AMA - Supervisors expect that AMA banking groups will continue efforts to develop increasingly risk-sensitive operational risk allocation techniques, notwithstanding initial approval of techniques based on gross income or other proxies for operational risk.

By using the most sophisticated approach of three methods, namely the AMA, banks are given unprecedented amount of flexibilities to develop their own models for

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concerning the stochastic models used. Loss Distribution Approach (LDA), also known as Actuarial Model, is well known to actuaries because it is also a standard model for P&C insurers. LDA concerns the measurement of risk for random losses generated from a matrix whose element corresponds to a combination of business line and event type. LDA is a statistical method which has been used widely in actuarial science for computing aggregate loss distributions. Research by using LDA are as follows. Frachot, Georges, and Roncalli (2001) explores the LDA for computing the capital charge of a bank for its operational risk. This paper uses the LDA which follows Lognormal with parameter mu=8 and variance=2.2, the frequency distribution which follows Poisson with parameter lambda = 50 and Normal copula which

describes the correlated aggregate loss distributions. Chapelle et al. (2008) develops a comprehensive LDA framework for the measurement of operational risk. In this paper, it chooses Pareto, Weibull, Lognormal distribution to model severity, negative binomial distribution to model frequency, and linear Spearman copula to model the dependence of aggregate losses. Warnung (2008) uses independence assumption and proposes that Weibull distribution, generalized Pareto distribution (GPD) to fit loss severity and negative binomial distribution, Poisson distribution to fit loss frequency. Fantazzini, Dalla Valle, and Giudici (2008) uses Gamma, Exponential, Pareto

distribution to model loss severity and negative binomial and Poisson to model loss frequency. Guégan, Hassani, and Naud (2011) proposes a two-pattern model to characterize loss distribution functions associated with operational risks: a lognormal on the corpus of the severity distribution and a generalized Pareto distribution (GPD) on the right tail.

The most commonly used copula in AMA is the Gaussian copula. Many copulas of modeling dependence are introduced by Embrechts, McNeil, and Straumann (2002) and Embrechts, Lindskog, and McNeil (2003). The dependence structure of these random losses within LDA is usually modeled through copulas (Chavez-Demoulin, Embrechts, and Nešlehová, 2006).For instance, Klüppelberg (2008) chooses the LDA, and uses Lévy copulas to model the dependence structure of operational loss events. Fantazzini, Dalla Valle, and Giudici (2008) uses normal copula and t copula to describe the dependence structure among the losses. Embrechts and Puccetti (2008)

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distribution.

Two major challenges are faced by LDA: data collection and usage, and

computation issue of getting an accurate risk measure for operational risk. Chapelle et al. (2008), Dahen and Dionne (2010) , and many other research discuss data collection and usage, but much less attention has been paid to the computation issue. The risk measure used for regulatory capital purposes reflects a holding period of one-year and a confidence level of 99.9% (BIS, 2006). It is almost infeasible to get an accurate estimate of such risk measure if naïve Monte Carlo approach is used Asmussen;Glynn (2007) Therefore, the main objective of this paper is to propose an efficient Monte Carlo simulation algorithm for computing such risk measure. In particular, we will use variance reduction techniques to speed up the computation.

Research Model

For each single risk cell (a combination of business line and event type), the total loss follows the standard LDA approach, is the sum of individual losses:

where is individual loss (severities) and is the number of losses

(frequency) in cell i , j. In (BIS 2004), it has defined eight business lines and seven event types as Table 1.

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Event types Business lines Internal fraud External fraud Employment practices Clients, Products, and Business Practice Damage to physical assets Business Disruption and Systems Failures Execution, Delivery, and Process Management Corporate finance Trading and sales Retail banking Commercial banking Payment and settlement Agency services Asset management Retail brokerage Source: BIS (2014)

The distributions for severities and frequency can be estimated from internal and external loss data. The dependence structure among is harder to model. Some researchers tried to use copulas function to direct link their relationship (e.g. Böcker and Klüppelberg (2010) and Chavez-Demoulin, Embrechts, and Nešlehová (2006)). In this paper, we propose to use factor copulas to model their dependence structure (Burtschell, Gregory, and Laurent (2009) and Hull (2009)).

Let the marginal distribution function of , and be denoted by (.), (.) and (.), respectively. According to fit different random variables, there are different distributions can be used. Therefore, we will choose the distribution to model loss severity and loss frequency from the past research (Table 2).

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Table 2. Marginal distribution function setting

Literature (.) (.)

Guégan, Hassani, and Naud (2011)

1. Lognormal 2. Generalized Pareto

Distribution

1. Poisson

Chapelle et al. (2008) 1. Pareto 2. Weibull 3. Lognormal

1. Negative Binomial 2. Poisson

Embrechts and Puccetti (2008)

1. Pareto 2. Lognormal

1. Poisson

Klüppelberg (2008) 1. Generalized Pareto Distribution

1. Poisson

Fantazzini, Dalla Valle, and Giudici (2008) 1. Gamma 2. Exponential 3. Pareto 1. Negative Binomial 2. Poisson

According to the past research about operational risk, we can sum up the

distributions which are suitable to loss severity or loss frequency. The distributions are used to model loss severity are cited from Klugman, Panjer, and Willmot (2012) as follows:

1. Pareto Distribution

The Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes and other financial variables. The Pareto distribution is a one parameter continuous distribution with PDF

, 1 ∞

where a is the scale parameter. 2. Weibull Distribution

The Weibull distribution is a two parameter continuous distribution with PDF

⁄

where a is the scale parameter and b is the location parameter. 3. Lognormal Distribution

The Lognormal distribution is a continuous distribution with PDF 1

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4. Exponential Distribution

The probability density function of an exponential distribution is , 0

Here 0 is the parameter of the distribution, often called the rate parameter.

5. Gamma Distribution

The probability density function using the shape-scale parameterization is

; ,

Γ

Where is shape parameter and is scale parameter. The distributions are used to model loss severity are as follows:

1. Poisson Distribution

Poisson distribution is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed time window. The probability mass function is defined as

!

The parameter λ is estimated based on desired frequency. If the Poisson distribution is supposed to generate a yearly frequency, λ is simply the yearly average of the sample.

1

2. Negative Binomial Distribution

Negative binomial distribution is a two parametric discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number r of failures. The success probability is denoted p and the PDF is

1

where q = 1 - p and if r is not an integer the binomial coefficient in the expression is replaced by

Γ

Γ Γ 1

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implement dependence in the model:

1. The frequency distribution between cells are dependent. 2. The severities between cells are dependent.

3. The aggregated loss between cells are dependent.

Assume that the joint behaviors of total loss within cell i, j can be described by Gaussian factor copulas that include normal copulas and t copulas (Asmussen and Glynn (2007), page 53). As suggested by Asmussen and Glynn (2007), copulas provide a possible approach for modeling multivariate distributions in which one has a well-defined idea of the marginal distributions but a vague one on the dependence structure. Burtschell, Gregory, and Laurent (2009) selected normal factor copulas to model the dependence structure of the times to the defaults for a bond portfolio subject to credit risk. Table 3 shows the copula setting of loss severity and loss frequency from the past research.

