Mortality Risk Modeling

Fall 2016 Actuarial Science Undergraduate Research

Professor Shu Li Jeffrey Chen

Fatihah Ahmad Fauzi Yijing Chen

Section I: Introduction As seen in historical data, mortality has been decreasing over the past century due to numerous factors such as improvements in medicine. While lower mortality rates are certainly a positive sign for the population, this increase in life expectancy creates difficulties in accurately pricing products relating to mortality. Without adjusting to this trend, pension plans and retirement benefits will generally be underfunded causing corporations, insurance companies, the Social Security Administration, and other related organizations to suffer losses. The idea of increasing life expectancy is well known to actuaries and analysts; however, obtaining accurate forecasts from a reliable model seem to be a more difficult task. This report will analyze the mortality trends across all ages based on Americans male and female mortality by involving the reduction factors, rx, forecasts mortality using various models such as the reduction factor model and the Lee-Carter model and manages the longevity risk through a pension plan pricing example based on the models mentioned. According to Renshaw and Haberman, the reduction factor model is useful to monitor existing reduction factors’ effectiveness and to predict the possible behaviors of regression patterns, which are used for extrapolation.1 The Lee-Carter model on the other hand is a method to forecast mortality patterns in the long run by adopting statistical time series methods. This model is known for its wide use in the U.S and also other countries in dealing with mortality age distribution2. In this report, in order to ensure the accuracy of our calculation and get the most reliable conclusion, we use the data from the U.S Social Security Administration and the Human mortality database (HMD)3 that was done by two teams of researchers in the USA and Germany. In Section 2, we discussed mortality curves behavior in a year through Gompertz and the assumptions portrayed through the reduction factor model and Lee-Carter model. These assumptions are used in Section 3 where the research provides discussion on forecasting mortality rates and life expectancy based on the reduction factor model and Lee-Carter model. In section 4, we compared two models through a pension plan with defined contribution or defined benefit to give the reader a better idea of how the models would impact in real life. For the last part, we draw a conclusion on the advantage and problems of reduction factor model and Lee-Carter model that we see through the research.

1 On the Forecasting of Mortality Reduction Factors (Renshaw and Haberman) 2 The Lee-Carter Method for Forecasting Mortality, with Various Extensions and Applications (Lee) 3 Data extracted from The Human Mortality Database: http://www.mortality.org/

Section II: Models and Assumptions 2.1 Mortality trends

In order to understand the mortality curve in one year, our initial assumption is to use Makeham’s law to fit the historical data from the U.S Social Security Administration:

𝜇x = A + Bcx for x ≥ 0, A ≥ -B, B > 0, c > 1 (2.1)

However, there were difficulties in extracting parameters A, B and c. We cannot

simply fit them into a linear regression model. Instead, we attempted to find the parameters by using the {fmsb} package in R4 (See Appendix A).

In this case A= -0.003596724 whereas B=0.0001428244. Here, A is less than -B which violate our conditions. We realize a more realistic and approachable method by fitting the data using Gompertz Law for a single year through regression since its force of mortality does not consider the estimation for parameter A:

𝜇x = Bcx for x ≥ 0, B > 0, c > 1 (2.2)

Therefore, in order to find the parameters for Gompertz Law, we fit the

probability of death of age after 30 into the simple linear regression model in R based on the equation (2.2) (See Appendix A).

ln(𝜇x ) = ln(B)+ xln(c) (2.2)

4 Gompertz-Makeham's model mortality for u(x) and its fitting: http://minato.sip21c.org/msb/man/GompertzMakeham.html

This results to:

Figure 2.1 Force of Mortality Trend Figure 2.1 shows the fitted force of Mortality with the observed force of Mortality. We can see that the model fits well for bigger ages. Note that when we plot 𝜇x of one year for people at each age in Figure 2.1, mortality rate increases exponentially after age 30. However for age before 30, we find it hard to generate a trend. It should be noted that modeling annual mortality rate using Gompertz Law is relatively simple, which thus making it a less credible method to provide a good fit over the ages. We could observe, however, that Gompertz law tends to be more useful at later ages as mortality rates behaved better at older periods. 2.2 Reduction Factor Model To identify mortality trends, we are required to analyze the mortality patterns over a period of years. Afterwards, we project mortality with the assumption that mortality rates decrease annually depending on the individual’s age and sex.5 However, here we only take account to the dependency on the age of the individual. In order to involve the 5 Actuarial mathematics for life contingent risks Section 3.11(2nd ed.)

idea of mortality trends, we look for the mortality reduction factors following the equation:

q(x,s) = q(x,0)rsx where 0 < rx ≤ 1 (2.3)

q(x,s) is the mortality rate for an individual at age x in s years away from the base year, which tells us that q(x,0) is the mortality of age x at the base year. The reduction factor is denoted by the rx parameter.

