The Annuity Duration Puzzle
N. Charupat, M. Kamstra and M. A. Milevsky1
Version: 29 February 2012
1 Narat
Charupat is faculty member at the DeGroote School of Business, McMaster University,
Mark Kamstra is a faculty member at the Schulich School of Business, York University, and Moshe
A. Milevsky is Executive Director of the IFID Centre as well as a faculty member at the Schulich
School of Business, York University, Toronto, CANADA. The contact author (Milevsky) can be
reached via email at: [email protected], or at Tel: 416-736-2100 x 66014. The authors would like
to acknowledge CANNEX Financial Exchanges for access to the annuity data and the IFID Centre
at the Fields Institute for financial support.
Abstract
We test whether life annuity prices promptly and fully adjust to changes in interest rates.
The immediate response of annuity prices to interest rate changes is a standard assumption
that is typically made, implicitly or explicitly, in a growing annuity literature. Using a
unique database consisting of over 3 million weekly quotes, we find that real-life annuity
prices do not behave as assumed in theory. Rather, insurance companies adjust annuity
prices gradually and over a period of several weeks, in response to certain interest rate
changes. In addition, we find that the sensitivity to interest rates (a.k.a annuity duration) is
asymmetric. Annuity prices react more rapidly and with greater sensitivity to an increase in
interest rates compared to a decrease in interest rates. Overall our findings are inconsistent
with a standard annuity-pricing model used by many researchers. As such, our results have
wide-ranging implications for the existing literature, including portfolio choice decisions,
the optimal timing of an annuity purchase, the money’s worth ratio calculations and the
inference of mortality expectations from observed annuity prices.
1
Introduction
There has recently been a considerable increase in investor and academic interest in survivalcontingent financial products, particularly life annuities. This increase in interest is due
mainly to two factors. First, during the past decade, employers have gradually shifted their
pension plans from a defined-benefit (DB) type, whose payments are typically constant and
last for life, to a defined-contribution (DC) type, whose payments depend on how quickly
the individual wants to decumulate his/her retirement savings. As a result, more and more
employees now bear market risk and their own longevity risk. Secondly, life expectancy has
consistently improved, while returns on traditional investments have been low and/or more
volatile. Consequently, investors have become increasingly concerned about their late-life
consumption and the possibility that they may outlive their investment. A product such
as life annuities, whose payoff lasts as long as their holders are alive, can be used to hedge
investors’ longevity risk.
The study of life annuities has a long and rich history, and can be divided into four
major strands. The first strand examines the optimal portfolio decisions in the presence of
life annuities and the welfare gains from annuitization. Yaari (1965) is the first person who
extended the life-cycle model (Modigliani and Brumberg (1954)) to include an uncertainty in
the length of life. He showed that under some restrictive assumptions, risk-averse individuals
should annuitize all of their wealth at retirement provided that annuity prices are actuarially
fair. Subsequent studies extended Yaari’s model to incorporate more realistic assumptions
such as incomplete markets and the irreversibility of annuity purchase. These studies derived
the conditions under which full/partial annuitization is optimal and also the optimal timing
of the annuity purchase (See, for example, Brown, Mitchell and Poterba (2002), Davidoff,
Brown and Diamond (2005), Milevsky and Young (2007), and Horneff et al (2009)).
The second, and closely related, strand of studies attempts to explain the so-called annuity puzzle, which refers to the observation that in real life, voluntary demand for life annuities
is much lower than predicted by the models, both in terms of the number of people who
purchase life annuities and the value of the transactions. Various reasons have been offered
including strong bequest motives, habit formation in preferences, pre-annuitized wealth such
as private DB pensions and social security benefits, medical expense uncertainty, and behavioral factors (see, for example, Friedman and Warshawsky (1990), Brown, Mitchell and
Poterba (2001), Dushi and Webb (2004), Turra and Mitchell (2004), Ameriks et al (2011)
and Benartzi, Previtero and Thaler (2011)). More recently, a study by Inkmann, Lopes and
Michaelides (2011) questions how deep the annuity puzzle really is. Its authors show that
the lack of annuity demand can be explained by a mild bequest motive and preference parameters that are consistent with existing empirical estimates. Yet, one thing is for certain,
life annuities are not very popular.
A third strand of the literature estimates the “money’s worth” of life annuities, which is
1
the expected present value of the annuity payments taking into account the term structure of
interest rates and mortality probabilities. That is, the money’s worth calculations attempt
to estimate the annuities’ actuarially fair values so that they can be compared with the
purchase prices. Typically, the comparison is done in terms of the ratio between the estimated
fair value and the purchase price. The calculations have been done for various markets
including the US, UK, Canadian and German markets (see, for example, Mitchell et al
(1999), Finkelstein and Poterba (2004) and Cannon and Tonks (2009)). The findings show
that the money’s worth ratios are generally only slightly less than unity. This is particularly
true when the mortality probabilities used in the calculations reflect the adverse selection
problem that is common in voluntary annuity purchases (i.e., voluntary annuity buyers tend
to live longer than the average population). In other words, the evidence suggests that
mark-ups (or loadings) on life annuity policies do not appear to be excessive.
Finally, the fourth strand applies annuity pricing techniques to the valuation of corporate
and governmental DB pension liabilities, which can be thought of as deferred life annuities
(i.e., annuities whose payments do not commence immediately, but rather when the employees retire). Example of these studies include Bodie (1990), Sundaresan and Zapatero (1997),
Ippolito (2002), Brown and Wilcox (2009) and Novy-Marx and Rauh (2011).
In most of the above studies, it is assumed, explicitly or implicitly, that annuity prices
fully reflect the concurrent term structure of interest rates and mortality probabilities. In
other words, it is generally assumed that annuity prices respond promptly and proportionally
to changes in the values of the two factors. Since changes in mortality probabilities occur
very gradually, the assumption implies that in the short run, changes in annuity prices can
be explained primarily by movements in interest rates. In this paper, we test whether this
assumption holds in reality, using a database that consists of over three million annuity
quotes from twenty-five US annuity providers over the period from 2004 to 2010. The quotes
are for single premium immediate life annuities (i.e., ones whose purchase prices are paid
once at the start, and whose income payments start right away), and are classified along
five dimensions - ages of annuitants, gender, guarantee periods, payout options (i.e., single
life, joint life and term certain), and income tax treatment (i.e., qualified vs. non-qualified).
The extensive nature of this unique database allows us to obtain reliable estimates of the
sensitivity (to interest rate changes) of prices of various types of life annuities.
Our findings show that real-life annuity prices do not behave in the assumed manner. Annuity prices do not promptly and fully respond to changes in interest rates. Rather, annuity
providers appear to make gradual price changes over several weeks in order to incorporate
changes in interest rates over a longer period of time into annuity prices. This is in contrast
to market-traded bond prices and other fixed income securities that respond instantaneously
to changes in rates. As such, life annuities should therefore not be treated as fixed income
bonds with additional longevity insurance. Their dynamics are quite different. This may be
2
due to annuity providers’ desire to smooth out the price changes and/or to wait in order to
get a better sense of the trend of interest rates. Alternatively, since annuity providers from
time to time have unbalanced books (i.e., mismatches of durations of assets and liabilities),
they may deliberately offer uncompetitive annuity prices or delay price adjustments in order
to allow themselves time to rebalance their books. Although, we carefully shy away from
explaining why we observe these effects. In addition, we find that the response of annuity
prices to changes in interest rates is asymmetric. Annuity prices react more rapidly and with
greater sensitivity to an increase in interest rates than to a decrease in interest rates. That
is, when rates increased, insurance companies reduce their annuity prices (by increasing the
amount of monthly annuity income per the same principal amount) more quickly and in a
larger magnitude than what they do in the opposite direction when rates declined. Needless
to say, market-traded bond prices do not react this way.
Our findings are inconsistent with the implications of the standard annuity-pricing model.
As a result, they have wide-ranging implications on the existing literature. The fact that
annuity prices are slow to adjust to new interest-rate information suggests that there are periods during which annuities are temporarily “mispriced” relative to their theoretical values.1
The mispricing can persist for a period of time because it is not possible to create a riskless
arbitrage transaction to eliminate it due to the fact that life annuities cannot be replicated
(because they are person-specific) or sold short. The temporary mispricing suggests that
contrary to the theoretical results, the timing of the annuity purchase can affect the individual’s optimal allocation decisions and the magnitude of welfare gains from annuitization.
This is particularly important considering that an annuity purchasing decision is irreversible
and typically involves a substantial portion of an individual’s wealth. In addition, our results
imply that the money’s worth of an annuity can vary depending on the timing of the calculations (even if every other factor remains the same). Similarly, caution should be exercised
when attempting to infer mortality expectations from observed annuity price; for example
Philips and Becker (1998).
Our findings may also be of interest to researchers in the area of interest rate passthrough. That strand of literature examines how quickly and how completely changes in
interest rates (typically official rates such as the Federal Reserve policy rate) are passed
through to retail interest rates (e.g., deposit rates and prime lending rates). In our case,
we can interpret our findings as showing that annuity rates (which are inversely related to
annuity prices) are sluggish to respond to movements in interest rates, and that annuity rates
are more rigid downward than upward. That is, when market interest rates increase, annuity
rates go up more quickly than they go down when market interest rates decline. Obviously,
such an asymmetric response is to the benefit of annuity buyers. This is quite unexpected
1
Since our tests are based on the speed of price adjustments, the existence of the temporary mispricing is not conditional upon the assumption on the underlying mortality distribution as long as mortality
probabilities are not correlated with interest rates, which is what one intuitively expects
3
and in contrast to the results of pass-through studies done on consumer interest rates such
as Hannan and Berger (1991) and Neuman and Sharpe (1992), which find that US banks
are slower to raise deposit rates than to reduce them.2 Our findings are consistent with
the customer reaction hypothesis (Hannan and Berger (1991)), which, if adapted to in our
context, would state that insurance companies may fear a negative reaction from potential
annuity buyers if annuity rates are reduced.
To our knowledge, the sensitivity of annuity prices to interest rates has not been examined
before. The closest mention that we find is in Cannon and Tonks (2009) who calculate the
correlation between UK annuity rates and the UK government 10-year bond yields. They
report that the correlation was very high (i.e., 0.98) during the period from 1994 to 2000, but
declined substantially to only 0.57 during the period from 2001 to 2007. The low correlation
in the latter period (which overlaps with our sample period) is consistent with our findings
here.
The paper is organized as follows. In the next section, we discuss the theoretical annuity
pricing equation and the duration of a life annuity, which is our measure of annuity price
sensitivity. Section 3 discusses our database and the structure of the life annuity markets.
We present our empirical results in Section 4. Finally, Section 5 concludes the paper.
2
The Pricing and Duration of a Life Annuity
In this section, we present the pricing equation for a single-life immediate annuity, which is
an annuity whose payments are constant, start immediately and last as long as the annuitant
is alive. This is the most common type of life annuity and is the type whose quotes we will
examine in the next section. We will also derive the expression for the annuity’s duration,
which is our measure of the sensitivity of its price to changes in interest rates.
The traditional and most common annuity pricing principle is the equivalence principle.
