1. INTRODUCTION
5.5 H EDGING THE RISKS OF THE RATCHET EIA PRODUCTS
Our formulas will also be useful in hedging the risks of the ratchet EIA products.
Insurers can hedge the risks introduced by embedded options using a passive approach or the dynamic-hedging approach (Boyle and Hardy, 1997). Under the passive method the insurance company offsets the liability associated with the embedded option by purchasing appropriate options in an exchange and/or from another financial institution. For instance, the insurer may purchases call options with the same underlying stock indexes in an exchange to hedge the embedded call options in the ratchet EIA products. These exchange-traded options have short maturities only, but the insurer may roll them over to provide longer-term protections. If the insurer is concerned with the basis risk resulted from the complex contract features of the ratchet EIA products (e.g., return averaging), it may purchase average rate options in an over-the-counter market. It may even arrange an equity swap with an investment bank. Our formulas will help insurers to assess the due prices/costs of the above hedging arrangements.
Under the dynamic-hedging approach, the assets of the portfolio are adjusted on an ongoing basis so that the fund at maturity provides the minimum guaranteed amount when the guarantee is operative and the value of the assets otherwise. The insurer can employ our formulas to derive the compositions of the replicating portfolios that will be adjusted dynamically to reflect the changing indexes and time to maturity. Due to the existence of transactions costs, the insurer has to adjust the replicating portfolios discretely rather than continuously and will incur hedging errors.
It therefore faces the tradeoff between discrete hedging errors and transaction costs.
Hardy (2003; chapter 8) provides detailed descriptions and assessments on this dynamic-hedging approach. Her results, in general, showed that the pricing formulas derived under simple Black-Scholes assumptions can have good hedging capacity for more general assumptions about linked-index and interest rate, which provide another justification for using the B-S framework.
6 Reference
1. Boyle, P. E. and Hardy, M. R. (1997) Reserving for maturity guarantees: two approaches, Insurance: Mathematics and Economics, 21, 113-127.
2. Gerber, Hans U., Elias S. W. Shiu, and Griselda Deelstra, 2003, “Pricing lookback options and dynamic guarantees,” North American Actuarial Journal 7, 48-67.
3. Baxter, M., and A. Rennie. 1996. Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press.
4. Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities.
Journal of Political Economy 81:637–59.
5. Bjork, T. 2004. Arbitrage Theory in Continuous Time, 2nd eds. Oxford University Press.
6. Bratley, P., B. L. Fox, and L. Schrage. 1983. A guide to simulation. New York:
Springer-Verlag.
7. Gerber, H., and E. Shiu. 2003. Pricing lookback options and dynamic guarantees.
North American Actuarial Journal 7(1):48–67.
8. Hardy, M. 2004. Ratchet equity indexed annuities. In 14th Annual International AFIR Colloquium.
9. Hardy, M. R. 2003. Investment guarantees: Modelling and risk management for equity-linked life insurance.Wiley, New York.
10. Harrison, J. M., and D. M. Kreps. 1979. Martingales and arbitrage in multiperiod security markets. Journal of Economics Theory 20:381–408.
11. Harrison, J. M., and S. R. Pliska. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11:215–60.
12. Hsieh, M.-H., and Y.-F. Chiu. 2007. Monte carlo methods for valuation of ratchet
equity indexed annuities. In Proceedings of the 2007 Winter Simulation Conference, ed. S. Henderson, B. Biller, M.-H. Hsieh, and J. Shortle, 998–1003.
Piscataway, New Jersey, USA: Institute of Electrical and Electronics Engineers, Inc.
13. Hull, J. C. 2006. Options, futures, and other derivatives securities, 6th edition.
Prentice Hall International Editions.
14. Jaimungal, S. 2004. Pricing and hedging equity indexed annuities with Variance-Gamma deviates.
Http://www.utstat.utoronto.ca/sjaimung/papers/eiaVG.pdf.
15. Kemma, A. G. Z. and A. C. F. Vorst, 1990, “A Pricing Method for Options Based on Average Asset Values,” Journal of Banking and Finance 14, 113–129.
16. Kijima, M., and T. Wong. 2007. Pricing of ratchet equity-indexed annuities under stochastic interest rates. Insurance: Mathematics and Economics 41: 317-338.