Based on the equation (2.3), we first began with the mortality data from the Human Mortality Database and selected years 1980 to 2000 by arranging it into columns of years and rows of individual’s ages for our testing in R. In order to perform a regression, we then manipulated equation (2.3) into the following equation:

ln(q(x,s)/(q(x,0)) = s ln(rx) (2.4)

With q(x,0) set as the mortality rate of 1980, q(x,s) as the mortality rate of the following years after 1980 (s being the number of years away from 1980). We were then able to set up a 20 by 1 matrix, x, consisting of s values ranging from 0 to 20. Similarly, we set up a 20 by 111 matrix, y, of logarithm (q(x,s)/(q(x,0)) with rows of s values and columns of ages. We then ran a multiple regression (between x and y) in order to find a 1 by 111 matrix, which would have ln(rx) for each age x. From here, we obtain the reduction factors by taking the exponent of the matrix. For any reduction factors that exceed 1.00, we adjusted them to 1.00, so as to keep the assumption that rx is typically within the range of 0.95 to 1, inclusive (refer to Appendix B).

Figure 2.2 Adjusted reduction factors, rx, based on American male and female mortality

2.3 Lee-Carter Model

Other than forecasting future mortality rates, the Lee-Carter model is meant to provide probabilistic confidence intervals, although this report does not provide a discussion of it. However, the model’s age-specific death rates are to diminish exponentially without limit. The model also has no requirement on using additional information on behavioral, medical, or social influences on mortality change. Using the central death rates of 5-year age groups from 1933 to 1987 in a matrix, Lee and Carter were able to fit this data this equation:

ln(mx,t) = ax + bxkt + εx,t (2.5)

In the Lee-Carter model, mx,t represents the central death rate for age group x in year t. {ax} and {bx} are age-specific constants and {kt} is the time-varying index dependent on year. {ax}, or more specifically eax, represents the general shape across age of the mortality schedule. The {bx} profile indicates how rapidly the central death rate declines in response to changes in {kt}. This implies that age groups change at its own constant exponential rate. This model prevents negative death rates, which is an advantage for forecasting. After extracting the 5-year age groups of central death rates for age x in year t (mx,t ) data from the Human Mortality Database, the {ax} values are found by calculating the average of ln(mx,t) from years 1933 to 1987. For the age-specific constant {bx} and index kt, we used both singular value decomposition and an alternative method mentioned in the original paper. We used the ax values for both methods mentioned to estimate {bx} and {kt}. It should be noted that we followed the following principles from the original paper: {kt} values sum up to 0 whereas {bx} values is up to unity.

For SVD, we arranged values of ln(mx,t) - ax into a matrix as an input to an SVD function in R, in which we obtained a left matrix and a right matrix. The left matrix, after dividing by its sum to make it sum to unity, was our estimates for {bx}. The right matrix, after multiplied by the sum of the left matrix, represented the {kt} estimates index obtained through SVD.

Lee-Carter also provides an alternative method to SVD in the original paper’s Appendix A; given that if SVD is not available. For research purposes, we followed the procedures of the alternative method. Here, kt values are estimated as approximately close to

equal the sum over age of the difference ln(mx,t) and ax. We continue on obtaining each bx through regression. The resulted index of {kt} we obtained from the alternative method has the same pattern as the one through SVD; however, we realized the alternative method produces significantly higher values than the SVD method. This is likely due to the difference in calculating assumptions, where the alternative method provides a more simplified assumptions. When comparing the age-specific constants, {ax} and {bx}, with the original papers constants, we noticed that the SVD results were nearly identical and decided to continue using the SVD results rather than the alternative method.

