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Does mortality improvement increase equity risk premiums? A risk perception perspective
Journal of Empirical Finance 22 (2013) 67–77
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Does mortality improvement increase equity risk premiums? A risk perception perspective Rachel J. Huang a, d,⁎, Jerry C.Y. Miao b, 1, Larry Y. Tzeng c, d, 2 a b c d
Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei, Taiwan Department of Insurance, Tamkang University, Taipei, Taiwan Department of Finance, National Taiwan University, Taipei, Taiwan Risk and Insurance Research Center, National Chengchi University, Taipei, Taiwan
a r t i c l e
i n f o
Article history: Received 22 July 2010 Received in revised form 15 January 2013 Accepted 4 March 2013 Available online 21 March 2013
a b s t r a c t Using data for G7 countries over the period from 1950 to 2007, this paper finds that an unexpected shock to the mortality rate is significantly negatively correlated with the equity premium. A one basis point unexpected negative shock to the mortality rate increases both the one-year and five-year equity premiums by 0.54% and 1.66%, respectively. We also demonstrate how financial institutions could use our findings to hedge the risk of mortality-linked securities. © 2013 Elsevier B.V. All rights reserved.
JEL classification: G12 D03 J10
Keywords: Mortality risk Equity risk premium Demography Risk perception
1. Introduction It has been widely-recognized both in the literature and in the insurance industry that mortality rates have improved and are continuing to improve all over the world. For example, Oeppen and Vaupel (2002) found that life expectancy in the United States is increasing by 2.43 years each decade. 3 The trend toward mortality improvement has attracted the interest of many researchers who have analyzed various related issues. 4 One of the important issues concerns the impact on the equity risk premium. The literature has asserted that mortality improvement is correlated with the equity risk premium from a demographic structural change point of view. Researchers argue that the change in the participants' age structure reflects the change in the degree of risk aversion and investment needs in the life cycle, and will cause price fluctuations in the capital markets. For example, by using data from the United States, Bakshi and Chen (1994) found that, when the population ages, the demand for ⁎ Corresponding author. Tel.: +886 2 2730 1272. E-mail addresses: [email protected] (R.J. Huang), [email protected] (J.C.Y. Miao), [email protected] (L.Y. Tzeng). 1 Tel.: +886 2 2621 5656x3515. 2 Tel.: +886 2 2363 8597. 3 According to Turner's (2006) estimation, the mortality rate for a 65-year-old UK male in 2013 will fall by 46% in the twenty years between 1983 and 2003. 4 In addition to the risk premium, other issues include the relationship between aversion to financial risk and the willingness to pay to reduce mortality risk (e.g., Eeckhoudt and Hammitt, 2004), the hedging and pricing of life insurance policies (e.g., Bayraktar and Young, 2007; Cairns et al., 2006; Cox et al., 2010; Lin and Tzeng, 2010; Tsai et al., 2010; Young, 2008), the design of mortality derivatives (see Denuit et al., 2007; Lin and Cox, 2008), and the demand for annuity products and retirement planning (Davidoff et al., 2005; Horneff et al., 2008). 0927-5398/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jempfin.2013.03.002
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financial investment increases and the demand for housing decreases. They also found that the change in average age is positively correlated with the risk premium. Goyal (2004) found that outflows from the United States stock market are positively correlated with the change in the fraction of retired people and negatively correlated with the change in the fraction of middle-aged people (aged 45 to 64). He found that an increase in the average age will increase next year's excess stock return. By pooling international data, Ang and Maddaloni (2005) found that the higher the growth rate of the fraction of population in the retired group, the lower the risk premium. Instead of focusing on population changes among alternative age groups as in the previous literature, our paper adopts a different approach by directly examining the relationship between the unexpected shock to the mortality rate and the equity premium. There are three reasons for doing so. First, while demographic variables could proxy for changes in risk aversion, the impact of an unexpected shock to the mortality rate on the equity premium could be interpreted as a change in risk perception. Both risk aversion and risk perception are essential for individuals to determine their demand for risky assets. When the mortality rate for all the participants in the capital markets changes, the demographic structure may not change right away. However, the investors will immediately modify their risk perception regarding life expectancy when they recognize the unexpected shocks in the mortality rate. Thus, the marginal impact of the unexpected shock to the mortality rate on the risk premium can only represent the effect of a change in risk perception as opposed to that of a change in risk aversion. In addition, as noted by Della Vigna and Pollet (2007), the changes in demographics will affect the future consumption plans and further influence the profits and returns among industries. When observing an unexpected shock to the mortality rate, the investors will anticipate the future changes in revenues for different industries, and will immediately modify their investment strategies among industries. Thus, the changes in portfolio choices will affect the equity premium right away. Second, financial institutions can use the correlation between the unexpected mortality rate shock and the equity premium to hedge the mortality-linked securities 5 which have been actively introduced by major investment banks and insurers in recent years. A change in demographic structure is determined by several factors, such as the current demographic structure, the mortality rate, the fertility rate, and the immigration policies. Separating the impact of the mortality rate from the impacts of other factors could provide a more accurate measure of the sensitivity of the equity premium with respect to the mortality rate and could reduce the basis risk faced by financial institutions when hedging mortality-linked securities. Finally, the relationship between the change in the mortality rate and the equity premium has been proposed in theory but has never been examined empirically. Athanasoulis (2006) adopted a general equilibrium overlapping generations model and showed that the mortality rate is positively correlated with the equity premium. Although Athanasoulis (2006) has provided theoretical predictions and supports his results by simulation, whether his results can be supported by real world empirical evidence deserves a further study. Hence, this paper employs G7 country data to provide the empirical evidence. The methodology underlying our empirical model is similar to that in Ang and Maddaloni (2005) who examined the relationship between the equity risk premium and demographic variables. In addition to Ang and Maddaloni's (2005) dependent variables, we further employ the unexpected mortality rate shock in the regression models. Our empirical evidence finds strong support for the view that an unexpected shock to the mortality rate is significantly negatively correlated with the risk premiums. By using a one-year horizon, we find that the magnitude of the increase in the risk premium is about 0.54% per year when the mortality rate is one basis point lower than expected. We also find similar results when testing the long-run relationship. For the five-year horizon, the magnitude of the increment in the risk premium due to a one basis point unexpected negative shock to the mortality rate is around 0.331% per year. We further provide an example to demonstrate the applications of our empirical findings in hedging a mortality-linked product. We derive the immunization strategy for an annuity issuer. By simulation, we further demonstrate that our strategy could improve the performance in terms of hedging the unexpected shocks to the mortality rate. The remainder of this paper is organized as follows. Section 2 reports our data. The methodology is introduced in Section 3. Section 4 presents our empirical findings. Section 5 illustrates how we use our findings to hedge the longevity risk of annuity issuers. Section 6 concludes the paper.
2. Data The sample comprises the G7 countries. Our data are merged from three sources, with financial data being collected from the Global Financial Data (GFD), consumption data being gathered from the Center for International Comparisons (CIC), and demographic and mortality data being obtained from the Human Mortality Database (HMD). The data periods are 1950–2007 for Canada, Italy, the United Kingdom and the United States, 1960–2007 for France and Japan, and 1970–2007 for Germany. The data consist of annual observations for excess aggregate equity returns, dividend yields, term spreads, consumption growth, demographic variables, and mortality rates. The definitions of all variables are provided in Table 1. Besides the explanatory variables used in Ang and Maddaloni (2005), we additionally employ two independent variables: the unexpected shock to the mortality rate and the dividend yield. 6 5
For example, longevity bonds, annuities, survival bonds, mortality swaps and forwards, etc. Fama and French (1988, 1989) have shown that the dividend yield is important in predicting the future risk premium. Thus, we include the dividend yield as one of the independent variables. 6
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Table 1 Definitions and data sources of variables. Variable
Definitions
excess
The difference between the continuously compounded total return on the equity index and short-term risk-free investment. The database is provided by Global Financial Data (GFD). Equity returns represent price plus dividend returns (total returns) from GFD, which are compiled from Canada's S&P/TSX-300 total return index, France's SBF-250 total return index, Germany's CDAX total return index, Italy's BCI Global return index, Japan's Nikko securities composite total return, the UK's FTSE all-share return index, and the S&P 500 total return index annual. The returns of short-term risk-free assets are the yields of 3-month treasury bills in G7 countries. %DIVY The dividend yield. The database is also from GFD. The observations are compiled from Canada's S&PTSX-300 dividend yield, France's dividend yield, Germany's dividend yield, Italy's dividend yield, the Tokyo SE dividend yield, the FT-Actuaries dividend yield, and the S&P 500 dividend yield. term The difference between the long bond yield and the short bond yield. The database is from GFD. The long bond yields are Canadian government bonds of 10-year maturity, France's 10-year government bond yield, Germany's 10-year government bond yield, Italy's 10-year government bond yield, Japan's 7-year government bond yield, the U.K.'s 20-year government bond yield, and the USA's 10-year bond constant maturity yield. The short term bond yield returns are the yields of 3-month treasury bills in G7 countries. dcons The continuously compounded change in aggregate consumption. Consumption data are gathered from the Center for International Comparisons (CIC). Annual aggregate consumption is estimated as the difference between the total value of the real GDP and the share of GDP invested. d%age65 The log change in the fraction of adults over 65 years old. The data are obtained from the Human Mortality Database (HMD). MOR The mortality rate of adults over 25 years old. The data are obtained from the HMD. MORSHOCK The unexpected change in the mortality rate of adults over 25 years old, which is the residual of the following AR(1) model: MORt = γ0 + γ1MORt − 1 + εt
The mortality rate, denoted by MORt, is defined as DEATHt/EXPOSUREt, where DEATHt is the number of people who die over the age of 25 at time t, and EXPOSUREt is the total number of people over 25 years old at time t. We focus on the mortality rate for adults above 25 years old rather than the mortality rate of the retired group because the participants in the capital market include the young, middle-aged and retired group. Thus, the mortality rate for adults above the age of 25 could reflect the general risk perception for the overall capital markets. Since we argue that the unexpected shock to the mortality rate will change the investors' risk perception and further affect the demand for risky assets, a model to estimate the unexpected shock to the mortality rate is necessary. For each country, we adopt the following first-order autoregressive, AR(1), model: MORt ¼ γ 0 þ γ 1 MORt−1 þ εt
ð1Þ
where εt represents the unexpected shock to MORt, and is referred to as MORSHOCKt hereafter. We will use MORSHOCKt as the major independent variable in our regressions. 7 Table 2 offers descriptive statistics for the variables. The basic statistics for our financial variables and demographic variables are consistent with those in Ang and Maddaloni (2005). The average equity premium (denoted as excess) is 6.4% per annum with a standard deviation of 15.9% for the United States, which has a relatively high excess return compared to the other G7 countries. Turning to the demographic variables, Japan has the fastest rate of increase in the fraction of retired people compared to the other G7 countries. The average MOR is 1.4% for the United States. Japan has the smallest values for MOR and Canada has the 2nd lowest mortality rate among the G7 countries. The average MORSHOCK for all countries is small since it is the residual obtained from the AR(1) model. Canada and the United States have the highest standard deviations of MORSHOCK, whereas France has the lowest standard deviation of MORSHOCK. The selected correlation matrices are reported in the Appendix, Table A1. It is shown that MORSHOCK and d%age65 are negatively correlated in Canada and the United States, whereas they are positively correlated in other countries. Table A1 further indicates that, for all G7 countries, the simple correlations between excess and d%age65 differ from those between excess and MORSHOCK. For the United States, excess is positively correlated with d%age65, but negatively correlated with MORSHOCK. On the other hand, excess is negatively correlated with d%age65, but positively correlated with MORSHOCK for Germany, Italy, Japan and the United Kingdom. These results partially support the view that MORSHOCK might provide additional information to d%age65 regarding estimating the equity premium. In general, the excess variables are positively correlated with the variables for the yield spread between long-term bonds and short-term bonds (denoted as term) and MOR, and negatively correlated with other variables for Canada and France. For Germany and Italy, the excess variables are positively correlated with the variables for term and MORSHOCK, and negatively correlated with the other variables. The international correlations are listed in the Appendix, Table A2. The mortality rates for those over 25 years old are highly correlated among countries except for Japan. We further use Fig. 1 to demonstrate the time trends of MOR for the G7 countries. Panel A in Fig. 1 shows that Germany, Italy, the United Kingdom and the United States exhibit similar broad trends, and the lines of the mortality rates go up slightly before 1970. After the mid-1970s, the mortality rates decrease. Panel B in Fig. 1 shows that the mortality rates for Canada and France experience a downward trend from the late 1960s onwards. The pattern of Japan's 7 We also use the differences between MORt and MORt-1 to proxy the unexpected shock to the mortality rate. The results are consistent with the results using MORSHOCK, and are therefore unreported.