Table 3. Copula setting of loss severity and loss frequency

Literature Copula Model

(Klüppelberg 2008) Lévy copula to describe dependence in

frequency and severity between different cells (Embrechts and Puccetti 2008) Gumbel copula and Gaussian copula to describe

dependence in severity distribution (Fantazzini, Dalla Valle, and

Giudici 2008)

Normal copula and t copula to describe the dependence structure among the losses

We introduce several copulas used in past research, including Lévy copula, Gumbel copula, Gaussian factor copulas (in particular, normal and factor copulas) to model the dependence structures of . The definitions of Gaussian and t copula are as follows. Assume , , … , follow univariate standard uniform distribution.

1. Lévy copula

Klüppelberg (2008) had a detailed description about Lévy copula as follows. As a matter of fact, the definition of 56 different cells based on seven loss event types and eight business lines as suggested by the (BIS 2004) is quite arbitrary. For . ∑ . and ,

. . . ∈ compound Poisson processes. Above holds true, whenever the vector of all marginal t , … , t

constitutes an n-dimensional compound Poisson process. Therefore, choosing an appropriate distributional copula C at , we could write

, … , , … , .

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is completely determined by the frequency parameter λ > 0 and the distribution function of the cell’s severity, namely Π 0,

λ λ for ∈ 0, ∞ . A one-dimensional tail integral is simply the expected number of losses per unit time that are above a given threshold :

Π Π , ∞ , ∈ 0, ∞ .

In the dynamic framework of a multivariate Lévy process the multivariate Lévy measure controls the joint jump behaviour (per unit time) of all univariate components and contains all information of dependence between the components. Now, similarly to the fact that a multivariate distribution can be built from marginal distributions via a distributional copula, a multivariate tail integral

Π , … , Π , ∞ … , ∞ , ∈ 0, ∞ .

This is the content of Sklar’s theorem for Lévy processes with positive jumps, which basically says that every multivariate tail integral Π can be decomposed into its marginal tail integrals and a Lévy copula according to

Π , … , Π … Π , ∈ 0, ∞ .

2. Gumbel copula

(Embrechts and Puccetti 2008) had a detailed description about Gumbel copula as follows.

, , … , exp ⋯ ln

The Gumbel copula interpolates between independence ( Π) and comonotonic dependence ( M). The parameter θ can easily be calibrated using Kendall’s tau 1 and exhibits upper tail dependence 2 2 as explained in Embrechts, Frey, and McNeil (2005).

3. Gaussian Copula

Embrechts and Puccetti (2008) had a detailed description about Gaussian copula as follows.

, , … , Φ Φ , … , Φ

Here Φ denotes the joint distribution of a zero-mean Gaussian random vector with equicorrelation matrix Σ, with ones on the main diagonal and off-diagonal elements equal to . Φ is the quantile function of a standard normal random variable. Analogously to the Gumbel, the Gaussian copula

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case ( ). In contrast however to , for < 1 is

asymptotically tail independent, 0; see Embrechts, Frey, and McNeil (2005). Also the Gaussian copula family is closed under coordinate projections and simulation of Gaussian copula data is straightforward. Gaussian copula is very popular at first in practice due to the common

assumption of normality in many financial modeling applications and ease of use and understanding. However, it does not allow for tail dependence, which is a major drawback and as such becomes a less favorable candidate for capital applications

4. t Copula

Fantazzini, Dalla Valle, and Giudici (2008) had a detailed description about t copula as follows.

, , … , | , , , … ,

Where _, denotes the distribution function with degree of freedom and correlation coefficient matrix and denotes the inverse function of

distribution function. We are currently seeing growing literature on the usefulness of the t copula as an alternative to the Gaussian copula for

modeling financial risks. The main impetus for the t copula’s rise to notoriety is associated with its ability to incorporate tail dependence. Theoretically, the lower the degree of freedom, the heavier the tail dependence for a t copula. Gaussian copula is in fact a limiting case of t copula as the degree of freedom approaches ∞. By contrast, t copula exhibits the heaviest tail dependence as the degree of freedom approaches 1.

Due to above descriptions of copula, we know that there are many methods that we can model the dependence of losses. Then, we choose the normal factor copula to model the dependence of losses. It is clear that depends on and . The function form of and can be estimated from data and are assumed to be given. If the fitted distributions of and are common used distributions, then it is likely that the distributions of are also known. If is unknown, then we can generate independent empirical distributions of first.

When the dependence structure can be described by normal factor copulas, we may express

where N(.) is the cumulative distribution function (CDF) of the standard normal random variable and are the latent variables used to model the joint distributions of . The dependence among is induced through common factors and as follows:

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1

where , , , ..., are independent standard normal random variables and a and b denote constant factor loadings. The common factor represents the

common factor within business line, while represents the common factor within event type. On the other hand, , ..., are specific factors pertaining to each risk cell.

We also model the dependence structure via factor copulas. In particular

where (.) is the CDF of the t distributed random variable with v degrees of freedom. The t-copula is more powerful in terms of capturing tail dependence (Klugman, Panjer, and Willmot, 2012). The dependence among is induced through common factors and as follows:

1

where is an independent Chi-square random variable with degrees of freedom. The aggregate operational loss is then defined by

L ,

The empirical distribution of the aggregate operational loss L can be repeated sampled from above stochastic model through naïve Monte Carlo sampling

procedure. For most complex insurance and finance models, Monte Carlo method usually is the only viable numerical technique. However, the converge rate of Monte Carlo method is ⁄ , which is slow if each replication is expensive to generate. Therefore, the technique of variance reduction can be used to accelerate the Monte Carlo method. Variance reduction typically involves a fair amount of both theoretical study of the problem in question and additional programming effort. Since Basel II plans to compute the risk measures of operation risk with high confidence level, the simulation problem becomes rare event simulation problem and makes the

computation even harder. Then, we will use Matlab to solve simulation and computation problems.

In this paper, we choose Exponential, Gamma, Pareto, Lognormal distribution to model loss severity and choose Poisson, Negative Binomial distribution to model loss frequency. Through Monte Carlo method by simulating 5,000,000 times, we can get the data of loss. Then, we use one factor normal copula to model the dependence of loss.

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Empirical Result

Because total loss of the combination, Exponential-Negative Binomial, is too small, we choose two thresholds. Version 1 is based on Exponential-Negative Binomial and Version 2 is based on Exponential-Poisson. We set all distributions with the same first moment.

Version1

We use Monte Carlo to choose threshold through 2,000,000 simulations. The threshold value of 95%, 99%, and 99.9% confidence level are 1630478.84 (threshold95), 2185175.635 (threshold99), and 2914924.281 (threshold99.9) respectively.

