投資型保險商品評價方法研究

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行政院國家科學委員會專題研究計畫成果報告

投資型保險商品評價方法研究

(完整版

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完整版

計畫類別

:

□

個別型計畫

□

整合型計畫

計畫編號

: NSC 97-2410-H-004-041-MY2

執行期間

:

中華民國

97 年

8 月

1 日至

99 年

7 月

31 日

執行單位

:

國立政治大學風險管理與保險學系

中華民國

99 年

10 月

31 日

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INDEX

中文摘要... 1

ABSTRACT ... 3

1. INTRODUCTION ... 5

1.1 VARIABLE ANNUITY AND EQUITY-LINKED ANNUITY ... 5

1.2 EQUITY-INDEXED ANNUITIES (EIAS) ... 6

1.3 RRATCHET EIAS ... 7

2 PURPOSE OF RESEARCH ... 8

3 PAPER REVIEW ... 11

3.1 VARIABLE ANNUITY CONTRACTS WITH CLIQUET OPTIONS IN ASIA MARKETS 11 3.2 VALUATION FRAMEWORK OF RATCHET EIAS ... 12

3.2.1 Pricing and hedging of ratchet EIAs ... 12

3.2.2 Product Specification ... 13

3.2.3 Risk-Neutral Valuation ... 16

3.2.3.1 Valuation of Ratchet EIAs via the Risk-Neutral Valuation Principle ... 16

3.2.3.2 Risk-Neutral Valuation of Contracts with quantos ... 18

4 RESEARCH METHODOLOGY... 21

4.1 VA CONTRACT VALUATION ... 21

4.1.1 Valuation formula of VA contract considered ... 21

4.1.2 Monde Carlo method of VA contract considered ... 22

4.2 PRICING FORMULAS OF RATCHET EIAS ... 28

4.2.1 . Compound and Simple Ratchet EIAs with the First Type of Geometric Averaging Scheme ... 28

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4.2.2 Compound and Simple Ratchet EIAs with the Second Type of Geometric

Averaging Scheme ... 33

4.2.3 Numerical Examples ... 36

4.2.3.1 Impact of Return Cap ... 36

4.2.3.2 Impact of Return Averaging ... 39

4.2.3.3 Impact of Return Accumulation Method ... 42

4.3 PRICING FORMULA OF CONTRACTS WITH QUANTO FEATURES ... 45

4.3.1 Quanto Ratchet EIAs with no Return Averaging ... 45

4.3.2 Quanto Ratchet EIAs with G1 Return Averaging ... 48

4.3.3 Quanto Ratchet EIAs with G2 Return Averaging ... 49

4.3.4 Special Cases of No Quanto Feature ... 52

4.3.5 Numerical illustrations ... 54

4.3.6 Contract Feature/Parameter Analyses ... 55

4.3.7 Market Parameter Analyses ... 59

5 RESULT AND CONCLUSION ... 65

5.1 IMPROVING THE COMPUTATION EFFICIENCY OF VALUING THE VARIABLE ANNUITY CONTRACT HAS QUANTO FEATURE AND EMBEDDED CLIQUET OPTIONS ... 65

5.2 PRICING AND NUMERICAL ANALYSIS OF QUANTO RATCHET EIAS ... 65

5.3 DERIVED PRICING FORMULAS FOR VARIOUS RATCHET EIA CONTRACTS UNDER THE BLACK-SCHOLES ASSUMPTIONS ... 66

5.4 ANALYZE HOW THE MARKET PARAMETERS AND CONTRACT FEATURES AFFECT THE CONTRACT VALUE... 67

5.5 HEDGING THE RISKS OF THE RATCHET EIA PRODUCTS ... 68

6 REFERENCE ... 70

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7.1 JOURNAL PAPER ... 72 7.2 CONFERENCE PAPER ... 73

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投資型保險商品評價方法研究

Valuation methods for equity-linked insurance products

計畫編號:

NSC 97-2410-H-004-041-MY2

執行期間:

中華民國

97 年

8 月

1 日至

99 年

7 月

31 日

主持人: 謝明華 E-mail: [email protected] 執行單位: 國立政治大學風險管理與保險學系

中文摘要

受金融風暴影響，近年來金融市場的景氣起伏波動亦較大，能同時保有獲利並控制風險的商品就成為市場的主流，而指數連動保險就成為市場上非常受歡迎的商品，其主要可以分為兩種型態：變額年金(VA)以及指數連動年金(EIA)。然而，由於這二種商品報酬計算結構亦相當的複雜，對保險公司而言，該如何進行這二種商品的評價與風險管理一直是非常重要的課題。在本研究中，我們主要就是專注在變額年金以及涵蓋不同商品條款的指數連動年金的評價方法上。首先，我們會介紹亞洲市場的連結棘輪選擇權(cliquet option)的匯率變額年金商品(quanto variable annuity)，此商品能提供對於下方風險(downside risk)的保護，並同時保有高獲利的潛力，故相當受投資人的喜愛。在本研究的第一個主題，我們提出一個有效率的蒙地卡羅演算法進行此商品的評價，透過數值實例的實驗進行驗證，結果顯示我們的方法相當成功的提高評價匯率變額年金的計算效率。本計畫的第二個主題是專注在指數連動年金(EIAs)的評價，指數連動年金是銷售成長率相當高的金融創新商品，在幾種具有不同商品條款的指數連動年金裡，獲利保障型的指數連動年金(Ratchet EIAs)是最受歡迎的商品，主要是因為其提供

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定期配息並有最低報酬的保證，同時亦保證帳戶價值絕對不會減少。因為隱含匯率連動的獲利保障指數連動年金是連結外幣投資，並提供類似選擇權的屬性。在過去文獻中，僅探討不含匯率性質的指數連動年金的定價。因此，本研究的第二個主題，就是探討匯率指數連動年金的定價。在本文中，我們結合交換利率模型與外幣的無風險利率模型做為後續對此商品進行 Black and Scholes 定價的基礎架構，本文提出的定價公式涵蓋匯利獲利保障指數連動年金的單利與複利計算方式，這可能會存在一個報酬上限，並包含兩種不同類型的幾和平均報酬。在本研究中，我們更進一步地利用數值實例進行實驗分析，並提出不同商品條款和市場參數對商品價值的影響

在本研究的第三個部份，我們針對獲利保障型指數連動年金的封閉解進行更深入的探討，Hardy (2004)宣稱單利型的Ratchet EIAs沒有封閉解，Kijima and Wong (2007) 也認為有報酬上限的複利型Ratchet EIAs 很難求得封閉解。本研究在Black-Scholes的架構下推導出目前為止涵蓋最多商品條款的獲利保障型指數連動年金的封閉解。這些條款包含複利型與單利型、有報酬率上限、以及兩類型的幾何平均報酬率。這些封閉解可以使精算人員很容易地分析各商品條款對保單價值的影響，並架構適當的避險組合。此外，本研究數值分析結果顯明各商品條款對此種指數連動年金價值的個別影響以及各條款彼此間的關連。關鑑字：變額年金、指數連動年金、選擇權定價、Black-Scholes 模型

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Abstract

Since the recent turmoil in financial markets, products that eliminate the downside risk while still providing upside potential are in great demand. Equity-indexed annuities (EIAs) are such products. Equity-indexed annuities (EIAs) are one recent innovative product and the sales have been growing rapidly. Equity-linked insurance contracts have become very popular in the market.