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Table 2 Descriptive statistics — means and standard deviations.
Canada France Germany Italy Japan U.K. U.S.
excess
%DIVY
term
dcons
d%age65
MOR
MORSHOCK
0.047 (0.153) 0.031 (0.221) 0.059 (0.230) 0.032 (0.246) 0.047 (0.231) 0.054 (0.228) 0.064 (0.159)
0.034 (0.011) 0.039 (0.016) 0.032 (0.011) 0.035 (0.013) 0.029 (0.028) 0.047 (0.016) 0.034 (0.013)
0.012 (0.014) 0.009 (0.014) 0.020 (0.012) 0.012 (0.015) 0.017 (0.014) 0.006 (0.019) 0.012 (0.012)
0.024 (0.024) 0.042 (0.037) 0.006 (0.020) 0.059 (0.042) 0.039 (0.055) 0.042 (0.026) 0.021 (0.019)
0.010 (0.008) 0.006 (0.010) 0.013 (0.013) 0.016 (0.008) 0.026 (0.009) 0.007 (0.006) 0.008 (0.006)
0.012 (0.001) 0.016 (0.002) 0.016 (0.002) 0.014 (0.001) 0.011 (0.002) 0.017 (0.001) 0.014 (0.001)
0.0000 (0.0003) −0.0001 (0.0007) 0.0000 (0.0004) 0.0000 (0.0005) −0.0001 (0.0006) 0.0000 (0.0006) 0.0000 (0.0003)
The table reports the means and standard deviations (shown in parentheses) of the equity premium (excess), dividend yields (%DIVY), the yield spread between the long bond and the short bond (term), the continuously compounded change in aggregate consumption (dcons), the log change in the fraction of adults over 65 years old (d%age65), and the unexpected shock in the mortality rate of adults over 25 years old (MORSHOCK) for the G7 member nations. The data periods are 1950–2007 for Canada, Italy, the United Kingdom and the United States, 1960–2007 for France and Japan, and 1970–2007 for Germany.
A) Mortality rates for Germany, Italy, the U.K. and the U.S. 0.02
0.015
0.01
0.005 UK
0 1950
1955
1960
1965
US
1970
Germany
1975
1980
Italy
1985
1990
1995
2000
2005
1990
1995
2000
2005
B) Mortality rates for Canada, Japan and France. 0.025
0.02
0.015
0.01
0.005 Japan
0 1950
1955
1960
1965
1970
Canada
1975
1980
France
1985
Fig. 1. Mortality rates, MOR, for G7 countries. These figures plot the mortality rates of adults over 25 years old for each country, which equals the number of people who die over the age of 25 divided by the total number of people over 25 years old.
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mortality rate, however, is quite different. MOR exhibits a clear downward trend before the late 1980s and a slight increase after the late 1980s. The differences in the mortality rate patterns among countries might be due to immigration laws. Among the G7 countries, Canada has the highest per capita net immigration rate. 8 Japan has relatively tight immigration laws. In the early 1990s, to reduce the labour shortage, Japan started to allow special entry permits for foreigners, and the change in immigration laws might have caused certain fluctuations in the mortality rates. Fig. 2 shows the pattern of MORSHOCK, from which it can be seen that there is no significant upward or downward time trend for MORSHOCK in the G7 countries. All of the G7 countries share a similar pattern in that MORSHOCK varies substantially year by year before 1970 and becomes stable after 1970. In addition, we find no significant upward or downward time trend for excess, which provides partial evidence that the time series of the unexpected shock to the mortality rate and the equity premium may not have unit roots. To further detect whether the relationship between the unexpected shock to the mortality rate and the equity premium might be spurious, we run the Dickey–Fuller tests to examine the unit roots. Table A3 in the Appendix shows that the time series of the unexpected shocks in the mortality rate and the equity premium are stationary. 3. Empirical methodology We first use a panel data methodology to examine the relationship between the risk premium and the unexpected shock to the mortality rate. 9 Panel data enjoy more degrees of freedom, and hence improve the efficiency of the econometric estimates. Furthermore, the use of panel data in our study presents a more reliable picture than that arising from purely cross-sectional studies. Thus, we can make more accurate inferences using the model parameters. Because the number of observations for each country is not identical, an unbalanced panel results. 10 Secondly, since cross-country studies may further provide interesting insights into the differences across countries, we also analyze the relationship between MORSHOCK and excess for one country at a time. In the panel regressions, we empirically investigate the relationship between expected excess returns and the unexpected shock to mortality rates in the short run and in the long run as follows: excessitþ1 ¼ β0 þ β1 %DIVY it þ β2 termit þ β3 dconsit þ β4 d%age65it þ β5 MORSHOCK it þ θitþ1 ;
ð2Þ
where θt + 1 denotes the residual. MORSHOCK is our main explanatory variable. As explained in the Introduction and based on the preliminary evidence from the previous section, MORSHOCK could provide additional information to d%age65. Thus, we examine the effect of MORSHOCK by controlling for the effect of d%age65 and the regression is referred to as Model 1. 11 To isolate the effects of MORSHOCK and d%age65 in the regression model, we also include only one of these two variables at a time. In Model 2, the variable d%age65 is not included. In Model 3, we do not control for the variable MORSHOCK. Similar to the regressions in Ang and Maddaloni (2005), we include the consumption growth and the term spread as the control variables. We further employ the dividend yield as another control variable because the literature (see Fama and French, 1988, 1989) has documented that the dividend yield has significant predictive power in regard to the future risk premium. For the long-run correlation in the panel regressions, we use non-overlapping windows of data as in Poterba (2001) to construct the long-run excess return. Although non-overlapping windows of data will dramatically decrease the number of observations, this methodology can avoid the autocorrelation problem in the error terms of the regressions. For the cross-country studies, we only investigate the short-run relationship because the number of observations for each country is quite small, while the non-overlapping windows of data are adopted for the long-run models. 4. Empirical results The results of the panel analyses are presented in Table 3. 12 For the 1-year horizon, Model 1 in Table 3 shows that the coefficient of MORSHOCK is significantly negative. 13 It means that if mortality rates are lower than expected by one basis point, the equity premium will rise by 0.54% and vice versa. If there is an unexpected negative shock to the mortality rate by one standard deviation, which is 0.051%, then the risk premium increases by 2.74%. 14 This evidence shows that our results are not only statistically significant, but may also be economically significant. The negative relationship between MORSHOCK and excess is also 8
Please see Dolin and Young (2004). We estimate the unbalanced panel regression by using the generalized least squares random effects estimators. 10 For a robustness check, we also construct a balanced panel data set which contains the countries with the longest time series data, i.e., Canada, Italy, the United Kingdom and the United States. The balanced panel data also provide support for the view that the unexpected mortality shock is significantly negatively correlated with the equity premium for the 1-year horizon. Since the results from the balanced panel are similar to the results from the unbalanced panel, we only report the results from the unbalanced panel. 11 Ang and Maddaloni (2005) found that the average age of the population has no forecasting power in international data, but the change in the proportion of retired persons is a significant predictor of the excess return. Thus, in our regression, we only control for d%age65. 12 In Table 3, the standard errors of the pooled estimates have been corrected for autocorrelation and/or heterogeneity across countries. 13 Note that the negative relationship between the equity premium and the unexpected shock to the mortality rate remains when the unexpected shock to the mortality rate for adults over 65 years old is adopted. 14 The average excess return for the pooled data is 4.82%. 9
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A) Unexpected mortality shocks for Germany, Italy, the U.K. and the U.S. 0.004 0.003 0.002 0.001 0 -0.001 UK -0.002 1950
1955
1960
1965
US 1970
1975
Germany
Italy
1980
1990
1985
1995
2000
2005
2000
2005
B) Unexpected mortality shocks for Canada, Japan and France. 0.004 0.003 0.002 0.001 0 -0.001 Japan -0.002 1950
1955
1960
1965
1970
1975
Canada 1980
France 1985
1990
1995
Fig. 2. Unexpected shocks in the mortality rate, MORSHOCK. This figure plots the residuals of the AR(1) model for the mortality rate of adults over 25 years old for each country.
found in Model 2, where the average change in the proportion of retired persons is absent. The magnitude of the effect in Model 2 is similar to that in Model 1. Note that our paper empirically finds a negative relationship between the equity premium and the unexpected shock to the mortality rate, whereas Athanasoulis (2006) numerically predicted a positive relationship between the equity premium and the expected change in the mortality rate. The difference between Athanasoulis (2006) and our paper mainly arise because we analyze the impact of the unexpected change in the mortality rate on the equity premium, while Athanasoulis (2006) focused on the impact of the change in the expected mortality rate. In his model, he assumed that the mortality rate is constant over time. He then provided a simulation to show how the equity premium changes when the expected mortality rate is at a different level. When the mortality rates are different from the expected, the individuals will change their risk perception and further change their consumption plans. For example, when there is an unexpected negative shock to the mortality rate for the adults over 25 years old, the demand for housing and life insurance may increase right away. Furthermore, the demand for pharmaceuticals, nursing houses, funeral houses and the goods needed in the corresponding life cycle will also increase in the future. Thus, the equity premium on those industries may change since investors anticipate that the revenues of those industries will increase due to an unexpected negative shock in the mortality rate. Della Vigna and Pollet (2007) used US data and showed that an additional 1% increase in the annualized demand growth due to demographics enabled them to predict a 5% to 10% increase in annual abnormal industry stock returns. If these demographic industries play significant roles in the overall stock market, then the equity premium will also increase. As a result, we will observe a negative relationship between the equity premium and the unexpected mortality shocks. Thus, for future theoretical analyses, the investor should be assumed to be able to invest among different industries rather than only invest in an aggregate equity market. A consequent assumption is that the investors will update their beliefs regarding
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Table 3 Pooled regressions on equity premiums across G7 countries. One-year horizon
Constant %DIVY term dcons d%age65 MORSHOCK 2
Adj-R Num. of obs.