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F (.) - G (.) Exp - Poisson Exp - NB Gamma-Poisson Gamma- NB Pareto - Poisson Pareto - NB LN - Poisson LN - NB p (s.e.) 0.999993 0.047722 0.998205 0.113961 1 0.031372 0.031372 0.031372 2.65E-06 0.000213 4.23E-05 0.000318 0 0.000174 0.000174 0.000174 p (se) 0.999905 0.00944 0.992843 0.058125 0.999989 0.007592 0.007592 0.007592 9.75E-06 9.67E-05 8.43E-05 0.000234 3.32E-06 8.68E-05 8.68E-05 8.68E-05 p .

(s.e.)

0.998871 0.000911 0.974842 0.023893 0.999743 0.002009 0.002009 0.002009 3.36E-05 3.02E-05 0.000157 0.000153 1.60E-05 4.48E-05 4.48E-05 4.48E-05 E(L,L>threshold95) (s.e.) 7449191 1975635 7443721 2457494 7469782 2107258 2107258 2107258 1798.172 325.8218 2769.414 818.0701 1715.917 1143.03 1143.03 1143.03 E(L,L>threshold99) (s.e.) 7449671 2505578 7473346 3009988 7469842 2956383 2956383 2956383 1797.523 305.1423 2747.283 815.3478 1715.833 2092.276 2092.276 2092.276 E(L,L>threshold99.9) (s.e.) 7454602 3210554 7563151 3737069 7471015 4364734 4364734 4364734 1791.893 287.6131 2690.973 811.5117 1714.41 3707.017 3707.017 3707.017

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Table 5. Simulation Result ( = 0.5) F (.) - G (.) Exp - Poisson Exp - NB Gamma-Poisson Gamma- NB Pareto - Poisson Pareto - NB LN - Poisson LN - NB p (s.e.) 0.999994 0.019724 0.997441 0.066378 0.999999 0.011798 0.011798 0.011798 2.45E-06 0.000139 5.05E-05 0.000249 1.00E-06 0.000108 0.000108 0.000108 p

(s.e.)

0.999927 0.002599 0.989726 0.029647 0.999993 0.002371 0.002371 0.002371 8.54E-06 5.09E-05 1.01E-04 0.00017 2.65E-06 4.86E-05 4.86E-05 4.86E-05 p .

(s.e.)

0.998693 0.000153 0.962479 0.010328 0.999774 0.000603 0.000603 0.000603 3.61E-05 1.24E-05 0.00019 0.000101 1.50E-05 2.45E-05 2.45E-05 2.45E-05 E(L,L >threshold95) (s.e.) 6915466 1910052 6596146 2325777 6980788 2036389 2036389 2036389 1571.773 267.3916 2385.204 691.5924 1449.312 870.5259 870.5259 870.5259 E(L,L >threshold99) (s.e.) 6915795 2452220 6632284 2883326 6980817 2861440 2861440 2861440 1571.312 252.187 2358.922 691.3629 1449.268 1682.6 1682.6 1682.6 E(L,L>threshold99.9) (s.e.) 6921030 3163799 6746178 3612965 6981746 4060499 4060499 4060499 1565.191 257.0963 2291.216 689.8648 1448.065 3016.611 3016.611 3016.611

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F (.) - G (.) Exp - Poisson Exp - NB Gamma-Poisson Gamma- NB Pareto - Poisson Pareto - NB LN - Poisson LN - NB p (s.e.) 1 0.240607 1 0.490936 1 0.1594 0.1594 0.1594 0.00E+00 0.000427 0.00E+00 0.0005 0 0.000366 0.000366 0.000366 p (s.e.) 1 0.029162 1 0.270314 1 0.0239 0.0239 0.0239

0.00E+00 1.68E-04 0.00E+00 0.000444 0.00E+00 1.53E-04 1.53E-04 1.53E-04 p .

(s.e.)

1 0.000514 1 0.097598 1 0.003191 0.003191 0.003191

0.00E+00 2.27E-05 0 0.000297 0.00E+00 5.64E-05 5.64E-05 5.64E-05 E(L,L >threshold95) (s.e.) 9847438 1906673 11508953 2430589 9632114 1936037 1936037 1936037 1033.623 234.8532 1934.199 661.9563 1108.433 342.6457 342.6457 342.6457 E(L,L >threshold99) (s.e.) 9847438 2387049 11508953 2865780 9632114 2559184 2559184 2559184 1033.623 181.5665 1934.199 594.6023 1108.433 453.2181 453.2181 453.2181 E(L,L>threshold99.9) (s.e.) 9847438 3072078 11508953 3495968 9632114 3423679 3423679 3423679 1033.623 149.5574 1934.199 531.8879 1108.433 669.9469 669.9469 669.9469

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Version2

We use Monte Carlo to choose threshold through 2,000,000 simulations. The threshold value of 95%, 99%, and 99.9% confidence level are 10669435

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F (.) - G (.) Exp - Poisson Exp - NB Gamma-Poisson Gamma- NB Pareto - Poisson Pareto - NB LN - Poisson LN - NB p (s.e.)

0.051742 0 0.135328 0 0.042932 6.00E-05 6.00E-05 6.00E-05

2.22E-04 0 3.42E-04 0 0.000203 7.75E-06 7.75E-06 7.75E-06

p (s.e.)

0.010346 0 0.06325 0 0.009892 3.70E-05 3.70E-05 3.70E-05

1.01E-04 0.00E+00 2.43E-04 0 9.90E-05 6.08E-06 6.08E-06 6.08E-06

p _. (s.e.)

0.000983 0 0.022313 0 0.002213 2.60E-05 2.60E-05 2.60E-05

3.13E-05 0.00E+00 0.000148 0 4.70E-05 5.10E-06 5.10E-06 5.10E-06

E(L,L>threshold95) (s.e.) 11614733 NaN 12558115 NaN 11831595 18492662.28 18492662 18492662 863.2502 NaN 1719.884 NaN 1694.496 9778.54468 9778.545 9778.545 E(L,L>threshold99) (s.e.) 13014064 NaN 13933041 NaN 13753332 22819347.18 22819347 22819347 759.1951 NaN 1600.502 NaN 2660.873 10316.15127 10316.15 10316.15 E(L,L>threshold99.9) (s.e.) 14779856 NaN 15649104 NaN 16850747 27112629.72 27112630 27112630 672.9459 NaN 1484.373 NaN 4294.873 9415.990743 9415.991 9415.991

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Table 8. Simulation Result ( = 0.5) F (.) - G (.) Exp - Poisson Exp - NB Gamma-Poisson Gamma- NB Pareto - Poisson Pareto - NB LN - Poisson LN - NB p (s.e.)

0.031654 0 0.115676 0 0.028095 6.00E-05 6.00E-05 6.00E-05

1.75E-04 0 3.20E-04 0 1.65E-04 7.75E-06 7.75E-06 7.75E-06

p (s.e.)

0.004504 0 0.048583 0 0.006104 3.60E-05 3.60E-05 3.60E-05

6.70E-05 0.00E+00 2.15E-04 0 7.79E-05 6.00E-06 6.00E-06 6.00E-06

p _. (s.e.)