There are two major types of equity-linked insurance contracts: variable annuity (VA) and equity-linked annuity (EIA). However, due to their complicated payoff structure, their valuation and risk management are challenges to the insurers. In our research, we focus on the valuation methods of variable annuity and equity-linked annuity with different contract features.

First, we introduce a variable annuity contract embedded cliquet options in Asia markets and has quanto feature. Such contracts are attractive to the investor because of their protection against downside risk and significant upside potential. We propose an efficient Monte Carlo method to value the contract. Numerical examples suggest our approach is quite efficient.

Second, among several product designs of EIAs, ratchet EIAs are the most popular because returns are credited periodically with a guaranteed minimum and the account value never decreases once the return is credited. Because Quanto Ratchet EIAs link to foreign investments and provide options-like properties. The literature covers the pricing of the EIAs that are not quantos. We intends to fill the hole. To derive the pricing formulas, we added an exchange rate model as well as a foreign risk-free rate model to the pricing framework of Black and Scholes. Our formulas cover quanto ratchet EIA products for both compound and simple versions that may have a return cap and employ two types of geometric return averaging. We provide

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numerical analyses on how contract features and market parameters affect the contract value too.

In further, pricing ratchet EIAs, however, is challenging. Hardy (2004) claimed that the value of the simple version of ratchet EIAs is not analytically tractable even under the standard Black-Scholes framework. Kijima and Wong (2007) could not obtain closed-form solutions for the compound version that has a return cap. In our research, we derive pricing formulas that cover more contract features of ratchet EIAs than the literature did. We obtain closed-form solutions in the Black-Scholes framework for both compound and simple versions of annual-reset ratchet products that may have a return cap and employ two types of geometric return averaging. Via these pricing formulas, actuaries can analyze easily the impacts of various contract features on the contract value and construct appropriate hedging portfolios for the product. Our numerical results demonstrate not only the impacts of individual contract features but also how they affect each other.

Keywords: variable annuity, equity-indexed annuities; option pricing; foreign exchange Black-Scholes model

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1. Introduction

1.1 Variable Annuity and Equity-Linked Annuity

The markets of VA and EIA are very large. According to LIMRA International and Advantage Compendiums, VA and EIA contracts have very high sales volume (Table 1) in United States. Compared to the traditional annuity product, VA and EIA have the advantage of offering additional return when the linked equities perform well. Although VA and EIA share this commonality, they are different in various aspects: I. A typical (non-registered) EIA is an insurance company “general account” product.

This means that EIA is treated as a liability item on insurance company’s balance sheet. On the other hand, VA is a “separate account” product. Except the guarantees embedded in the contract, any gains or losses to the underlying assets in the VA separate account are reflected directly and immediately in the VA contract owners’ accumulation values.

II.Owners of variable annuities are generally allowed to modify their investment allocations periodically, sometimes as often as daily, and VA contract values change according to the performance of the selected investment portfolios. Therefore, VA contracts provide their owners with considerably greater investment flexibility than do EIA contracts. The linked indices in EIA contracts are usually the well-known indices such as S&P 500. So, EIA owners basically cannot have direct influence on the values of linked indices.

III. VAs are considered to be securities and must be registered with the Securities and Exchange Commission (SEC). On contrary, EIAs are not required to register with SEC and most EIAs are not registered.

IV. At maturity, the guarantees embedded in EIA contracts are usually in-the-money. On the other hand, VA’s guarantees are usually out-of-the-money.

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Table 1-1 VA and EIA sales volume in billion dollars in United States, 2003-2007.

In order to be a qualified provider for such structure products, the investment bank must maintain a high credit rating. These structure products are classified as variable annuities in Taiwan, because they are separate account products. However, their financial property are more like EIA products in United States. Following section, we introduce EIAs in more detailed to analysis the features of EIAs.

1.2 Equity-indexed annuities (EIAs)

Since the recent turmoil in financial markets, products that eliminate the downside risk while still providing upside potential are in great demand. Equity-indexed annuities (EIAs) are such products. An EIA is a hybrid between a variable and a fixed annuity that allows the policyholder to participate in the potential appreciation of the stock market while eliminating the downside risk by a minimum return guarantee. The sales in 2009 is $30.1 billion, a 15.4% increase over 2008, and the fourth quarter sales in 2008 were up 13% when compared with the same period one year ago.1

The product designs of EIAs are diverse, but can be divided into three major categories: point-to-point, ratchet, and look-back (including the high-water-mark and Asian-end designs). The return of the point-to-point EIA is determined by the realized return of the linked index between two time points. Ratchet EIAs are more

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favorable because returns are credited periodically with a guaranteed minimum and the account value never decreases once the return is credited. A popular design of the look-back EIA is the high-water-mark that earns the highest return on the index attained during the life of the contract.

1.3 Rratchet EIAs

Among the three categories, ratchet EIAs are the most popular in the markets. Ratchet EIAs may vary in contract features such as reset frequency, return accumulation, return cap, and return averaging. Most ratchet EIAs have the annual-reset feature meaning that the return is credited to the contract annually. The annual return may be accumulated in two ways. The simple version of ratchet EIAs add the annual returns up to give the final payout while the returns in the compound version are accumulated compounded. To reduce the costs of EIAs, the insurer may place a fixed upper limit, also called ceiling or cap, on the annual return. It may also employ an averaging scheme in calculating the annual return to reduce the volatility of credited returns and thus the costs of guarantees. For instance, an insurer may calculate the geometric average of the index return over several sub-periods as the credited return of the period.

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2 Purpose of Research

I. The variable annuity contract considered has quanto feature and embedded cliquet options:

We focus on one of such structure products. The variable annuity contract considered has quanto feature and embedded cliquet options. Such contracts are attractive to the investor because of their protection against downside risk and significant upside potential. We discuss the Monte Carlo valuation approach and then propose efficient algorithms. In particular, we applied variance reduction technique of control variates to improve the performance of Monte Carlo approach.