Five-year horizon
(1)
(2)
(3)
(1)
−0.086** (−2.12) 3.146*** (4.25) 1.473** (1.97) 0.004 (0.10) −0.065 (−0.27) −53.768** (−2.12) 0.056 359
−0.087*** (−3.05) 3.120*** (4.41) 1.464* (1.95) −0.004 (−0.17)
−0.085*** (−2.86) 3.201*** (4.27) 1.458* (1.92) 0.015 (0.04) −0.031 (−0.13)
−0.122 (−1.36) 5.370** (2.39) −0.273 (−0.13) −0.017 (−0.40) 0.158 (0.58) −165.708** (−2.14) 0.058 67
−53.326** (−2.10) 0.056 359
0.044 359
(2) −0.103 (−1.22) 4.992** (2.33) −0.295 (−0.14) 0.001 (0.02)
−173.983** (−2.28) 0.054 67
(3) −0.118 (−1.27) 5.152** (2.21) −1.335 (−0.64) −0.027 (−0.63) 0.255 (0.91)
−0.005 67
The table lists the coefficients for the pooled regressions with AR(1) disturbances across G7 countries over the sample period 1950–2007. The t-statistics are reported in parentheses with those significant at the 10% (5%, 1%) level denoted by * (** and ***). The dependent variable for the first three columns is the one-year excess return which is the difference between the continuously compounded total return on the equity index and the short-term risk-free investment. The dependent variable for the last three columns is the 5-year equity premium from non-overlapping window data. The explanatory variables are %DIVY, the dividend yields, term, the difference between the long bond yield and the short bond yield, dcons, the continuously compounded change in aggregate consumption, d%age65, the log change in the fraction of adults over 65 years old, and MORSHOCK, the residual of the AR(1) model for the mortality rate of adults over 25 years old.
the future revenues in different industries after they observe the unexpected mortality shocks in the previous period. With this setting, the theoretical prediction might be consistent with our empirical findings. In both Models 1 and 3, the short-run equity premium is not significantly affected by the average change in the proportion of retired persons. This finding is, however, not consistent with the results of Ang and Maddaloni (2005). This might be due to the differences between our paper and theirs in terms of the data period and the countries included. 15 The results for %DIVY coincide with the results of most studies in the literature. This variable is significantly positively correlated with excess. The signs of the coefficients of term are consistent with the findings of the pooled regression across the G5 countries in Ang and Maddaloni (2005). The variable is also significantly positively correlated with the short-run equity premium. The coefficients of dcons are insignificant in all models. Table 3 further demonstrates that the results for 5-year horizons are similar to those for one-year horizons: the coefficients of MORSHOCK are significantly negative. 16 These results provide us with evidence that an unexpected shock to the mortality rate gives rise to an accumulated effect in the long run. The coefficient of MORSHOCK in Model 1 shows that the five-year equity premium increases by 0.331% per year when there is a one basis point decrease in MORSHOCK. In the long run, we find that the coefficients of d%age65 are insignificantly positive in all regressions, whereas the coefficients are significantly negative in Ang and Maddaloni's (2005) regressions. 17 The coefficients of %DIVY are significantly positive at the 5% level. However, the coefficients of both term and dcons are insignificant. Table 4 shows the results for individual countries for a 1-year horizon. The coefficient of MORSHOCK is significantly negative for Germany in Model 1. Although the coefficients of MORSHOCK are negative for Canada, France, Italy, the United Kingdom and the United States in Model 1, they are insignificant. For Japan, we find that the coefficient of MORSHOCK in Model 1 is insignificantly positive. Generally speaking, the results in Table 4 demonstrate that the estimated correlation between the unexpected shock to the mortality rate and the equity premium is negative, which is consistent with our findings in Table 3. Probably due to the limitation of the numbers of observations, the empirical results are not statistically significant. 18 From the results of individual countries, one reasonable conjecture is that the large coefficients of MORSHOCK in the pooled regressions might be due to the inclusion of Germany. Thus, the regressions for the pooled data excluding Germany are examined and the results are shown in the Appendix, Table A4. 19 When Germany is not included, the coefficients of MORSHOCK are still
15 When Ang and Maddaloni's (2005) regression is adopted for the G5 countries, our data reveal evidence of an insignificant negative relationship between d%age65 and excess. 16 We also check a three-year period by using non-overlapping data. However, in the regression, only the dividend yield is significant. All other variables are insignificant. 17 Note that our non-overlapping window data still predict an insignificant positive relationship between d%age65 and excess for the G5 countries. 18 In unreported analyses, we further divide countries to respectively examine the effects of financial market development, social security benefits and geography on the relationship between the unexpected shock to the mortality rate and the equity premium. We find that the unexpected shock to the mortality rate and the equity premium are still significantly negatively correlated, and the degree of financial market development, social security benefits and geography do not significantly make the negative correlation different. 19 In Table A4, the standard errors of the pooled estimates have also been corrected for autocorrelation and/or heterogeneity across countries.
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Table 4 Regressions for individual countries — one-year horizon. Constant Panel A: Canada (1) −0.063 (−0.95) (2) −0.074 (−1.26) (3) −0.064 (−0.97) Panel B: France (1) 0.011 (0.15) (2) −0.069 (−0.99) (3) 0.017 (0.24) Panel C: Germany (1) 0.005 (0.05) (2) −0.057 (−0.48) (3) 0.059 (0.46) Panel D: Italy (1) 0.185 (1.33) (2) −0.040 (−0.41) (3) 0.191 (1.40) Panel E: Japan (1) 0.287 (1.61) (2) −0.074 (−1.01) (3) 0.278 (1.58) Panel F: UK (1) −0.244*** (−3.50) (2) −0.226*** (−2.94) (3) −0.241*** (−3.40) Panel G: US (1) −0.235** (−2.42) (2) −0.073 (−0.92) (3) −0.234** (−2.42)
%DIVY
term
dcons
d%age65
MORSHOCK
−0.999 (−0.46)
−1.050 (−0.06) 0.254 (0.01)
4.275*** (2.76) 4.312*** (2.82) 4.290*** (2.84)
2.895** (2.52) 3.039*** (2.73) 2.898** (2.56)
−2.590*** (−3.29) −2.632*** (−3.40) −2.593*** (−3.39)
2.915 (1.67) 3.507** (2.21) 3.016* (1.71)
1.287 (0.73) 1.829 (1.00) 1.383 (0.79)
−2.038** (−2.25) −1.557 (−1.61) −2.222** (−2.60)
0.842 (0.32) 2.162 (0.70) 0.027 (0.01)
3.535 (1.65) 1.193 (0.66) 2.584 (1.00)
−3.241** (−2.34) −2.461* (−1.90) −2.540 (−1.53)
−6.377** (−2.26)
5.555** (2.18) 4.830* (1.97) 5.557** (2.22)
−2.891 (−1.52) −1.312 (−0.60) −2.974 (−1.67)
−2.277** (−2.33) −1.394* (−1.70) −2.322** (−2.44)
−11.005** (−2.12)
8.826** (2.25) 10.550** (2.55) 8.557** (2.15)
−0.931 (−0.59) 0.153 (0.11) −0.880 (−0.56)
−3.091*** (−3.42) −2.711*** (−2.79) −3.017*** (−3.44)
−10.608** (−2.32)
−8.037* (−1.73)
9.532*** (5.26) 8.680*** (4.77) 9.552*** (5.38)
1.076 (0.82) 0.604 (0.55) 1.092 (0.83)
−2.441** (−2.06) −3.151** (2.51) −2.429** (−2.09)
13.292*** (3.01) 3.980* (1.95) 13.293*** (3.04)
2.637* (1.84) 2.452* (1.73) 2.654 (1.86)
−1.384 (−1.46) −1.391 (−1.66) −1.402* (−1.78)
−0.978 (−0.47) −4.798** (−2.23)
−11.802 (−0.87) −9.440 (−0.84)
−4.556** (−2.32) −62.502** (−2.31) −37.331 (−1.55)
−4.179 (−1.56) −1.923 (−0.27) −5.710 (−0.80)
−11.086** (−2.18) 8.764 (0.69) 6.346 (0.46)
−10.426** (−2.33) −3.578 (−0.32) −6.460 (−0.60)
−8.361** (−2.00) −20.053*** (−2.90)
−20.019*** (−2.87)
−1.206 (−0.06) 3.442 (0.18)
Adj. R2
Num. of obs.
0.1445
57
0.1587
57
0.1613
57
0.034
47
0.010
47
0.045
47
0.047
37
−0.041
37
−0.024
37
0.1156
57
0.0135
57
0.1325
57
0.131
47
0.077
47
0.147
47
0.333
57
0.320
57
0.345
57
0.182
57
0.066
57
0.198
57
Note: The table lists the coefficients of regression with AR(1) disturbances for the G7 member nations over the sample period 1950–2007. The t-statistics are reported in parentheses with those significant at the 10% (5% and 1%) level denoted by * (** and ***). The dependent variable of excess is the difference between the continuously compounded total return on the equity index and the short-term risk-free investment. The explanatory variables are %DIVY, the dividend yields, term, the difference between the long bond yield and the short bond yield, dcons, the continuously compounded change in aggregate consumption, d%age65, the log change in the fraction of adults over 65 years old, and MORSHOCK, the residual of the AR(1) model for the mortality rate of adults over 25 years old.
significantly negative in all models. For the remaining G6 countries, if mortality rates are one basis point lower than expected, the equity premium will rise by 0.46%. In a way that differs from the findings in the panel regressions, d%age65 is significantly negative for all countries except Canada. Among the G7 countries, d%age65 has a higher effect on the equity premium in the United States, as evidenced by a one percent increase in d%age65 which will cause the equity premium to increase by 20% in the United States. We further find that the coefficients of %DIVY are insignificantly positive for Germany, but they are significantly positive for the rest of the G7 countries. The coefficients of term are significantly positive in Canada and the United States. However, for other countries, the effect of term on the equity premium is significant. The findings regarding the relationship between the consumption growth rate and the equity premium across countries are different from the findings in the literature. We find that the coefficients of dcons are negative for all G7 countries. In Model 1, the coefficients are significant except for the United States.
