A new option for mortality–interest rates

• Published date: 01 February 2023
• Author: Tzuling Lin,Cary Chi‐Liang Tsai
• Date: 01 February 2023
• DOI: http://doi.org/10.1002/fut.22390

Received: 28 March 2022

|

Accepted: 26 November 2022

DOI: 10.1002/fut.22390

RESEARCH ARTICLE

A new option for mortality–interest rates

Tzuling Lin

1

|Cary Chi‐Liang Tsai

2

1

Department of Finance, National Chung

Cheng University, Minhsiung, Taiwan

2

Department of Statistics and Actuarial

Science, Simon Fraser University,

Burnaby, British Columbia, Canada

Correspondence

Tzuling Lin, Department of Finance,

National Chung Cheng University,

168 University Rd., Minhsiung,

Chiayi 62102, Taiwan.

Email: tzuling@ccu.edu.tw

Funding information

National Science and Technology Council

of Taiwan, Grant/Award Number: 109‐

2410‐H‐194‐017‐MY3

Abstract

We propose a new type of mortality–interest option related to a new random

variable, the force of mortality–interest denoted as

μ

*

, the addition of the force

of mortality and the force of interest. We assume

μ

*

moves approximately

linearly, design the new mortality–interest option, and then derive closed‐form

formulas for its expected values. We show that using the new

mortality–interest options, an annuity provider and a life insurer can,

respectively, hedge the longevity and mortality risks with interest rate risk;

a financial intermediary selling the new options can benefit from natural

hedges resulted from two‐side businesses with the annuity provider and life

insurer.

KEYWORDS

interest rate risk, longevity risk, mortality risk, risk‐neutral valuation

JEL CLASSIFICATION

G13, G22

1|INTRODUCTION

The mortality improvement of worldwide populations for decades, reflected in increased life expectancies, has aroused

considerable attention. The trend in most developed countries toward a gradual increase in life expectancy implies that

individuals, annuity providers, retirement programs, and long‐term care systems face a significant financial challenge

from this longevity. However, as we see, the mortality uncertainty cuts both ways; outbreak of a pandemic, such as

COVID‐19, human disasters, such as wars, or environmental risks due to extreme weather, such as heat waves,

earthquakes, and so on can substantially increase mortality rates, which causes individuals and enterprises to face the

loss of personal risk, and causes life insurers, social security, and health related systems to face financial losses of

mortality risk. In view of the worldwide phenomenon, we propose to build and trade standardized securities of

longevity and mortality risks in the capital markets.

In the past decades, the type of mortality‐linked securities (e.g., longevity bonds, q‐forwards, survivor swaps,

annuity futures, mortality options, and survivor caps) to manage longevity or mortality risks has been proposed, for

example, Milevsky and Promislow (2001), Menoncin (2008), Dawson et al. (2009), Lin and Tzeng (2010), Li et al.

(2011), Lin and Tsai (2016), and Schmeck and Schmidli (2021). In the practitioner community, JP Morgan has set up

LifeMetrics for the purpose of building a liquid market for longevity derivatives since 2007. The first derivative

transaction, a q‐forward contract between J. P. Morgan and the UK pension fund buy‐out company Lucida, occurred in

January 2008. The first longevity swap was executed in July 2008 by Canada Life to hedge £500m of its UK‐based

annuity book purchased from the defunct UK life insurer Equitable Life. In 2010, Swiss Re successfully issued a

longevity bond, called Kortis. In November 2013, Deutsche Bank structured the Longevity Experience Option (LEO) as

an out‐of‐the‐money call option spread on 10‐year forward survival rates with a 10‐year maturity.

J Futures Markets. 2023;43:273–293. wileyonlinelibrary.com/journal/fut © 2022 Wiley Periodicals LLC.

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However, until now we still cannot see a liquid and transparent market for longevity risk and mortality risk. Loeys

et al. (2007) suggested that homogeneous and transparent contracts must be used to permit exchanges between agents

in order for a new market to succeed. Hedging instruments of longevity risk have been structured so far mainly as

insurance contracts which indemnify the hedgers against their own mortality experiences rather than make payments

based on a referred longevity index (Biffis et al., 2016). This impedes product standardization and liquidity, which the

market is slowly trying to overcome through indexed solutions (Fetiveau & Jia, 2014). Lin et al. (2022) proposed to

attach a mortality index to a conventional government bond, called survival‐mortality (SM) bond, strip the survival‐

coupon bond, called survival (S) bond, and the mortality‐coupon bond, called mortality (M) bond, and further split it

into S, M, and SM zero‐coupon STRIPS

1

to construct a complete term structure for mortality rates.