This principal is based on the notion that insurance companies can diversify mortality risk
in a large portfolio. As a result, without considering the expenses that insurance companies
incur, the actuarially fair price of a single-premium annuity is equal to the expected present
value of annuity payments.3 That is, an insurance company will set the annuity price such
that the price and future investment return on it exactly cover annuity payouts that the
company expects to make.
2
Similar incomplete and upward-rigid pass through of deposit rates is found in other countries such as
the UK and the eurozone countries (See, for example, Fuertes, Heffernan and Kalotychou (2010) and de
Bondt, Mojon and Valla (2005) and the references contained therein). Note that more recent studies have
found that the speed of adjustments of bank deposit rates in the US markets has substantially improved,
especially after the mid 1990’s (See, for example, Sellon (2005)).
3
For more information about the equivalence principle and other actuarial pricing principles, see Dickson,
Hardy and Waters (2009).
4
For ease of exposition, we will present the continuous-time version of the annuity pricing
equation. Consider a single-life immediate annuity that pays $1 per year (in continuous
time), starting immediately and lasting as long as the holder is alive. For an x-year-old
individual, the actuarially fair price of this life annuity is:
Z ∞
e−rt p(x, t) dt,
(1)
a(x, r) =
0
where p(x, t), 0 ≤ p(x, t) ≤ 1 is the probability that an individual who is now x years old
will survive to at least age x + t, and r is the interest rate, which is assumed to be constant
for all terms.4
Some life annuities come with a guarantee period whereby the payments will be made
for at least a certain number of years (i.e., the guarantee period). If the annuitant dies
before the end of the guarantee period, his or her estate or beneficiary will continue to
receive the payments for the remaining length of the guarantee period. On the other hand,
if the annuitant outlives the guarantee period, the payments will continue until he or she
dies. Accordingly, this type of annuity can be thought of as consisting of two separate
components. The first component is the guaranteed component, while the second component
is the life-contingent component that will come into existence only if the annuitant survives
the guarantee period.
Consider a single-life immediate annuity that pays a lifetime income of $1 per year (in
continuous time) with a guarantee period of g years. The actuarially fair price of this annuity
for an x-year-old individual is:
Z
a(x, g, r) =
g
e
−rt
Z
dt +
0
∞
e−rt p(x, t)dt.
(2)
g
The first term on its right-hand side of equation (2) is the price of the guarantee component,
which is the same as the price of a bond that pays $1 per year (in continuous time) for g
years. The second term on the right-hand side of equation (2) is the price of a life annuity
that pays $1 per year (in continuous time), starting at age x + g (if the holder survives
to that age) and lasting until the holder dies. Indeed, it is the price of a deferred life
annuity. Obviously, if g = 0 (i.e., no guarantee period), equation (2) collapses to equation
(1). Therefore, equation (2) is a more general expression for the actuarially fair price of a
single-life immediate annuity.
To obtain the sensitivity of annuity price to changes in interest rates, we will use the
concept of (modified) duration, which is commonly used in the fixed-income literature to
4
This assumption is made for ease of exposition. In Appendix A, we present a more general equation for
annuity price where interest rates can vary depending on the terms. We then derive the duration measure
based on this more general equation. Under reasonable assumptions on possible changes in the term structure,
the resulting duration values are generally very close to those obtained under this assumption.
5
measure how bond prices adjusts to changes in yields. Following the convention in that
literature, we define the duration, D, of a life annuity as:
D=−
1
∂a(x, g, r)
·
a(x, g, r)
∂r
(3)
The exact, closed-form expression for D will depend on the expression for a(x, g, r), which,
in turn, depends on the mortality distribution (which determines survival probabilities) that
one assumes. A common assumption used in the insurance and biology literature is to use
the Gompertz law of mortality, which has been shown to provide a reasonably accurate
description of human mortality, particularly for individuals who are between middle age and
early old age.5 Under this law, the instantaneous force of mortality (i.e., the hazard rate) at
age x, λ(x), is:
1
(4)
λ(x) = e(x−m)/b ,
b
where m > 0 is the modal value of life span and b > 0 is a dispersion parameter. This leads
to the conditional survival probability of:
p(x, t) = exp bλ(x) 1 − et/b ,
(5)
which is decreasing exponentially in the time horizon t.
Accordingly, we can obtain the expression for annuity price by solving equation (2) as
follows. The first term on the right-hand side of equation (2) is not contingent on mortality
and can be solved to obtain:
Z g
1
−rt
e dt =
1 − e−rg .
(6)
r
0
Next, we can use equations (4) and (5) to solve the second term on the right-hand side of
equation (2) to obtain:
Z
∞
−rt
e
g
bΓ −br, exp x−m+g
b
,
p(x, t)dt =
exp (m − x) r − exp x−m
b
(7)
where Γ (a, c) is the upper incomplete gamma function:
∞
Z
e−t ta−1 dt.
Γ (a, c) =
(8)
c
Finally, combining equations (6) and (7), we have the following closed-form expression for
the actuarially fair price of a life annuity for an x-year-old individual with a g-year guarantee
period under the Gompertz law of mortality:
5
For the use of the Gompertz law in insurance mortality modeling, see, for example, Finkelstein, Poterba
and Rothschild (2009). For empirical evidence that the Gompertz law provides a fairly good fit to mortality
data, see, for example, Horiuchi and Coale (1982) and Haybittle (1998).
6
bΓ −br, exp x−m+g
1
−rg
b
.
+
a(x, g, r) =
1−e
r
exp (m − x) r − exp x−m
b
(9)
With this expression, we can then calculate the duration for a life annuity as:
1
∂a(x, g, r)
·
a(x, g, r)
∂r


bΓ(−br,exp{ x−m+g
})(m−x)
ge−rg
1−e−rg
b
− r2 + r − exp (m−x)r−exp x−m
1
{
{ b }}
·
=−
 ,
x−m+g
a(x, g, r)
+Meijer G [[ ], [1, 1]] , [[0, 0, −br], [ ]] , exp
D=−
(10)
b
where the last term in the square bracket on the right-hand side is the Meijer G function.
The duration expression in equation (10) can be obtained once the values of all relevant
parameters are specified. In Table 1, we present the actuarially fair prices for male and
female annuitants (Panels A and B respectively) and corresponding duration values (Panels
C and D respectively) for single-life immediate annuities for various combinations of ages of
annuitants (55 to 80 years old in increments of 5 years) and lengths of guarantee periods
(0 to 20 years in increments of 5 years). These are the same combinations as the ones we
will be using in our empirical tests in Section 4 below. The current interest rate is assumed
to be 4.35% p.a., which is the same as the average 10-year swap interest rates during our
sample period.6 The parameters of the Gompertz distribution are m = 88.18 years and
b = 10.50 years for males, and m = 92.63 years and b = 8.78 years for females. These values
are taken from Milevsky and Young (2003), who calibrate the Gompertz distribution to the
Individual Annuity Mortality (IAM) 2000 table, with projection scale G. The IAM 2000
table was published by the Society of Actuaries. It reflects the mortality statistics of people
who buy life annuities (and who tend to have longer lifespans than the average population).
The higher m value for females reflects the fact that women generally live longer than men
do.
Insert Table 1 here
There are a few facts to note from Table 1. First, both the annuity prices and duration
values are higher for female annuitants than male annuitants. That is, a life annuity held by
a woman is more expensive and more sensitive to interest rate changes (i.e., have a longer
duration) than a life annuity held by a man of the same age. This is due to lower mortality rates of women, which translates to a longer expected stream of payments. Secondly,
everything else being equal, the older the annuitant, the less expensive and less sensitive
to interest rate movements the annuities are. This is consistent with the fact that survival
probabilities declines with age. Thirdly, the length of the guarantee period has a positive
effect on annuity prices, but a mixed effect on the duration values. Duration first declines as
the length of the guarantee period increases, but, after a certain length, starts to increase.
6
The 10-year swap rates are the best match to the average duration of the annuities we look at, and are
thus the rates we will use in our empirical test in the next section.
7
To see why the effect on prices is positive, note from equation (2) that a life annuity can be
thought of as a portfolio of a guarantee component and a deferred-life-annuity component.
A longer guarantee period increases the value of the guarantee component, but lowers the
value of the life-contingent component. However, the increase in the former always more
than offsets the decline in the latter. This is because under the Gompertz law, the force of
mortality increases at a faster rate as one ages, which is consistent with reality (until age
100 or so). This is why the effect of the guarantee periods on annuity prices is particularly
strong for older annuitants, whose survival probabilities are progressively smaller. To see
why the effect on duration values is mixed, note that the duration of a life annuity is simply
the weighted average of the durations of the guarantee component and the life-contingent
component7 It can be shown that both of these durations increase with the guarantee period.
However, their rates of increase are such that when they are weighted averaged (note that
the weights also change as the values of the two components change), the resulting annuity
duration has an ambiguous relationship with the length of the guarantee period.
In sum, we are confident that we have properly estimated the theoretical duration (i.e. interest rate sensitivity) values, assuming annuity prices are determined by contemporaneously
observable interest rates.
3
The US Annuity Markets, Data Description and Summary Statistics
The US annuity markets offer a wide variety of annuity products for both people who are
in their working years (i.e., the accumulation phase) and people who are in their retirement
(i.e., the payout phase). These products may provide income that is fixed or variable (i.e.,
linked to an equity index). The income may be provided for a certain term (e.g., 10 years),
for life or for life with a guarantee period. Life annuities can also be based on one life or two
lives (i.e., jointly with a spouse).
It can be challenging to quantify the size of the US annuity markets. This is because
many products that are sold as annuities may not lead to streams of annuity income payments. For example, an individual who purchases an annuity in her working years can choose
to receive the payoff at retirement in one lump sum, rather than conventional annuity payments.8 According to LIMRA (previously known as Life Insurance Marketing and Research
Association), the total sale figure for fixed immediate life annuities (which provide fixed
income payments for life starting immediately, and which are the subject of this paper) was
7
This is similar to the fact that the duration of a bond portfolio is the weighted average of the durations
of the individual bonds in the portfolio.
8
In fact, according to LIMRA, most people who buy annuities in their working years choose to receive a
lump sum at their retirement.
8
$5.7 billion in 2010, while the total sale of all annuity product in the same year was $221
billion.
Our data set consists of weekly annuity quotes for several types of single premium immediate life annuities that were offered in the United States during the period from September
2004 to December 2010 (332 weeks). The quotes are in terms of annuity payouts (i.e.,
amounts of monthly income) per $100,000 of annuity principal.9 The annuities are classified along the following five dimensions; ages of annuitants (55, 60, 65, 70, 75 and 80 years
old); gender (male and female); guarantee periods, ranging from 0 to 25 years in increments
of 5 years; payout options, including single life, joint life and term certain; and income
tax treatment of the annuity income, where annuities are classified as either qualified or
non-qualified10
Every week, the incoming quotes were validated and checked for irregularities. In total,
the data set contains over three million quotes from twenty-five life insurers (Not all of them
offered every annuity type). To our knowledge, this is the most comprehensive and accurate
collection of quotes of US annuity payouts available.