0.000269 0 0.014749 0 0.001731 3.10E-05 3.10E-05 3.10E-05

1.64E-05 0.00E+00 0.000121 0 4.16E-05 5.57E-06 5.57E-06 5.57E-06

E(L,L>threshold95) (s.e.) 11458955 NaN 12345272 NaN 11852280 19840254.04 19840254 19840254 733.5049 NaN 1535.341 NaN 1911.724 10099.99272 10099.99 10099.99 E(L,L>threshold99) (s.e.) 12876146 NaN 13729061 NaN 14107463 25510088.52 25510089 25510089 652.3682 NaN 1431.633 NaN 3116.801 9440.748597 9440.749 9440.749 E(L,L>threshold99.9) (s.e.) 14684181 NaN 15469619 NaN 17258198 27522935.05 27522935 27522935 622.4213 NaN 1331.485 NaN 4447.908 8587.135924 8587.136 8587.136

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Figure 6. Simulation Result ( = 0.9) (.) - (.) Exp - Poisson Exp - NB Gamma-Poisson Gamma- NB Pareto - Poisson Pareto - NB LN - Poisson LN - NB ̂ (s.e.) 0.000278 0 0.035983 0 0.004283 0 0 0

1.67E-05 0 1.86E-04 0 6.53E-05 0 0 0

̂ (s.e.)

1.00E-06 0 0.004702 0 0.000866 0 0 0

1.00E-06 0.00E+00 6.84E-05 0 2.94E-05 0.00E+00 0.00E+00 0.00E+00 ̂ _.

(s.e.)

0 0 0.000237 0 0.000231 0 0 0

0.00E+00 0.00E+00 1.54E-05 0 1.52E-05 0.00E+00 0.00E+00 0.00E+00 E(L,L>threshold95)

(s.e.)

10945253 NaN 11431529 NaN 11711416 NaN NaN NaN

261.7337 NaN 698.4829 NaN 1320.303 NaN NaN NaN

E(L,L>threshold99) (se.)

12263935 NaN 12839346 NaN 13710083 NaN NaN NaN

0 NaN 597.2168 NaN 1720.912 NaN NaN NaN

E(L,L>threshold99.9) (s.e.)

NaN NaN 14536462 NaN 16016781 NaN NaN NaN

(23)

Reference

Asmussen;Glynn. 2007. Stochastic Simulation: Algorithms and Analysis: Algorithms and Analysis. Vol. 57: Springer.

Böcker, Klaus, and Claudia Klüppelberg. 2010. Multivariate models for operational risk. Quantitative Finance 10 (8):855-869.

BIS. 2004. International Convergence of Capital Measurement and Capital Standard. ———. 2006. International convergence of capital measurement and capital

standards.

Burtschell, Xavier, Jon Gregory, and Jean-Paul Laurent. 2009. A comparative analysis of CDO pricing models. The Journal of Derivatives 16 (4):9-37.

Chapelle, Ariane, Yves Crama, Georges Hübner, and Jean-Philippe Peters. 2008. Practical methods for measuring and managing operational risk in the financial sector: A clinical study. Journal of Banking & Finance 32 (6):1049-1061. Chavez-Demoulin, V., P. Embrechts, and J. Nešlehová. 2006. Quantitative models for

operational risk: Extremes, dependence and aggregation. Journal of Banking & Finance 30 (10):2635-2658.

Cope, Eric, and Gianluca Antonini. 2008. Observed correlations and dependencies among operational losses in the ORX consortium database. Journal of Operational Risk 3 (4):47-74.

Dahen, Hela, and Georges Dionne. 2010. Scaling models for the severity and

frequency of external operational loss data. Journal of Banking & Finance 34 (7):1484-1496.

Embrechts, Paul, Rdiger Frey, and Alexander McNeil. 2005. Quantitative risk management. Princeton Series in Finance, Princeton 10.

Embrechts, Paul, Filip Lindskog, and Alexander McNeil. 2003. Modelling

dependence with copulas and applications to risk management. Handbook of heavy tailed distributions in finance 8 (1):329-384.

Embrechts, Paul, Alexander McNeil, and Daniel Straumann. 2002. Correlation and dependence in risk management: properties and pitfalls. Risk management: value at risk and beyond:176-223.

Embrechts, PAUL, and GIOVANNI Puccetti. 2008. Aggregating risk across matrix structured loss data: the case of operational risk. Journal of Operational Risk 3 (2):29-44.

Fantazzini, Dean, Luciana Dalla Valle, and Paolo Giudici. 2008. Copulae and

operational risks. International Journal of Risk assessment and management 9 (3):238-257.

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Guégan, Dominique, Bertrand K. Hassani, and Cédric Naud. 2011. An efficient

threshold choice for the computation of operational risk capital. The Journal of Operational Risk.

Hull, John. 2009. Options, futures and other derivatives: Pearson education. Klüppelberg, Klaus Böcker;Claudia. 2008. Modeling and measuring multivariate

operational risk with Lévy copulas. The Journal of Operational Risk.

Klugman, Stuart A, Harry H Panjer, and Gordon E Willmot. 2012. Loss models: from data to decisions. Vol. 715: John Wiley & Sons.

Warnung, Grigory Temnov;Richard. 2008. A Comparison of Loss Aggregation Methods for Operational.

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Outline Valuation of a VA contract with GMWB Variance Reduction Techniques

Efficient Valuation of GMWB Annuities:

A Variance Reduction Approach

Jennifer L. Wang Ming-hua Hsieh

Dept. of of Risk Management and Insurance National Chengchi University, TAIWAN

Yu-Fen Chiu

Dept. of Financial Engineering and Actuarial Mathematics, Soochow University, TAIWAN

August 21, 2013

(26)

Valuation of a VA contract with GMWB Variance Reduction Techniques

1 Valuation of a VA contract with GMWB

Discrete Withdrawal GMWB GMWB Option

Valuation of GMWB option

2 Variance Reduction Techniques

Control Variates Numerical examples

(27)

Outline

Valuation of a VA contract with GMWB

Variance Reduction Techniques

Contract Specification

We consider the following contract: Single premium f0

The initial value of the sub-account F0 equals f0.

Annually a certain percentage (g ) of the guaranteed amount f0 will be withdrawn from the sub-account for T years (gT is

usually equals 1).

At the beginning of year t (t = 0, 1, . . . , T − 1), guarantee fee (α times Ft) and a fixed management fee K are withdrawal

from the sub-account by the insurer.

(28)

GMWB Option

Contract Specification

Above GMWB contract provides the following cash-flows to the policy holder

ct = gf0, t = 1, 2, . . . , T − 1;

(29)

Outline

GMWB Option

The cash-flow received at time T can be decomposed into cT = max(gf0, FT) = gf0+ max(FT − gf0, 0)

These cash-flows can be decomposed into a term annuity with annual payment gf0 and an option-like payment max(FT − gf0, 0).

We call the option-like payment the GMWB option.

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GMWB Option

The fair value of a VA contract with GMWB

The fair value of a VA contract with GMWB is the sum of fair values of the term annuity and the GMWB option

The fair value of the term annuity is easy to compute, since its value only depends on the current term structure of interest rates.

The problem of fair valuation of a VA contract with GMWB reduced to the valuation problem of GMWB option.