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5. An application The empirical evidence in this study shows that an unexpected negative shock to the mortality rate significantly increases the equity premium. As mentioned in the Introduction, understanding the relationship between the unexpected shock to the mortality rate and the equity premium could help the financial institutions hedge the risk associated with holding mortality-linked securities. In what follows, we will provide an illustration of a hedging strategy for an artificial insurance company. Assume that an insurance company with equity St issues an annuity at time t, which pays a one dollar benefit at time t + 1 if the policyholder is alive. The insurer receives an insurance premium Pt. Assume that the annuity policy is the only insurance contract the insurer sells and the liability of the insurer, Lt, is recognized as the unearned premium reserve of the annuity, which happens to be Pt. Thus, the total assets of the insurer are equal to At = Pt + St. Assume that the insurer prices the annuity using the actuarially fair premium, which is equal to the present value of the expected coverage. Let us assume that the insurance company does not observe MORSHOCKt when pricing the annuity. Thus, the insurer considers the process of the mortality rate to follow Eq. (1) and sets −r P t ¼ 1−E MORtþ1 e f ;
ð3Þ
where rf denotes the continuous compounded risk-free rate and E is the expectation operator. Note that E(MORt + 1) = γ0 + γ1E(MORt) from Eq. (1). Assume that MORSHOCKt will be realized immediately after the insurer issues the annuity. Since the equity premium is correlated with MORSHOCKt as shown by our empirical results, the insurer could hedge the unexpected mortality shock by constructing a portfolio in capital markets. Let the insurer choose α, the proportion of its investment in risky assets, and 1 − α, the proportion in risk-free assets. Thus, after asset allocation, the assets of the insurer at t + 1 can be evaluated by: h i r~tþ1 r~ r r ~ þ ð1−α ÞðP t þ St Þe f ¼ α ðP t þ St Þe tþ1 −rf þ ð1−α ÞðP t þ St Þ e f A tþ1 ¼ α ðP t þ St Þe excesstþ1 r ¼ α ðP t þ St Þe þ ð1−α ÞðP t þ St Þ e f ;
ð4Þ
~ tþ1 is also a random variable after the where ~r tþ1 is a random variable and denotes the equity return at time t + 1. Note that A investment decision since the stock return is random. excesst + 1 denotes the equity premium at time t + 1 and is a function of MORSHOCKt as indicated in our empirical results. If the unexpected mortality shock occurs, the liability of the insurer at t + 1 becomes L~tþ1 ¼ 1−γ1 MORSHOCK t −MORSHOCK tþ1 :
ð5Þ
Note that L~tþ1 is also a random variable since MORSHOCKt + 1 is random. ~ tþ1 −L~tþ1 . As proposed by the immunization strategy, the The equity of the insurer at t + 1 can then be rewritten as S~tþ1 ¼ A objective of the insurance company is to ensure that the expected equity at time t + 1, E S~tþ1 , does not vary based on the unexpected mortality shock MORSHOCKt. Thus, the insurer can set α such that dE S~tþ1 dMORSHOCK t
¼ 0;
ð6Þ
where ~ ~ E S~tþ1 ¼ E A tþ1 −E L tþ1
r excesstþ1 ¼ α ðP t þ St ÞE e þ ð1−α ÞðP t þ St Þ e f
− 1−γ1 MORSHOCK t −E MORSHOCK tþ1 :
Since E(MORSHOCKt + 1) = 0, and Eðeexcesstþ1 Þ are functions of MORSHOCKt, Eq. (6) will become excesstþ1 rf α ðP t þ St ÞE e e
∂excesstþ1 þ γ 1 ¼ 0: ∂MORSHOCK t
Rearranging the above equation yields α¼
−γ1 ∂excess
tþ1 ðP t þ St Þ ∂MORSHOCK Eðeexcesstþ1 Þerf
:
ð7Þ
t
∂excesstþ1 It is very important to recognize that ∂MORSHOCK can be retrieved from an appropriate empirical model which regresses t excesst + 1 on MORSHOCKt such as our Eq. (2). To test empirically these findings, we take returns data from the US financial market during 1950 to 2007 to demonstrate that investing in a portfolio of the stock index and a risk-free asset in the proportions α and (1 − α), as estimated in Eq. (7), will yield a return to make the payment due for the above-constructed annuity.
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Table 5 Calibration Results.
Panel A: OLS regression Constant MORSHOCKt Adj. R2 Num. of obs. Panel B: basic statistics Mean Standard deviation
St + 1 under the hedging strategy
St + 1 without the hedging strategy
0.0118*** (29.77) 0.6299 (0.46) −0.0143 57
0.0105*** (225.61) 0.8433*** (5.30) 0.32590 57
0.0118 0.0030
0.0105 0.0004
Note: The table shows our calibration results using US data covering the period from 1950 to 2007. In each year, there is an insurance company with a capital of 0.01 that issues an annuity that will pay one dollar in the next year. The actuarially fair premium is calculated by Eq. (3). The hedging strategy of the insurance company is to invest the proportion α of its assets in the stock market and the proportion 1 − α in risk-free assets. α is obtained by Eq. (7). The t-statistics are reported in parentheses with those significant at the 10% (5% and 1%) level denoted by * (** and ***), respectively. The dependent variable St + 1 is the equity value of the insurance company after the annuity is paid at time t + 1. The independent variable MORSHOCKt is the unexpected shock in the mortality rate, which is the residual of the AR(1) model for the mortality rate of adults over 25 years old.
The calibration is as follows. Let us assume that there is an insurance company with a capital of 0.01 in the year 1950. The insurance company will calculate the premium according to Eq. (3) and the investment proportion α from Eq. (7). Note that since the annuity is a one-year contract, we use the coefficients estimated from Model 1 in Table 3 to calculate α. By using the capital market data for the year 1951, we can obtain the asset value of this insurance company after adopting the hedging strategy. From the real mortality rate for the year 1951, we are able to obtain the liability of the insurance company. Thus, if the equity, which is the difference between total assets and total liabilities, is positive, then it means that the hedging strategy could yield a return to make the payment due for the annuity. We further repeat the above steps by assuming that there is an insurance company with a capital of 0.01 in the year 1951, and we stop the steps in the year 2007. The calibration results are as follows. For these insurance companies, we obtain an average α of 1.87% and a standard deviation of 0.09%. Panel A in Table 5 indicates that the equity of the insurance companies is not sensitive to the unexpected mortality shock when the hedging strategy is employed. However, the equity of the insurance companies is significantly positively correlated with the unexpected mortality shock when the insurance company does not hedge the unexpected shock through asset allocation, i.e., α = 0. This result demonstrates that the hedging strategy can indeed help the insurance companies to hedge mortality risk. Panel B in Table 5 further shows that, under the hedging strategy, the insurance companies are able to pay for the annuity. After adopting the hedging strategy, the average equity of the insurance companies is 0.0118. If these insurance companies do not use the hedging strategy and set α = 0, then the equity will become 0.0105 on average, which is smaller than the equity value under the hedging strategy. However, since the insurance companies do not participate in the stock market when α = 0, they will not face the risk associated with the stock market. Thus, the standard deviation of the equity without the hedging strategy is only 0.004, which is much smaller than the standard deviation of the equity under the hedging strategy. It is worth noting that although the insurance companies will face additional risk from the stock market under a hedging strategy, the risk in the stock market will not cause the insurance companies to go bankrupt in our calibration when the insurance companies have some capital at the beginning. The initial capital we assume is 0.01, which means that the equity-to-debt ratio is about 1%. In reality, the capital-to-debt ratio is much higher than 1%. Thus, the insurance companies will be able to face the risk from the stock market when the hedging strategy is employed. In other words, although the risk from the investment in the stock market does exist, the risk might not give rise to a severe insolvency problem. The above analytical solution of α serves to demonstrate the application of our empirical results. However, the reality could be more complicated than the artificial case that we assumed. Several practical issues should be taken into consideration. First, the annuity could be sold to different age groups. Thus, the cohort effect should be included when the insurer estimates the mortality process. 20 Second, annuity policies vary in both the method of premium payment and the schedule of annuity benefits. 21 The impact of a shock to the unexpected mortality rate on the insurer's liability requires the details of the annuity policies. Third, most annuities are long-term contracts. The investment strategies of the insurer could be long-term and dynamic rather than short-term and static. Adding all these practical concerns could provide a fruitful result and deserves a future study.
20
For example, see Lee and Carter (1992). The benefit of an annuity can be designed as index-linked, escalating, as well as guaranteed, capital-protected, etc. On the other hand, the payment schedule of the premium is quite different for the immediate annuity, deferred annuity, variable annuity, etc. 21
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6. Conclusions The literature has found that mortality improvement is correlated with the equity risk premium from a demographic change point of view. However, there is no direct evidence to show the relationship between the equity premium and the unexpected shock to the mortality risk. This paper empirically tests this relationship by using G7 data. We find that excess returns are negatively correlated with the unexpected shock to mortality rates across the G7 countries. Specifically, we find that the risk premium could increase by 0.54% per year when there is a one basis point unexpected negative shock to the mortality rate. Long-run horizon returns are also negatively correlated with the unexpected shock to mortality rates. In the long run, the risk premium could increase by 0.331% per year where there is a one basis point unexpected negative shock to the mortality rate over the five-year horizon. Our paper contributes to the literature in terms of the way in which the empirical results have implications for the management of longevity risk. For example, when issuing annuities, many insurance companies face tremendous longevity risk due to the mortality improvement. Our findings could provide them with a potential tool to hedge longevity risk in annuities through asset allocation. In this paper, an example is provided and practical concerns are further discussed. On the other hand, our paper contributes to the literature in terms of the development of the theoretical models in asset pricing. Athanasoulis (2006) provided a theory predicting a positive relationship between the equity premium and the mortality rate. Our empirical study finds that the unexpected shock to the mortality rate is negatively correlated with the equity premium. A further theoretical model to reconcile Athanasoulis's (2006) model and our empirical findings could be fruitful. In addition, we suggest that one reason for our findings might be that an unexpected shock to the mortality rate could change the demand for the goods needed in the corresponding life cycle and further affect the equity premium of the demographically sensitive industries. Our paper only examines the overall stock market. Thus, an empirical study that examines the impact of the unexpected shock to the mortality rate on the equity premium among different industries could further contribute to the literature. Furthermore, from the analyses of individual countries, we find that the unexpected mortality shock is significantly negative in Germany but not in other countries. A one basis point unexpected negative shock to the mortality rate increases the one-year equity premiums by 0.63% in Germany which is a very large impact. A future study including a longer time period for Germany would be fruitful to explain the different behavior of the German series. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.jempfin.2013.03.002. References Ang, A., Maddaloni, A., 2005. Do demographic changes affect risk premiums? Evidence from international data. J. Bus. 78, 341–379. Athanasoulis, S.G., 2006. Asset prices and consumption in a model of perpetual youth. J. Bus. 79, 2835–2868. Bakshi, G., Chen, Z., 1994. Baby boom, population aging, and capital markets. J. Bus. 67, 165–202. Bayraktar, E., Young, V.R., 2007. Hedging life insurance with pure endowments. Insur.: Math. Econ. 40, 435–444. Cairns, A.J.G., Blake, D., Dowd, K., 2006. A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. J. Risk Insur. 73, 687–718. Cox, S.H., Lin, Y., Pedersen, H., 2010. Mortality risk modeling: applications to insurance securitization. Insur.: Math. Econ. 46, 242–253. Davidoff, T., Brown, J.R., Diamond, P.A., 2005. Annuities and individual welfare. Am. Econ. Rev. 95, 1573–1590. Della Vigna, S., Pollet, J., 2007. Demographics and industry returns. Am. Econ. Rev. 97, 1167–1702. Denuit, M., Devolder, P., Goderniaux, A., 2007. Securitization of longevity risk: pricing survivor bonds with Wang transform in the Lee–Carter framework. J. Risk Insur. 74, 87–113. Dolin, B., Young, M., 2004. Law and Government Division. Canada's Immigration Program. http://www.parl.gc.ca/information/library/PRBpubs/bp190-e.htm. Eeckhoudt, L.R., Hammitt, J.K., 2004. Does risk aversion increase the value of mortality risk? J. Environ. Econ. Manag. 47, 13–29. Fama, E.F., French, K.R., 1988. Dividend yields and expected stock returns. J. Financ. Econ. 22, 3–25. Fama, E.F., French, K.R., 1989. Business conditions and expected returns on stocks and bonds. J. Financ. Econ. 25, 23–49. Goyal, A., 2004. Demographics, stock market flows, and stock returns. J. Financ. Quant. Anal. 39, 115–142. Horneff, W.J., Maurer, R.H., Mitchell, O.S., Dus, I., 2008. Following the rules: integrating asset allocation and annuitization in retirement portfolios. Insur.: Math. Econ. 42, 396–408. Lee, R.D., Carter, L.R., 1992. Modeling and forecasting U.S. mortality. J. Am. Stat. Assoc. 87, 659–671. Lin, Y., Cox, S.H., 2008. Securitization of catastrophe mortality risks. Insur.: Math. Econ. 42, 628–637. Lin, T., Tzeng, L.Y., 2010. An additive stochastic model of mortality rates: an application to longevity risk in reserve evaluation. Insur.: Math. Econ. 42, 423–435. Oeppen, J., Vaupel, J.W., 2002. Broken limits to life expectancy. Science 296, 1029–1031. Poterba, J.M., 2001. Demographic structure and asset returns. Rev. Econ. Stat. 83, 565–584. Tsai, J.T., Wang, J.L., Tzeng, L.Y., 2010. On the optimal product mix in life insurance companies using conditional value at risk. Insur.: Math. Econ. 42, 235–241. Turner, A., 2006. Pensions, risks, and capital markets. J. Risk Insur. 73, 559–574. Young, V.R., 2008. Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio. Insur.: Math. Econ. 42, 691–703.