For the purposes of this paper, we focus on qualified annuities, which are purchased with
savings inside a retirement savings account (e.g., a DB pension plan or an IRA). Thus, the
tax treatment of all annuity income would be treated as ordinary interest and we would not
be at risk of observing tax-effects. Out of all qualified annuities in the data set, we limit our
attention to six age groups (55, 60, 65, 70, 75 and 80 years old) and five guarantee periods (0,
5, 10, 15 and 20 years). Therefore, our sample has in total 30 different annuity combinations
for male annuitants, and another 30 combinations for female annuitants. We believe that
these 60 combinations are sufficiently representative of all qualified annuities in our data set.
For every annuity combination, we use the following three steps to transform the quoted
payouts from various providers into annuity prices. First, for every week in the sample
period, we average the quoted payouts after we remove the highest and the lowest quotes (in
order to limit the effects of outliers).11 Secondly, we annualize the payouts by multiplying
them by 12. Finally, we divide the annualized payouts into $100,000 to get the annuity
prices based on one dollar of payout per year (henceforth referred to as “annuity factors”).
These three steps provide us with a time series of weekly annuity factors for each annuity
combination.
The summary statistics of these time series are presented in Table 2 (Panel A for male
9
For example, the quote of 650 means that if one buys $100,000 worth of this life annuity, one would get
monthly income of $650 for life.
10
A qualified annuity is purchased as part of a retirement plan which is set up by an employer (e.g., a DB
pension plan) or by the annuitant (e.g., an Individual Retirement Account (IRA)). Contributions made to
qualified annuities are typically deductible from the gross income of the individual for income tax purposes.
On the other hand, a non-qualified annuity is purchased with after-tax dollars. As a result, the tax treatment
of annuity payouts will be different between the two types.
11
Weeks that don’t have enough quotes to generate reliable averages are removed from the data set.
9
and Panel B for female). The empirical annuity factors display similar properties to those
of the theoretical factors in Table 1, namely (i) the factors are higher for women than for
men; (ii) the older the annuitant, the lower the factors are; and (iii) the longer the guarantee
period, the higher are the factors. Also, we note that the average annuity factors are quite
close to the theoretical values in Table 1 (Panels A and B), suggesting that the assumptions
used for the numbers in Table 1 are reasonable.
Insert Table 2 here
To obtain more intuition on the annuity factors in Table 2, consider, for example, the
annuity for a 60-year-old male with no guarantee period. The annuity factor is $14.2. This
is the price that he has to pay in exchange for a lifetime income of $1 per year. Suppose that
he buys $100,000 worth of this annuity. He will then receive $100, 000/14.2 = $7, 042.25 per
year (or about $587 per month) for life. Alternatively, we can present the same information
in terms of annuity rates, which are the inverse of annuity factors. That is, annuity rates
the rates of income on the annuities (analogous to bonds’ coupon rates). In this case, the
annuity rate is 7.04% p.a.
The annuity rate is much-higher than prevailing bond rates because of the mortality
credits or risk pooling under which those who die prior to life expectancy subsidize those
who life beyond life expectancy.
Figure 1 displays the time series of annuity rates over the sample period for selected
annuities (i.e., men aged 60, 65 and 70 with no guarantee period, in Panel A, women in
Panel B). For comparison, the 10-year swap interest rates are also plotted as a representative
long-term interest rates (see the next section for a discussion on the choice of rates). As can
be seen, the three series of annuity rates move together, which is to be expected because in
the short run, changes in annuity prices/rates should depend mainly on the movements in
interest rates. The annuity rates are very similar for men and women, with women having
lower rates (higher annuity prices). The annuity rates were trending up slightly between
2004 and 2008, and declined afterwards. The 10-year swap interest rates have a similar
up-and-then-down trend. However, they appear to have much more variability. That is,
annuity rates/factors do not fluctuate as much as the 10-year swap rates do. This provides
us with the first clue that annuity factors may not be as responsive to changes in interest
rates as generally assumed in the literature.
Insert Figure 1 here
4
Sensitivity Estimations
To measure properly the sensitivity of observed annuity factors to changes in interest rates,
we estimate the durations of the annuities in our sample. This is done by regressing the
10
negative of the percentage changes in the annuity factors over a specified period of time on
the changes in interest rates over the same time period. The resulting slope coefficients will
be the estimated sensitivity (i.e., empirical durations).12
Let ai (x, g) be the annuity factor observed in week i for an x year-old individual with a
guarantee period of g years. The percentage change in the value of this factor over a k-week
period (i.e., from week (i − k) to week i, k ≥ 1) is denoted by:
ai (x, g) − ai−k (x, g)
∆ai (x, g|k)
=
.
ai−k (x, g)
ai−k (x, g)
(11)
Let ri (T ) denote the interest rate for the term of T years observed in week i. The change
in the rate from week (i − k) to week i, is denoted by:
∆ri (T |k) = ri (T ) − ri−k (T )
(12)
Our regression equation is:
−
∆ai (x, g|k)
= βk + β∆rt (10 |k ) ∆ri (T |k) + εi .
ai−k (x, g)
(13)
If annuity factors adjust promptly and fully to the changes in interest rates, βk should
be equal to zero and β∆rt (10 |k ) should be close to the theoretical duration values in Table 1.
There are two reasons why the estimated β∆rt (10 |k ) may not be exactly equal to the theoretical
durations in Table 1. First, duration as a measure of sensitivity is at its most accurate when
the interest rate changes only by an infinitesimal amount, while actual changes in interest
rates are typically larger. Secondly, the theoretical durations in Table 1 were calculated
under specific assumptions on mortality probabilities and interest rates. However, as noted
earlier, the theoretical annuity factors in Table 1 are quite close to the observed ones in
Table 2, suggesting that the assumptions that were used are not unreasonable. As a result,
the theoretical durations in Table 1 can be used as benchmarks for comparison. If the
estimated values of β∆rt (10 |k ) are far away from the theoretical durations, it will be obvious
that the adjustment process is slow and incomplete. The low R2 value will provide further
evidence that the adjustment process is lagged and slow.
Table 3, Panel A, displays the summary statistics of the change in the 10-year swap rates
(i.e., ∆ri (10|1)), the change in the 30-year conventional mortgage rates, and the levels of
these two variables. Panels B and C of Table 3 display the percentage changes in the annuity
factors for males and females.13 During the sample period, the average weekly change in
12
Based on equation (3), the negative of the percentage change in the annuity factor is approximately
equal to the change in interest rate multiplied by the duration. See equation (13) below.
13
Note that this is not the dependent variable of the regression. As defined in equation (13), the dependent
variable of this regression is the negative of the rate of change in annuity factors (without making it a
percentage).
11
the 10-year swap rates is −0.0033% p.a., with a standard deviation of 0.1394% p.a. Since
changes can occur in either direction, their average may not give a proper idea of the average
absolute magnitude of the changes during the sample period. To get a better idea, we take
absolute values of the changes and average them. The average is 0.1044% p.a.. That is, on
average, the magnitude of weekly changes in the 10-year swap interest rates is about 10 basis
points. Out of the 332 weeks in our sample period, 137 of them (or 41.3% of the sample)
experienced a change of at least 0.10% p.a. in either direction (not shown in the table).
Similarly, 26 of the weeks (or 7.8% of the sample) experienced a change of at least 0.25%
p.a. in absolute value.
As for the weekly percentage changes in the annuity factors, the average weekly changes
are typically between -0.01% and -0.02%. For older annuitants (i.e., age 65 and up), the
weekly changes appear to be smaller and less volatile, as evidenced by lower standard deviations and narrower ranges between the minimum and the maximum values. We believe
that the markets for older annuitants are not very active, and so insurance companies may
not adjust their quotes as frequently as they do for younger annuitants.
Insert Table 3 here
For all annuitant ages, large changes in the annuity factors did not occur as frequently as
changes in the 10-year swap rates would suggest. To see this, we will use the ten annuities
for a 60-year-old person (i.e., five guarantee periods each for male and female) as examples.
Recall again that equation (3) approximates the percentage changes in the annuity factor
as:
∆a(x, g, r)
= D · ∆r,
(14)
−
a(x, g, r)
where the equation is re-arranged and presented in a discrete form. As the theoretical
durations of annuities for a 60-year-old person is around 10 years (see Panels C and D of
Table 1), a 0.10% change in interest rates should translate to a change in the annuity factor
of approximately 1%. During the sample period, none of the ten annuities had a change of
this magnitude (or greater) more than thirty times during the sample period. This is despite
the fact that, as stated earlier, a change of 10 basis points or larger in interest rates occurred
over 130 times during the same period.14 This provides us with another hint that annuity
factors are not adjusted as immediately or fully as generally assumed in the literature.
4.1
Initial Regressions
As an exploratory step, we run the regressions in equation (13) for every annuity combination
in our sample, where changes in the annuity factors and changes in the interest rates are
14
Larger changes in annuity factors are even more rare. Of all the ten annuities for a 60-year-old person,
none had a change of larger than 2.50% more than twice. This is despite that fact that a 25-basis-point
change in the 10-year swap rates happened 26 times during the sample period.
12
measured over a period of one week (i.e., measurement period k = 1). One decision that
we have to make is regarding the proxy for interest rates. As a first attempt, we choose the
10-year swap interest rates as the proxy. While swap rates are not entirely default-free (as
they reflect the borrowing rates among large financial institutions), they are considered by
many market participants to be better proxies for risk-free rates than Treasury rates (Hull,
Predescu and White (2005)). Although insurance companies can become insolvent, holders
of life annuities are typically protected by state law up to a certain limit.15 Therefore, up to
that limit, cash flows from life annuities can be regarded as close to being default-free. The
choice of the 10-year term reflects the fact that the theoretical durations of the annuities in
our sample are around 10 years (see Table 1) and the assumption that insurance companies
will try to match the durations of their assets and liabilities.
The regression results for all annuity combinations are presented in Table 4 (Panel A for
male and Panel B for female). All the intercept (βk ) estimates are zero, and so we only show
the estimates of the slope coefficients (β∆rt (10 |k ) ), which are the empirical durations. All the
numbers are well below (and statistically different from) the theoretical durations in Table 1.
In fact, most of the estimates are less than 1.16 The highest empirical duration of all annuity
combinations is only 1.41 (for 55-year-old female with no guarantee period). This suggests
that empirical annuity factors are far from being responsive to the weekly movements in the
10-year swap interest rates. The magnitude of the adjustments from week to week are only a
small fraction (about 1/10th) of what the theory predicts. In other words, the results appear
to suggest that insurance companies do not change their weekly quotes to correspond fully
to changes in risk-free interest rates.
Insert Table 4 here
The fact that the R-squares of the regressions are all very low - most of which are around
6 − 7% - indicates that the observed weekly movements in annuity factors may also be
attributable to other reasons. From our discussion with insurance practitioners, these other
reasons include annuity providers’ desire to smooth out the price changes and/or to wait in
order to get a better sense of the trend of interest rates. Another possible reason is that
annuity providers from time to time have unbalanced books (i.e., mismatches of durations of
assets and liabilities). As a result, they may deliberately offer uncompetitive annuity prices
or delay price adjustments in order to allow themselves time to rebalance their books. For
these reasons, annuity quotes can in the short run be unresponsive to interest rate changes.