Finally, we had to re-estimate the {kt} index due to a discrepancy between the observed, or actual, number of deaths and the number of deaths calculated using the Lee-Carter model. Lee and Carter explains “this is because when fitting the log-transformed rates, the low death rates receive the same weight as the high death rates of the older ages, yet the contribute far less to the total deaths”. We re-estimated the {kt} index by keeping the same age-specific constants and using an iterative search by solving equation (2.6).

D(t) = ∑ [N(x,t) * eax+k(t)*bx] (2.6)

D(t) is the total number of deaths in year t and N(x,t) is population age distribution for age group. We used newton's method in R to solve this equation, giving us an accurate index of {kt}. Section III: Data analysis 3.1 Reduction factor model Forecasting mortality rates

After obtaining our reduction factor for each individual age x (Section 2.2), we were able to apply the reduction factor on the mortality rates of our base year, 1980. The generalized formula (2.3) was then used to derive forecasted mortality tables for each year.

Calculating life expectancy

After assembling the mortality tables for each year after the base year, we are able to calculate the forecasted life expectancy. We felt that comparing life expectancy with and without the reduction factor would provide an idea of how much the reduction factor affected calculations. All life expectancy calculations were done based on the equation (3.1).

ex = ∑ kPx (3.1)

For our example, we chose to find the life expectancy in 2014. 2014 was chosen because the data for 2014 is the latest set of data available so we can compare the forecasted life expectancy compared to the current life expectancy. We decided to calculate life expectancy for the reduction factor with two different assumptions. The first life expectancy calculation is done using only the forecasted 2014 mortality rates with equation (3.2). This is done to evaluate the accuracy of the reduction factor model.

kPx = P(x,2014) * P(x+1,2014) * P(x+2,2014) ... (3.2)

We also calculated life expectancy using a prospective method. When forecasting life expectancy of 2014 using the prospective method, each kPx was calculated using equation (3.3). This method is used to demonstrate the flaw with assuming mortality rates will not change beyond 2014.

kPx = P(x,2014) * P(x+1,2015) * P(x+2,2016) … (3.3) 3.2 Lee-Carter Model Modeling and forecasting the mortality index, k After re-estimating the {kt} index for years 1933 to 1987, we had to forecast the {kt} index for years beyond 1987. Lee and Carter explained “after carrying out the standard model identification procedures, we found that a random walk with drift describes k well“. We decided to follow this approach; by assuming that the {kt} followed a random walk, we could assume that the expected decrease in {kt} per year was the average decrease per year in years 1933 to 1987. One key benefit of modeling the mortality index {kt} as a random variable is the capability of obtaining confidence intervals for forecasts of mortality rates and life expectancy. (see Appendix C) Forecasts of death rates and life expectancy Now that we have the forecasted {kt} index, we can forecast future central death rates using equation (2.4). Converting these forecasted 5-year age group central death rates into single age mortality rates was a little more complicated. First, we converted the central death rates for 5-year age groups into mortality rates in 5 year intervals (5qx). Next was to interpolate these death rates to derive individual death rates (qx). While there are numerous ways to approach this, we decided to follow the Social Security Administration's approach using an osculatory formula. The SSA mentions how they used a “fourth degree osculatory formula developed by H.S. Beers to the natural logs of

the complements of 5qx”. While the mathematics behind the equation was beyond our scope, we are still able to apply and use it to calculate our forecasted individual rates. (see appendix D) Once we calculated these individual mortality rates, we could calculate the forecasted life expectancy under the Lee-Carter model. This was done using the same generalized equations (3.1), (3.2), and (3.3) from section 3.1. 3.3 Comparison Below is the calculated life expectancy for 2014 under their respective forecast assumptions and model. This calculation was done using only forecasted 2014 mortality rates.

We chose to compare life expectancy to observe how much forecasting mortality has an effect on calculations. We felt that comparing individual mortality rates would not display the overall effect, as the difference between the mortality rates was small. It seems the observed life expectancy of 2014 lies between the forecasted expectancies of the reduction factor model and the Lee Carter model. The reduction factor model has a closer estimate compared to the Lee Carter model indicating that it may be more accurate. This may also be due to the use of more recent data in the reduction factor. Another interesting observation is the consistent underestimation from the reduction factor and overestimation from the Lee Carter Model. Now that we have an understanding of the accuracy of the two models, we can observe the impact of using poor mortality assumptions. Below is the life expectancy calculated using the prospective method mentioned in equation (3.2)