(31)

Outline

Valuation of GMWB option

Based on risk-neutral valuation principle, the fair value of GMWB option can be expressed as

EQ

max(FT − gf0, 0)

B(T )

where EQ denote the expectation under risk neutral measure and

B(T ) denotes the account value of a money market account with initial account value 1.

(32)

GMWB Option

Dynamics of the sub-account

The value of the sub-account depends on the annual returns of the invested mutual fund.

Let S (t) be NAV of the invested mutual fund at year t. Then the annual return of the invested mutual fund over the t-th year would be:

Rt =

S (t) S (t − 1)

Let us denote F_t− the account value at year t before withdraws and F_t+ the account value at year t after withdraws.

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Outline

Dynamics of the sub-account

The process of the account value can then be described

F₀− = f0,

F₀+ = max((1 − α)F₀−− K , 0), F_t− = RtF_t−1+ , t = 1, 2, · · · , T

F_t+ = max((1 − α)F_t−− K − gf0, 0), t = 1, 2, · · · , T − 1

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GMWB Option

Dynamics of the invested mutual fund and MMA

The value of GMWB option only depends on FT and FT in

turns only depends on the joint distribution of (R1, · · · , RT).

Therefore, the dynamic of S (t) can be very flexible.

For simulation based method, the only restriction is that the sample of (R1, · · · , RT) is easy to generate.

The dynamic of B(t) can be derived from the selected interest rate model.

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Outline Valuation of a VA contract with GMWB

Monte Carlo method

Suppose that we wish to estimate β = EX , where X is the output of a complex stochastic process. In our case,

X = max(FT− gf0, 0)

B(T ) .

A naive Monte Carlo procedure would generate n independent copies of X , and produce the standard estimate

βnaive = 1_n

n

X

i =1

Xi

where X1, . . . , Xn are independent copies of X

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Variance Reduction Techniques Numerical examples

Control Variates

Let Y be a d by 1 random vector and each component of Y is correlated with X . Let (µ, Σ) denote the mean vector and

covariance matrix of Y . The mean vector is known. Suppose that the covariance between X and Yi is ci and c = (c1, . . . , cd)T.

Define control variates

C = Y − µ

It is clear that the mean vector of C = 0, covariance matrix of C = Σ, and the covariance between X and Ci is ci.

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Control Variates

Let λ ∈ <d and define

XC(λ) = X − λTC

It is obvious that E [XC(λ)] = β and

Var[XC(λ)] = σ2X − 2λTc + λTΣλ

The minimizer of above formula λ∗= Σ−1c and

Var[XC(λ∗)] = σX2 − 2(Σ−1c)Tc + (Σ−1c)TΣ(Σ−1c)

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Control Variates

Hence

Var[XC(λ∗)] = σ2X − cTΣ−1c < σ2X

Let X_C(i )(λ∗), i = 1, . . . , n, be independent copies of XC(λ∗). Then

it is obvious that βcontrol = 1_n n X i =1 X_C(i )(λ∗) is a more efficient estimate for β.

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Estimators with control variates

Assume S (t) is a Levy process and then R1, . . . , RT are

independent. We propose two estimators with control variates. Define H1 = ((1 − α)f0− K )R1, Ht = ((1 − α)Ht−1− K − gf0)Rt, t = 2, · · · , T and set C1 = HT− E [HT], C2= T Y t=1 Rt− E "_T Y t=1 Rt #

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Numerical example I

Table: Point estimates ×105 (f0= 1000000, r = 0.04, T = 20, g =

0.05, σ = 0.16, n = 1000000, α = 0.008) K _βnaive βC1 βC2 βC1,C2 1000 2.5517 2.5585 2.5566 2.5584 2000 2.4681 2.4666 2.4679 2.4667 3000 2.3783 2.3757 2.3777 2.3758 4000 2.2856 2.2881 2.2886 2.2882

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Numerical example I

Table: Standard errors (f0= 1000000, r = 0.04, T = 20, g = 0.05, σ =

0.16, n = 1000000, α = 0.008) K _βnaive βC1 βC2 βC1,C2 1000 440.7616 64.4768 146.5464 63.5819 2000 434.3924 67.3809 148.4233 66.3715 3000 426.0773 70.3953 149.9440 69.2145 4000 419.0520 73.3947 151.6640 72.0314

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Numerical example I

Table: Variance Ratios (f0= 1000000, r = 0.04, T = 20, g = 0.05, σ =

0.16, n = 1000000, α = 0.008) K βC1 βC2 βC1,C2 1000 46.7305 9.0460 48.0552 2000 41.5615 8.5656 42.8352 3000 36.6345 8.0746 37.8950 4000 32.5992 7.6343 33.8448

(43)

Numerical example II

Table: Point estimates ×105 (f0= 1000000, r = 0.04, T = 20, g =

0.05, σ = 0.16, n = 1000000, α = 0.005) K _βnaive βC1 βC2 βC1,C2 1000 2.8568 2.8607 2.8604 2.8608 2000 2.7634 2.7629 2.7620 2.7628 3000 2.6650 2.6652 2.6638 2.6651 4000 2.5683 2.5713 2.5708 2.5713

(44)

Numerical example II

Table: Standard errors (f0= 1000000, r = 0.04, T = 20, g = 0.05, σ =

0.16, n = 1000000, α = 0.005) K _βnaive βC1 βC2 βC1,C2 1000 472.6329 62.7985 151.2676 62.0168 2000 470.0813 65.8974 154.2422 65.0286 3000 462.5048 69.1576 156.3422 68.1162 4000 454.2942 71.8611 157.7726 70.7243

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Numerical example II

Table: Variance Ratios (f0= 1000000, r = 0.04, T = 20, g = 0.05, σ =

0.16, n = 1000000, α = 0.005) K βC1 βC2 βC1,C2 1000 56.6435 9.7624 58.0804 2000 50.8874 9.2884 52.2561 3000 44.7252 8.7515 46.1033 4000 39.9656 8.2911 41.2607

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Conclusions and extensions

The selected control variates are effective from the numerical examples.

The algorithm is easy to generalize to more complex S (t) and B(t) processes

The algorithm is easy to generalize to life-long GMWB (model T driven by a specific mortality model)

The algorithm can extend to value contracts with g is time dependent.

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9

LONGEVITY

NINE

September 6 and 7, 2013

Beijing, China

The Ninth International Longevity Risk &

Capital Markets Solutions Conference

第九届长寿风险与资本市场国际研讨会

Host: 主办方

China Institute for Actuarial Science, Central University of Finance & Economics 中央财经大学中国精算研究院

Co-hosts: 协办方

China Association of Actuaries 中国精算师协会

Waterloo Research institute in Insurance, Securities and Quantitative finance, University of Waterloo, Canada

加拿大滑铁卢大学保险，证券和定量金融研究所 Edmondson-Miller Chair, Illinois State University, USA

美国伊利诺斯州立大学 Edmondson-Miller Chair

Pensions Institute, Cass Business School, City University London, UK 英国伦敦城市大学卡斯商学院养老金协会

Risk Management and Insurance Department, National Cheng-Chi University, Taiwan 台湾国立政治大学风险管理与保险学系

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Thursday September 5, 2013

16:00 – 21:00 Conference Registration (Pre-Function Hall)

Friday September 6, 2013

08:00 – 09:00 Opening Ceremony (Grand Ballroom C)

09:00 – 10:30 Plenary Session I (Grand Ballroom C)

Sponsored by Société Générale

10:30 – 11:00 Refreshment Break (Pre-Function Hall)

11:00 – 12:15 Parallel Session I

12:15 – 13:15 Lunch (City Wall Ballroom)

Sponsored by SCOR Global Life

13:15 – 13:45 Plenary Session II (Grand Ballroom C)

13:45 – 15:00 Roundtable Discussion Forum I (Grand Ballroom C)

Sponsored by Cathay Life Insurance Co.

