Annuitization and aggregate mortality risk

• DOI: http://doi.org/10.1111/jori.12313
• Date: 01 March 2021
• Author: Marias H. Gestsson,Torben M. Andersen
• Published date: 01 March 2021

J Risk Insur. 2021;88:79–99. wileyonlinelibrary.com/journal/JORI

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79

Received: 5 April 2019

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Accepted: 26 April 2020

DOI: 10.1111/jori.12313

ORIGINAL ARTICLE

Annuitization and aggregate mortality risk

Torben M. Andersen

1,2,3,4

|Marias H. Gestsson

5

1

Department of Economics and Business

Economics, Aarhus University, Aarhus,

Denmark

2

CEPR, London, England

3

CESifo, Munich, Germany

4

IZA, Bonn, Germany

5

Department of Economics, University of

Iceland, Reykjavík, Iceland

Correspondence

Marias H. Gestsson, Department of

Economics, University of Iceland,

101 Reykjavik, Iceland.

Email: marias@hi.is

Funding information

Danish Research Council; Independent

Research Fund Denmark,

Grant/Award Number: DFF –6109‐00043

Abstract

It is well established that annuities can fully diversify

idiosyncratic mortality risks. However, survival rates at

the cohort level are changing, raising the question what

is the scope of annuities in the presence of aggregate

mortality risk? In an overlapping generations setting, we

show that risk free annuities exist, but offer a return

below the (fair) certainty equivalent return, and agents

do not fully annuitize their savings. Higher aggregate

mortality risk increases savings and thus the mean level

of the capital stock. This lowers the mean rate of return

on capital, the survival premium on annuities and the

share of individual savings in annuities.

KEYWORDS

aggregate mortality risk, annuitization, annuity markets, dynamic

efficiency, overlapping generations models

JEL CLASSIFICATION

D15; D81; E21; E44; G11

1|INTRODUCTION

Mortality rates have displayed significant changes in advanced economies.

1

In a first phase, child

mortality decreased to very low levels, to be followed by declining old‐age mortality (a right‐ward shift

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© 2020 American Risk and Insurance Association

[Correction added on 24 June 2020, after first online publication: the deletion of 0 on the left hand side of the first

inequality in eq 5]

1

Oeppen and Vaupel (2002) show that female life expectancy at birth has exhibited a linear trend increasing life‐

expectancy by about 2.4 years by decade. For a discussion of demographic trends, see for example, Lee (2003).

in age‐dependent mortality rates), see for example, OECD (2016a). Mortalities at higher ages are

particularly important for retirement‐income systems, and there has been much focus on the trend in

mortality rates. However, there is substantial risk in the developments of mortality rates

2

having an

independent influence on the scope to diversify mortality risk via annuities acquired by adults to

protect their consumption possibilities when older/retired from the labour market.

3

The observed

aggregate risk in mortality rates has large potential implications for households, insurance companies

and pension funds, see for example, Friedberg and Webb (2007), Dushi, Friedberg, and Webb (2010),

Fong, Piggott, and Sherris (2015), and OECD (2016b). Although forecasting methods have been

improved, a substantial aggregate longevity risk is present;

4

that is, future mortality rates are highly

uncertain at the cohort level, see for example, Haberman and Renshaw (2013) and Mitchell, Brockett,

Mendoza‐Arriaga, and Muthuraman (2013). The importance and relevance of aggregate longevity risk

is further emphasized by OECD's (2015) report on aggregate longevity risk in relation to global

pension liabilities and by numerous papers on pricing and managing aggregate longevity risk (see e.g.,

Blake & Burrows, 2001; Dowd, Blake, Cairns, & Dawson, 2006; Xu, Sherris, & Ziveyi, 2019).

There is a large—both macro and pension—literature analysing idiosyncratic or nonsyste-

matic mortality risk; that is, individuals face a mortality risk, but at the aggregate level (due to

the law of large numbers) a deterministic fraction of the population survives. Insurance com-

panies can fully diversify such risk by offering annuities, which under perfect competition have

fair returns. Risk averse households save for old‐age consumption in such annuities only,

thereby eliminating the importance of mortality risk for savings/consumption decisions, leaving

no unintentional bequests,

5

see Yaari (1965), Sheshinski (2008), and Davidoff, Brown, and

Diamond (2005). On the supply side, insurance companies operate as intermediaries trans-

forming standard financial assets (stocks) into annuities, investing their capital in stocks. Under

perfect competition the fair price of the annuity is higher than the market return on stocks,

reflecting that not all customers survive, and profits are (deterministically) equal to zero. It is a

less discussed implication that this setting implies an equilibrium portfolio structure where

households save in annuities only, and insurance companies hold all capital in society.

The case of aggregate or systematic mortality risk is much more complicated. In the words

of Gatzert and Wesker (2014, p. 63) “Systematic mortality risk is the risk that cannot be

diversified through enlarging the insurance portfolio; that is, it is the risk of unexpected de-

viations from the expected mortality rates applying to all individuals, which can result, for

example, from a common factor unexpectedly impacting mortality at all ages …and thus

destroy diversification benefits of large pool sizes.”Even though aggregate shocks cannot be

fully diversified, some scope for risk diversification remains,

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but several complications arise.

The standard annuity product with fair pricing is not viable in equilibrium, since insurance

companies are unable to honour their obligations in states of nature where the survival rate

2

In 2010–2015 women (men) aged 65 can expect to live an additional 20.8 (17.4) years, and in 2060–2065 25.8 (21.9)

years, compare OECD (2015).

3

Most annuities tied to pension savings are deferred annuities; that is, outpayment starts when reaching some statutory

retirement age.

4

This is captured by the well‐known Lee‐Carter methodology and later refinements hereof, see Lee and Carter (1992).

Over a long period, there has been a systematic tendency to underpredict changes in mortality, see for example,

Keilman (2001).

5

For macroeconomic analyses, see for example, Blanchard (1985), Maurer (2018), and Gârleanu and Panageas (2020).

While Maurer (2018) assumes stochastic mortality risk, the continuous time setup of the model implies that the size of a

cohort is deterministic and, hence, the aggregate survival rate is nonstochastic.

6

In practice there are several instruments by which longevity risk can be spread across various actors, see for example,

Maurer, Mitchell, Rogalla, and Kartashov (2013), but the aggregate risk has to be absorbed somehow.

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ANDERSEN AND GESTSSON