II. Pricing Rratchet EIAs with the quanto feature

One of our contributions is that we derive the pricing formulas for ratchet EIAs with the quanto feature. A contract is a quanto or cross-currency if the linked index is dominated in a different currency (e.g., Baxter and Rennie, 1996; Hull, 2006). For instance, the contracts pay off in Australian dollar and the linked index is S&P 500 that is dominated in US dollar. The quanto feature is common in the derivatives market. Many variable (also called unit-linked) products of life insurance and annuities also have this feature. Target customers include the people interested in international diversification for their portfolios and the people who live in the countries with less-developed capital markets and want to invest in more-developed markets. Quanto ratchets EIAs are particularly popular in areas such as Asia and Australia.

We derive the pricing formulas for ratchet EIAs with the quanto feature. To incorporate the quanto feature, we add an exchange rate model as well as a

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foreign risk-free rate model to the pricing framework of Black and Scholes.2 Based on the Girsanov’s theorem and the martingale representation theorem , we rewrite the processes of the linked index and the exchange rate so that we may apply the risk-neutral valuation principle to obtain closed-form solutions. Our pricing formulas cover quanto ratchet EIA products with various features including both compound and simple versions that may have a return cap and employ two types of geometric return averaging. Via these formulas, actuaries can easily analyze the impacts of various contract features as well as market parameters on the contract value and construct appropriate hedging portfolios for the product. Moreover, our formulas apply also to the ratchet EIA products that have no quanto features with easy de-generalizations.

III. The valuation method of Rratchet EIAs

We focus on the valuation of ratchet EIAs, the most popular product design of EIAs. Ratchet EIAs may vary in contract features such as reset frequency, return accumulation, return cap, and return averaging. Most ratchet EIAs have the annual-reset feature meaning that the return is credited to the contract annually. The annual return may be accumulated in two ways. The simple version of ratchet EIAs adds up the annual returns to give the final payout while the returns in the compound version are accumulated compoundedly. To reduce the costs of EIAs, the insurer may place a fixed upper limit, also called ceiling or cap, on the annual return. The insurer may also employ an averaging scheme in calculating the annual return to reduce the volatility of credited returns and thus the costs of guarantees. For instance, an insurer may calculate the geometric average of the index return over several sub-periods as the credited return of the period. We derive the pricing formulas for

2

Note that the typical B-S assumptions are commonly seen in the insurance literature, e.g., Tiong (2000), Gerber and Shiu (2003), Lee (2003), Hardy (2004), and Chui, Hsieh, and Tsai (2010).

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both compound and simple versions of annual-reset ratchet EIAs that have a return cap and can employ two types of geometric return averaging. The cases of no return averaging and/or no return cap are then presented as special cases. These closed-form solutions enable us to see through the impacts of these contract features on the value of the contract.

IV. To analyze how contract features and market parameter may affect the contract value

We employed the derived formulas to analyze numerically how contract features and market parameters may affect the contract values. Their effects intertwine with each other. We further learned from these numerical analyses the importance of the quanto feature in determining the contract value. The price of a quanto ratchet EIA might deviate from that of a non-quanto one by XX% under normal market conditions. The deviation could reach XX% when the foreign exchange market exhibit high volatilities and high correlations with the linked investment market. Insurance companies therefore should pay close attention to the cost and risk of the quanto feature. They show that the value of the contract increases with the return cap, decreases with the frequency of averaging, and is higher with the compound accumulation method. The analyses further illustrate that the impacts of contract features are affected by each other. The impact of return cap is the most significant when returns are accumulated compoundedly and when there is no return averaging. The impact of return averaging is reduced significantly by return cap, and the impact of return accumulation is reduced by both return cap and return averaging. Actuaries should therefore bear in mind not only the individual impacts of contract features but also their joint effects when controlling guarantee costs and product risks.

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3 Paper Review

3.1 Variable annuity contracts with cliquet options in Asia

markets

The payoff of the variable annuity contracts under consideration is very similar to simple ratchet EIA contracts with maturity guarantee; see, Hardy (2003) and Hsieh and Chiu (2007) for more information. The VA contracts might also be called cliquet options with global floor (Kjaer 2006).

However, there is one major difference between the VA contracts considered and simple ratchet EIA contracts: the VA contracts considered are quantos. That is, the linked index is dominated in a different currency. For example, the contracts pay off in Australian dollar and the linked-index is S&P 500 which is dominated in United States dollar. The index participation is evaluated annually. Let T be the maturity of a VA contract and S(t) be the linked-index at time t T. We set

(1) which are the annual returns of linked-index. The effective annual returns of the VA contract are defined as

(2) where f is the annual guarantee rate, c is the annual cap rate, and a is the participation rate in the linked-index. Treat these effective annual returns as simple returns and sum them arithmetically, then the total return at maturity is

(3) In addition to annual guarantee rate f, the VA contract also provides guarantee at

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maturity. This type of guarantee is sometimes called global guarantee. Let the initial investment be P. If the maturity guarantee promises a maturity guarantee rate G, then the payoff of the VA contract is

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3.2 Valuation framework of ratchet EIAs

3.2.1 Pricing and hedging of ratchet EIAs

The pricing and hedging of EIAs have been studied by several researchers, and many of them adopted the Black-Scholes (B-S) assumptions (Black and Scholes, 1973). Tiong (2000) derived closed-form solutions for the three major product designs by means of Esscher transforms. Gerber and Shiu (2003) provided closed-form formulas for lookback options and dynamic guarantees embedded in EIAs. Lee (2003) proposed four designs of EIAs to increase participation rates and derived the associated pricing formulas. Hardy (2004) presented a lattice method for valuing ratchet EIAs. Extending the B-S assumption of constant risk-free rate to stochastic interest rates, Lin and Tan (2003) determined the fair participation rates for the three major designs of EIAs numerically under Vasicek (1977) short rate model. Jaimungal (2004) assumed that the underlying index followed a geometric Variance-Gamma process and developed closed-form expressions for prices of point-to-point and ratchet EIAs. Kijima and Wang (2007) adopted the extended Vasicek model and derived explicit pricing formulas for several ratchet EIA products. Recently, Chui, Hsieh, and Tsai (2010) filled a hole left by Kijima and Wang (2007) through deriving the pricing formulas for the capped compound version with two geometric return averaging schemes in the B-S framework.