Now we can observe just how much forecasting can have on calculations based on mortality. We can see that under the reduction factor, earlier ages benefit from the reduction factor compared to the later ages. Surprisingly, the reduction factor and Lee-Carter estimates for ages 40 to 70 are very similar but diverge before and after this range. Once again the Lee Carter model has higher expectations compared to the reduction factor. Section IV: Example - Pension Plan Pricing 4.1 Assumption

We decided to take a look at the longevity risk for both defined contribution and defined benefit plans. Here we assume that interest rates, salary rates, and fund accrual rates remain constant throughout the individual’s life. We also assume that death is the only exit from the company, contributions and benefits are paid out annually, and that salary increases once a year. While it would have been more accurate to use monthly contributions and monthly benefits, we felt that using the annual equivalents would still illustrate the effects of the longevity risk.

For defined contribution plans, we created a function in R that takes the following inputs: age, retirement age, target replacement ratio, interest, current salary, salary rate, and a kPx matrix. This function returns the expected liability at time 0 and the required contribution as a percentage of salary. Since the target replacement ratio is the annual benefit of the plan divided by the final salary, we can find the annual benefit by multiplying the target replacement ratio with the final salary. The final salary was forecasted using the current salary and the salary rate. Now that we have the annual benefit required to satisfy the target replacement ratio, we can calculate the expected liability at time 0 by multiplying the annual benefit with a deferred life annuity. Finally, to find the required contribution as a percentage of salary, we need to calculate a term annuity to represent the expected value of the fund at time 0. This was done with standard term annuity calculations but with the discount factor as (1+ salary rate)*(1 +

interest rate)-1 instead of (1 + interest rate)-1. In other words, the calculation is equivalent of a geometrically increasing annuity. Once we have this term annuity, we can calculate the percentage of salary needed each year by taking the expected liability at time 0, dividing by the term annuity, and dividing again by the current salary.

Similarly, we also created a function for defined benefit plans which uses the following inputs: age, number of years worked, retirement age, target replacement ratio, interest rate, current salary, salary rate, and a kPx matrix. This function returns the expected liability at time 0, the necessary accrual rate for each year of service, and the normal contribution required that year to fund this plan. The calculation for the expected liability is the same as the defined contribution plans. Next, we assume that the annual benefit would be equal to the number of years of service times the accrual rate times the 3-year final average salary. We can then find the necessary accrual rate with the target replacement ratio times the final salary divided by the final average divided by the number of years of service. Finally, since projected unit crediting was used, we can calculate the normal contribution by taking the expected liability at time 0 and dividing it by the number of years of service.

4.2 Defined Contribution/ Defined Benefit When deciding on which examples to analyze and compare, we chose ages 25 and 30 to generate an idea on how the longevity risk affects financially funding a pension plan. We chose these ages because we feel most workers enter their pension plans around these ages; however, if one wants to compare the longevity risk at other ages or with other variables, they may refer to appendix D for the base R code. We decided to analyze both defined contribution and defined benefit plans to see what kind of effects mortality assumptions would have. We chose to look at both plans to see if lower mortality had a larger effect on one plan compared to the other. One key difference to keep in mind between the two plans is how defined benefit plans are reliant on the number of years already worked. Defined benefit plans are funded by making a yearly contribution to a fund which promises to pay a yearly benefit, usually based on the number of years worked multiplied by an accrual rate multiplied by an measure of salary. Defined contribution plans are funded by making a yearly contribution based on a percentage of current salary. At retirement, the fund is used to pay out yearly benefits. For both plans, we set the retirement age at age 70, target replacement ratio at 0.8, annual effective interest at 5%, current salary at $100000, and the salary increase rate at 2% per year. With a target replacement ratio, the two plans become very similar as we can forecast the final salary and multiply it to the target replacement ratio to calculate the required annual benefit. In both plans, this calculated required annual benefit would be the same due to having the same current salary, salary scale, and target replacement ratio.

We will begin by comparing the defined contribution plan calculated using the reduction factor, Lee-Carter model, and 2014 observed mortality rates.

