Valuation and Hedging of the Ruin‐Contingent Life Annuity (RCLA)

• Published date: 01 June 2014
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01509.x
• Date: 01 June 2014
• Author: H. Huang,T. S. Salisbury,M. A. Milevsky

©

DOI: 10.1111/j.1539-6975.2012.01509.x

367

VALUATION AND HEDGING OF THE RUIN-CONTINGENT

LIFE ANNUITY (RCLA)

H. Huang

M. A. Milevsky

T. S. Salisbury

ABSTRACT

We analyze an insurance instrument called a ruin-contingent life annuity

(RCLA), which is a stand-alone version of the option embedded inside a

variable annuity (VA) but without the buyer having to transfer investments

to the insurance company. The annuitant’s payofffrom an RCLA is a dollar

of income per year for life, deferred until a certain wealth process hits zero.

We derive the partial differential equation (PDE) satisfied by the RCLA

value assuming no arbitrage, describe efficient numerical techniques, and

provide estimates for RCLA values. The practical motivation is twofold.

First, numerous insurance companies are now offering similar contingent

deferred annuities (CDAs). Second, the U.S. Treasury and Department of

Labor have encouraged DC plans to offer longevity insurance to participants

and the RCLA might be the ideal product.

INTRODUCTION

As its name suggests, a mortality-contingent claim is a derivative product whose

payoff is dependent or linked to the mortality status of an underlying reference life

or pool of lives. The simplest and perhaps the most trivial mortality-contingent claim

is a personal life insurance policy with a face value of $1 million, for example. In

this case, the underlying state variable is the (binary) life status of the insured. If and

when it jumps from the value of one (alive) to the value of zero (dead) the beneficiary

of the life insurance policy receives a payout of $1 million. Another equally trivial

example is a life or pension annuity policy that provides monthly income until the

H. Huang is Professor of Mathematics and Statistics at York University. M. A. Milevsky is

Associate Professor of Finance, YorkUniversity, and Executive Director of the IFID Centre. T.S.

Salisbury is Professor of Mathematics and Statistics at YorkUniversity, all in Toronto, Canada.

The second author can be contacted via e-mail: milevsky@yorku.ca. The authors acknowledge

the helpful comments from a JRI referee,as well as from seminar participants at the Department

of Risk Management and Insurance at The Wharton School, seminar participants at Monash

University,Melbourne, The University of New South Wales, and the University of Technology,

Sydney. In particular the authors would like to acknowledge helpful comments from Carl

Chiarella, Neil Doherty,Olivia Mitchell, and Eckhard Platen. Huang’s and Salisbury’s research

is supported in part by NSERC and MITACS.

The Journal of Risk and Insurance, 2013, Vol. 81, No. 2, 367–395

368 THE JOURNAL OF RISK AND INSURANCE

annuitant dies. Payment for these options can be made up front, as in the case of a

pension income annuity, or by installments, as in the case of a life insurance policy.

Indeed, the analogy to credit default swaps is obvious and it is said that much of the

technology—such as Gaussian copulas and reduced-form hazard rate models—that

are (rightfully or wrongfully) used for pricing credit derivatives can be traced to the

actuarial science behind the pricing of insurance claims.

Yet, in the past these puremortality-contingent claims have been (perhaps rightfully)

ignored by the mainstream quant community primarily because of the law of large

numbers. It dictates that a large-enough portfolio of policies held by a large insur-

ance company should diversify away all risk. Under this theory pricing collapsed to

rather trivial time-value-of-money calculations based on cash flows that are highly

predictable in aggregate.

However, this conventional viewpoint came into question when, in the early part of

the previous decade, a number of largeinsurance companies began offering equity put

options with rather complex optionality that was directly tied to the mortality status

of the insured. These variable annuity (VA) policies, as they are commonly known,

have been the source of much public and regulatory consternation in late 2008 and

early 2009, as the required insurance reserves mushroomed. An additional source of

interest, not directly addressed in this article, is the emergence of actuarial evidence

that mortality itself contains a stochastic component. See, for example, Dawson et al.

(2010) or Schulze and Post (2010).

Motivated by all of this, in this article we value and provide hedging guidance on a

type of product called a ruin-contingent life annuity (RCLA). The RCLA provides the

buyer with a type of insurance against the joint occurrence of two separate (and likely

independent) events; the two events are under average investment returns and above

average longevity. The RCLA behaves like a pension annuity that provides lifetime

income but only in bad economic scenarios. In the good scenarios, properly defined,

it pays nothing. The RCLA is obviously cheaper than a generic life annuity, which

provides income under all economic scenarios.

The RCLA is a fundamental mortality-contingent building block of all VA “income

guarantees” in the sense that it is not muddled by tax frictions and other institutional

issues. At the same time it retains many of the real-world features embedded within

these policies. At the very least this article should provide an introduction to what

we label finsurance—products that combine financial and insurance options in one

package.

The RCLA falls into the broader category of (what practitioners in the United States

have called) contingent deferred annuities (CDA), which are being sold and promoted

by insurance companies as alternatives to VAs with guaranteed lifetime withdrawal

benefits (GLWBs.) Recall that with a VA + GLWBthe buyer of the product must invest

a (substantial) sum of money into the insurance company’s investment accounts in

order to gain access to the put option. The CDA, in contrast, would enable the buyer to

retain control of the funds and invest in any investment they want (exchange-traded

funds [ETFs], mutual funds, or individual stocks). At the same time, the CDA would

provide a guaranteed lifetime annuity payout at some point in the future if those