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3.2.2 Product Specification

The fundamental variable in pricing ratchet EIAs is the annual return calculated based

on the linked index. Let T be the maturity of an EIA contract and S(t) be the linked

index at t≤T . Then the annual return of the linked index over the tth year would

be: T t t S t S Rt , 1,2 , ) 1 ( ) ( L = − = . (5)

Insurers often take averages of the index returns over sub-periods of a year

when calculating the annual return to reduce the guarantee costs through dampening

the return volatility. We analyze two types of geometric averaging in this paper.3

In the first case (which we refer as G1 hereafter), the annual return over the tth year,

1 ,G

t

R , is taken as the geometric average of index sampled at an interval of 1/m. That

is, m m i G t m i t S m i t S R 1 1 0 1 , ) 1 ( ) 1 1 (           + − + + − =

∏

− = . (6)

3_{We do not consider arithmetic averaging for two reasons. Firstly, the annual return calculated using}

the arithmetic averaging scheme is the sum of lognormal random variables. It is well known that the options based on the sum of lognormal random variables have no closed-form pricing formulas under the B-S model (Kemma and Vorst, 1990). Secondly, the closed-form pricing formulas for options based on lognormal random variables can serve as effective control variates in pricing

arithmetic-averaging-based options using the Monte Carlo algorithm (Kemma and Vorst, 1990). The pricing formulas derived later in this paper can hence be used in pricing the EIAs with arithmetic averaging.

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In the second case (referred as G2 hereafter), the annual return over the tth year is denoted by R_t_,G₂as follows: m m i G t t S m i t S R 1 1 2 , ) 1 ( ) 1 1 (           − + + − =

∏

= . (7)

The next step after calculating the annual return is to calculate the return to

be credited to the contract each year. The general formula is as follows:

(

)

(

)

(

R f c

)

R~_t =1+minmax

α

_t_,_⋅−1, , , (8) where R_t_,_⋅denote the annual return over the tth year with or without geometric averaging, α is the participation rate in the linked index, f denotes the minimum

guaranteed return rate (also called the floor rate), and c represents the cap rate. The

participation rate is usually less than 100%, which is reasonable in the sense that

investors sacrifice some of the upside potential for the downside protection of the

minimum guarantee. When f = 0, the product provides a principal/premium

guarantee. The cap rate or ceiling rate c is the maximum rate that can be credited

each year. Placing a cap on the credited return is a direct way to reduce the product

cost.

The annual return credited to the policy can then be accumulated in two ways.

For the compound version of ratchet EIAs, the total return at maturity T is calculated

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∏

= = T t t CR R R 1 ~ . (9)

The version without compounding but with simple adding up, which often referred as

simple ratchet EIAs, would pay out:

( )

∑

= = + − = − + = T t t T t t SR R T R R 1 1 ~ 1 1 ~ 1 , (10)

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3.2.3 Risk-Neutral Valuation

3.2.3.1Valuation of Ratchet EIAs via the Risk-Neutral Valuation Principle

We follow Hardy (2004), Lee (2003), Gerber and Shiu (2003), and Tiong (2000) in adopting the B-S assumptions for the linked index and interest rate. More

specifically, we assume that the linked index S(t) follows the geometric Brownian motion and the short rate of interest r is constant. Therefore, under the risk-neutral measure,

dS (t) = rS (t)dt +σS(t)dz(t) , and (11)

dB(t) = rB(t)dt , (12)

where z(t) is a Wiener process, σ represents the volatility of the linked index (assumed to be constant), and B(t) denotes the risk-free money market account.

There are several reasons why we do not consider stochastic interest rates in our pricing framework. Firstly, our aim is to provide simple pricing formulas similar to B-S pricing formula that facilitates analyzing the contract features and parameters that may have significant impact on the contract value. Interest rates have little impact on contract value because the payoffs of ratchet EIAs do not depend on interest rates, and this statement is partly confirmed by the numerical results in Kijima and Wong (2007).

Secondly, the pricing formulas will be computationally inefficient when considering stochastic interest rates. For instance, the annual returns and discount factor 1/B(T) are (T+1)-dimensional lognormally distributed random variables under risk neutral measure under the pricing framework of Kijima and Wong (2007). With suitable chosen numeraire (zero coupon bond matured at time T), the annual returns and discount factor 1/B(T) become T-dimensional lognormally distributed random

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variables. This approach (forward measure approach) reduces the problem dimension by 1 only and thus cannot break the curse of dimensionality. A rule of thumb in high-dimensional integral problems is to use Monte Carlo type algorithm when T is great than 3. Therefore, it is more suitable to use numerical methods for valuation of ratchet EIAs under stochastic interest rates.

Thirdly, it is questionable to assume the index return and short rate have constant correlation during the whole lifetime of EIAs from the econometric point of view. If the index return and short rate are driven by independent Brownian motions, then the models with stochastic short rates and the models with constant interest rates give the same pricing formulas.

More specifically, let g(S(t):t≦T) be the payoff of a ratchet EIA, rt be the short rate process and P(0,T) be the price of zero-coupon bond paying a unit amount at time T. The price of a ratchet

EIA product, V, under stochastic interest rates is equal to:

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On the other hand, V under the constant interest rate assumption is equal to:

(13) Here we made a common assumption (see Hull (2006) for more detail) that both interest rate models calibrate their parameters to fit the current price of P(0,T), i.e.,

(14) In other words, the price of a ratchet EIA product has the same pricing formula under both assumptions of constant and stochastic interest rates. We therefore choose to adopt the constant interest rate assumption.

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Since the market is arbitrage-free and complete under the Black-Scholes model, the prices of the EIA products with contract features specified as in chapter 3.2 can be represented as expectations according to the risk neutral valuation principle (e.g., Harrison and Kreps, 1979; Harrison and Pliska, 1981). More specifically, the price of the above ratchet EIA contract can be expressed under the risk neutral measure as:

(15) where ver may be C (compound version) or S (simple version).

3.2.3.2Risk-Neutral Valuation of Contracts with quantos

Since the contracts we considered in this paper are quantos, we add an exchange

rate model as well as a foreign risk-free rate model to the pricing framework of Black

and Scholes. The linked index S(t) and exchange rate C(t) are assumed to follow

geometric Brownian motions, and the interest rate r (for local currency) and rf ( for

foreign currency) are assumed to be constants. More specifically,

( )

[

( )

]

( )

, , , , 2 1 1 dt r t D t dD rdt t B t dB t dz t dz dt t C t dC t dz dt t S t dS f C C S S = = + + = + =

ρ

σ

µ

σ

µ

(16)

where

σ

_S is the volatility of the linked index,

σ

_C is the volatility of the exchange

rate,

ρ

is the correlation coefficient of log

( )

S

( )

t _and log

( )

C

( )

t , ρ = 1−ρ2 is the orthogonal complement of

ρ

, and z_i

( )

t , i =1, 2 are independent Brownian

(23)

motions. B

( )

t andD

( )

t denote the domestic and foreign money market accounts, respectively.