Next, we further investigate the sensitivity of annuity factors by looking at their responsiveness over a longer period than one week. That is, we examine the relationships between
15
The limit is usually $100,000 worth of annuity principal. Some states, however, have limits as high as
$500,000. The protection is provided by each state’s Guaranty Association.
16
It should be noted that the estimated durations are particularly low for a few annuity combinations (i.e.,
old annuitants with long guarantee periods).
13
the changes in the annuity factors and interest rates over a measurement period of two weeks
or longer (i.e., k ≥ 2). A longer period also reduces the non-synchronous problem (if any)
in the observations.
4.2
4.2.1
Alternative Regressions
Changes over Longer Periods
We repeat the regressions in the previous subsection, with the independent variable (∆ri (T |k))
and dependent variable measured over periods of two weeks up to thirty weeks (i.e., 2 ≤ k ≤
30). In these regressions, we use overlapping weekly observations.17 The standard errors of
the estimates are calculated using the Newey-West procedure.
We compare the estimated durations and the R-squares of those regressions across all
ages, guarantee periods and measurement periods (k). Since there are numerous regression
combinations and their results tend to be similar, we will only present the results for annuities
for a 65-year-old male. This is the age at which most people start their retirement. The
regression estimates are displayed in Table 5, Panel A, and in Figures 2 (R-squares) and 3
(duration).
Insert Table 5 here
Insert Figures 2 and 3 here
As can be seen from the two figures, the estimated durations and the R-squares increase
when changes are measured over longer periods. Both of them reach their maximums at the
measurement period of k = 5 weeks and then start to decline gradually. This pattern is true
for both male and female annuities. At their maximums, the estimated durations are 2.54
years for male and 2.61 years for female. While both numbers are higher than in the case
where k = 1 week, they are still far below the theoretical values of 9.56 years (from Table 1).
The R-squares peak at around 33%.
The fact that the estimated sensitivity and the R-squares increase when measured over
longer periods (up to k = 5 weeks) support the notion that insurance companies do not
instantaneously and fully adjust the annuity factors to respond to every change in risk-free
interest rates (as proxied by the swap rates). Rather, the adjustments are gradual and
infrequent as they let some time pass before incorporating changes in interest rates over a
longer period of time into annuity prices. Note, however, that the estimated durations and
the R-squares start to decline as the measurement period lengthens beyond five weeks. We
attribute this to the fact that annuity factors do not change as much or as often as swap rates
do. Therefore, beyond a certain measurement horizon (5 weeks in this case), the correlation
17
For example, for the measurement period of k = 4 weeks, the first observation is the change from week
1 to week 5, while the second observation is the change from week 2 to week 6, and so on.
14
between the two changes become lower and lower. This causes the observed declines in the
sensitivity and the explanatory power of the independent variable.
To account for the possibility that a linear regression may not capture the convexity
in the relationship between interest rates and annuity factors (similar the convexity in the
relationship between interest rates and bond prices), we also perform the above regressions
with the squares of changes in interest rates as an additional independent variable. That is,
the regression equation becomes:
−
∆ai (x, g|k)
= βk + β∆rt (10 |k ) ∆ri (10|k) + β2 ∆ri2 (10|k) + εi .
ai−k (x, g)
(15)
The regression estimates are shown in Table 5, Panel B. As before, we only present the
results for annuities for a 65-year-old male and a 65-year-old female. As can be seen, the
estimated coefficients for the squares of changes in interest rates are significantly different
from zero for all lags presented, suggesting that the relationship between annuity factors and
interest rates are indeed non-linear. However, the estimated durations and the R-Squares are
only slightly higher than before. For example, when the measurement period is k = 5 weeks,
the new estimated duration is 2.91 years for male and 2.99 years for female. While both
estimates are higher than their counterparts without the squared term, they are again still
far short of the theoretical durations. Similarly, the improvement in R-squares is minimal
(36% vs. 33%). Accordingly, we conclude that the low estimated sensitivity in Table 5 is
not the result of model misspecification.
4.2.2
30-Year Mortgage Rates
Next, we investigate the possibility that it may not be appropriate to use swap rates, which
are only slightly higher than the risk-free rates, in the regressions. That is, annuity providers
typically base their prices on the rates of return on their investment portfolios. According
to a report by the National Association of Insurance commissioners (NAIC), the four largest
types of assets held by an average life insurance companies as of July, 2011 are corporate
bonds (43.6%), US government bonds (18.7%), structured securities such as mortgage-backed
and asset-backed bonds (18.5%) and commercial mortgage loans (8.5%).18 As a result, the
average investment portfolio is not risk-free and certainly more risky than the risk of swap
transactions. The use of a more risky proxy may provide an additional insight into the
movements of annuity factors.
To test this conjecture, we substitute the 30-year mortgage interest rates for the 10-year
swap rates in the regressions. Mortgage rates are determined primarily by the rates of return
on mortgage-backed securities (see, for example, Roth (1988)), which are more risky than
swap transactions. As a result, mortgage rates should be more representative of the rates of
18
The information can be found at http://www.naic.org/capital markets archive/110819.htm
15
return on insurance companies’ investment portfolios, which also include mortgage-backed
securities.
These rates are obtained from the Federal Home Loan Mortgage Corporation’s Primary
Mortgage Market Survey, which is conducted every week. The survey collects information
from various mortgage lenders on the rates for several kinds of mortgage products. The
30-year mortgage rates are the rates that borrowers can expect to be offered if they request
30-year fixed-rate mortgage loans on the survey day. Since the rates can vary across lenders,
the number reported for each week is the weighted average of all the survey responses in
that week, where the weights are based on the dollar volume of mortgage transactions.19
Table 3 , Panel A, displays the summary statistics for the 30-year mortgage rates. The
30-year mortgage rates are less volatile than the 10-year swap rates (see Table 3). The
average weekly change in the rates during the sample period is −0.0029% p.a., while the
standard deviation is 0.1041% p.a., both of which values are lower than their counterparts
for the 10-year swap rates. The range between the minimum and the maximum changes
is also narrower. Although not shown, the average of the absolute values of the changes is
0.071% p.a., which again is lower than the 10-year-swap-rate counterpart. The frequencies
of large changes are also lower, with changes larger than 0.10% (0.25%) p.a. happening only
in 70 (13) of the weeks. These observations are consistent with past findings that mortgage
rates move more slowly than market interest rates (see, for example, Allen, Rutherford and
Wiley (1999) and Payne (2006)).
As before, we run the regressions for various measurement periods, and examine the
pattern of the R-squares and durations. Since the patterns are similar across ages, we show
only the R-squares and durations for annuities for a 65-year-old male (Figures 4 and 5, Panel
A) and a 65-year-old female (Figures 4 and 5, Panel B). Similar to the previous case, Rsquares are less than 10% when changes are measured over a period of one week. They then
increase rapidly to approximately 50% when the measurement period is beyond five weeks.
We interpret this result to mean that changes in annuity factors occur even more slowly than
changes in mortgage rates. However, in contrast to the previous case, the R-squares do not
decline beyond that length. Rather, they remain stable for all longer measurement periods.
This pattern of larger impact and stable values as the measurement period stretches out to
20 weeks is also apparent in durations , Figure 5.
Insert Figures 4 and 5 here
Since the R-squares remain approximately the same beyond k = 8 weeks, we choose to
present the results for the measurement period of k = 8 weeks (Table 6). For comparison,
we also report the estimates for the measurement period of k = 16 weeks (Table 7).
Insert Table 6 and Table 7here
19
For more details about how the survey is conducted and how the averages are calculated, see the web
site of the survey at: http://www.freddiemac.com/pmms/abtpmms.htm.
16
The estimated durations in Table 6 (k = 8 weeks) are well below the theoretical values,
but now much closer than we saw for the 10 year swap rate. Consistent with the theoretical
results, duration generally declines as guarantee period drops and as age increases. Model
fit is also much better when making use of the 30 year mortgage rates.
4.2.3
30-Year Mortgage Rates and Asymmetric Responses
We now re-estimate the empirical durations using the 30-year mortgage rates and in addition,
we want to account for the possibility that the manner in which insurance companies react
to changes in interest rates can be different between a rate increase and a rate decrease.
Studies on consumer interest rates such as bank deposit rates have shown that US banks
are slower to raise deposit rates than to reduce them (See, for example, Hannan and Berger
(1991) and Neuman and Sharpe (1992)). We want to know whether such an asymmetric
response also occurs in the annuity markets. To this end, we specify the regression equation
as follows:
−
∆ai (x, g|k)
= βk + β∆rt,Neg.Sign (30 |k ) · ∆ri (30|k) · I∆r,neg + β∆rt (30 |k ) · ∆ri (30|k) + εi , (16)
ai−k (x, g)
where ai (x, g|k) is as previously defined, ∆ri (30|k) is the change in the 30-year mortgage
rates over the measurement period k, and I∆r,neg is an indicator variable that takes on the
value of 1 when interest rates decline (i.e., ∆ri (T |k) < 0), and zero otherwise. Accordingly,
β∆rt,Neg.Sign (30 |k ) + β∆rt (30 |k ) is the estimated duration when ∆ri (T |k) < 0, and β∆rt (30 |k ) is the
estimated duration when ∆ri (T |k) > 0.
We again choose to present the results for the measurement period of k = 8 weeks
(Table 8) and for the measurement period of k = 16 weeks (Table 9). Our qualitative results
are not sensitive to this choice, as comparison of Table 8 and Table 9 verify.
Insert Table 8 here
Consider first the estimated durations in Table 8 (k = 8 weeks). For every annuity
combination (for both male and female), the estimated duration is much higher when interest
rates increase (Panels A and C) than when interest rates decline (Panels B and D). When
rates increase, the durations are generally about 5 − 7 years, which are much higher than
the durations in the previous set of regressions (i.e., with the 10-year swaps rates). They
are also much closer to the theoretical values in Table 1 although the differences between
them are still significant. On the other hand, when rates decline, the estimated durations
are only about 3 − 4 years, which nevertheless are higher than the results when the 10-year
swap rates were used. These results suggest that annuity factors correspond more closely
to changes in mortgage rates than to changes in the 10-year swap rates. They also indicate
that the response is asymmetric. When interest rates increase, insurance companies reduce
17
annuity prices (by increasing the amounts of payouts) faster and by a greater magnitude than
what they do in the opposite direction when interest rates decline. While this downwardsticky pattern of actions is beneficial to the customers, it clearly violates the conventional
annuity-pricing models. It also raises the possibility that potential annuitants can time their
purchases based on the observed movements in interest rates.20
The downward-sticky response is consistent with the customer reaction hypothesis mentioned in Hannan and Berger (1991). In our context, the hypothesis states that insurance
companies, especially those operating in a highly competitive market, may fear a negative
reaction from potential annuity buyers if annuity payout rates are reduced. Therefore, they
will try to delay a rate reduction as long as they can. This will also allow them to observe the
trend of the interest rate movements before they commit to a change. In the meantime, they
absorb the loss through the profit margin that they build into the annuity prices. While the
US annuity markets are dominated by only a few large providers, the results of the money’s
worth studies indicate that these few providers do not abuse their oligopolistic power, as
evidenced by the fact that the money’s worth ratios are generally not far below unity (see
Mitchell et al (1999) and James and Song (2001)).21 That is, the US annuity markets appear
to operate in a competitive manner.