Contents lists available at SciVerse ScienceDirect
Journal of Empirical Finance journal homepage: www.elsevier.com/locate/jempfin
Does mortality improvement increase equity risk premiums? A risk perception perspective Rachel J. Huang a, d,⁎, Jerry C.Y. Miao b, 1, Larry Y. Tzeng c, d, 2 a b c d
Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei, Taiwan Department of Insurance, Tamkang University, Taipei, Taiwan Department of Finance, National Taiwan University, Taipei, Taiwan Risk and Insurance Research Center, National Chengchi University, Taipei, Taiwan
a r t i c l e
i n f o
Article history: Received 22 July 2010 Received in revised form 15 January 2013 Accepted 4 March 2013 Available online 21 March 2013
a b s t r a c t Using data for G7 countries over the period from 1950 to 2007, this paper finds that an unexpected shock to the mortality rate is significantly negatively correlated with the equity premium. A one basis point unexpected negative shock to the mortality rate increases both the one-year and five-year equity premiums by 0.54% and 1.66%, respectively. We also demonstrate how financial institutions could use our findings to hedge the risk of mortality-linked securities. © 2013 Elsevier B.V. All rights reserved.
JEL classification: G12 D03 J10
Keywords: Mortality risk Equity risk premium Demography Risk perception
1. Introduction It has been widely-recognized both in the literature and in the insurance industry that mortality rates have improved and are continuing to improve all over the world. For example, Oeppen and Vaupel (2002) found that life expectancy in the United States is increasing by 2.43 years each decade. 3 The trend toward mortality improvement has attracted the interest of many researchers who have analyzed various related issues. 4 One of the important issues concerns the impact on the equity risk premium. The literature has asserted that mortality improvement is correlated with the equity risk premium from a demographic structural change point of view. Researchers argue that the change in the participants' age structure reflects the change in the degree of risk aversion and investment needs in the life cycle, and will cause price fluctuations in the capital markets. For example, by using data from the United States, Bakshi and Chen (1994) found that, when the population ages, the demand for ⁎ Corresponding author. Tel.: +886 2 2730 1272. E-mail addresses: [email protected] (R.J. Huang), [email protected] (J.C.Y. Miao), [email protected] (L.Y. Tzeng). 1 Tel.: +886 2 2621 5656x3515. 2 Tel.: +886 2 2363 8597. 3 According to Turner's (2006) estimation, the mortality rate for a 65-year-old UK male in 2013 will fall by 46% in the twenty years between 1983 and 2003. 4 In addition to the risk premium, other issues include the relationship between aversion to financial risk and the willingness to pay to reduce mortality risk (e.g., Eeckhoudt and Hammitt, 2004), the hedging and pricing of life insurance policies (e.g., Bayraktar and Young, 2007; Cairns et al., 2006; Cox et al., 2010; Lin and Tzeng, 2010; Tsai et al., 2010; Young, 2008), the design of mortality derivatives (see Denuit et al., 2007; Lin and Cox, 2008), and the demand for annuity products and retirement planning (Davidoff et al., 2005; Horneff et al., 2008). 0927-5398/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jempfin.2013.03.002
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financial investment increases and the demand for housing decreases. They also found that the change in average age is positively correlated with the risk premium. Goyal (2004) found that outflows from the United States stock market are positively correlated with the change in the fraction of retired people and negatively correlated with the change in the fraction of middle-aged people (aged 45 to 64). He found that an increase in the average age will increase next year's excess stock return. By pooling international data, Ang and Maddaloni (2005) found that the higher the growth rate of the fraction of population in the retired group, the lower the risk premium. Instead of focusing on population changes among alternative age groups as in the previous literature, our paper adopts a different approach by directly examining the relationship between the unexpected shock to the mortality rate and the equity premium. There are three reasons for doing so. First, while demographic variables could proxy for changes in risk aversion, the impact of an unexpected shock to the mortality rate on the equity premium could be interpreted as a change in risk perception. Both risk aversion and risk perception are essential for individuals to determine their demand for risky assets. When the mortality rate for all the participants in the capital markets changes, the demographic structure may not change right away. However, the investors will immediately modify their risk perception regarding life expectancy when they recognize the unexpected shocks in the mortality rate. Thus, the marginal impact of the unexpected shock to the mortality rate on the risk premium can only represent the effect of a change in risk perception as opposed to that of a change in risk aversion. In addition, as noted by Della Vigna and Pollet (2007), the changes in demographics will affect the future consumption plans and further influence the profits and returns among industries. When observing an unexpected shock to the mortality rate, the investors will anticipate the future changes in revenues for different industries, and will immediately modify their investment strategies among industries. Thus, the changes in portfolio choices will affect the equity premium right away. Second, financial institutions can use the correlation between the unexpected mortality rate shock and the equity premium to hedge the mortality-linked securities 5 which have been actively introduced by major investment banks and insurers in recent years. A change in demographic structure is determined by several factors, such as the current demographic structure, the mortality rate, the fertility rate, and the immigration policies. Separating the impact of the mortality rate from the impacts of other factors could provide a more accurate measure of the sensitivity of the equity premium with respect to the mortality rate and could reduce the basis risk faced by financial institutions when hedging mortality-linked securities. Finally, the relationship between the change in the mortality rate and the equity premium has been proposed in theory but has never been examined empirically. Athanasoulis (2006) adopted a general equilibrium overlapping generations model and showed that the mortality rate is positively correlated with the equity premium. Although Athanasoulis (2006) has provided theoretical predictions and supports his results by simulation, whether his results can be supported by real world empirical evidence deserves a further study. Hence, this paper employs G7 country data to provide the empirical evidence. The methodology underlying our empirical model is similar to that in Ang and Maddaloni (2005) who examined the relationship between the equity risk premium and demographic variables. In addition to Ang and Maddaloni's (2005) dependent variables, we further employ the unexpected mortality rate shock in the regression models. Our empirical evidence finds strong support for the view that an unexpected shock to the mortality rate is significantly negatively correlated with the risk premiums. By using a one-year horizon, we find that the magnitude of the increase in the risk premium is about 0.54% per year when the mortality rate is one basis point lower than expected. We also find similar results when testing the long-run relationship. For the five-year horizon, the magnitude of the increment in the risk premium due to a one basis point unexpected negative shock to the mortality rate is around 0.331% per year. We further provide an example to demonstrate the applications of our empirical findings in hedging a mortality-linked product. We derive the immunization strategy for an annuity issuer. By simulation, we further demonstrate that our strategy could improve the performance in terms of hedging the unexpected shocks to the mortality rate. The remainder of this paper is organized as follows. Section 2 reports our data. The methodology is introduced in Section 3. Section 4 presents our empirical findings. Section 5 illustrates how we use our findings to hedge the longevity risk of annuity issuers. Section 6 concludes the paper.