We call the model defined in (7) the Black-Scholes quanto model (Baxter and

Rennie, 1996). To make the model more concrete, we could assume a case in which

the local currency is Australian dollar and the linked index is denominated in US

dollar. The model thus have three tradable assets in Australian dollar: the Australian

dollar cash bondB

( )

t , the Australian dollar worth of the US-dollar denominated bond

( ) ( )

t Dt

C , and the Australian dollar worth of the linked indexC

( ) ( )

t S t .

Based on the Girsanov’s theorem and the martingale representation theorem

(see, for example, Bjork (2004)), there exists a unique measure Q under which both

the discounted processes C

( ) ( ) ( )

t Dt B t and C

( ) ( ) ( )

t S t Bt are martingales. The processes S(t) and C(t) under Q can then be written as:

( )

(

)

( )

(

)

[

( )

]

, , 2 1 1 t z d t z d dt r r t C t dC t z d dt r t S t dS C f S C S f ρ ρ σ σ σ ρσ + + − = + − = (17)

where z₁

( )

t and z₂

( )

t are independent standard Brownian motions under measure Q.

According to the risk-neutral valuation principle (see, for example, Harrison

and Kreps (1979) and Harrison and Pliska (1981)), the no-arbitrage price of the EIA

(24)

[

*

]

* R e E V = _Q −rT , (18)

where E_Q

[]

⋅ denotes the expectation operator under measure Q and the asterisk may be CR or SR.

(25)

4 Research Methodology

4.1 VA contract valuation

4.1.1 Valuation formula of VA contract considered

Most of the previous related research in insurance field (Hardy (2004), Lee (2003), Tiong (2000), and Gerber and Shiu (2003)) adopted the Black-Scholes assumptions (Black and Scholes 1973) for the linked-index and interest rate. We follow these assumptions. But, since the VA contract is quanto, we need to add exchange rate model. More specifically, the linked-index S(t) and exchange rate C(t) follow geometric Brownian motions and the interest rate r(local currency) and r f (foreign currency) are constants.

(19) where (t), i = 1,2 are independent standard Brownian motions, is the volatility of the linked index, is the volatility of the exchange rate, r is the correlation coefficient between log(S(t)) and log(C(t)), ̅ 1 is the orthogonal complement of r and B(t) and D(t) denote the local and foreign money market accounts, respectively. We call the models defined in (19) Black-Scholes quanto model (Baxter and Rennie 1996). To make the model more concrete, we might assume that the local current currency is Australian dollar and the foreign currency is US dollar. So, there are three tradables in Australian dollar: B(t), C(t)D(t), and

(26)

Rennie (1996), for example), there exists a unique measure Q such that, under which, both C(t)D(t)/B(t) and C(t)S(t)/B(t)

(20) where and  are independent standard Brownian motions under measure Q. If a derivative’s payoff depends only on S(t) and/or C(t), then its fair value can be

represented as expectation under measure Q (Baxter and Rennie 1996). This fair value is actually equal to the initial value of a (dynamic) replicating portfolio. Such

valuation approach is usually called risk neutral valuation (see, for example, Harrison and Kreps (1979) and Harrison and Pliska (1981)) and measure Q is called risk neutral measure. Therefore, the fair value of the VA contract defined in Section 2 can be written as

(21) where [·] denotes the expectation operator under measure Q.

4.1.2 Monde Carlo method of VA contract considered

It is well known that, under the risk neutral measure Q, log () are independent normal random variables with common mean

and common variance (Hull 2006). Since RM is just a function of , t = 1, . . . ,T. Equation (21) implies Monte Carlo approach can be used to price the

contract. We shall use control variates (see, e.g., Bratley, Fox, and Schrage (1983) and Law and Kelton (2000)) to speed up the Monte Carlo method. From Equation (21), it is easy to see that RM can be a good control variate. To make a valid control variate, we need to compute . To this end, we follow the idea in Hsieh and

(27)

Chiu (2007). Using (2) we can get

(22) where 1 / and 1 /. Set

(23) Then it is easy to see that ’s are independent censored lognormal random variables with censored values and

We use (3) to obtain

(24) Therefore, we reduce the task of computing to the task of computing. To compute , we first write

Then, by representing as and letting (25) and (26) we obtain and

(28)

where f(·) and F(·) are the density function and the cumulative distribution function of standard normal random variable, respectively. Combining these three terms, we get the explicit formula for:

(27) With (13) and (10), the following proposition is straightforward.

Proposition 1

where d1, d2 are defined in (25) and (26). Now, we are ready to test some numerical examples. We begin with a description of the parameters of Black-Scholes quanto model for the VA contracts: the contract maturity T = 5, initial investment P = 100, floor rate f = 1%, the volatility of the linked-index = 25%, the volatility of the exchange rate = 10%, the correlation coefficient r = −0.1, local currency interest rates r = 6%, foreign currency interest rates rf = 4%, and the global guarantee rate G = 110%. We simulate 1000 independent runs of (, · · · , ). From these 1000 simulated paths, we can easily obtain 1000 independent replications of !"#_max(

,G). Based on these independent copies, standard point estimates of

(29)

Table 2: Fair value of the VA contract computed by naïve Monte carlo method. The upper table contains point estimates and the lower table contains their standard errors.

The accuracy of the point estimates in Table 2 are not very satisfactory. We shall apply the variance reduction technique of control variate. In particular, we take advantage of Proposition 1 and select control variate

(28) Using the same 1000 replications of ( , · · · , )., we can also obtain 1000

independent replications of C. Let λ1 and λ2 be any real numbers and set

Since = 0, it is easy to see that $ equal to the fair value of the VA contract. Therefore, it provides alternative mean of computing the price. It is well known that the optimal (variance-minimizing) weight l_ of the control variate is Cov(Y,C)/Var(C) (Bratley, Fox, and Schrage 1983). The quantities of Cov(Y,C) and Var(C) are estimated by the sample covariance and variance. It turns out that control variate Y(l) is quite effective. Table 3 show the results. These results indicate that the accuracy of the estimates has been improved significantly with control variate C.

(30)

Table 3: Fair value of the VA contract computed by Monte carlo method with control variate C = −. The upper table contains point estimates and the lower table

contains their standard errors.

To further quantify the effectiveness of control variate C, we define variance reduction ratio as follows

(29) Because most of the computational effort was used to generate the sample paths of ( , · · · , ). the additional work needed to compute C is minor. Therefore, it seems reasonable to use VRR as a proxy of computational gain. Table 4 shows that the quantities of variance reduction ratio range from 88 to 2070. This indicates control variate C is very effective in reducing the estimator’s variance.