15:30 – 16:30 Plenary Session III (Grand Ballroom C)

16:30 – 17:45 Parallel Session II

18:15 – 19:30 Reception (Pre-Function Hall B)

19:30 – 22:00 Gala Dinner (City Wall Ballroom)

Sponsored by Institute and Faculty of Actuaries

Saturday September 7, 2013

08:20 – 10:00 Parallel Session III

10:30 – 12:00 Plenary Session IV (Grand Ballroom C)

12:00 – 13:00 Roundtable Discussion Forum II (Grand Ballroom C)

13:00 – 13:15 Closing Ceremony (Grand Ballroom C)

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Thursday September 5, 2013

08:00 – 09:00 Opening Ceremony (Grand Ballroom C)

Moderator: Xiaolin Li (Professor, China Institute for Actuarial Sci-ence, CUFE)

Hu Shuxiang (Chairman of University Council, Central University of Finance and Economics)

David Blake (Director, Pensions Institute, Cass Business School, City University London)

Wei Yingning (President, China Association of Actuaries)

09:00 – 10:30 Plenary Session I (Grand Ballroom C)

Sponsored by Société Générale

Moderator: David Blake

"How and Why Mortality Change Varies Between Sexes and

Countries"

Shripad (Tulja) Tuljapurkar (Dean & Virginia Morrison Professor of Population Studies, Stanford University)

"Mortality Assumptions and Longevity Risk”

Pablo Antolin (Principal Economist, Financial Affairs Division,

“Enhanced Annuities in Asia – A Case Study”

Cord-Roland Rinke (Managing Director, Hannover Life Re.)

11:00 – 12:15 Parallel Session I

12:15 – 13:15 Lunch (City Wall Ballroom)

Sponsored by SCOR Global Life

13:15 – 13:45 Plenary Session II (Grand Ballroom C)

Moderator: Ken Seng Tan (Associate Director. WatRISQ, Univer-sity of Waterloo & Hon. Director, China Institute for Actuarial Sci-ence, CUFE)

“China’s Three-Pillar Pension Facing with the Challenge of

Longevity Risk”

Bingwen Zheng (Director General, Center for International Social Security Studies and Institute of Latin American Studies, Chinese Academy of Social Science)

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3

13:45 – 15:00 Roundtable Discussion Forum I: “Longevity Issues in Asia: Challenges

and Opportunities” Sponsored by Cathay Life Insurance Co.

Moderator: Xiaolin Li

Yu Cai (Director, Policy and Research Department, China Insurance

Regula-tory Commission)

"How China’s insurance industries play a greater role in longevity risk

transferring"

Meipan Tian (Chief Actuary, China Re Group)

“The Challenges of Longevity Risk in Taiwan and Future Policy Develop-ments”

Jennifer Wang (Vice Chairperson, Financial Supervisory Commission and

Distinguished Professor of NCCU, Taiwan)

“The Impacts of Longevity Risk on Taiwan Insurance Industry: Risk

Man-agement and Product Innovation”

Chun-Hung Wu (Senior Vice President, Cathay Life Insurance Co.)

“Longevity Risk Challenges in China’s Public Pension & Private

Pen-sions: in the Perspective of Financial Markets Context”

Bingwen Zheng

15:00 – 15:30 Refreshment Break II (Pre-Function Hall)

15:30 – 16:30 Plenary Session III (Grand Ballroom C)

Moderator: Richard MacMinn (Edmondson-Miller Chair in Insurance and

Fi-nancial Services, Illinois State University)

“The role of the SOA in Addressing Longevity Risk” Tonya Manning (President, Society of Actuaries)

“Reinsurance of Longevity: Risk Transfer and Capital Market

Manage-ment Solutions”

Daria Ossipova-Kachakhidze (Head of Longevity and Mortality R&D Centre,

SCOR)

16:30 – 17:45 Parallel Session II

18:15 – 19:30 Reception (Pre-Function Hall B)

19:30 – 22:00 Gala Dinner (City Wall Ballroom)

Sponsored by Institute and Faculty of Actuaries

Moderator: Ken Seng Tan

Pre-Dinner Presentation:

“The Challenges for Actuaries in Dealing with Longevity Predictions”

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4

Saturday September 7, 2013

08:00 – 13:15 Conference Registration(Pre-Function Hall)

08:20 – 10:00 Parallel Session III

10:30 – 12:00 “Accelerating the Development of a Liquid Risk Transfer Market for Longevity”

Moderator: Guy Coughlan (Managing Director, Pacific Global

Speakers:

Jeff Mulholland (sion Solutions in the Americas, Société GénéraleManaging Director and Head of Insurance and Pen-)

Chris Madsen (Head of Risk Structuring & Transfer, Aegon NV) Chris Hornsby (Longevity Risk Model Manager, RMS LifeRisks) Andrew Coburn (Senior Vice President, RMS LifeRisks)

Peter Schliebs (Challenger Life Company Limited)

Plenary Session IV (Grand Ballroom C)

“Anatomy of a Successful Transaction”

Transaction Sponsor Perspective - Madsen

Bank Perspective - Mulholland

Modeling Firm Perspective - Hornsby & Coburn Investor Perspective - Schliebs

12:00 – 13:00 Roundtable Discussion Forum II (Grand Ballroom C)

“Lessons Learned and How to Generate More and Larger Trans-actions”

What are the lessons learned from the transaction? What were the most effective elements of the transaction?

What could be improved for future transactions to create a more rep-licable structure and process?

What can be improved to produce more and larger transactions?