Insert Table 9 here
Similar results are obtained when the measurement period is k = 16 weeks (Table 9).
The estimated duration is indeed slightly higher for every annuity combination. Again,
we observe the asymmetric responsiveness to changes in interest rates. The results from
both measurement periods confirm that annuity prices are more closely related to mortgage
rates than to swap rates, which are proxies for risk-free rates. This is in contrast to what
is traditionally assumed in the literature. In addition, since mortgage rates tend to move
more slowly than risk-free rates, the high R-Squares and high estimated durations add more
support to our argument that insurance companies do not quickly and fully adjust annuity
prices in response to every change in the risk-free rates.
5
Conclusion and Implications
In this paper, we test whether annuity prices promptly and fully adjust to changes in interest
rates. The prompt and full response of prices to interest rate changes is a standard assump20
For example, potential annuitants who expect market interest rates to rise can afford to wait. If,
as expect, interest rates increase, they will benefit by getting higher payout rates. If, contrary to their
expectation, interest rates move down, annuity prices do not quickly react to the decline and so they will
likely still be able to get a similar payout rate as before. Of course, making this statement more precise would
require a dynamic model of annuity prices and would require some sort of utility function, but hopefully the
intuition is clear.
21
This argument follows a similar one made by Cannon and Tonks (2005).
18
tion that is made, implicitly or explicitly, in most annuity studies. The database that we
use consists of over three million annuity quotes for single premium immediate life annuities
over the period from 2004 to 2010.
We find that real-life annuity prices do not behave as assumed. Rather, annuity providers
adjust annuity prices only gradually over a period of several weeks to respond to interest
rate changes that occur over the same number of weeks. We conjecture that this is due to
annuity providers’ desire to smooth out the price changes and/or to wait in order to get a
better sense of the trend of interest rates. Alternatively, since annuity providers from time
to time have unbalanced books (i.e., mismatches of durations of assets and liabilities), they
may deliberately offer uncompetitive annuity prices or delay price adjustments in order to
allow themselves time to rebalance their books.
In addition, we find that the response of annuity prices to changes in interest rates is
asymmetric. Annuity prices react more rapidly and with greater sensitivity to an increase
in interest rates than to a decrease in interest rates. That is, when interest rates increased,
insurance companies reduce their annuity prices (by increasing the amount of monthly annuity income per the same principal amount) more quickly and in a larger magnitude than
what they do in the opposite direction when rates declined. In other words, annuity rates
(which are inversely related to annuity prices) are more rigid downward than upward. This is
quite unexpected and in contrast to the results of several studies done on consumer interest
rates pass through, which find that US banks are slower to raise deposit rates than to reduce
them. Our findings are consistent with the customer reaction hypothesis whereby insurance
companies may fear a negative reaction from potential annuity buyers if annuity rates are
reduced.
Our findings are inconsistent with the implications of the standard annuity-pricing model.
As a result, they have wide-ranging implications on the existing literature. The fact that
annuity prices are slow to adjust to new interest-rate information suggests that there are
periods during which annuities are temporarily “mispriced” relative to their theoretical values. The temporary mispricing suggests that contrary to the theoretical results, the timing
of the annuity purchase can affect the individual’s optimal allocation decisions and the magnitude of welfare gains from annuitization. This is particularly important considering that
an annuity purchasing decision is irreversible and typically involves a substantial portion of
an individual’s wealth. In addition, our results imply that the money’s worth of an annuity
can vary depending on the timing of the calculations (even if every other factor remains the
same). Similarly, caution should be exercised when attempting to infer mortality expectations from observed annuity prices.
Either way, we have identified yet another puzzle in the annuity market, one that we
have labeled a duration puzzle.
19
6
Appendix 1: General Derivation of Annuity Factor
and Duration
In the body of the paper we derived an expression for the (theoretical) duration of a life
annuity assuming the (pricing) yield curve was flat and equal to a constant r. In that simple
case the duration was defined as the (negative) derivative of the annuity factor a(x, r) with
respect to the pricing rate r, and then scaled by the same annuity factor a(x, r). In this
appendix we derive a more general duration measure D, assuming the annuity is priced with
a non-constant yield curve denoted by rt . The duration is properly defined in terms of a
parallel shift in the yield curve. Arguably, this is more realistic compared to assuming a
constant rate.
Our objective is not to (better) price life annuities. Rather, it is to show that even under
a more general definition of interest rate sensitivity, the theoretical values for the duration
of a life annuity are very similar to the values obtained under the more restrictive (constant)
case. In other words, regardless of how exactly the interest rate sensitivity is defined for a
life annuity, the theoretical values are much larger than the empirically observed ones. The
puzzle doesn’t go away.
In general the annuity factor can be represented as follows:
Z ∞
e−(rt )t p(x, t)dt,
(17)
a(x, g, r̃) =
0
where rt is the entire yield curve, and r̃ is short-hand notation for the function itself. The
survival probability function p(x, t) is defined in the following way:
p(x, t) = e−
Rt
0
λgs ds
,
where λgs is a (truncated) mortality rate defined by:
(
0 for s ≤ g
λgs =
λs for s > g
(18)
(19)
Using this notation, g is the guarantee (certain) period in years and λs is any continuous
mortality rate. Since payments are guaranteed for g years, the effective mortality rate is
zero until time s = g.
The derivative of the annuity factor with respect to a parallel shift in the yield curve in
the amount ε, can be defined and expressed as follows:
Z ∞
Z ∞
∂
∂a(x, g, r̃+ε)
−(rt +ε)t
=
e
p(x, t)dt =
−te−(rt +ε)t p(x, t)dt
(20)
∂ε
∂ε 0
0
The ability to switch the integral and the derivative sign follows directly from Fubini’s
Theorem, since our functions p(x, t) and (rt + ε)t are well-behaved and satisfy the necessary
differentiability conditions.
20
Now, if the interest rate (yield) curve rt is continuous over the interval (0, t), then the
composite function rt t is differentiable and by the fundamental theorem of calculus can be
written as:
Z t
ξs ds
rt t =
(21)
0
As an example, consider the interest rate function rt = r − c(t + 1)−1 . It starts at a value
of (r − c) for maturity zero, and then asymptotes to r as t → ∞. The function is obviously
continuous, and the product with time is rt t = rt − ct(t + 1)−1 . So, the function ξs :=
r − c((t + 1)−1 − t(t + 1)−2 ).
A by-product of this decomposition is that the derivative of the annuity factor can now
be expressed as:
Z ∞
Z ∞
Rt
R
g
∂a(x, g, r̃+ε)
−εt − 0t ξs ds
(22)
p(x, t)dt =
−te−εt e− 0 (ξs +λs )ds dt
=
−te e
∂ε
0
0
Z ∞
=
−te−εt p̂(x, t)dt,
(23)
0
where the new survival probability curve p̂(x, t) is defined structurally as before, but with a
new underlying hazard rate:
(
ξs
for s ≤ g
λ̂gs =
(24)
λs + ξs for s > g
Intuitively, this is a mortality model in which the mortality rate is increased by the relevant
interest rate. Think of someone that is older. All else being equal, the (simple) duration will
be lower, but will satisfy the same structural equation derived within the body of the paper,
but with different mortality rate parameters.
At this point we (finally) let ε → 0, so that e−εt → 1 and the duration formally becomes:
Z ∞
−∂a(x, g, r̃+ε)
1
1
=
tp̂(x, t)dt.
(25)
D(x, g) :=
∂ε
a(x, g, r̃) ε=0 a(x, g, r̃) 0
This expression can be computed numerically (fairly easily.) For example, consider the
following yield curve model rt = 0.046 − (0.0275)(t + 1)−1 , which is a non-constant upward
sloping yield curve. The zero-maturity rate is r0 = 0.046 − 0.0275 = 0.018 5, which is 1.85%.
The ten-year maturity rate is r10 = 0.046 − (0.0275)(11)−1 = 0.0435, which is 4.35% – the
same rate we used within the body of the paper. The asymptotic rate is r∞ = 4.6% This is
ad hoc, but our point is as follows.
Assuming a x = 65 and x = 75 year-old female with Gompertz mortality (parameters
m = 92.63 and b = 8.78), and the above interest rate curve, the annuity factor’s are:
a(65, 0, r̃) = $14.1284
a(75, 0, r̃) = $10.7470
21
and the generalized duration will be:
D(65, 0) = 10.3764
(26)
D(75, 0) = 7.7975
(27)
which are very close, but slightly lower than the values presented in Table 1, which assumed
a flat 4.35% pricing curve. In other words, a more robust yield curve model doesn’t result
in (much) lower durations and can’t explain our findings.
22
References
[1] Allen, M. T., Rutherford, R. C. and Wiley, M. K. (1999), The relationships between
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26
Table 1
Theoretical Annuity Factors ($) and Duration (Years)
Panel A: Annuity Factor, Male
Guarantee Period (Years)
20
15
10
5
Age
55
16.4375 16.0483 15.7793 15.6229
60
15.5394 14.9462 14.5260 14.2777
65
14.7217 13.8449 13.1999 12.8092
14.0685 12.8308 11.8671 11.2606
70
75
13.6400 12.0020 10.6198 9.6976
80
13.4318 11.4354 9.5650 8.2071
Panel B: Annuity Factor, Female
55
17.3398 17.1439 17.0201 16.9532
60
16.3545 16.0199 15.8047 15.6875
65
15.3703 14.8128 14.4439 14.2395
70
14.4940 13.6032 12.9846 12.6319
75
13.8459 12.5149 11.5159 10.9178
80
13.4922 11.6904 10.1722 9.1880
Panel C: Duration, Male
55
12.04
11.90
11.89
11.93
60
11.00
10.74
10.68
10.74
65
10.07
9.59
9.44
9.50
70
9.34
8.54
8.21
8.23
75
8.86
7.68
7.04
6.97
80
8.64
7.10
6.03
5.75
Panel D: Duration, Female
55
13.00
12.95
12.95
12.97
60
11.86
11.74
11.73
11.76
65
10.76
10.50
10.44
10.48
70
9.79
9.27
9.11
9.15
75
9.08
8.17
7.79
7.79
8.70
7.34
6.56
6.43
80
0
15.5719
14.1959
12.6786
11.0530
9.3701
7.6965
16.9327
15.6513
14.1757
12.5200
10.7228
8.8515
11.96
10.78
9.56
8.33
7.10
5.91
12.98
11.78
10.51
9.20
7.87
6.55
Notes: Panels A and B display the theoretical annuity factors for male and female annuitants, respectively, of various ages and with various guarantee periods. Panels C and D
display the theoretical durations of life annuities for male and female annuitants, respectively, of various ages and with various guarantee periods.