We can quickly notice that assuming mortality will remain the same for years beyond 2014 will cause defined contribution plans to be heavily underfunded. The difference between forecasting using the reduction factor and simply using the 2014 mortality table is almost $100000, or a year's salary. Furthermore, the reduction factor forecasts around a 4% higher salary contribution necessary to fund this plan while the Lee-Carter model has around a 3% higher salary contribution required. This increase in expected liability demonstrates how lower mortality rates will have a considerable effect on the necessary funds needed.

We see a similar effect in defined benefit plans. For this example, we assumed an individual age 30 has worked for the company for 5 years already. The required accrual rate is constant regardless of mortality assumption as it is calculated using the target replacement ratio, final salary, and number of years already worked.

Once again, we notice large differences between the expected liability and normal contribution required. It seems like the reduction factor model is more conservative and has lower expected mortality compared to the Lee-Carter model. This is noticeable as both defined contribution and defined benefit plans under the reduction factor have higher expected liability than the same calculation under the Lee-Carter model.

Section V: Conclusion While the purpose of this paper is not to compare and determine what is the best method of forecasting mortality rates, there are key aspects for each model that should be noted. The reduction factor is conceptually easy to apply and understand. This allows for fast calculations and could be used as an introduction to mortality trends in a classroom environment. One difficulty we noticed with the reduction factor is how accuracy is reliant on both the base year and size of data. Choosing a closer base year would in theory increase accuracy; however, this also forces the size of the data to be smaller which reduces accuracy. Furthermore, the reduction factor model does not allow room for error like the Lee-Carter model does. Due to the kt index being a random variable, confidence intervals are able to be calculated providing a better idea or range of how mortality should be in the future. While the mathematical concepts behind the Lee-Carter model are more difficult to understand, it incorporates more aspects including ideas from previous models. This makes the Lee-Carter a more powerful forecasting model in comparison with the reduction factor model. One component we disliked about the Lee-Carter model was the use of 5-year age groups and central death rates instead of individual age mortality rates. Although Lee-Carter report intends to present its forecasts in a way for readers to be able “to calculate life tables functions and their confidence intervals for each year of the forecast”, the 5-year age groups arrangement made it more difficult for us to understand and apply the Lee-Carter model in our calculations. Through the reduction factor model, comparisons of life expectancies with the application of reduction factors and no reduction factors to the mortality rates were made. This is particularly important as we could observe significant effects on expected future lifetimes, especially on earlier ages, when reduction factors are applied. As for the Lee-Carter model, kt values were forecasted by following a random walk before we forecast the central death rates using the osculatory formula. Forecasting kt following a random walk indicates that the mortality of each age group declines exponentially at its own age-specific rate. After finding and calculating meaningful measures, it becomes apparent mortality trends have a heavy influence on pricing mortality related products. The large differences in life expectancy, expected liability for pension, and required contributions demonstrates the necessity of forecasting mortality. Assuming that mortality will remain the same would result in heavily underfunded pension plans. This conclusion is based on our analysis of historical data and key assumptions within each model. While our research material is derived from reputable sources, there may be slight differences between other predictions and opinions. The accuracy of our predictions could be improved using newer data; however, we chose to use 1980-2000

for the reduction factor and 1933-1987 for the Lee-Carter model as an attempt to replicate and compare the original papers to our own research. For example, The use of data beyond year 2000 would likely increase the accuracy of forecasts. With advancements in computational power, it may even be possible to build a superior model capable of updating or adjusting itself with new data. It is important to understand the risks involved when we predict the future using historical data.

References

1. Lee, R. D., & Carter, L. R. (Sep. 1992). Modeling and Forecasting U. S. Mortality. Journal of the American Statistical Association, 87, 659-671. Retrieved September 23, 2016, from http://www.jstor.org/stable/2290201

2. Dickson, D. C., Hardy, M., & Waters, H. R. (2013). Actuarial mathematics for life

contingent risks (2nd ed.). New York: Cambridge University Press.

3. Life Table Methods. (n.d.). https://www.ssa.gov/oact/NOTES/as116/as116_IV.html

4. The Lee-Carter Model. (n.d.).http://data.princeton.edu/eco572/LeeCarter.html

5. Renshaw, A.e., and S. Haberman. "On the forecasting of mortality reduction

factors." Insurance: Mathematics and Economics 32, no. 3 (2003): 379-401. http://www.cassknowledge.com/sites/default/files/article-attachments/402~~stevenhaberman_on_the_forecasting_of_mortality_reduction_factors_-_2003.pdf.