In this paper, we propose a new type of mortality–interest option which is related to a new random variable, the

force of mortality–interest denoted as

μ

*

, the addition of

μ

(the force of mortality) and

δ

(the force of interest). Here, for

the simplicity, we first assume μ

μ

*,, and

δ

are all constant within

k

years. Then since ek

δ

−is the present value of $1 in

the future

k

years and

e

k

μ

−

is the probability that one survives

k

years, ekμ−

*

means the present value of $1 paid at the

end of the next

k

years conditional on whether one survives

k

years. When the capital markets worldwide are in a low‐

interest rate environment (i.e.,

≈δ0

) and individuals, enterprises, and institutions face challenges from personal risk

and mortality risk, pricing the uncertainty of the force of mortality will become more important. In the past decades,

there is a plentiful literature about the term structure and the derivatives about

δ

(see, e.g., Cox et al., 1985; Heath et al.,

1992; Hull & White, 1990; Jeffrey, 1995; Vasicek, 1977). We propose a new type of mortality–interest option for the

development of term structure and valuation about mortality rates and mortality–interest rates, which is helpful to

hedge personal risk, longevity risk, and mortality risk. Motivated by the linear relationship between two sequences of

μ

*

, we assume that

μ

*

moves approximately linearly, design the new type of mortality–interest option, and then derive

closed‐form formulas for its expected values.

Besides, we also discuss the marketable potential by analyzing whether the new option can provide hedge

effectiveness. We illustrate hedges for a financial intermediary, an annuity provider, and a life insurer. We assume

there is an annuity provider (life insurer) who will buy the optimal units of the new options to hedge longevity

(mortality) and interest rate risks of his annuity (life) exposures, and a financial intermediary who sells the new options

to the annuity provider and the life insurer simultaneously and then creates a natural hedge opportunity. The annuity

provider and life insurer can buy the optimal units of new options and the financial intermediary can seek the optimal

weights of two‐side businesses with the annuity provider and the life insurer. In the numerical examples, we calculate

the optimal units/weights and the hedge costs/premiums, and evaluate the hedging performances for the annuity

provider, the life insurer, and the financial intermediary, respectively.

The remainder of this paper proceeds as follows. In Section 2, we propose a simple linear regression to modeling

mortality and interest rates, and then structure new options of mortality and interest rates based on the linear model.

Section 3values the new options. In Section 4, we illustrate hedges for an annuity provider, a life insurer, and a

financial intermediary. Section 5provides numerical illustrations, and Section 6concludes this paper.

2|MODELING MORTALITY–INTEREST RATES

We first give some notations. Assume that the force of mortality μxistis++,++ for age

xis++

in time

t

is++is constant

for

∈s[0 , 1)

,where

xt,

,and

i

are all nonnegative integers. Denote

δ

is+

the continuously compounded risk‐free rate at time

i

s+

with the assumption that the force of interest

δ

is+

is constant for

∈s[0 , 1)

.Thus,

μμ=

xistis xiti++,++ +,+

and δδ=

is

i

+

(i.e., both μxistis++,++ and

δ

is+

are piecewise constant) for

∈s[0 , 1)

. Then the force of mortality–interest,

μμδ

*=+

xiti xiti i

+,+ +,+ (the addition of the forces of mortality and of interest) for

i

=0,1,

…

,canbecreated.

With the original force of mortality

μ

xiti+,+, we obtain the

k

‐year survival probability that an individual currently

aged

x

at the beginning of year

t

(we use “in year

t

”hereafter) survives

k

years to age x

k

+in year

tk

+

, denoted as

1

STRIPS is the acronym of Separate Trading Registered Interest and Principal Securities. Before 1985, a number of brokerage firms created their own

zero‐coupon securities by stripping the coupons from Treasury bills and bonds, implying that the coupons become separate investments sold

separately. Nowadays Treasury STRIPS are issued by the US Treasury and backed by the US government since the STRIPS system was introduced

in 1985.

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