Age
55
[
60
[
65
[
70
[
75
80
Table 2, Panel A
Summary Statistics of Observed Annuity Factor
Male Annuitant
Guarantee Period (Years)
20
15
10
5
0
16.156
15.830
15.594
15.430
15.375
( 0.96)
( 0.94)
( 0.93)
( 0.94)
( 0.93)
14.6, 19.0] [ 14.3, 18.6] [ 14.1, 18.4] [ 14.0, 18.2] [ 13.9, 18.2]
15.371
14.853
14.484
14.249
14.168
( 0.87)
( 0.83)
( 0.81)
( 0.82)
( 0.81)
14.0, 18.0] [ 13.5, 17.3] [ 13.2, 16.9] [ 13.0, 16.7] [ 12.9, 16.6]
14.648
13.846
13.248
12.871
12.748
( 0.79)
( 0.72)
( 0.69)
( 0.69)
( 0.69)
13.4, 16.9] [ 12.7, 16.0] [ 12.1, 15.3] [ 11.8, 14.9] [ 11.7, 14.8]
13.963
12.925
12.006
11.387
11.190
( 0.87)
( 0.64)
( 0.60)
( 0.59)
( 0.59)
13.0, 16.3] [ 11.9, 14.7] [ 11.1, 13.7] [ 10.5, 13.1] [ 10.3, 12.9]
12.089
10.847
9.884
9.553
N/A
( 0.70)
( 0.52)
( 0.50)
( 0.52)
[ 11.3, 13.9] [ 10.1, 12.4] [ 9.2, 11.4] [ 8.9, 11.0]
9.769
8.432
7.899
N/A
N/A
( 0.53)
( 0.41)
( 0.44)
[ 9.2, 11.2]
[ 7.9, 9.6]
[ 7.3, 9.0]
Notes: This table displays the summary statistics of annuity factors for male annuitants
of various ages and with various guarantee periods, over the period 2004-2010. Standard
errors in round paretheses. Minimum and maximum in square paretheses.
Age
55
[
60
[
65
[
70
[
75
80
Table 2, Panel B
Summary Statistics of Observed Annuity Factor
Female Annuitant
Guarantee Period (Years)
20
15
10
5
0
16.627
16.412
16.254
16.155
16.150
( 0.99)
( 0.97)
( 0.97)
( 0.97)
( 0.98)
15.0, 19.7] [ 14.8, 19.4] [ 14.7, 19.2] [ 14.6, 19.1] [ 14.5, 18.9]
15.809
15.461
15.218
15.066
15.034
( 0.90)
( 0.87)
( 0.86)
( 0.86)
( 0.87)
14.3, 18.5] [ 14.0, 18.1] [ 13.8, 17.8] [ 13.6, 17.6] [ 13.6, 17.6]
14.986
14.416
14.021
13.784
13.711
( 0.81)
( 0.76)
( 0.73)
( 0.74)
( 0.74)
13.6, 17.4] [ 13.1, 16.7] [ 12.8, 16.2] [ 12.5, 16.0] [ 12.5, 15.9]
15.222
14.226
13.375
12.728
12.328
( 1.02)
( 0.86)
( 0.66)
( 0.63)
( 0.62)
14.1, 18.0] [ 13.2, 16.6] [ 12.3, 15.3] [ 11.7, 14.6] [ 11.3, 14.1]
13.674
12.382
11.398
10.721
N/A
( 0.88)
( 0.65)
( 0.54)
( 0.53)
[ 12.7, 16.0] [ 11.6, 14.2] [ 10.6, 13.0] [ 9.9, 12.3]
11.698
10.121
9.051
N/A
N/A
( 0.70)
( 0.50)
( 0.41)
[ 10.9, 13.5] [ 9.5, 11.6] [ 8.4, 10.2]
Notes: This table displays the summary statistics of annuity factors for female annuitants
of various ages and with various guarantee periods, over the period 2004-2010. Standard
errors in round paretheses. Minimum and maximum in square paretheses.
Table 3, Panel A
Summary Statistics
Variable
10 Year Swap Rate (%)
∆ 10 Year Rate (%): ∆ri (10|1)
30 Year Mortgage Rate (%)
∆ 30 Year MortgageRate (%): ∆ri (30|1)
N
333
332
332
330
Mean
Std
4.3687 0.8763
-0.0033 0.1394
5.7198 0.6826
-0.0025 0.1043
Min
Max
2.4700 5.8400
-0.7100 0.5200
4.1700 6.8000
-0.4400 0.5200
Skew Kurt
-0.45 -0.82
-0.16 2.49
-0.47 -0.93
0.43 5.47
Notes: This table displays the summary statistics for weekly levels and changes in the
10-year swap interest rate and 30 year conventional mortgage rate, over 2004-2010. The
mortgage rate is sourced from the Board of Governors of the Federal Reserve System, series
id WRMORTG.
Table 3, Panel B
Summary Statistics of Weekly Percentage Changes in Annuity Factors,
Male Annuitant
Guarantee Period (Years)
Age
20
15
10
5
0
55
-.0180
-.0241
-.0253
-.0215
-.0189
(0.6503)
(0.6790)
(0.6469)
(0.6649)
(0.6346)
[-2.53,3.204] [-2.56,3.737] [-2.62,3.222] [-2.45,4.163]
[-2.52,3.275]
60
-.0202
-.0202
-.0193
-.0227
-.0149
(0.5856)
(0.5766)
(0.5946)
(0.6000)
(0.5732)
[-2.18,2.853] [-2.20,2.803] [-2.28,2.818] [-2.08,3.671]
[-2.13,2.888]
65
-.0145
-.0156
-.0201
-.0140
-.0111
(0.6447)
(0.5210)
(0.5317)
(0.5346)
(0.5154)
[-2.78,2.778] [-2.01,2.389] [-2.09,2.390] [-1.89,3.178]
[-1.90,2.505]
70
-.0388
-.0134
-.0112
-.0107
-.0099
(0.6366)
(0.4655)
(0.4633)
(0.4706)
(0.4543)
[-2.34,3.187] [-1.60,1.972] [-1.89,1.979] [-1.64,2.680]
[-1.45,2.175]
75
-.0359
-.0124
-.0137
-.0123
N/A
(0.5356)
(0.4313)
(0.4371)
(0.4095)
[-2.11,1.951] [-2.04,1.416] [-1.93,1.956]
[-1.43,1.649]
80
-.0201
-.0158
-.0127
N/A
N/A
(0.5226)
(0.3887)
(0.3991)
[-2.50,2.782] [-1.73,1.208]
[-1.50,1.333]
Notes: This table displays the summary statistics of weekly changes in annuity factors
(male annuitant). Standard errors in round parentheses. Minimum and maximum in square
parentheses.
Table 3, Panel C
Summary Statistics of Weekly Percentage Changes in Annuity Factors,
Female Annuitant
Guarantee Period (Years)
Age
20
15
10
5
0
55
-.0073
-.0105
-.0175
-.0196
-.0264
(0.6777)
(0.6660)
(0.6597)
(0.6780)
(0.6743)
[-2.71,2.982] [-2.73,2.889] [-2.80,2.841] [-2.67,3.613]
[-2.92,2.910]
60
-.0250
-.0233
-.0271
-.0256
-.0103
(0.6416)
(0.6332)
(0.6081)
(0.6134)
(0.5982)
[-3.39,2.647] [-2.64,2.465] [-2.35,2.361] [-2.21,2.991]
[-2.32,2.324]
65
-.0133
-.0125
-.0286
-.0234
-.0133
(0.6479)
(0.5396)
(0.5595)
(0.5475)
(0.5283)
[-2.80,3.022] [-2.07,2.055] [-2.13,1.856] [-1.97,2.313]
[-2.04,1.774]
70
-.0470
-.0144
-.0115
-.0111
-.0121
(0.8078)
(0.6646)
(0.4778)
(0.4731)
(0.4745)
[-3.03,3.562] [-2.43,3.256] [-1.61,1.679] [-1.88,1.401]
[-1.65,1.607]
75
-.0169
-.0150
-.0084
-.0107
N/A
(0.6683)
(0.5698)
(0.4356)
(0.4338)
[-2.35,3.863] [-2.16,3.485] [-2.09,1.363]
[-1.97,1.281]
80
-.0256
-.0193
-.0059
N/A
N/A
(0.6615)
(0.4601)
(0.3433)
[-3.92,3.515] [-1.71,2.894]
[-1.13,1.039]
Notes: This table displays the summary statistics of weekly changes in annuity factors
(female annuitant). Standard errors in round parentheses. Minimum and maximum in
square parentheses.
Table 4, Panel A, Male Annuitant
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 1 week
Using Risk-free 10 Year Swap Rates
Age
Parameter Estimates
Guarantee Period (Years)
20
15
10
5
55
60
65
70
75
80
1.21
1.14
1.29
0.32
N/A
N/A
1.21
1.10
1.01
0.85
0.16
N/A
0
1.24
1.10
0.96
0.78
0.61
0.48
Age
1.25 1.31
1.01 1.12
0.92 0.99
0.80 0.83
0.68 0.68
-0.10 0.52
2
R s
Guarantee Period (Years)
20
15
10
5
55
60
65
70
75
80
0.069 0.064
0.076 0.072
0.080 0.074
0.005 0.067
N/A 0.002
N/A N/A
0.078
0.074
0.071
0.059
0.045
0.030
0.075
0.058
0.060
0.060
0.050
0.001
0.078
0.070
0.069
0.063
0.049
0.037
0
Notes: This table displays the estimated durations of life annuities for male annuitants of
various ages and with various guarantee periods. The independent variable of the regressions
is the weekly changes in the 10-year swap rates. The dependent variable is the negative of the
weekly percentage changes in the annuity factors. Bolded coefficients denote significance
at the 0.05 level, based on two-sided tests, of a difference between the estimated Duration
coefficient and the theoretical value.
Table 4, Panel B, Female Annuitant
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 1 week
Using Risk-free 10 Year Swap Rates
Age
55
60
65
70
75
80
Age
55
60
65
70
75
80
Parameter Estimates
Guarantee Period (Years)
20
15
10
5
1.22 1.22 1.27 1.34
1.17 1.15 1.17 1.22
1.22 1.04 0.97 1.06
0.49 0.39 0.91 0.86
N/A 0.16 0.10 0.70
N/A N/A -0.00 -0.11
R2 s
Guarantee Period (Years)
20
15
10
5
0.065 0.067
0.065 0.066
0.072 0.075
0.007 0.007
N/A 0.001
N/A N/A
0.074
0.074
0.060
0.073
0.001
0.000
0.079
0.079
0.075
0.066
0.052
0.001
0
1.41
1.19
1.02
0.89
0.72
0.51
0
0.088
0.079
0.075
0.071
0.055
0.045
Notes: This table displays the estimated durations of life annuities for female annuitants
of various ages and with various guarantee periods. The independent variable of the regressions is the weekly changes in the 10-year swap rates. The dependent variable is the negative
of the weekly percentage changes in the annuity factors. Bolded coefficients denote significance at the 0.05 level, based on two-sided tests, of a difference between the estimated
Duration coefficient and the theoretical value.