2. Data The sample comprises the G7 countries. Our data are merged from three sources, with financial data being collected from the Global Financial Data (GFD), consumption data being gathered from the Center for International Comparisons (CIC), and demographic and mortality data being obtained from the Human Mortality Database (HMD). The data periods are 1950–2007 for Canada, Italy, the United Kingdom and the United States, 1960–2007 for France and Japan, and 1970–2007 for Germany. The data consist of annual observations for excess aggregate equity returns, dividend yields, term spreads, consumption growth, demographic variables, and mortality rates. The definitions of all variables are provided in Table 1. Besides the explanatory variables used in Ang and Maddaloni (2005), we additionally employ two independent variables: the unexpected shock to the mortality rate and the dividend yield. 6 5
For example, longevity bonds, annuities, survival bonds, mortality swaps and forwards, etc. Fama and French (1988, 1989) have shown that the dividend yield is important in predicting the future risk premium. Thus, we include the dividend yield as one of the independent variables. 6
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Table 1 Definitions and data sources of variables. Variable
Definitions
excess
The difference between the continuously compounded total return on the equity index and short-term risk-free investment. The database is provided by Global Financial Data (GFD). Equity returns represent price plus dividend returns (total returns) from GFD, which are compiled from Canada's S&P/TSX-300 total return index, France's SBF-250 total return index, Germany's CDAX total return index, Italy's BCI Global return index, Japan's Nikko securities composite total return, the UK's FTSE all-share return index, and the S&P 500 total return index annual. The returns of short-term risk-free assets are the yields of 3-month treasury bills in G7 countries. %DIVY The dividend yield. The database is also from GFD. The observations are compiled from Canada's S&PTSX-300 dividend yield, France's dividend yield, Germany's dividend yield, Italy's dividend yield, the Tokyo SE dividend yield, the FT-Actuaries dividend yield, and the S&P 500 dividend yield. term The difference between the long bond yield and the short bond yield. The database is from GFD. The long bond yields are Canadian government bonds of 10-year maturity, France's 10-year government bond yield, Germany's 10-year government bond yield, Italy's 10-year government bond yield, Japan's 7-year government bond yield, the U.K.'s 20-year government bond yield, and the USA's 10-year bond constant maturity yield. The short term bond yield returns are the yields of 3-month treasury bills in G7 countries. dcons The continuously compounded change in aggregate consumption. Consumption data are gathered from the Center for International Comparisons (CIC). Annual aggregate consumption is estimated as the difference between the total value of the real GDP and the share of GDP invested. d%age65 The log change in the fraction of adults over 65 years old. The data are obtained from the Human Mortality Database (HMD). MOR The mortality rate of adults over 25 years old. The data are obtained from the HMD. MORSHOCK The unexpected change in the mortality rate of adults over 25 years old, which is the residual of the following AR(1) model: MORt = γ0 + γ1MORt − 1 + εt
The mortality rate, denoted by MORt, is defined as DEATHt/EXPOSUREt, where DEATHt is the number of people who die over the age of 25 at time t, and EXPOSUREt is the total number of people over 25 years old at time t. We focus on the mortality rate for adults above 25 years old rather than the mortality rate of the retired group because the participants in the capital market include the young, middle-aged and retired group. Thus, the mortality rate for adults above the age of 25 could reflect the general risk perception for the overall capital markets. Since we argue that the unexpected shock to the mortality rate will change the investors' risk perception and further affect the demand for risky assets, a model to estimate the unexpected shock to the mortality rate is necessary. For each country, we adopt the following first-order autoregressive, AR(1), model: MORt ¼ γ 0 þ γ 1 MORt−1 þ εt
ð1Þ
where εt represents the unexpected shock to MORt, and is referred to as MORSHOCKt hereafter. We will use MORSHOCKt as the major independent variable in our regressions. 7 Table 2 offers descriptive statistics for the variables. The basic statistics for our financial variables and demographic variables are consistent with those in Ang and Maddaloni (2005). The average equity premium (denoted as excess) is 6.4% per annum with a standard deviation of 15.9% for the United States, which has a relatively high excess return compared to the other G7 countries. Turning to the demographic variables, Japan has the fastest rate of increase in the fraction of retired people compared to the other G7 countries. The average MOR is 1.4% for the United States. Japan has the smallest values for MOR and Canada has the 2nd lowest mortality rate among the G7 countries. The average MORSHOCK for all countries is small since it is the residual obtained from the AR(1) model. Canada and the United States have the highest standard deviations of MORSHOCK, whereas France has the lowest standard deviation of MORSHOCK. The selected correlation matrices are reported in the Appendix, Table A1. It is shown that MORSHOCK and d%age65 are negatively correlated in Canada and the United States, whereas they are positively correlated in other countries. Table A1 further indicates that, for all G7 countries, the simple correlations between excess and d%age65 differ from those between excess and MORSHOCK. For the United States, excess is positively correlated with d%age65, but negatively correlated with MORSHOCK. On the other hand, excess is negatively correlated with d%age65, but positively correlated with MORSHOCK for Germany, Italy, Japan and the United Kingdom. These results partially support the view that MORSHOCK might provide additional information to d%age65 regarding estimating the equity premium. In general, the excess variables are positively correlated with the variables for the yield spread between long-term bonds and short-term bonds (denoted as term) and MOR, and negatively correlated with other variables for Canada and France. For Germany and Italy, the excess variables are positively correlated with the variables for term and MORSHOCK, and negatively correlated with the other variables. The international correlations are listed in the Appendix, Table A2. The mortality rates for those over 25 years old are highly correlated among countries except for Japan. We further use Fig. 1 to demonstrate the time trends of MOR for the G7 countries. Panel A in Fig. 1 shows that Germany, Italy, the United Kingdom and the United States exhibit similar broad trends, and the lines of the mortality rates go up slightly before 1970. After the mid-1970s, the mortality rates decrease. Panel B in Fig. 1 shows that the mortality rates for Canada and France experience a downward trend from the late 1960s onwards. The pattern of Japan's 7 We also use the differences between MORt and MORt-1 to proxy the unexpected shock to the mortality rate. The results are consistent with the results using MORSHOCK, and are therefore unreported.
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Table 2 Descriptive statistics — means and standard deviations.
Canada France Germany Italy Japan U.K. U.S.
excess
%DIVY
term
dcons
d%age65
MOR
MORSHOCK
0.047 (0.153) 0.031 (0.221) 0.059 (0.230) 0.032 (0.246) 0.047 (0.231) 0.054 (0.228) 0.064 (0.159)
0.034 (0.011) 0.039 (0.016) 0.032 (0.011) 0.035 (0.013) 0.029 (0.028) 0.047 (0.016) 0.034 (0.013)
0.012 (0.014) 0.009 (0.014) 0.020 (0.012) 0.012 (0.015) 0.017 (0.014) 0.006 (0.019) 0.012 (0.012)
0.024 (0.024) 0.042 (0.037) 0.006 (0.020) 0.059 (0.042) 0.039 (0.055) 0.042 (0.026) 0.021 (0.019)
0.010 (0.008) 0.006 (0.010) 0.013 (0.013) 0.016 (0.008) 0.026 (0.009) 0.007 (0.006) 0.008 (0.006)
0.012 (0.001) 0.016 (0.002) 0.016 (0.002) 0.014 (0.001) 0.011 (0.002) 0.017 (0.001) 0.014 (0.001)
0.0000 (0.0003) −0.0001 (0.0007) 0.0000 (0.0004) 0.0000 (0.0005) −0.0001 (0.0006) 0.0000 (0.0006) 0.0000 (0.0003)
The table reports the means and standard deviations (shown in parentheses) of the equity premium (excess), dividend yields (%DIVY), the yield spread between the long bond and the short bond (term), the continuously compounded change in aggregate consumption (dcons), the log change in the fraction of adults over 65 years old (d%age65), and the unexpected shock in the mortality rate of adults over 25 years old (MORSHOCK) for the G7 member nations. The data periods are 1950–2007 for Canada, Italy, the United Kingdom and the United States, 1960–2007 for France and Japan, and 1970–2007 for Germany.
A) Mortality rates for Germany, Italy, the U.K. and the U.S. 0.02
0.015
0.01
0.005 UK
0 1950
1955
1960
1965
US
1970
Germany
1975
1980
Italy
1985
1990
1995
2000
2005
1990
1995
2000
2005
B) Mortality rates for Canada, Japan and France. 0.025
0.02
0.015
0.01
0.005 Japan
0 1950
1955
1960
1965
1970
Canada
1975
1980
France
1985
Fig. 1. Mortality rates, MOR, for G7 countries. These figures plot the mortality rates of adults over 25 years old for each country, which equals the number of people who die over the age of 25 divided by the total number of people over 25 years old.
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mortality rate, however, is quite different. MOR exhibits a clear downward trend before the late 1980s and a slight increase after the late 1980s. The differences in the mortality rate patterns among countries might be due to immigration laws. Among the G7 countries, Canada has the highest per capita net immigration rate. 8 Japan has relatively tight immigration laws. In the early 1990s, to reduce the labour shortage, Japan started to allow special entry permits for foreigners, and the change in immigration laws might have caused certain fluctuations in the mortality rates. Fig. 2 shows the pattern of MORSHOCK, from which it can be seen that there is no significant upward or downward time trend for MORSHOCK in the G7 countries. All of the G7 countries share a similar pattern in that MORSHOCK varies substantially year by year before 1970 and becomes stable after 1970. In addition, we find no significant upward or downward time trend for excess, which provides partial evidence that the time series of the unexpected shock to the mortality rate and the equity premium may not have unit roots. To further detect whether the relationship between the unexpected shock to the mortality rate and the equity premium might be spurious, we run the Dickey–Fuller tests to examine the unit roots. Table A3 in the Appendix shows that the time series of the unexpected shocks in the mortality rate and the equity premium are stationary. 3. Empirical methodology We first use a panel data methodology to examine the relationship between the risk premium and the unexpected shock to the mortality rate. 9 Panel data enjoy more degrees of freedom, and hence improve the efficiency of the econometric estimates. Furthermore, the use of panel data in our study presents a more reliable picture than that arising from purely cross-sectional studies. Thus, we can make more accurate inferences using the model parameters. Because the number of observations for each country is not identical, an unbalanced panel results. 10 Secondly, since cross-country studies may further provide interesting insights into the differences across countries, we also analyze the relationship between MORSHOCK and excess for one country at a time. In the panel regressions, we empirically investigate the relationship between expected excess returns and the unexpected shock to mortality rates in the short run and in the long run as follows: excessitþ1 ¼ β0 þ β1 %DIVY it þ β2 termit þ β3 dconsit þ β4 d%age65it þ β5 MORSHOCK it þ θitþ1 ;
ð2Þ
where θt + 1 denotes the residual. MORSHOCK is our main explanatory variable. As explained in the Introduction and based on the preliminary evidence from the previous section, MORSHOCK could provide additional information to d%age65. Thus, we examine the effect of MORSHOCK by controlling for the effect of d%age65 and the regression is referred to as Model 1. 11 To isolate the effects of MORSHOCK and d%age65 in the regression model, we also include only one of these two variables at a time. In Model 2, the variable d%age65 is not included. In Model 3, we do not control for the variable MORSHOCK. Similar to the regressions in Ang and Maddaloni (2005), we include the consumption growth and the term spread as the control variables. We further employ the dividend yield as another control variable because the literature (see Fama and French, 1988, 1989) has documented that the dividend yield has significant predictive power in regard to the future risk premium. For the long-run correlation in the panel regressions, we use non-overlapping windows of data as in Poterba (2001) to construct the long-run excess return. Although non-overlapping windows of data will dramatically decrease the number of observations, this methodology can avoid the autocorrelation problem in the error terms of the regressions. For the cross-country studies, we only investigate the short-run relationship because the number of observations for each country is quite small, while the non-overlapping windows of data are adopted for the long-run models. 4. Empirical results The results of the panel analyses are presented in Table 3. 12 For the 1-year horizon, Model 1 in Table 3 shows that the coefficient of MORSHOCK is significantly negative. 13 It means that if mortality rates are lower than expected by one basis point, the equity premium will rise by 0.54% and vice versa. If there is an unexpected negative shock to the mortality rate by one standard deviation, which is 0.051%, then the risk premium increases by 2.74%. 14 This evidence shows that our results are not only statistically significant, but may also be economically significant. The negative relationship between MORSHOCK and excess is also 8
Please see Dolin and Young (2004). We estimate the unbalanced panel regression by using the generalized least squares random effects estimators. 10 For a robustness check, we also construct a balanced panel data set which contains the countries with the longest time series data, i.e., Canada, Italy, the United Kingdom and the United States. The balanced panel data also provide support for the view that the unexpected mortality shock is significantly negatively correlated with the equity premium for the 1-year horizon. Since the results from the balanced panel are similar to the results from the unbalanced panel, we only report the results from the unbalanced panel. 11 Ang and Maddaloni (2005) found that the average age of the population has no forecasting power in international data, but the change in the proportion of retired persons is a significant predictor of the excess return. Thus, in our regression, we only control for d%age65. 12 In Table 3, the standard errors of the pooled estimates have been corrected for autocorrelation and/or heterogeneity across countries. 13 Note that the negative relationship between the equity premium and the unexpected shock to the mortality rate remains when the unexpected shock to the mortality rate for adults over 65 years old is adopted. 14 The average excess return for the pooled data is 4.82%. 9
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A) Unexpected mortality shocks for Germany, Italy, the U.K. and the U.S. 0.004 0.003 0.002 0.001 0 -0.001 UK -0.002 1950
1955
1960
1965
US 1970
1975
Germany
Italy
1980
1990
1985
1995
2000
2005
2000
2005
B) Unexpected mortality shocks for Canada, Japan and France. 0.004 0.003 0.002 0.001 0 -0.001 Japan -0.002 1950
1955
1960
1965
1970
1975
Canada 1980
France 1985
1990
1995
Fig. 2. Unexpected shocks in the mortality rate, MORSHOCK. This figure plots the residuals of the AR(1) model for the mortality rate of adults over 25 years old for each country.