13:00 – 13:15 Closing Ceremony (Grand Ballroom C)

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5 PARALLEL SESSIONS

PARALLEL

SESSIONS

OVERVIEW AND LOCATIONS

Panel A Panel B Panel C Panel D Panel E Room D1 Room D2 Room D3 Meeting Room 1 Meeting Room 6 Session I Fri 6 Sept

11:00-12:15 Reverse Mortgages Mortality-linked Securities 1 Pensions Plans Insurance-linked Securities/Personal In-jury Modelling 1

Session II Fri 6 Sept, 16:30-17:45 Mortality-linked securities 2 Pension Systems Mortality Forecasting 1

Longevity Risk Modelling 2

Session III Sat 7 Sept, 08:20-10:00

Annuities Mortality

Forecasting 2

Modelling 3 Modelling 4

Parallel Session I - Panel A - Reverse mortgages

Session Chair – Deng Yinglu

"Reverse Mortgage Pricing and Risk Analysis Allowing for Idiosyncratic House Price Risk and Longevity Risk"

Adam Wenqiang Shao, Michael Sherris, Katja Hanewald

"A Study on Pricing of the Reverse Mortgage with an Embedded Redeemable Option: An

analysis based on China's market"

Bingzheng Chen, Yinglu Deng, Peng Qin

“Contributors to the Potential Demand for Reverse Mortgage in China —An Empirical In-vestigation Based on a Questionnaire Survey of Residents in Beijing”

Bingzheng Chen, Yinglu Deng, Xiaofei Liu, Lihong Zhang

Parallel Session I - Panel B - Mortality-linked securities 1

Session Chair – Ning Zhang

"Price Bounds of Mortality-Linked Security in Incomplete Insurance Market"

Jeffrey Tzuhao Tsai, Yu Lieh Huang, Sharon S. Yang

"Modelling Infectious Mortality Risk with Application to Mortality Security Pricing"

Fen-Ying Chen, Hong-Chih Huang, Sharon S.Yang

"Mortality Decomposition Model and its Application in the Graded Longevity Bond Build-ing in China"

(53)

6

Parallel Session I - Panel C - Pensions Plans

Session Chair –Jing Ai

"De-risking Defined Benefit Plans"

Yijia Lin, Richard D MacMinn, Ruilin Tian

"The Analysis On The Optimal Investment Return For Chinese Personal Pension Account In The View Of Longevity Risk"

Xiaoqian Yuan, Ning Zhang

"A New Approach to Pension Risk Management"

Jing Ai, Patrick L Brockett, Allen Jacobson

Parallel Session I - Panel D - Insurance-linked Securities/Personal Injury

Session Chair - Wai-Sum Chan

"Heterogeneous Expectations and Speculative Behavior in Insurance-linked Securities"

Min Zheng

"Disaggregation of Chinese Life Tables from National to Provincial Level, with an Appli-cation to Assessing Personal Injury Liabilities in China"

Felix W.H. Chan, Johnny S.H. Li, Wai-Sum Chan

Parallel Session I - Panel E - Modelling 1

Session Chair - Andrew Cairns

"An Extensions of the Lee-Carter Model For Mortality Projections"

Udi Makov

"A Two-Stage Linear Regression Approach To Modeling Mortality Rates Of Different Forms"

Cary Chi-Liang Tsai

"Mortality Modelling: Living With Unreliable Data"

(54)

7

Parallel Session II - Panel A - Mortality-linked securities 2

Session Chair – Hong-Chih Huang

"A Bayesian Pricing of Longevity Derivatives under Stochastic Interest Rates"

Takahiro Fushimi, Atsuyuki Kogure, Yoshimitsu Takamatsu

"Modeling Multi-Country Mortality Dependence and its Application in Pricing Survivor Swaps: a Dynamic Copula Approach”

Sharon S. Yang, Hong-Chih Huang, Chou-Wen Wang

Parallel Session II - Panel B - Pension Systems

Session Chair - Zaigui Yang

"Public Pension with Longevity and Population Growth Rate in China"

Qiang Cui

"Rural Public Pension, Uncertain Lifetime, Family Transfers And Endogenous Growth"

Zaigui Yang

Parallel Session II - Panel C - Mortality Forecasting 1

Session Chair - Andrew Hunt

"The Choice of Sample Size for Mortality Forecasting: A Bayesian Learning Approach"

Anja De Waegenaere, Hong Li, Bertrand Melenberg

"Forecasting ESRD Incidence and Mortality Based on Taiwan's Population Data"

Shing-Her Juang, Chih-Hua Chiao, Yin-Ju Lin, Hsin-Yi Li

"Projecting Mortality: Coherence and Co-integration in Multi-Population Projections"

Andrew Hunt, David Blake

Parallel Session II - Panel D - Longevity Risk

Session Chair - Tom Terry

"Longevity Risk Existence Analysis: Case in Beijing"

Xiaoting Gao, Zan Zhao

“Longevity Risk in China and its Financial Impact: Evidence from Model Test”

Wei Xiao, Chenzhe Liu

"Communicating Longevity Risk: Beyond the Definitions"

(55)

8

Parallel Session II - Panel E - Modelling 2

Session Chair – Johnny Li

"Comparison and Evaluation of Fuzzy Regression Methods in Lee-Carter Model"

Diheng Huang

"Methods of Improving Lee-Carter Model - Based on China's Mortality Data"

Lifeng Yang, Xiuye Meng

"A Step-by-Step Guide to Building Two-Population Stochastic Mortality Models"

Johnny Li, Rui Zhou, Mary R. Hardy Saturday September 7, 2013

Parallel Session III - Panel A - Annuities

Session Chair - Ralph Rogalla

"A Regime-Switching Framework for the Valuation of a Guaranteed Annuity Option"

Huan Gao, Rogemar Mamon, Anton Tenyakov, Xiaoming Liu

"Efficient Valuation of GMWB Annuities: A Variance Reduction Approach"

Jennifer L. Wang, Ming-hua Hsieh, Yu-Fen Chiu

"Research on Annuity Puzzle in China from Institutional Perspective"

Wei Wang, Ning Zhang

"Optimal Life Cycle Portfolio Choice with Variable Annuities Offering Liquidity and In-vestment Downside Protection",

Vanya Horneff, Raimond Maurer, Olivia S. Mitchell, Ralph Rogalla Parallel Session III - Panel B - Mortality Forecasting

Session Chair - Julien Tomas

"Prediction Intervals for Future Mortality Rates"

Ah Pooi, W.Y. Pan and Y.C. Wong

"Developing longevity de-risking solutions for Swiss pension funds: An application with

Swiss coherent mortality model"

Ljudmila Bertschi, Cheng Wan

"Forecasting Mortality in the Presence of Missing Data: An Application to Chinese Popu-lation"

Ping An, Ken Seng Tan, Johnny Li

"A New Non-Parametric Approach to Smoothing and Forecasting of Mortality"

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9

Parallel Session III - Panel C - Modelling 3

Session Chair - Jack C Yue

"A Study of Mortality Compression and Longevity Risk”

Jack C Yue

"Bayesian Modelling of Longevity and Lifespan Extension"

Ramona Meyricke, Michael Garratt

“Longevity Bonds Pricing Model Driven by OU Jump Process of Mortality Index and Af-fine Stochastic Interest Rate Model”

Jianwei Gao, Le Yang, Lizhi Wang

"A Relational Model Approach to Deal with the Longevity Risk"

Hsin Chung Wang, Jack C Yue

Parallel Session III - Panel D - Modelling 4

Session Chair - Hua Chen

"A Common Age Effect Model for the Mortality of Multiple Populations"

Torsten Kleinow

"Love and Death: A Freund Model with Frailty"

C Gourieroux , Yang Lu

"Longevity Hedges across Populations: A Two-Factor Model Approach"