Table 5, Panel A
Estimated Durations: Age 65 and Guarantee Period 0 Years
Measurement Period k = 1, ...20 weeks, Using Risk-free 10 Year Swap Rates
k
1
N
277
2
273
3
272
4
270
5
268
6
267
7
265
8
266
9
266
10
263
11
262
12
264
13
262
14
259
15
261
16
257
17
257
18
257
19
254
20
253
Male
β∆rt (10 |k )
0.96
( 0.24)
1.74
( 0.22)
2.21
( 0.30)
2.40
( 0.38)
2.54
( 0.44)
2.46
( 0.49)
2.42
( 0.54)
2.39
( 0.58)
2.30
( 0.63)
2.25
( 0.66)
2.22
( 0.74)
2.20
( 0.81)
2.15
( 0.83)
2.16
( 0.84)
2.30
( 0.85)
2.30
( 0.86)
2.33
( 0.87)
2.31
( 0.85)
2.22
( 0.84)
2.08
( 0.83)
R2
0.07
N
279
0.19
275
0.26
274
0.29
274
0.33
271
0.32
269
0.31
267
0.29
269
0.26
269
0.25
265
0.23
263
0.21
267
0.20
264
0.20
264
0.22
263
0.21
260
0.21
259
0.21
258
0.20
257
0.18
256
Female
β∆rt (10 |k )
1.02
( 0.24)
1.83
( 0.22)
2.27
( 0.30)
2.48
( 0.38)
2.61
( 0.43)
2.55
( 0.50)
2.51
( 0.53)
2.48
( 0.58)
2.42
( 0.62)
2.35
( 0.66)
2.33
( 0.74)
2.31
( 0.81)
2.27
( 0.82)
2.30
( 0.84)
2.38
( 0.84)
2.36
( 0.84)
2.41
( 0.87)
2.37
( 0.86)
2.35
( 0.87)
2.24
( 0.87)
R2
0.07
0.20
0.27
0.30
0.33
0.33
0.32
0.30
0.28
0.26
0.25
0.22
0.21
0.22
0.22
0.22
0.22
0.21
0.21
0.19
Notes: This table displays the estimated durations of life annuities for male and female annuitants, age
65 and 0 year guarantee. The independent variable of the regressions is the weekly changes in the 10-year
swap rates. The dependent variable is the negative of the weekly percentage changes in the annuity factors.
Bolded coefficients denote significance at the 0.05 level, based on two-sided tests, of a difference between
the estimated Duration coefficient and the theoretical value. Standard errors in parentheses.
Table 5, Panel B
Estimated Durations: Age 65 and Guarantee Period 0 Years
Measurement Period k = 1, ...20 weeks, Using Risk-free 10 Year Swap Rates
k
1
N
277
2
273
3
272
4
270
5
268
6
267
7
265
8
266
9
266
10
263
11
262
12
264
13
262
14
259
15
261
16
257
17
257
18
257
19
254
20
253
Male
β∆rt (10 |k ) β∆rt2 (10 |k )
0.97
14.26
( 0.24)
(73.14)
1.77
26.89
( 0.22)
(24.96)
2.37
72.09
( 0.31)
(32.78)
2.74
112.9
( 0.37)
(36.65)
2.91
107.4
( 0.42)
(42.08)
2.79
86.25
( 0.53)
(50.67)
2.69
70.04
( 0.56)
(48.32)
2.67
67.57
( 0.62)
(53.20)
2.58
63.56
( 0.67)
(59.70)
2.51
57.38
( 0.72)
(56.80)
2.50
63.69
( 0.79)
(58.65)
2.58
90.13
( 0.83)
(61.97)
2.60
96.69
( 0.88)
(58.11)
2.61
96.30
( 0.93)
(66.29)
2.66
78.27
( 0.96)
(78.30)
2.75
81.11
( 1.01)
(72.31)
2.77
74.72
( 1.08)
(78.64)
2.73
67.26
( 1.08)
(78.39)
2.46
36.96
( 1.15)
(84.36)
2.10
2.92
( 1.16)
(86.92)
R2
0.07
N
279
0.19
275
0.27
274
0.32
274
0.36
271
0.35
269
0.33
267
0.31
269
0.28
269
0.27
265
0.25
263
0.24
267
0.23
264
0.23
264
0.23
263
0.23
260
0.23
259
0.22
258
0.20
257
0.18
256
Female
β∆rt (10 |k ) β∆rt2 (10 |k )
1.02
11.84
( 0.24)
(75.87)
1.88
34.68
( 0.23)
(25.47)
2.44
74.31
( 0.30)
(31.80)
2.83
115.5
( 0.37)
(36.43)
2.99
109.0
( 0.42)
(41.31)
2.89
87.42
( 0.53)
(50.93)
2.78
69.09
( 0.56)
(48.33)
2.75
63.70
( 0.62)
(53.45)
2.68
59.51
( 0.68)
(60.70)
2.59
52.98
( 0.74)
(57.85)
2.58
57.60
( 0.81)
(60.10)
2.67
85.45
( 0.86)
(63.29)
2.70
92.29
( 0.90)
(58.78)
2.71
87.40
( 0.96)
(68.68)
2.69
70.32
( 1.00)
(79.38)
2.76
73.42
( 1.04)
(73.83)
2.81
67.26
( 1.13)
(81.06)
2.70
53.56
( 1.16)
(82.82)
2.51
24.88
( 1.22)
(88.31)
2.22
-3.03
( 1.23)
(89.56)
R2
0.07
0.20
0.28
0.32
0.36
0.36
0.33
0.32
0.29
0.27
0.26
0.25
0.24
0.24
0.24
0.23
0.23
0.21
0.21
0.19
Notes: This table displays the estimated durations of life annuities for male and female annuitants, age
65 and 0 year guarantee. The regression equation is:
−
∆ai (x, g|k)
= βk + β∆rt (10 |k ) ∆ri (10|k) + β∆rt2 (10 |k ) ∆ri2 (10|k) + εi .
ai−k (x, g)
(28)
The independent variable of the regressions is the weekly changes in the 10-year swap rates. The dependent variable is the negative of the weekly percentage changes in the annuity factors. Bolded coefficients
denote significance at the 0.05 level, based on two-sided tests, of a difference between the estimated Duration
coefficient and the theoretical value. Standard errors in parentheses.
Table 6
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 8 week
Male Annuitant, Using Risk-free 10 Year Swap Rates
Age
55
60
65
70
75
80
55
60
65
70
75
80
Panel A
Estimated Durations
Guarantee Period (Years)
20
15
10
5
0
5.03 5.03 5.03 5.12 5.02
4.76 4.65 4.65 4.79 4.66
4.38 4.18 4.08 4.16 4.18
3.72 3.78 3.63 3.69 3.74
N/A 3.08 3.16 3.14 3.19
N/A N/A 2.25 2.57 2.61
Panel B
R2 s
0.475 0.466 0.473 0.470 0.480
0.483 0.478 0.476 0.484 0.480
0.459 0.469 0.460 0.460 0.468
0.420 0.460 0.447 0.450 0.456
N/A 0.360 0.416 0.408 0.410
N/A N/A 0.260 0.341 0.313
Notes: This table displays the estimated durations of life annuities, together with the model R2 s, derived
from the parameter estimates of models of the form:
x,g
−∆at (x, g|8)/at−k (x, g) = β x,g + β∆r
∗ ∆rt (30 |8 ) + x,g
t .
t (30 |8 )
The independent variable of the regressions is the changes in the 30-year mortgage rates measured over
a period of eight weeks, ∆rt (30|8). The dependent variable is the negative of the percentage changes in
the annuity factors over the same measurement period for male annuitants of various ages and with various
guarantee periods. Bolded coefficients denote significance at the 0.05 level, based on two-sided tests, of a
difference between the estimated Duration coefficient and the theoretical value.
Table 6, Continued
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 8 week
Female Annuitant, Using Risk-free 10 Year Swap Rates
55
60
65
70
75
80
55
60
65
70
75
80
Panel C
Estimated Durations
Guarantee Period (Years)
5.12 5.08 5.10 5.17 5.25
4.83 4.87 4.82 4.86 4.78
4.52 4.30 4.32 4.33 4.34
3.78 3.73 3.88 3.79 3.83
N/A 3.46 3.04 3.30 3.29
N/A N/A 2.79 2.34 2.66
Panel D
R2 s
0.471 0.473 0.476 0.476 0.492
0.451 0.478 0.478 0.479 0.491
0.475 0.474 0.474 0.479 0.491
0.334 0.418 0.470 0.470 0.479
N/A 0.367 0.355 0.440 0.442
N/A N/A 0.282 0.287 0.378
Notes: This table displays the estimated durations of life annuities, together with the model R2 s, derived
from the parameter estimates of models of the form:
x,g
−∆at (x, g|8)/at−k (x, g) = β x,g + β∆r
∗ ∆rt (30 |8 ) + x,g
t .
t (30 |8 )
The independent variable of the regressions is the changes in the 30-year mortgage rates measured over
a period of eight weeks, ∆rt (30|8). The dependent variable is the negative of the percentage changes in
the annuity factors over the same measurement period for male annuitants of various ages and with various
guarantee periods. Bolded coefficients denote significance at the 0.05 level, based on two-sided tests, of a
difference between the estimated Duration coefficient and the theoretical value.
Table 7
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 16 week
Male Annuitant, Using 30-Year Mortgage Rates
Panel A
Estimated Durations When Interest Rates Increase
Guarantee Period (Years)
20
15
10
5
0
Age
55
5.58 5.51 5.62 5.75
5.66
5.29 5.18 5.30 5.36
5.24
60
65
4.94 4.75 4.73 4.78
4.80
70
4.46 4.35 4.28 4.33
4.43
75
N/A 3.75 3.85 3.86
3.98
N/A N/A 3.08 3.36
3.53
80
Panel B
R2 s
55
0.488 0.482 0.498 0.502
0.501
60
0.493 0.488 0.494 0.499
0.495
0.513 0.488 0.487 0.487
0.490
65
70
0.498 0.480 0.482 0.484
0.489
75
N/A 0.441 0.465 0.460
0.459
N/A N/A 0.374 0.402
0.373
80
Notes: This table displays the estimated durations of life annuities, together with the model R2 s, derived
from the parameter estimates of models of the form:
x,g
−∆at (x, g|16)/at−k (x, g) = β x,g + β∆r
∗ ∆rt (30 |16 ) + x,g
t .
t (30 |16 )
The independent variable of the regressions is the changes in the 30-year mortgage rates measured over
a period of sixteen weeks, ∆rt (30|16). The dependent variable is the negative of the percentage changes in
the annuity factors over the same measurement period for male annuitants of various ages and with various
guarantee periods. Bolded coefficients denote significance at the 0.05 level, based on two-sided tests, of a
difference between the estimated Duration coefficient and the theoretical value.
Table 7, Continued
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 16 week
Female Annuitant, Using 30-Year Mortgage Rates
Panel C
Estimated Durations When Interest Rates Increase
Guarantee Period (Years)
5.75
55 5.70 5.67 5.68 5.76
60 5.29 5.27 5.34 5.44
5.40
5.00
65 5.00 4.92 4.98 5.02
70 4.53 4.38 4.50 4.46
4.55
4.04
75 N/A 4.14 3.78 3.99
80 N/A N/A 3.53 2.90
3.55
Panel D
R2 s
55 0.491 0.492 0.494 0.496
0.503
60 0.478 0.487 0.501 0.512
0.503
65 0.502 0.497 0.501 0.508
0.509
0.514
70 0.454 0.494 0.496 0.501
75 N/A 0.428 0.445 0.483
0.488
80 N/A N/A 0.347 0.349
0.428
Notes: This table displays the estimated durations of life annuities, together with the model R2 s, derived
from the parameter estimates of models of the form:
x,g
−∆at (x, g|16)/at−k (x, g) = βkx,g + β∆r
∗ ∆rt (30 |16 ) + x,g
t .
t (30 |16 )
The independent variables of the regressions are the changes in the 30-year mortgage rates measured over
a period of sixteen weeks, ∆rt (30|16). The dependent variable is the negative of the percentage changes in
the annuity factors over the same measurement period for male annuitants of various ages and with various
guarantee periods. Bolded coefficients denote significance at the 0.05 level, based on two-sided tests, of a
difference between the estimated Duration coefficient and the theoretical value.