found in Model 2, where the average change in the proportion of retired persons is absent. The magnitude of the effect in Model 2 is similar to that in Model 1. Note that our paper empirically finds a negative relationship between the equity premium and the unexpected shock to the mortality rate, whereas Athanasoulis (2006) numerically predicted a positive relationship between the equity premium and the expected change in the mortality rate. The difference between Athanasoulis (2006) and our paper mainly arise because we analyze the impact of the unexpected change in the mortality rate on the equity premium, while Athanasoulis (2006) focused on the impact of the change in the expected mortality rate. In his model, he assumed that the mortality rate is constant over time. He then provided a simulation to show how the equity premium changes when the expected mortality rate is at a different level. When the mortality rates are different from the expected, the individuals will change their risk perception and further change their consumption plans. For example, when there is an unexpected negative shock to the mortality rate for the adults over 25 years old, the demand for housing and life insurance may increase right away. Furthermore, the demand for pharmaceuticals, nursing houses, funeral houses and the goods needed in the corresponding life cycle will also increase in the future. Thus, the equity premium on those industries may change since investors anticipate that the revenues of those industries will increase due to an unexpected negative shock in the mortality rate. Della Vigna and Pollet (2007) used US data and showed that an additional 1% increase in the annualized demand growth due to demographics enabled them to predict a 5% to 10% increase in annual abnormal industry stock returns. If these demographic industries play significant roles in the overall stock market, then the equity premium will also increase. As a result, we will observe a negative relationship between the equity premium and the unexpected mortality shocks. Thus, for future theoretical analyses, the investor should be assumed to be able to invest among different industries rather than only invest in an aggregate equity market. A consequent assumption is that the investors will update their beliefs regarding
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Table 3 Pooled regressions on equity premiums across G7 countries. One-year horizon
Constant %DIVY term dcons d%age65 MORSHOCK 2
Adj-R Num. of obs.
Five-year horizon
(1)
(2)
(3)
(1)
−0.086** (−2.12) 3.146*** (4.25) 1.473** (1.97) 0.004 (0.10) −0.065 (−0.27) −53.768** (−2.12) 0.056 359
−0.087*** (−3.05) 3.120*** (4.41) 1.464* (1.95) −0.004 (−0.17)
−0.085*** (−2.86) 3.201*** (4.27) 1.458* (1.92) 0.015 (0.04) −0.031 (−0.13)
−0.122 (−1.36) 5.370** (2.39) −0.273 (−0.13) −0.017 (−0.40) 0.158 (0.58) −165.708** (−2.14) 0.058 67
−53.326** (−2.10) 0.056 359
0.044 359
(2) −0.103 (−1.22) 4.992** (2.33) −0.295 (−0.14) 0.001 (0.02)
−173.983** (−2.28) 0.054 67
(3) −0.118 (−1.27) 5.152** (2.21) −1.335 (−0.64) −0.027 (−0.63) 0.255 (0.91)
−0.005 67
The table lists the coefficients for the pooled regressions with AR(1) disturbances across G7 countries over the sample period 1950–2007. The t-statistics are reported in parentheses with those significant at the 10% (5%, 1%) level denoted by * (** and ***). The dependent variable for the first three columns is the one-year excess return which is the difference between the continuously compounded total return on the equity index and the short-term risk-free investment. The dependent variable for the last three columns is the 5-year equity premium from non-overlapping window data. The explanatory variables are %DIVY, the dividend yields, term, the difference between the long bond yield and the short bond yield, dcons, the continuously compounded change in aggregate consumption, d%age65, the log change in the fraction of adults over 65 years old, and MORSHOCK, the residual of the AR(1) model for the mortality rate of adults over 25 years old.
the future revenues in different industries after they observe the unexpected mortality shocks in the previous period. With this setting, the theoretical prediction might be consistent with our empirical findings. In both Models 1 and 3, the short-run equity premium is not significantly affected by the average change in the proportion of retired persons. This finding is, however, not consistent with the results of Ang and Maddaloni (2005). This might be due to the differences between our paper and theirs in terms of the data period and the countries included. 15 The results for %DIVY coincide with the results of most studies in the literature. This variable is significantly positively correlated with excess. The signs of the coefficients of term are consistent with the findings of the pooled regression across the G5 countries in Ang and Maddaloni (2005). The variable is also significantly positively correlated with the short-run equity premium. The coefficients of dcons are insignificant in all models. Table 3 further demonstrates that the results for 5-year horizons are similar to those for one-year horizons: the coefficients of MORSHOCK are significantly negative. 16 These results provide us with evidence that an unexpected shock to the mortality rate gives rise to an accumulated effect in the long run. The coefficient of MORSHOCK in Model 1 shows that the five-year equity premium increases by 0.331% per year when there is a one basis point decrease in MORSHOCK. In the long run, we find that the coefficients of d%age65 are insignificantly positive in all regressions, whereas the coefficients are significantly negative in Ang and Maddaloni's (2005) regressions. 17 The coefficients of %DIVY are significantly positive at the 5% level. However, the coefficients of both term and dcons are insignificant. Table 4 shows the results for individual countries for a 1-year horizon. The coefficient of MORSHOCK is significantly negative for Germany in Model 1. Although the coefficients of MORSHOCK are negative for Canada, France, Italy, the United Kingdom and the United States in Model 1, they are insignificant. For Japan, we find that the coefficient of MORSHOCK in Model 1 is insignificantly positive. Generally speaking, the results in Table 4 demonstrate that the estimated correlation between the unexpected shock to the mortality rate and the equity premium is negative, which is consistent with our findings in Table 3. Probably due to the limitation of the numbers of observations, the empirical results are not statistically significant. 18 From the results of individual countries, one reasonable conjecture is that the large coefficients of MORSHOCK in the pooled regressions might be due to the inclusion of Germany. Thus, the regressions for the pooled data excluding Germany are examined and the results are shown in the Appendix, Table A4. 19 When Germany is not included, the coefficients of MORSHOCK are still
15 When Ang and Maddaloni's (2005) regression is adopted for the G5 countries, our data reveal evidence of an insignificant negative relationship between d%age65 and excess. 16 We also check a three-year period by using non-overlapping data. However, in the regression, only the dividend yield is significant. All other variables are insignificant. 17 Note that our non-overlapping window data still predict an insignificant positive relationship between d%age65 and excess for the G5 countries. 18 In unreported analyses, we further divide countries to respectively examine the effects of financial market development, social security benefits and geography on the relationship between the unexpected shock to the mortality rate and the equity premium. We find that the unexpected shock to the mortality rate and the equity premium are still significantly negatively correlated, and the degree of financial market development, social security benefits and geography do not significantly make the negative correlation different. 19 In Table A4, the standard errors of the pooled estimates have also been corrected for autocorrelation and/or heterogeneity across countries.
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R.J. Huang et al. / Journal of Empirical Finance 22 (2013) 67–77
Table 4 Regressions for individual countries — one-year horizon. Constant Panel A: Canada (1) −0.063 (−0.95) (2) −0.074 (−1.26) (3) −0.064 (−0.97) Panel B: France (1) 0.011 (0.15) (2) −0.069 (−0.99) (3) 0.017 (0.24) Panel C: Germany (1) 0.005 (0.05) (2) −0.057 (−0.48) (3) 0.059 (0.46) Panel D: Italy (1) 0.185 (1.33) (2) −0.040 (−0.41) (3) 0.191 (1.40) Panel E: Japan (1) 0.287 (1.61) (2) −0.074 (−1.01) (3) 0.278 (1.58) Panel F: UK (1) −0.244*** (−3.50) (2) −0.226*** (−2.94) (3) −0.241*** (−3.40) Panel G: US (1) −0.235** (−2.42) (2) −0.073 (−0.92) (3) −0.234** (−2.42)
%DIVY
term
dcons
d%age65
MORSHOCK
−0.999 (−0.46)
−1.050 (−0.06) 0.254 (0.01)
4.275*** (2.76) 4.312*** (2.82) 4.290*** (2.84)
2.895** (2.52) 3.039*** (2.73) 2.898** (2.56)
−2.590*** (−3.29) −2.632*** (−3.40) −2.593*** (−3.39)
2.915 (1.67) 3.507** (2.21) 3.016* (1.71)
1.287 (0.73) 1.829 (1.00) 1.383 (0.79)
−2.038** (−2.25) −1.557 (−1.61) −2.222** (−2.60)
0.842 (0.32) 2.162 (0.70) 0.027 (0.01)
3.535 (1.65) 1.193 (0.66) 2.584 (1.00)
−3.241** (−2.34) −2.461* (−1.90) −2.540 (−1.53)
−6.377** (−2.26)
5.555** (2.18) 4.830* (1.97) 5.557** (2.22)
−2.891 (−1.52) −1.312 (−0.60) −2.974 (−1.67)
−2.277** (−2.33) −1.394* (−1.70) −2.322** (−2.44)
−11.005** (−2.12)
8.826** (2.25) 10.550** (2.55) 8.557** (2.15)
−0.931 (−0.59) 0.153 (0.11) −0.880 (−0.56)
−3.091*** (−3.42) −2.711*** (−2.79) −3.017*** (−3.44)
−10.608** (−2.32)
−8.037* (−1.73)
9.532*** (5.26) 8.680*** (4.77) 9.552*** (5.38)
1.076 (0.82) 0.604 (0.55) 1.092 (0.83)
−2.441** (−2.06) −3.151** (2.51) −2.429** (−2.09)
13.292*** (3.01) 3.980* (1.95) 13.293*** (3.04)
2.637* (1.84) 2.452* (1.73) 2.654 (1.86)
−1.384 (−1.46) −1.391 (−1.66) −1.402* (−1.78)
−0.978 (−0.47) −4.798** (−2.23)
−11.802 (−0.87) −9.440 (−0.84)
−4.556** (−2.32) −62.502** (−2.31) −37.331 (−1.55)
−4.179 (−1.56) −1.923 (−0.27) −5.710 (−0.80)
−11.086** (−2.18) 8.764 (0.69) 6.346 (0.46)
−10.426** (−2.33) −3.578 (−0.32) −6.460 (−0.60)
−8.361** (−2.00) −20.053*** (−2.90)
−20.019*** (−2.87)
−1.206 (−0.06) 3.442 (0.18)
Adj. R2
Num. of obs.
0.1445
57
0.1587
57
0.1613
57
0.034
47
0.010
47
0.045
47
0.047
37
−0.041
37
−0.024
37
0.1156
57
0.0135
57
0.1325
57
0.131
47
0.077
47
0.147
47
0.333
57
0.320
57
0.345
57
0.182
57
0.066
57
0.198
57
Note: The table lists the coefficients of regression with AR(1) disturbances for the G7 member nations over the sample period 1950–2007. The t-statistics are reported in parentheses with those significant at the 10% (5% and 1%) level denoted by * (** and ***). The dependent variable of excess is the difference between the continuously compounded total return on the equity index and the short-term risk-free investment. The explanatory variables are %DIVY, the dividend yields, term, the difference between the long bond yield and the short bond yield, dcons, the continuously compounded change in aggregate consumption, d%age65, the log change in the fraction of adults over 65 years old, and MORSHOCK, the residual of the AR(1) model for the mortality rate of adults over 25 years old.
significantly negative in all models. For the remaining G6 countries, if mortality rates are one basis point lower than expected, the equity premium will rise by 0.46%. In a way that differs from the findings in the panel regressions, d%age65 is significantly negative for all countries except Canada. Among the G7 countries, d%age65 has a higher effect on the equity premium in the United States, as evidenced by a one percent increase in d%age65 which will cause the equity premium to increase by 20% in the United States. We further find that the coefficients of %DIVY are insignificantly positive for Germany, but they are significantly positive for the rest of the G7 countries. The coefficients of term are significantly positive in Canada and the United States. However, for other countries, the effect of term on the equity premium is significant. The findings regarding the relationship between the consumption growth rate and the equity premium across countries are different from the findings in the literature. We find that the coefficients of dcons are negative for all G7 countries. In Model 1, the coefficients are significant except for the United States.
