The Annuity Duration Puzzle

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1 The Annuity Duration Puzzle N. Charupat, M. Kamstra and M. A. Milevsky 1 Version: 29 February 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 at: or at Tel: x 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.

2 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.

3 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

4 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 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

5 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

6 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 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

7 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: a(x, r) = 0 e rt p(x, t) dt, (1) 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: a(x, g, r) = g 0 e rt dt + g e rt p(x, t)dt. (2) 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

8 measure how bond prices adjusts to changes in yields. literature, we define the duration, D, of a life annuity as: 1 D = a(x, g, r) a(x, g, r) r Following the convention in that 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: (3) λ(x) = 1 b e(x m)/b, (4) 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: which is decreasing exponentially in the time horizon t. p(x, t) = exp { bλ(x) ( 1 e t/b)}, (5) 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: g 0 e rt dt = ( ) 1 (1 ) e rg. (6) r Next, we can use equations (4) and (5) to solve the second term on the right-hand side of equation (2) to obtain: e rt bγ ( br, exp { }) x m+g b p(x, t)dt = g exp { (m x) r exp { }}, (7) x m b where Γ (a, c) is the upper incomplete gamma function: Γ (a, c) = c e t t a 1 dt. (8) 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

9 a(x, g, r) = ( ) 1 (1 ) e rg + r bγ ( br, exp { }) x m+g b exp { (m x) r exp { x m b With this expression, we can then calculate the duration for a life annuity as: 1 a(x, g, r) D = a(x, g, r) r 1 = a(x, g, r) 1 e rg r 2 + ge rg r x m+g bγ( br,exp{ b })(m x) exp{(m x)r exp{ x m b }} +Meijer G ( [[ ], [1, 1]], [[0, 0, br], [ ]], exp { x m+g b }}. (9) 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. }), (10) 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 = years and b = years for males, and m = 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

10 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 component 7 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

11 $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-qualified 10 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 them by 12. Secondly, we annualize the payouts by multiplying 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

12 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, 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

13 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 a i (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: a i (x, g k) a i k (x, g) = a i(x, g) a i k (x, g). (11) a i k (x, g) Let r i (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: Our regression equation is: r i (T k) = r i (T ) r i k (T ) (12) a i(x, g k) a i k (x, g) = β k + β rt (10 k) r i (T k) + ε i. (13) If annuity factors adjust promptly and fully to the changes in interest rates, β k 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 R 2 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., r i (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

14 the 10-year swap rates is % p.a., with a standard deviation of % 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 % 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) a(x, g, r) = D r, (14) 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

15 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 (β k ) 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 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

16 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 Alternative Regressions Changes over Longer Periods We repeat the regressions in the previous subsection, with the independent variable ( r i (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

17 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: a i(x, g k) a i k (x, g) = β k + β rt (10 k) r i (10 k) + β 2 r 2 i (10 k) + ε i. (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 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 markets archive/ htm 15

18 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 % p.a., while the standard deviation is % 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, R- squares 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: 16

19 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 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: a i(x, g k) a i k (x, g) = β k + β rt,neg.sign (30 k) r i (30 k) I r,neg + β rt (30 k) r i (30 k) + ε i, (16) where a i (x, g k) is as previously defined, r i (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., r i (T k) < 0), and zero otherwise. Accordingly, β rt,neg.sign (30 k)+ β rt (30 k) is the estimated duration when r i (T k) < 0, and β rt (30 k) is the estimated duration when r i (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

20 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 assump- 20 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

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