(31)

(32)

4.2 Pricing Formulas of

Ratchet EIAs

4.2.1

. Compound and Simple Ratchet EIAs with the First Type of Geometric Averaging Scheme

Suppose that the linked index pays a continuous dividend yield at a constant rate

d per year.3 It is well known (e.g., Hull, 2006) that %&' (_,*(++ are are independent normal random variables with mean Suppose that the linked index pays a continuous

dividend yield at a constant rate d per year.4 It is well known (e.g., Hull, 2006) that

( )

( ) 1 ,

logRtmG are independent normal random variables with mean

) 2 ( 1 2 1 , σ µ = r−d − m G m and variance 2 2 2 1 , m G m σ

σ = under the risk neutral measure.

Rewrite equation (4) as

(

1 )

min(max(

,

),

~

( ) 1 , ) ( 1 ,

α

R

f

α

c

α

R

_tm_G

=

−

+

_tm_G

in which f_α =1+ f

α

and c_α =1+c

α

. Let

) ), , min(max( (, )1 ) ( 1 , R fα cα X tmG m G t = . (30)

Then X_t(_,m_G)₁ are independent censored lognormal random variables with censored values fα and cα.

Combining equations (9), (14), (15), and (30), we obtain

(

)

m T G t rT T t m G t rT m G C E e X e E X V 1 [1 ( (, )1)] 1 ) ( 1 , ) ( 1 , = − + ×      + − = − = −

∏

_α

. (31) Similarly,

4_{Note that dividends are excluded when calculating the annual return based on the index. This is one}

(33)

(

)

[

1 ( )

]

~ 1 (, )1 1 ) ( 1 , ) ( 1 , m G t rT T t m G t rT m G S E e T R e T T E X V = − + ×            + − = − = −

_∑

_α _α _. ₍₃₂₎

We can thus obtain the closed-form solutions as soon as we get the explicit formula of

) (X_t(_,m_G)₁ E .

Proposition 1 The closed-form solution for the compound ratchet EIA with the first

type of geometric averaging scheme is:

( )

[

(

) (

)

]

(

m

)

T G G m m G G m m G m G rT m G C e f d e d d c d V mG mG               − Φ + − Φ − − Φ + Φ + − = − + ( ) 1 , 2 1 , ) ( 1 , 1 1 , ) ( 1 , 2 2 1 ) ( 1 , 1 ) ( 1 , 2 1 , 1 , 1 α α _α µ σ σ σ _α (33) , where 1 , 1 , ) ( 1 , 1 log G m G m m G f d σ µ α − = , 1 , 1 , ) ( 1 , 2 log G m G m m G c d σ µ α − = , ) 2 ( 1 2 1 , σ µ = r−d− m G m , and m G m σ σ , 1= .

Proof: We first write

[

α α

]

α

(

α

)

α αP R f E R f R c c P R c f X E tmG m G t m G t m G t m G t = ≤ + ≤ ≤ + ≥ ) ( 1 , ) ( 1 , ) ( 1 , ) ( 1 , ) ( 1 , ) ( ) ; ( _. Representing R_t(_,m_G₁) as _eµ +m σmN(0,1) , we obtain

(

)

(

( )

) ( )

( ) 1 , 1 ) ( 1 , 1 ) ( 1 , 0,1 m G m G m G t f P N d d R P ≤ _α = ≤ =Φ ,

(

)

(

( )

) (

( )

)

1 , 2 ) ( 1 , 2 ) ( 1 , 0,1 m G m G m G t c P N d d R P ≥ _α = ≥ =Φ− , and

[

]

( )

(

) (

)

[

]

, ; ) ( 1 , 1 ) ( 1 , 2 2 1 ) ( 1 , ) ( 1 , 2 1 , 1 , ) ( 1 , 2 ) ( 1 , 1 1 , 1 ,

σ

φ

σ µ σ µ α α − Φ − − Φ = = ≤ ≤ + × +

∫

m G m G d d z m G t m G t d d e dz z e c R f R E G m G m m G m G G m G m

(34)

where

φ

( )

⋅ and Φ

( )

⋅ denote the density function and the cumulative distribution function of the standard normal random variable respectively. We thus get the

explicit formula for E(Xt) as:

( )

[

(

) (

)

]

(

( )

)

1 , 2 1 , ) ( 1 , 1 1 , ) ( 1 , 2 2 1 ) ( 1 , 1 ) ( 1 , 2 1 , 1 , ) (X_tm_G f d m_G e d m_G _m_G d m_G _m_G c d m_G E = _αΦ + µmG+ σmG Φ −

σ

−Φ −

σ

+ _αΦ− . Substitute E(X_t(_,m_G)₁) into equation (31), and the proposition follows.

Proposition 2 The closed-form solution for the simple ratchet EIA with the first

( )

[

(

) (

)

]

(

)

              − Φ + − Φ − − Φ + Φ + − = − + ( ) 1 , 2 1 , ) ( 1 , 1 1 , ) ( 1 , 2 2 1 ) ( 1 , 1 ) ( 1 , 2 1 , 1 , 1 mG mG m G G m m G m G rT m G S e T T f d e d d c d V

α

_α µmG σmG

σ

_α (32)

, where d₁(_,m_G)₁, d₂(_,m_G)₁, µm,G1, and

σ

m,G1 are as defined in Proposition 1.

Proof: The proof follows the same lines as the proof of Proposition 1.

Ratchet EIAs without return averaging are equivalent to the case of m = 1.

Therefore, two corollaries follow straightforwardly.

Corollary 3 The closed-form solution for the compound ratchet EIA with no return

(35)

( )

[

(

) (

)

]

( )

T rT C e f d e d d c d V               − Φ + − Φ − − Φ + Φ + − = − + 2 1 2 2 1 1 2 1

α

_α µ σ

σ

_α , (33) where σ µ α − = f d₁ log , σ µ α − = c d₂ log , and 2 2 σ µ =r−d− .

Corollary 4 The closed-form solution for the simple ratchet EIA with no return

averaging scheme is:

( )

[

(

) (

)

]

(

)

              − Φ + − Φ − − Φ + Φ + − = − + 2 1 2 2 1 1 2 1 T T f d e d d c d e VS rT α σ µ α

σ

α

, (34)

where d₁, d₂, and µ are as defined in Corollary 3.

Furthermore, ratchet EIAs without ceiling rates are special cases of the

associated capped products with c→∞ . We thus have the following four

corollaries.