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5. An application The empirical evidence in this study shows that an unexpected negative shock to the mortality rate significantly increases the equity premium. As mentioned in the Introduction, understanding the relationship between the unexpected shock to the mortality rate and the equity premium could help the financial institutions hedge the risk associated with holding mortality-linked securities. In what follows, we will provide an illustration of a hedging strategy for an artificial insurance company. Assume that an insurance company with equity St issues an annuity at time t, which pays a one dollar benefit at time t + 1 if the policyholder is alive. The insurer receives an insurance premium Pt. Assume that the annuity policy is the only insurance contract the insurer sells and the liability of the insurer, Lt, is recognized as the unearned premium reserve of the annuity, which happens to be Pt. Thus, the total assets of the insurer are equal to At = Pt + St. Assume that the insurer prices the annuity using the actuarially fair premium, which is equal to the present value of the expected coverage. Let us assume that the insurance company does not observe MORSHOCKt when pricing the annuity. Thus, the insurer considers the process of the mortality rate to follow Eq. (1) and sets −r P t ¼ 1−E MORtþ1 e f ;
ð3Þ
where rf denotes the continuous compounded risk-free rate and E is the expectation operator. Note that E(MORt + 1) = γ0 + γ1E(MORt) from Eq. (1). Assume that MORSHOCKt will be realized immediately after the insurer issues the annuity. Since the equity premium is correlated with MORSHOCKt as shown by our empirical results, the insurer could hedge the unexpected mortality shock by constructing a portfolio in capital markets. Let the insurer choose α, the proportion of its investment in risky assets, and 1 − α, the proportion in risk-free assets. Thus, after asset allocation, the assets of the insurer at t + 1 can be evaluated by: h i r~tþ1 r~ r r ~ þ ð1−α ÞðP t þ St Þe f ¼ α ðP t þ St Þe tþ1 −rf þ ð1−α ÞðP t þ St Þ e f A tþ1 ¼ α ðP t þ St Þe excesstþ1 r ¼ α ðP t þ St Þe þ ð1−α ÞðP t þ St Þ e f ;
ð4Þ
~ tþ1 is also a random variable after the where ~r tþ1 is a random variable and denotes the equity return at time t + 1. Note that A investment decision since the stock return is random. excesst + 1 denotes the equity premium at time t + 1 and is a function of MORSHOCKt as indicated in our empirical results. If the unexpected mortality shock occurs, the liability of the insurer at t + 1 becomes L~tþ1 ¼ 1−γ1 MORSHOCK t −MORSHOCK tþ1 :
ð5Þ
Note that L~tþ1 is also a random variable since MORSHOCKt + 1 is random. ~ tþ1 −L~tþ1 . As proposed by the immunization strategy, the The equity of the insurer at t + 1 can then be rewritten as S~tþ1 ¼ A objective of the insurance company is to ensure that the expected equity at time t + 1, E S~tþ1 , does not vary based on the unexpected mortality shock MORSHOCKt. Thus, the insurer can set α such that dE S~tþ1 dMORSHOCK t
¼ 0;
ð6Þ
where ~ ~ E S~tþ1 ¼ E A tþ1 −E L tþ1
r excesstþ1 ¼ α ðP t þ St ÞE e þ ð1−α ÞðP t þ St Þ e f
− 1−γ1 MORSHOCK t −E MORSHOCK tþ1 :
Since E(MORSHOCKt + 1) = 0, and Eðeexcesstþ1 Þ are functions of MORSHOCKt, Eq. (6) will become excesstþ1 rf α ðP t þ St ÞE e e
∂excesstþ1 þ γ 1 ¼ 0: ∂MORSHOCK t
Rearranging the above equation yields α¼
−γ1 ∂excess
tþ1 ðP t þ St Þ ∂MORSHOCK Eðeexcesstþ1 Þerf
:
ð7Þ
t
∂excesstþ1 It is very important to recognize that ∂MORSHOCK can be retrieved from an appropriate empirical model which regresses t excesst + 1 on MORSHOCKt such as our Eq. (2). To test empirically these findings, we take returns data from the US financial market during 1950 to 2007 to demonstrate that investing in a portfolio of the stock index and a risk-free asset in the proportions α and (1 − α), as estimated in Eq. (7), will yield a return to make the payment due for the above-constructed annuity.
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Table 5 Calibration Results.
Panel A: OLS regression Constant MORSHOCKt Adj. R2 Num. of obs. Panel B: basic statistics Mean Standard deviation
St + 1 under the hedging strategy
St + 1 without the hedging strategy
0.0118*** (29.77) 0.6299 (0.46) −0.0143 57
0.0105*** (225.61) 0.8433*** (5.30) 0.32590 57
0.0118 0.0030
0.0105 0.0004
Note: The table shows our calibration results using US data covering the period from 1950 to 2007. In each year, there is an insurance company with a capital of 0.01 that issues an annuity that will pay one dollar in the next year. The actuarially fair premium is calculated by Eq. (3). The hedging strategy of the insurance company is to invest the proportion α of its assets in the stock market and the proportion 1 − α in risk-free assets. α is obtained by Eq. (7). The t-statistics are reported in parentheses with those significant at the 10% (5% and 1%) level denoted by * (** and ***), respectively. The dependent variable St + 1 is the equity value of the insurance company after the annuity is paid at time t + 1. The independent variable MORSHOCKt is the unexpected shock in the mortality rate, which is the residual of the AR(1) model for the mortality rate of adults over 25 years old.
The calibration is as follows. Let us assume that there is an insurance company with a capital of 0.01 in the year 1950. The insurance company will calculate the premium according to Eq. (3) and the investment proportion α from Eq. (7). Note that since the annuity is a one-year contract, we use the coefficients estimated from Model 1 in Table 3 to calculate α. By using the capital market data for the year 1951, we can obtain the asset value of this insurance company after adopting the hedging strategy. From the real mortality rate for the year 1951, we are able to obtain the liability of the insurance company. Thus, if the equity, which is the difference between total assets and total liabilities, is positive, then it means that the hedging strategy could yield a return to make the payment due for the annuity. We further repeat the above steps by assuming that there is an insurance company with a capital of 0.01 in the year 1951, and we stop the steps in the year 2007. The calibration results are as follows. For these insurance companies, we obtain an average α of 1.87% and a standard deviation of 0.09%. Panel A in Table 5 indicates that the equity of the insurance companies is not sensitive to the unexpected mortality shock when the hedging strategy is employed. However, the equity of the insurance companies is significantly positively correlated with the unexpected mortality shock when the insurance company does not hedge the unexpected shock through asset allocation, i.e., α = 0. This result demonstrates that the hedging strategy can indeed help the insurance companies to hedge mortality risk. Panel B in Table 5 further shows that, under the hedging strategy, the insurance companies are able to pay for the annuity. After adopting the hedging strategy, the average equity of the insurance companies is 0.0118. If these insurance companies do not use the hedging strategy and set α = 0, then the equity will become 0.0105 on average, which is smaller than the equity value under the hedging strategy. However, since the insurance companies do not participate in the stock market when α = 0, they will not face the risk associated with the stock market. Thus, the standard deviation of the equity without the hedging strategy is only 0.004, which is much smaller than the standard deviation of the equity under the hedging strategy. It is worth noting that although the insurance companies will face additional risk from the stock market under a hedging strategy, the risk in the stock market will not cause the insurance companies to go bankrupt in our calibration when the insurance companies have some capital at the beginning. The initial capital we assume is 0.01, which means that the equity-to-debt ratio is about 1%. In reality, the capital-to-debt ratio is much higher than 1%. Thus, the insurance companies will be able to face the risk from the stock market when the hedging strategy is employed. In other words, although the risk from the investment in the stock market does exist, the risk might not give rise to a severe insolvency problem. The above analytical solution of α serves to demonstrate the application of our empirical results. However, the reality could be more complicated than the artificial case that we assumed. Several practical issues should be taken into consideration. First, the annuity could be sold to different age groups. Thus, the cohort effect should be included when the insurer estimates the mortality process. 20 Second, annuity policies vary in both the method of premium payment and the schedule of annuity benefits. 21 The impact of a shock to the unexpected mortality rate on the insurer's liability requires the details of the annuity policies. Third, most annuities are long-term contracts. The investment strategies of the insurer could be long-term and dynamic rather than short-term and static. Adding all these practical concerns could provide a fruitful result and deserves a future study.
20
For example, see Lee and Carter (1992). The benefit of an annuity can be designed as index-linked, escalating, as well as guaranteed, capital-protected, etc. On the other hand, the payment schedule of the premium is quite different for the immediate annuity, deferred annuity, variable annuity, etc. 21
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6. Conclusions The literature has found that mortality improvement is correlated with the equity risk premium from a demographic change point of view. However, there is no direct evidence to show the relationship between the equity premium and the unexpected shock to the mortality risk. This paper empirically tests this relationship by using G7 data. We find that excess returns are negatively correlated with the unexpected shock to mortality rates across the G7 countries. Specifically, we find that the risk premium could increase by 0.54% per year when there is a one basis point unexpected negative shock to the mortality rate. Long-run horizon returns are also negatively correlated with the unexpected shock to mortality rates. In the long run, the risk premium could increase by 0.331% per year where there is a one basis point unexpected negative shock to the mortality rate over the five-year horizon. Our paper contributes to the literature in terms of the way in which the empirical results have implications for the management of longevity risk. For example, when issuing annuities, many insurance companies face tremendous longevity risk due to the mortality improvement. Our findings could provide them with a potential tool to hedge longevity risk in annuities through asset allocation. In this paper, an example is provided and practical concerns are further discussed. On the other hand, our paper contributes to the literature in terms of the development of the theoretical models in asset pricing. Athanasoulis (2006) provided a theory predicting a positive relationship between the equity premium and the mortality rate. Our empirical study finds that the unexpected shock to the mortality rate is negatively correlated with the equity premium. A further theoretical model to reconcile Athanasoulis's (2006) model and our empirical findings could be fruitful. In addition, we suggest that one reason for our findings might be that an unexpected shock to the mortality rate could change the demand for the goods needed in the corresponding life cycle and further affect the equity premium of the demographically sensitive industries. Our paper only examines the overall stock market. Thus, an empirical study that examines the impact of the unexpected shock to the mortality rate on the equity premium among different industries could further contribute to the literature. Furthermore, from the analyses of individual countries, we find that the unexpected mortality shock is significantly negative in Germany but not in other countries. A one basis point unexpected negative shock to the mortality rate increases the one-year equity premiums by 0.63% in Germany which is a very large impact. A future study including a longer time period for Germany would be fruitful to explain the different behavior of the German series. Appendix A. 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