17. Kjaer, M. 2006. Fast pricing of cliquet options with global floor. Journal Of Derivatives 14 (2): 47–60.
18. Law, A. M., and W. D. Kelton. 2000. Simulation modeling & analysis. 3rd ed.
New York: McGraw-Hill, Inc.
19. Lee, H. 2003. Pricing equity-indexed annuities with pathdependent options.
Insurance, Mathematics, and Economics 33(3):677–690.
20. Lin, S. X., and K. S. Tan. 2003. Valuation of equity-indexed annuities under stochastic interest rates. North American Actuarial Journal 6: 72–91.
21. Tiong, S. 2000. Valuing equity-indexed annuities. North American Actuarial Journal 4:149–170.
22. Vasicek, O. 1977. An equilibrium characterization of the term structure. Journal of Financial Economics 5: 177-188.
7 Appendix
7.1 Journal Paper
1. 謝明華;邱于芬*;陳松男, 2010.09, "Fast Algorithms for Pricing Ratchet Equity Indexed Annuities," International Research Journal of Finance and Economic, No.48, pp.144-152. (EconLite)(*為通訊作者)
Valuation of Ratchet Equity-Indexed Annuities 獲利保障型指數連動年金之評價
邱于芬Yu-Fen Chiu
國立政治大學 National Chengchi University 謝明華 Ming-Hua Hsieh*
國立政治大學 National Chengchi University 蔡政憲 Chenghsien Tsai**
國立政治大學 National Chengchi University 陳威光Wei-Kuang Chen
國立政治大學 National Chengchi University
Abstract
Equity-indexed annuities (EIAs) are one recent innovative product and the sales have been growing rapidly. 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. 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 this paper 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: equity-indexed annuities; option pricing; Black-Scholes model
* The author is grateful to the National Science Council of Taiwan for its financial support (project number NSC 97-2410-H-004-041-MY2).
** Corresponding author. E-mail: [email protected]. Tel: +886-2-2936-9647. Fax: +886-2-2939-3864. The author is grateful to the National Science Council of Taiwan for its financial support (project number NSC 96-2918-I-004-006 and NSC 96-2416-H-004 -026 -MY3) and to Santa Clara University for the kind support during the author’s visit.
1
摘要
指數連動年金（EIAs）是近年來頗受歡迎的創新產品，其銷售量一直在快速地成長。
在眾多種指數連動年金的商品中，獲利保障型指數連動年金（Ratchet EIAs）可以說是最受 市場歡迎的，因為年金每年計算的報酬率有最低的保障。然而要評價此種獲利保障型指數 連動年金也是困難的。譬如，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。
本文在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，本文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。
關鍵字：指數連動年金、選擇權定價、Black-Scholes模型
2
Valuation of Ratchet Equity-Indexed Annuities 獲利保障型指數連動年金之評價
I. Introduction
Equity-indexed annuities (EIAs) are regarded as one of the most innovative products
brought into the insurance market in years (Jaimungal, 2004). 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 guarantee is usually high enough to meet the nonforfeiture laws so that the EIA
does not need to be registered with the Securities Exchange Commission (SEC) and may enjoy
tax deferring. EIAs thus have gained popularity. The sales increased from $1.5 billion in
1996 to more than $26 billion in 20081.
The three major product designs of EIAs, in the order of decreasing sales volumes, are
ratchet, point-to-point, and lookback (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 favorable because returns are credited
periodically with a guaranteed minimum and the account value never decreases once the return is
1 Please see online reports on Advantage Compendium (http://www.indexannuity.org).
3
credited. A popular design of the lookback EIA is the high-water-mark that earns the highest
return on the index attained during the life of the contract.
Pricing EIAs is a challenging problem due to the complex payoff structure and attracts
research attention. Tiong (2000) derived closed-form solutions for the values of the three major
types of EIAs in the standard Black-Scholes (B-S) framework2. Gerber and Shiu (2003)
presented closed-form formulas for pricing lookback options and dynamic guarantees. Lee
(2003) proposed four product designs of EIAs to increase participation rates and derived
associated pricing formulas. Extending the constant risk-free rate assumption in the previous
papers to the short rate model of Vasicek (1977), Lin and Tan (2003) determined the fair
participating rates numerically for the three major types of EIAs. Kijima and Wong (2007)
adopted the extended Vasicek model and derived closed-form price formulas for ratchet EIAs.