6. Alho, Juha M. "“The Lee-Carter Method for Forecasting Mortality, with Various

Extensions and Applications“, Ronald Lee, January 2000." North American Actuarial Journal 4, no. 1 (2000): 91-93. https://www.soa.org/library/journals/north-american-actuarial-journal/2000/january/naaj0001_5.pdf.

Appendix Appendix A – Makeham’s Law and Gompertz Law R-code

>#Makeham’s Law >library(fmsb) >fitGM(,mortality$𝜇x) >#Gompertz Law > fit = lm(ln.ux.~Age)

Appendix B - Reduction factors, rx, based on American male and female mortality in 10-year intervals.

Age Reduction Factor Adjusted Reduction Factor

0 0.971426175 0.971426175

10 0.97794191 0.97794191

20 0.988774066 0.988774066

30 0.989633407 0.989633407

40 0.998606422 0.998606422

50 0.984019757 0.984019757

60 0.984853334 0.984853334

70 0.987694771 0.987694771

80 0.992938246 0.992938246

90 0.998629645 0.998629645

100 1.002805107 1

110 1 1

Appendix C - kt forecasting

Year K Year K

1987 -9.82357917 2027 -24.17644626

1988 -10.18240085 2028 -24.53526794

1989 -10.54122252 2029 -24.89408962

1990 -10.9000442 2030 -25.2529113

1991 -11.25886588 2031 -25.61173297

1992 -11.61768756 2032 -25.97055465

1993 -11.97650923 2033 -26.32937633

1994 -12.33533091 2034 -26.68819801

1995 -12.69415259 2035 -27.04701968

1996 -13.05297427 2036 -27.40584136

1997 -13.41179594 2037 -27.76466304

1998 -13.77061762 2038 -28.12348472

1999 -14.1294393 2039 -28.48230639

2000 -14.48826098 2040 -28.84112807

2001 -14.84708265 2041 -29.19994975

2002 -15.20590433 2042 -29.55877142

2003 -15.56472601 2043 -29.9175931

2004 -15.92354768 2044 -30.27641478

2005 -16.28236936 2045 -30.63523646

2006 -16.64119104 2046 -30.99405813

2007 -17.00001272 2047 -31.35287981

2008 -17.35883439 2048 -31.71170149

2009 -17.71765607 2049 -32.07052317

2010 -18.07647775 2050 -32.42934484

2011 -18.43529943 2051 -32.78816652

2012 -18.7941211 2052 -33.1469882

2013 -19.15294278 2053 -33.50580988

2014 -19.51176446 2054 -33.86463155

2015 -19.87058614 2055 -34.22345323

2016 -20.22940781 2056 -34.58227491

2017 -20.58822949 2057 -34.94109658

2018 -20.94705117 2058 -35.29991826

2019 -21.30587285 2059 -35.65873994

2020 -21.66469452 2060 -36.01756162

2021 -22.0235162 2061 -36.37638329

2022 -22.38233788 2062 -36.73520497

2023 -22.74115955 2063 -37.09402665

2024 -23.09998123 2064 -37.45284833

2025 -23.45880291 2065 -37.81167

2026 -23.81762459

Appendix D - qx using Lee Carter Model

Age 1987 2020 2065

5 0.00034 0.00014 m 4.73892E-05

10 0.00012 0.00002 -1.21562E-05

15 0.00054 0.00029 0.000124812

20 0.00091 0.00049 0.000214835

25 0.00095 0.00048 0.000190811

30 0.00102 0.00050 0.000186797

35 0.00133 0.00065 0.000240228

40 0.00206 0.00108 0.000447229

45 0.00338 0.00196 0.000932893

50 0.00553 0.00343 0.001786457

55 0.00867 0.00578 0.003310152

60 0.01349 0.00958 0.005983048

65 0.01963 0.01413 0.008961816

70 0.02862 0.02064 0.013102897

75 0.04116 0.02924 0.018074499

80 0.06067 0.04500 0.029395536

85 0.08653 0.06820 0.048383359

90 0.11708 0.09536 0.070411664

95 0.14611 0.09153 0.068031993

100 0.18816 0.03924 0.029158566

105 0.24592 0.05128 0.03810908

110 0.32141 0.06702 0.049807043

Appendix E - Pension Calculation R Code