Valuation of Guaranteed Contracts Set Relative to Cross‐currency Stochastic Rates of Return

• Author: Son‐Nan Chen,Tsung‐Yu Hsieh,Chi‐Hsun Chou
• Date: 01 August 2014
• Published date: 01 August 2014
• DOI: http://doi.org/10.1111/ajfs.12059

Valuation of Guaranteed Contracts Set

Relative to Cross-currency Stochastic Rates

of Return*

Tsung-Yu Hsieh**

Department of Banking and Finance, Tamkang University

Chi-Hsun Chou

Department of Management, Fo Guang University

Son-Nan Chen

Shanghai Advanced Institute of Finance, Shanghai Jiao Tong University

Received 6 May 2013; Accepted 17 April 2014

Abstract

We derive the pricing formulas for guaranteed contracts with guaranteed minimum rates of

return linked to cross-currency stochastic rates of return under a cross-currency framework.

These rates are often embedded in contracts such as life and pension insurance policies, guar-

anteed investment contracts, and index-linked bonds. Valuation of such contracts has not

been investigated in the previous literature. Our research finds that the past valuation of

these rates via a single-currency framework causes significant underestimation under both

maturity and (especially) multi-period guarantees. Our pricing formulas are more suitable,

tractable, and feasible in practice than those in the previous literature.

Keywords Cross-currency; Interest rate; LIBOR; Rate of return guarantee; Stochastic

JEL Classification: G12, G13, G23

1. Introduction

A frequent way to reduce financial risk in financial contracts is to embed policies

with minimum rates of return guarantees to bind the returns from below. Such con-

tracts include life and pension insurance policies, guaranteed investment contracts

*Acknowledgements: The authors would like to thank the Editor, the Associate Editor and

the anonymous referees for their comments and suggestions. This research was supported by

the National Science Council, Taiwan.

**Corresponding author: Tsung-Yu Hsieh, Department of Banking and Finance, Tamkang

University, No. 151, Yingzhuan Road, Tamsui District, New Taipei City 25137, Taiwan. Tel:

886-2-26215656 ext 2859, Fax: 886-2-26209653, email: gtyhsieh@gmail.com.

Asia-Pacific Journal of Financial Studies (2014) 43, 589–619 doi:10.1111/ajfs.12059

©2014 Korean Securities Association 589

(GICs; Walker, 1992), index-linked bonds, and so on. Because these guarantees ar e

often embedded in policies issued by insurance companies, investment banks, or

governments, it is important for issuers to know the value of the policies they are

selling. Since these guarantees are usually surprisingly expensive, issuers may incor-

rectly charge very small premiums, thereby putting their financial stability at risk.

Moreover, insurance companies in some countries are required explicitly to inform

their customers of the economic value of embedded guarantees. As a result, a further

analysis of accurate pricing for rate-of-return guarantees is critical and warranted.

A variety of guarantee designs is used in financial contracts embedded with

guaranteed rates of return in practice. One class of these guarantees is the so-called

absolute guarantees, for which the minimum rates of return are set to be determin-

istic. The other is the so-called relative guarantees (Lindset, 2004), that is, guaran-

tees for which minimum guaranteed rates of return are linked to a stochastic rate

of return on an asset such as an index, a reference portfolio, or a specific asset

traded in a financial market. Relative guarantees arose from the low guaranteed

rates of absolute guarantees, which are unable to attract contract participants. How-

ever, using higher guaranteed rates to attract contract participants may cause con-

tract issuers to bear financial burdens, likely leading to higher financial risk.

Previous research on valuing guarantees for life insurance products or pension

funds has focused on absolute guarantees, which provide participants with a con -

stant or predetermined minimum rate of return. The existing literature on absolute

guarantees under deterministic interest rates includes the work of Brennan and Sch-

wartz (1976), Boyle and Schwartz (1977), Boyle and Hardy (1997), and Grosen and

Jorgensen (1997, 2000). Persson and Aase (1997) and Hansen and Miltersen (2002)

employed the Vasicek (1977) interest rate model. Miltersen and Persson (1999),

Lindset (2003), and Bakken et al. (2006) adopted the Heath-Jarrow-Morton (HJM)

framework (Heath et al., 1992).

Despite the popularity of relative rate of return guarantees, especially those

issued in Latin America, there is little research on these when compared to the

extensive literature on absolute guarantees. Only a few articles have been written on

relative rate-of-return guarantees. Ekern and Persson (1996) investigated unit-linked

life insurance contracts with different types of relative guarantees. Pennacchi (1999)

valued both the absolute and the relative guarantees provided for Chilean and Uru-

guayan pension plans by using a contingent claim analysis. Both papers assume that

the interest rate is deterministic. However, Lindset (2004) analyzed several kinds of

minimum guaranteed rates of return within the HJM framework. The guaranteed

rates of return examined in these three papers are set relative to the rates of return

on equity-market assets. Moreover, Yang et al. (2008) studied rate of retur n guaran-

tees for pension funds linked relative to a return measured by market-realized

d-year spot rates with the HJM framework. However, the limitation of their results

is that they are applicable only to a special case with a limited guarantee period

under multi-period guarantees.

T.-Y. Hsieh et al.

590 ©2014 Korean Securities Association