Linus Fang-Shu Chan, Cary Chi-Liang Tsai, Chenghsien Tsai

"A Multi-Population Mortality Model via the Factor Copula Approach"

(57)

Efficient Valuation of GLWB Annuities:

A Variance Reduction Approach

*

Jennifer L. Wang

Department of Risk Management and Insurance National Chengchi University, Taiwan

64, Sec. 2, Chihnan Rd, Taipei, Taiwan, 11605, R.O.C. E-mail: [email protected]

Yu-Fen Chiu

Department of Financial Engineering and Actuarial Mathematics Soochow University, Taiwan

56 Kueiyang Street, Section 1, Taipei, Taiwan 100, R.O.C. E-mail: [email protected]

Ming-hua Hsieh

Department of Risk Management and Insurance National Chengchi University, Taiwan

64, Sec. 2, Chihnan Rd, Taipei, Taiwan, 11605, R.O.C. E-mail: [email protected]

*_{The authors are grateful to the Risk and Insurance Research Center of National Chengchi University}

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Abstract

Facing volatile financial markets, policy holders of investment-linked insurance products demand the policies that can eliminate the downside risk while still providing upside potential. Variable annuities (VAs) with guarantees are such products for retirement saving and became very popular. In particular, VA contracts with Guaranteed Minimum Withdrawal Benefits (GMWB) and Guaranteed Lifelong Withdrawal Benefits (GLWB) dominate the market since their introduction.

According to LIMRA VA GLB Election Tracking Survey of the third quarter 2013, when any GLBs are available, rate of election for guaranteed living benefits (GLBs) was 81% and VA assets with GLBs increased during the third quarter from $712 billion to $750 billion. GLWB offers a lifelong withdrawal guarantee; therefore, it is a suitable replacement for traditional life annuities. Its popularity is increasing since its introduction in 2002. In light of the growing importance of GMLB market and GMWB is just a special case of GLWB by assuming the future lifetime of the insured is deterministic, the main purpose of this project is to develop a fast Monte Carlo algorithm for valuation of GLWB for flexible asset models and product specification.

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1. Introduction

Facing volatile financial markets, policy holders of investment-linked insurance products demand the policies that can eliminate the downside risk while still providing upside potential. Variable annuities (VAs) with guarantees are such products for retirement saving and became very popular Ledlie et al. (2008). In particular, VA contracts with Guaranteed Minimum Withdrawal Benefits (GMWB) and Guaranteed Lifelong Withdrawal Benefits (GLWB) dominate the market since their introduction (Piscopo 2009). According to LIMRA VA GLB Election Tracking Survey of the third quarter 2013 (LIMRA 2013), when any GLBs are available, rate of election for guaranteed living benefits (GLBs) was 81% and VA assets with GLBs increased during the third quarter from $712 billion to $750 billion. GMWB and GLWB allow the policyholders to participate in the potential appreciation of the stock market while eliminating the downside risk by a minimal withdrawal guarantee. GMWB and GLWB gives the insured the possibility to withdraw a pre-specified amount periodically, even if the sub-account value has fallen below this amount. Compared to traditional guarantee contracts, GMWB and GLWB are more suitable products for insurers to sell because the cost of the embedded options of GMWB and GLWB are cheaper compared to more traditional embedded options such as GMAB and GMIB. GLWB offers a lifelong withdrawal guarantee; therefore, it is a suitable replacement for traditional life annuities. It’s popularity is increasing since its

introduction in 2002 (LIMRA 2013; Piscopo and Haberman 2012). In light of the growing importance of GMLB market and GMWB is just a special case of GLWB when the future lifetime of the insured is deterministic, the main purpose of this project is to develop a fast valuation algorithm for valuation of GLWB.

Fair valuation and hedging strategies are very crucial for the risk management of GMWB and GLWB. The problems of fair valuation of GMWB have been discussed by (Bacinello 2003; Chen and Forsyth 2008; Chen, Vetzal, and Forsyth 2008; Holz, Kling, and Russ 2008; Milevsky and Salisbury 2006; Min Dai, Yue Kuen Kwok, and Zong 2008; Peng, Leung, and Kwok 2012; Yang and Dai 2013; Bacinello et al. 2011) and the problems of fair valuation of GLWB have been discussed by (Bernard 2010; Holz, Kling, and Ruß 2012; Piscopo 2009; Piscopo and Haberman 2012). However, the previous research consider only continuous withdrawal models (either assume that the amount of insurance fee or policyholder’s withdraw behavior is continuous) to simplify the complexity of the corresponding valuation problem. In this project, we assume a discrete withdrawal model which is more closed to the real practice of

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rather simple. Therefore, in this project, we extend asset models to include all Lévy processes.

When using Monte Carlo method to value such guarantees, risk neutral scenarios of the account value are often simulated. However, many scenarios turn out to be out-of-money. This implies crude Monte Carlo method is not efficient. In this project, we propose an efficient Monte Carlo valuation method by using the techniques of variance reduction. We will design a fast valuation algorithm. The project will further compare the efficiency of the proposed approach with traditional Monte Carlo method.

2. Product Specification

In an immediate GLWB contract, a certain percentage gt of the single premium (the guaranteed amount w0) can be withdrawn from the policy at the end of year t for Kx +1 years, where Kx is the curtate future lifetime of the insured and thus is random. The guaranteed annual percentage gt is usually setup as follows

1

to meet the expectation that the expected amount received by the policyholder is greater than the guaranteed amount w0. Under such setup, it is easy to incorporate the roll-up and step-up options commonly exist in GLWB (Piscopo and Haberman 2012) or the deferred period by assigning appropriate values to gt.

At time T = Kx +1, if the sub-account value of the policy WT is greater than gTw0, then the policy holder can receive WT; otherwise the insured can receive gTw0. To be more precise, a typical GLWB contract provides the following cash-flows to the policyholder,

, 1, 2, … , 1; max ,

The cash-flow received at time T can be decomposed into

損失分配法下作業風險值快速蒙地卡羅法的設計

科技部補助專題研究計畫成果報告

期末報告

損失分配法下作業風險值快速蒙地卡羅法的設計

中 華 民 國 104 年 04 月 27 日

An efficient Monte Carlo method

for estimation of operational VaR and ES

under loss distribution approach

Introduction

Research Model

Empirical Result

Version1

Version2

Reference

Efficient Valuation of GMWB Annuities:

A Variance Reduction Approach

Contract Specification

Contract Specification

GMWB Option

The fair value of a VA contract with GMWB

Valuation of GMWB option

Dynamics of the sub-account

Dynamics of the sub-account

Dynamics of the invested mutual fund and MMA

Monte Carlo method

Control Variates

Control Variates

Control Variates

Estimators with control variates

Numerical example I

Numerical example I

Numerical example I

Numerical example II

Numerical example II

Numerical example II

Conclusions and extensions

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Beijing, China

The Ninth International Longevity Risk &

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第九届长寿风险与资本市场国际研讨会

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PARALLEL SESSIONS

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OVERVIEW AND LOCATIONS

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中華民國 104 年 04 月 27 日