Corollary 5 The closed-form solution for the non-capped compound ratchet EIA

with the first type of geometric averaging scheme is:

( )

(

)

, 1 2 _, ₁ ₁(_, )₁ 1 ) ( 1 , 1 ) ( ' 1 , 2 1 , 1 , T m G G m m G rT m G C e f d e d V mG mG               − Φ + Φ + − = − _α _α µ + σ _σ α (35)

(36)

Corollary 6 The closed-form solution for the non-capped simple ratchet EIA with

the first type of geometric averaging scheme is:

( )

(

)

, 1 , 1 1(, )1 2 1 ) ( 1 , 1 ) ( ' 1 , 2 1 , 1 ,               − Φ + Φ + − = − + m G G m m G rT m G S e T T f d e d V

α

_α µmG σmG

σ

(36)

where d₁(_,m_G₁), µm,G1, and

σ

with no return averaging scheme is:

( )

(

)

, 1 2 ₁ 1 1 ' 2 T rT C e f d e d V               − Φ + Φ + − = − _α _α µ+ σ _σ α (37)

where d₁ and µ are as defined in Corollary 3.

no return averaging scheme is:

( )

(

)

              − Φ + Φ + − = − + 1 2 1 1 ' 2 1 T T f d e d e VS rT

α

σ

σ µ α , (38)

(37)

4.2.2 Compound and Simple Ratchet EIAs with the Second Type of Geometric Averaging Scheme

Equation (6) can be rewritten as

( )

[

]

m m k k m m m m m G t Y Y Y Y Y t S t S t S m t S t S m m t S t S m m t S R 1 1 1 1 2 1 1 ) ( 2 , ) 1 ( ) 1 ( 1 ) 1 ( 2 ) 1 ( 1       = ⋅ ⋅⋅ ⋅ ⋅ =             − −       − ⋅⋅ ⋅ −       − − ⋅ −       − − =

∏

= − . (39)

Each Yk follows the lognormal distribution with parameters )

2 ( 2 σ µ = r−d− m k y and 2 2 σ σ m k

y = . However, Yk are not independent with each other and are difficult to

analyze.

Let Z₁ ≡log Y

( )

₁ , Z₂ ≡log

( )

Y₂ −log

( )

Y₁ ,…,Z_n ≡log

( )

Y_m −log

( )

Y_m₋₁ . It is easy to show that Zi are Brownian motion increments and thus are independent and

normally distributed with mean ) 2 ( 1 σ2 µ = r−d− m Z and variance m Z 2 2 σ σ = .

Taking log on both sides of equation (39), we get:

(

)

(

)

[

m

]

m k k m G t Z Z Z Z Z m Y m R =

∑

= + + +L+ +L =1 1 1 2 1 ) ( 2 , 1 log 1 log . (40)

Then it follows that log

( )

Rt(_,mG)₂ are independent normal random variables with mean

) 2 ( 2 1 2 2 , σ µ = + r−d − m m G m and variance

(

)(

)

2 2 2 2 , 6 1 2 1 _σ σ m m m G m + + = . Define min(max( (, )2, ), ) ) ( 2 , R fα cα X tmG m G

t = . Employing the same logic in deriving

equations (31) and (32) and Propositions 1 and 2, we obtain the following two

(38)

Proposition 9 The closed-form solution for the compound ratchet EIA with the

second type of geometric averaging scheme is:

( )

[

(

) (

)

]

(

m

)

T G G m m G G m m G m G rT m G C e f d e d d c d V mG mG               − Φ + − Φ − − Φ + Φ + − = − + ( ) 2 , 2 2 , ) ( 2 , 1 2 , ) ( 2 , 2 2 1 ) ( 2 , 1 ) ( 2 , 2 2 , 2 , 1 α α _α µ σ σ σ _α , (41) , where 2 , 2 , ) ( 2 , 1 log G m G m m G f d σ µ α − = , 2 , 2 , ) ( 2 , 2 log G m G m m G c d

σ

µ

α − = , ) 2 ( 2 1 2 2 , σ µ = + r−d − m m G m , and

(

)(

)

2 2 2 2 , 6 1 2 1 _σ σ m m m G m + + = .

Proposition 10 The closed-form solution for the simple ratchet EIA with the second

( )

[

(

) (

)

]

(

)

              − Φ + − Φ − − Φ + Φ + − = − + ( ) 2 , 2 2 , ) ( 2 , 1 2 , ) ( 2 , 2 2 1 ) ( 2 , 1 ) ( 2 , 2 2 , 2 , 1 m_G m_G _m_G m_G _m_G m_G rT m G S e T T f d e d d c d V α α _α µmG σmG σ σ _α (42)

, where d₁(_,m_G)₂, d₂(_,m_G)₂, µm,G2, and

σ

Letting c→∞ in Propositions 9 and 10, we obtain the two corollaries

corresponding to the non-capped ratchet EIAs with the second type of geometric

(39)

with the second type of geometric averaging scheme is:

( )

(

m

)

T G G m m G rT m G C e f d e d V mG mG               − Φ + Φ + − = − + ( ) 2 , 1 2 , 2 1 ) ( 2 , 1 ) ( ' 2 , 2 2 , 2 , 1 α α _α µ σ σ , (43)

where d₁(_,m_G)₂, µm,G2, and

σ

the second type of geometric averaging scheme is:

( )

(

)

              − Φ + Φ + − = − + ( ) 2 , 1 2 , 2 1 ) ( 2 , 1 ) ( ' 2 , 2 2 , 2 , 1 m G G m m G rT m G S e T T f d e d V α α _α µmG σmG σ , (44)

(40)

4.2.3 Numerical Examples

To illustrate how various contract features affect the value of the contract, we set

up numerical examples using different combinations of feature specifications. The

benchmark case is that contract maturity T = 7 years, initial investment P = $100,

floor rate f = 0 (i.e., principal guarantee), the ceiling rate c = 20%, the participating

rate

α

= 100%, the volatility of the linked-index σ = 25%, the risk-free rate r = 6%,

the dividend rate of the linked-index d = 2%, and the number of averaging in a year m

= 2 (when applicable). We are interested in the impacts of return cap, return

averaging, and return accumulation on the contract value.5

4.2.3.1 Impact of Return Cap

The value of the contract increases with the return cap, as each line of Table 1

shows. The case of no return cap has the largest value in each line since the case is

5_{Other parameters, including the contract maturity, floor rate, participating rate, volatility and}

dividend rate of the linked index, and risk-free rate, will affect the value of the contract. We do not analyze how they affect the contract value in this paper mainly for the sake of the paper length but also because we may infer their effects either from the standard option pricing analysis or by intuition. The inner parenthesis of Equation (4), max(α(R_t(_,m_Gj)−1),f), defines a payoff function similar to that of holding a call option. We therefore can infer from Table 9.1 of Hull (2006) that the value of a ratchet EIA product will increase with the volatility of the linked index and the risk-free rate but decrease with the dividend rate of the index. The contract value will increase with the contract maturity since the annual-reset ratchet EIA product is like a series of one-year call options and longer maturities means more options. The contract value will increase with the participating rate and the floor rate for obvious reasons. Furthermore, the above inferences are further confirmed by numerical implementation of the propositions and corollaries established in section III.