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.
This paper focuses 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
2 Under the B-S framework, the linked index follows the geometric Brownian motion while the risk-free rate is assumed to be constant.
4
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 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.
Our pricing formulas cover more contract features than the literature did. Tiong (2000)
did not derive the pricing formulas for the capped compound version with return averaging.
Hardy (2004) claimed that the value of the simple version is not analytically tractable and
proceeded with numerical analyses in the B-S framework. The ratchet EIA considered by
Jaimungal (2004) under a geometric Variance-Gamma index process is the capped compound
version without return averaging. Under a stochastic interest rate environment, Kijima and
Wong (2007) derive closed-form solutions for both compound and simple versions of ratchet
5
EIAs with geometric averaging. They have to resort to simulation for valuation, however,
when return cap is considered in the compound version of EIAs. Our major contribution to the
literature is that we derive the pricing formulas for the capped compound version with two
geometric return averaging schemes in the B-S framework. In addition, our closed-form
solutions for comprehensive contract features, including return accumulation, return cap, and
return averaging, will be a more helpful guide for actuaries involved in ratchet EIAs than the
literature.
The numerical analyses utilizing our pricing formulas demonstrate how contract features
may affect the contract value. 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.
The paper is organized as follows. In section 2 we describe the ratchet EIA contracts
6
under consideration and set up the risk-neutral pricing formula in the B-S framework. The
pricing formulas for the EIAs contracts are derived in section 3. Section 4 contains the
numerical analyses on the impact of contract features on product prices. Conclusions and
remarks are presented in section 5.
II. Product Specification and Valuation
1. Product Specifications
The fundamental variable in pricing the ratchet EIA is the annual return calculated based on
the index. Insurers often take averages of the index returns in a certain way when calculating
the annual return to reduce the guarantee costs through dampening the volatility of credited
returns. In the first type of geometric averaging scheme, the annual return is calculated as
follows:
m m
i m G t
m t i
S
m t i
S R
1 1
0 ) (
1
, ]
) 1 (
1) 1 ( [
∏
−= − +
+ +
= − , (1)
in which S(t) represents the linked-index at time t, t = 1, 2,…, T and T denotes the maturity of a
ratchet EIA contract. The formula represents the geometric average of m sub-period returns
over the tth year. In the second type of geometric averaging scheme, the annual return is
calculated as:
7
The case of no return averaging is equivalent to setting m =1 in equations (1) and (2) and results
in the following annual return calculation:
).
Another popular return averaging scheme is arithmetic averaging scheme. In this case, the
annual return becomes sum of lognormal random variables. It is well known that options based
on the sum of lognormal random variables, under B-S model, have no closed-form pricing
formulas (Kemma and Vorst, 1990). On the other hand, the closed-form pricing formulas
derived for the options based on product of lognormal random variables, when exist, can serve as
effective control variates for pricing arithmetic-averaging-based options using Monte Carlo
algorithm (Kemma and Vorst, 1990). Therefore, our pricing formulas derived later in this
paper can be used for pricing EIAs using arithmetic averaging scheme.
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:
)
where Gj may be G1 or G2, α is the participation rate in the linked-index, f denotes the minimum
8
guaranteed return rate (also called floor), 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 product with no cap can be deemed as a special case of the capped
product with c→∞.
The annual return can then be accumulated in two ways. For the compound version of
ratchet EIAs, the payoff at maturity T based on an initial premium of $1 invested at time 0 is
calculated as:
The payoff at maturity T of the simple version based on an initial premium of $1 at time 0 is:
∑ ∑
2. Valuation 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
9
interest r is constant. Therefore, under the risk-neutral measure,
) ( ) ( )
( )
(t rS t dt S t dz t
dS = +σ , and (7)
dt t rB t
dB( )= ( ) , (8)
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 variables. This approach (forward measure
approach) reduces the problem dimension by 1 only and thus cannot break the curse of
10
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.
stochastic short rates and the models with constant interest rates give the same pricing formulas.