Table 8, Asymmetric Case
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 8 week
Male Annuitant, Using 30-Year Mortgage Rates
Panel A
Estimated Durations When Interest Rates Increase
Guarantee Period (Years)
20
15
10
5
0
Age
55
6.75 6.86 6.84 7.06
6.75
6.52 6.41 6.38 6.54
6.41
60
65
5.93 5.87 5.75 5.90
5.87
70
5.63 5.36 5.10 5.22
5.25
75
N/A 5.12 4.60 4.57
4.60
N/A N/A 4.02 3.66
3.65
80
Panel B
Estimated Durations When Interest Rates Decline
55
3.61 3.52 3.53 3.51
3.58
60
3.31 3.20 3.23 3.37
3.21
3.10 2.78 2.69 2.72
2.78
65
70
2.19 2.48 2.41 2.42
2.48
75
N/A
1.43 1.97 1.95
2.02
N/A N/A
0.82 1.66
1.75
80
Panel C
R2 s
55
0.493 0.486 0.493 0.493
0.499
0.504 0.501 0.498 0.504
0.502
60
0.478 0.494 0.486 0.486
0.493
65
70
0.454 0.486 0.472 0.476
0.481
75
N/A 0.410 0.445 0.436
0.437
80
N/A N/A 0.311 0.362
0.329
Notes: This table displays the estimated durations of life annuities when interest rates increase or
decrease, together with the model R2 s, derived from the parameter estimates of models of the form:
x,g
−∆at (x, g|8)/at−8 (x, g) = β x,g + β∆rt,Neg.Sign (30 |8 ) · ∆ri (30|8) · I∆r,neg + β∆r
∗ ∆rt (30 |8 ) + x,g
t .
t (30 |8 )
The independent variables of the regressions are the changes in the 30-year mortgage rates measured
over a period of sixteen weeks, ∆rt (30|16) and this variable interacted with a dummy variable that takes on
a value of 1 if the interest rate decreases, ∆ri (30|16) · I∆r,neg . The dependent variable is the negative of the
percentage changes in the annuity factors over the same measurement period for male annuitants of various
ages and with various guarantee periods. Bolded coefficients denote significance at the 0.05 level, based
on two-sided tests, of a difference between the estimated Duration coefficient and the theoretical value.
Table 8, Continued
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 8 week
Female Annuitant, Using 30-Year Mortgage Rates
Panel D
Estimated Durations When Interest Rates Increase
Guarantee Period (Years)
20
15
10
5
0
Age
55
7.01 6.94 6.91 7.05
6.96
7.02 6.74 6.66 6.81
6.54
60
65
6.08 6.10 6.02 6.09
6.02
70
5.49 5.98 5.55 5.34
5.40
75
N/A 5.71 4.95 4.85
4.74
N/A N/A 4.79 4.42
3.70
80
Panel E
Estimated Durations When Interest Rates Decline
55
3.59 3.57 3.63 3.65
3.86
60
2.88 3.37 3.35 3.30
3.33
3.25 2.85 2.94 2.91
2.96
65
70
2.49 2.01 2.54 2.53
2.57
75
N/A
1.73
1.49 2.05
2.10
N/A N/A
1.25
0.76
1.82
80
Panel F
R2 s
55
0.492 0.493 0.496 0.496
0.509
0.482 0.500 0.500 0.503
0.513
60
0.493 0.500 0.497 0.504
0.515
65
70
0.355 0.464 0.497 0.495
0.504
75
N/A 0.414 0.400 0.471
0.470
80
N/A N/A 0.326 0.356
0.397
Notes: This table displays the estimated durations of life annuities when interest rates increase or
decrease, together with the model R2 s, derived from the parameter estimates of models of the form:
x,g
−∆at (x, g|8)/at−8 (x, g) = β x,g + β∆rt,Neg.Sign (30 |8 ) · ∆ri (30|8) · I∆r,neg + β∆r
∗ ∆rt (30 |8 ) + x,g
t .
t (30 |8 )
The independent variables of the regressions are the changes in the 30-year mortgage rates measured
over a period of sixteen weeks, ∆rt (30|16) and this variable interacted with a dummy variable that takes on
a value of 1 if the interest rate decreases, ∆ri (30|16) · I∆r,neg . The dependent variable is the negative of the
percentage changes in the annuity factors over the same measurement period for male annuitants of various
ages and with various guarantee periods. Bolded coefficients denote significance at the 0.05 level, based
on two-sided tests, of a difference between the estimated Duration coefficient and the theoretical value.
Table 9, Asymmetric Case
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 16 week
Male Annuitant, Using 30-Year Mortgage Rates
Panel A
Estimated Durations When Interest Rates Increase
Guarantee Period (Years)
20
15
10
5
0
Age
55
7.19 7.08 7.22 7.32
7.24
6.87 6.77 6.83 6.99
6.89
60
65
6.35 6.21 6.16 6.22
6.31
70
5.44 5.67 5.56 5.58
5.56
75
N/A 4.56 5.07 5.03
4.94
N/A N/A
4.42
4.22
4.21
80
Panel B
Estimated Durations When Interest Rates Decline
55
4.51 4.50 4.59 4.73
4.62
60
4.25 4.13 4.31 4.33
4.14
4.06 3.78 3.78 3.83
3.81
65
70
3.81 3.47 3.43 3.51
3.68
75
N/A 3.23 3.04 3.08
3.35
N/A N/A
2.20 2.79
3.08
80
Panel C
R2 s
55
0.497 0.491 0.507 0.510
0.510
0.503 0.499 0.503 0.509
0.506
60
0.522 0.498 0.498 0.497
0.501
65
70
0.503 0.490 0.491 0.493
0.496
75
N/A 0.445 0.475 0.469
0.465
80
N/A N/A 0.389 0.408
0.376
Notes: This table displays the estimated durations of life annuities when interest rates increase or
decrease, together with the model R2 s, derived from the parameter estimates of models of the form:
x,g
−∆at (x, g|16)/at−16 (x, g) = β x,g + β∆rt,Neg.Sign (30 |16 ) · ∆ri (30|16) · I∆r,neg + β∆r
∗ ∆rt (30 |16 ) + x,g
t .
t (30 |16 )
The independent variables of the regressions are the changes in the 30-year mortgage rates measured
over a period of sixteen weeks, ∆rt (30|16) and this variable interacted with a dummy variable that takes on
a value of 1 if the interest rate decreases, ∆ri (30|16) · I∆r,neg . The dependent variable is the negative of the
percentage changes in the annuity factors over the same measurement period for male annuitants of various
ages and with various guarantee periods. Bolded coefficients denote significance at the 0.05 level, based
on two-sided tests, of a difference between the estimated Duration coefficient and the theoretical value.
Table 9, Continued
Estimated Durations: All Ages and Guarantee Periods
Measurement Period k = 16 week
Female Annuitant, Using 30-Year Mortgage Rates
Panel D
Estimated Durations When Interest Rates Increase
Guarantee Period (Years)
7.47
55 7.49 7.45 7.49 7.62
60 7.31 7.19 7.18 7.26
7.12
6.56
65 6.58 6.46 6.49 6.60
70 5.65 5.82 5.88 5.73
5.88
5.23
75 N/A 5.22 4.63 5.17
80 N/A N/A 4.05 4.05
4.16
Panel E
Estimated Durations When Interest Rates Decline
55 4.50 4.48 4.48 4.52
4.69
60 3.96 4.07 4.18 4.30
4.26
65 3.95 3.90 3.99 3.99
3.97
3.67
70 3.87 3.49 3.58 3.62
75 N/A 3.47 3.22 3.20
3.25
80 N/A N/A 3.22 2.19
3.14
Panel F
R2 s
55 0.502 0.503 0.505 0.508
0.513
60 0.494 0.501 0.513 0.524
0.515
0.520
65 0.513 0.508 0.512 0.519
70 0.459 0.504 0.507 0.510
0.524
75 N/A 0.433 0.450 0.493
0.497
80 N/A N/A 0.348 0.360
0.430
Notes: This table displays the estimated durations of life annuities when interest rates increase or
decrease, together with the model R2 s, derived from the parameter estimates of models of the form:
x,g
−∆at (x, g|16)/at−16 (x, g) = β x,g + β∆rt,Neg.Sign (30 |16 ) · ∆ri (30|16) · I∆r,neg + β∆r
∗ ∆rt (30 |16 ) + x,g
t .
t (30 |16 )
The independent variables of the regressions are the changes in the 30-year mortgage rates measured
over a period of sixteen weeks, ∆rt (30|16) and this variable interacted with a dummy variable that takes on
a value of 1 if the interest rate decreases, ∆ri (30|16) · I∆r,neg . The dependent variable is the negative of the
percentage changes in the annuity factors over the same measurement period for male annuitants of various
ages and with various guarantee periods. Bolded coefficients denote significance at the 0.05 level, based
on two-sided tests, of a difference between the estimated Duration coefficient and the theoretical value.
Figure 1:
10 Year Swap Rate Versus Annuity Payout Rates
Panel A: Males
Panel B: Females
Plot of Annuity Rates and 10 Year Swap across age groups for males and females, 0 year
guarantee. The 10 year swap is the lowest line and the annuity rates are plotted to increase
in thickness as age extends, ages 60, 65, 70.
Figure 2:
R as a Function of the Observation Lag
10 Year Swap Rate
2
Panel A: Males
Panel B: Females
Plot for males and females, age 65, qualified, across guarantee periods: Guarantee Period=
0: Guarantee Period= 5: ? Guarantee Period=10: Guarantee Period=15: 4
Guarantee Period=20: ◦ .
Figure 3:
Estimated Duration as a Function of the Observation Lag
10 Year Swap Rate
Panel A: Males
Panel B: Females
Plot of duration coefficients for males and females, age 65, qualified, across guarantee
periods: Guarantee Period= 0: Guarantee Period= 5: ? Guarantee Period=10: Guarantee Period=15: 4 Guarantee Period=20: ◦ .
Figure 4:
R as a Function of the Observation Lag
30 Year Mortgage Rate
2
Panel A: Males
Panel B: Females
Plot for males and females, age 65, qualified, across guarantee periods: Guarantee Period=
0: Guarantee Period= 5: ? Guarantee Period=10: Guarantee Period=15: 4
Guarantee Period=20: ◦ .
Figure 5:
Estimated Duration as a Function of the Observation Lag
30 Mortgage Rate
Panel A: Males
Panel B: Females
Plot of duration coefficients for males and females, age 65, qualified, across guarantee
periods: Guarantee Period= 0: Guarantee Period= 5: ? Guarantee Period=10: Guarantee Period=15: 4 Guarantee Period=20: ◦ .