(41)

like having an infinite return cap. The contract value increases with the return cap,

as it should, because capping the return that can be credited to the contract truncates

the upside potential.

The value increases at a diminishing rate when the return cap (c) rises steadily

from 0.1 to 0.4. For instance, the contract value increases 22%, 14%, and 8%

respectively as the return cap increases 0.1 at a time from 0.1. This is reasonable

because the probability of hitting the upper bound decreases at an increasing rate

when the upper bound rises as long as the probability densities of positive returns are

decreasing functions of returns.

We further observe that the impact of return cap is the most significant when

there is no return averaging and is the least significant when returns are averaged by

the first type of scheme. The underlying reason is that the first type of averaging

scheme has the most significant averaging effect. It averages over non-overlapping

sub-periods while the second type averages on cumulative returns of sub-periods.

The stronger return averaging effect decreases the probability of hitting the upper

bound more and thus reduces the impact of return cap.

The impact of return cap is more significant when returns are accumulated

compoundedly than the corresponding case when returns are accumulated additively

(42)

generates higher returns in our current parameter settings and thus is bounded more

by return caps.6

We employ the propositions and corollaries established in section III to examine

how return cap affects the contract value of a ratchet EIA product. The annual return

credited to a ratchet EIA product may be accumulated by the compound-interest way

or the simple-interest way (denoted as Compound and Simple respectively). In

calculating the annual return, the insurer may adopt averaging schemes. Section III

analyzes two types of geometric averaging schemes and treats no averaging as a

special case. They are denoted as 1st averaging, 2nd averaging, and no averaging

respectively. This table presents the values of the ratchet EIAs products with several

combinations of return accumulation methods and averaging schemes under five

return caps (c): 10%, 20%, 30%, 40%, and No Cap (i.e., c→∞). We further display

the incremental percentage change caused by the increase of the return cap. More

specifically, we calculate ) ( ) ( ) ' ( c CV c CV c CV −

in which CV(c) denotes the contract value

given the return cap of c.

Return Cap (c)

6

The higher return of the compound version can be inferred from the higher contract value. The compound version may not generate higher returns in other parameter settings. When the simple version produces higher returns and thus has a higher contract value, it will be bounded more by the return cap than the compound version.

(43)

Return Accumulation and Averaging Scheme 0.1 0.2 ) 1 . 0 ( ) 1 . 0 ( ) 2 . 0 ( CV CV CV − 0.3 ) 2 . 0 ( ) 2 . 0 ( ) 3 . 0 ( CV CV CV − 0.4 ) 3 . 0 ( ) 3 . 0 ( ) 4 . 0 ( CV CV CV − No Cap Compound; No Averaging 88.66 108.22 22% 122.89 14% 132.90 8% 148.18 Simple; No Averaging 85.82 99.69 16% 108.74 9% 114.40 5% 122.37 Compound; 1st Averaging 84.47 93.08 10% 95.77 3% 96.40 1% 96.54 Simple; 1st Averaging 82.51 89.17 8% 91.14 2% 91.59 0% 91.70 Compound; 2nd Averaging 87.45 103.41 18% 113.18 9% 118.42 5% 122.97 Simple; 2nd Averaging 84.88 96.49 14% 102.86 7% 106.08 3% 108.79

4.2.3.2Impact of Return Averaging

Tables 2A and 2B show that the contract value decreases with the frequency of

averaging. For both averaging schemes, the ratchet contract has the lowest value

when m = 4 and is the most valuable when there is no return averaging in each line of

Tables 2A and 2B. The impact of return averaging can be rather significant. For

instance, the value of the contract with the first type of averaging scheme decreases

about 35% for the compound contract with no cap when the minimal averaging

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quarterly, compared with the case of no return averaging. The frequency of return

averaging would decrease the contract value because higher frequencies produce

stronger averaging effects and reduce the volatilities of annual returns. The reduced

volatilities decrease the value of the options embedded in the ratchet EIA products.

The existence of return cap reduces the impact of return averaging significantly.

For instance, the value of the contract with the first type of averaging scheme

decreases by 14% (vs. 35%) and 27% (vs. 47%) respectively if m increases to 2 and 4

when the return is capped at 20%. For the compound version with the second type

of averaging, the 20% return cap decreases the impact of return averaging from 17%

to 4% and from 24% to 7% when m increases from 1 to 2 and 4 respectively. The

reason behind the above results is that both return cap and return averaging work to

reduce the value of the option embedded in the ratchet EIA products through

truncating the upside potential and reducing the volatility of annual returns

respectively. The effect of the upside potential truncation substitutes for that of

volatility reduction to some extent when pricing the contract and thus diminishes the

impacts of return averaging.

The impact of return averaging is more significant for the compound version

than for the simple version. For instance, the value of the simple contract with the

(45)

36% (vs. 47%) respectively when m increases to 2 and 4. The value of the simple

contract with the second type of averaging scheme and no return cap is reduced by

16% when m increases from 1 to 4 while the reduction of the corresponding

compounding case is 24%. These results originate from the compound version

having values than the simple version in our current parameter settings. Higher

values imply higher returns and/or higher volatility that will be affected by return

averaging to a larger extent.

We employ the propositions and corollaries established in section III to examine how return averaging affects the contract value of a ratchet EIA product. The annual return credited to a ratchet EIA product may be accumulated by the compound-interest way or the simple-interest way (denoted as Compound and Simple respectively). In calculating the annual return, the insurer may adopt averaging schemes. Section III analyzes two types of geometric averaging schemes and treats no averaging as a special case. They are denoted as 1st averaging, 2nd averaging, and no averaging respectively. This table presents the values of the ratchet EIAs products with several combinations of return accumulation methods and return caps under three return averaging frequencies (m): 1 (i.e., no averaging), 2, and 4 for the two types of return averaging schemes. We further display the incremental percentage change caused by the increase of the averaging frequency. More specifically, we calculate ) ( ) ( ) ' ( m CV m CV m CV −

in which CV(m) denotes the contract value

given the averaging frequency of m.

A: 1st Return Averaging