Understanding, Measuring & Managing Longevity Risk. Longevity Modelling Technical Paper

Transcription

1 Longevity Modelling Technical Paper

2 Table of Contents Table of Figures and Tables Introduction The Importance of Understanding Longevity Risk Deterministic vs. Stochastic Models Explaining the Concepts Defining Longevity Risk Pooling Risk Catastrophic Mortality Risk Longevity Risk Population dynamics: age, period and cohort effects Age Effect Period Effect Cohort Effect Deterministic Projections Explaining CMI s Deterministic Cohort Projections Short Cohort Projection Medium Cohort Projection Long Cohort Projection Implication of Deterministic Mortality Projections Future Cash Flows Present Value and Duration Introduction to Eight Stochastic Mortality Models Lee Carter model Lee Carter Extension APC Model and P Splines Two Factor CBD Model Three Factor CBD Model Three factor CBD Model Extension Four factor CBD Model Methodology Evaluation Criteria MSE Test January

3 4.1.2 Sign Test Outliers Test Discussion of Results MSE Test and Sign Test Outlier Test Compare Number of Outliers Fan Chart Analysis Longevity Risk Analysis based on Two-Factor CBD Model Stochastic Cash Flow & Value at Longevity Risk (VaLR) Stochastic Cash Flow & Deterministic Cash Flow Managing Longevity Risk Risk Components in DB Pension Fund VaR Road Map The Efficient Frontier LE Conclusions Appendix January

4 Table of Figures and Tables Figure 1.1: Life Expectancy for a 65 year old individual from Figure 1.2: Life expectancy assumptions reported in 2006 Males Retiring at age 60 on the accounting date (LCP 2007)... 8 Figure 1.3: Life expectancy assumptions reported in 2006 Males Retiring at age 60 in the future (LCP 2007)... 9 Figure 2.1: Pooling Risk Figure 2.2: Catastrophic Mortality Risk Figure 2.3: Longevity Risk Figure 2.4: Age Effect Figure 2.5: Period Effect Figure 2.6: Cohort Effect for England and Wales males by age group and year Figure 2.7: Cohort Effect for England and Wales females by age group and year Figure 2.8: 92 Series Mortality Improvement Projection Figure 2.9: Short Cohort Projection Figure 2.10: Medium Cohort Projection Figure 2.11: Long Cohort Projection Figure 2.12: Cashflows under ten mortality projection assumptions Figure 4.1: Mortality Rates for Age 65 in Period Figure 4.2: Mortality Rates for Age 65 in Period Figure 5.1: Lee Carter model in 40 years sample test Figure 5.2: Lee Carter extension 40 years sample test Figure 5.3: Two factor CBD model in 40 years sample test Figure 5.4: Three factor CBD model 40 years sample test Figure 5.5: Lee Carter 20 years sample test Figure 5.6: Lee Carter extension 20 years sample test Figure 5.7: Two factor CBD model 20 years sample test Figure5.8: Three factor CBD model 20 years sample test Figure 6.1: Cash Flow Structures (Two Factor CBD Model vs. CMI Deterministic Assumptions) Figure 6.2: Risk Components in DB Pension Funds Figure 6.3: Comparison of VaR Road Map Figure 6.4: Increase in VaR due to longevity risk Figure 6.5: Change in Efficient frontier by availability of longevity risk hedging products January

5 Table 2.1: PV and duration of liabilities under various deterministic mortality projections Table 2.2: Change in PV and duration of liabilities under various deterministic mortality projections Table 3.1: Formula for the stochastic mortality models Table 5.1: Overall Fitting Statistics Table 5.2: projection: Analysis of outliers by year Table 5.3: projection: Analysis of outliers by year Table 6.1: Key Pension Metrics based on Two Factor CBD Model Table 6.2: Comparison of Present Value derived from Two Factor CBD model and Cohort Projections Table 6.3: Sensitivity to 1 year rise in life expectancy for individual member Table 6.4: Age and salary distribution of pension Table 6.5: LE01 Table January

6 1.0 Introduction In a defined benefit (DB) pension scheme, benefit levels are determined in advance of retirement and guaranteed irrespective of how underlying funds are invested (Barr & Diamond, 2006; Barr, 2006). Benefits are usually calculated by accounting for the number of years an individual was employed to the salary level of that employee during part or all of the time employed (Barr & Diamond, 2006). However, the accumulated pension liabilities for DB schemes in the UK now represent a significant financial burden for many corporations due to the following developments: the decline in equity markets between 2001 and 2003, late recognition of significant increases in life expectancy and increasingly stringent pension regulations and accounting standard This paper starts by discussing the difference between traditional deterministic mortality assumptions and stochastic mortality modelling, arguing that stochastic modelling is needed to quantify and manage longevity risk. It then explains the concepts surrounding longevity, defining longevity risk and discussing age, period and cohort effects. The deterministic mortality assumptions produced by the Continuous Mortality Investigation (CMI) are then explained and analysed. This is followed by analysis of the stochastic models that were researched and tested, followed by a discussion of the research methodology and the data used in the tests. After presenting and analyzing the results from the model testing, this paper concludes that the Two Factor Cairns, Blake and Dowd (CBD) stochastic mortality model is the best way for pension funds to incorporate longevity risk into an overall risk management framework. Finally some possible applications based on the findings are discussed. This paper is the first in a series of papers on the topic of longevity risk. Subsequent papers will focus on how to integrate longevity risk into liability risk analysis and then overall Asset Liability Management (ALM), study the effectiveness of various longevity products in the market today (e.g. JPMorgan q forward, Credit Suisse longevity swaps) and research other potential ways of managing and transferring longevity risk (e.g. buy out solutions). 1.1 The Importance of Understanding Longevity Risk Unlike financial risks such as investment risk, interest rate and inflation risk, the analysis and understanding of longevity risk has only recently been examined by pension schemes. Longevity risk represents the risk that members of some reference population might live longer on average than January

7 anticipated (Blake et al, 2006). Until recently, longevity has been treated as a risk to which pension schemes are unable to hedge or reduce their exposure. However, the research findings presented in this paper illustrate that this is no longer the case. David Norgrove, Chairman of The Pensions Regulator (TPR) commented during the 2006 UK Pensions and Investment Summit that each year of extra life adds about 3 4% to pension scheme liabilities so, with 800 billion of liabilities across UK pension schemes, getting it wrong now could mean sharp rises in pension costs in the future (TPR 2006 Norgrove speech). It is crucial for pension schemes to understand their longevity risk exposure, and take action today to meet these future liabilities. People are living longer today than they were 20 years ago. Figure 1.1 shows that life expectancy for an individual living in England and Wales (E&W) improved at an increasing rate from 1920 through to While females have historically lived longer than males, Figure 1.1 also demonstrates that the gap between the life expectancy of a male and female aged 65 is now closing. Based on data from 1983 to 2003, the life expectancy for a 65 year old male increased on average 59 days per year while the life expectancy for a female of the same age increased at a rate of 37 days per year. Figure 1.1: Life Expectancy for a 65 year old individual from Growth in Life Expectancy 65 years old in England & Wales life expectancy YEARS OLD MALE 65 YEARS OLD FEMALE Source: The Human Mortality Database One of the reasons for the sharp increase in life expectancy in England and Wales in recent years is the cohort effect. The cohort effect, as defined by the Government Actuary Department (GAD), is January

8 a higher than average rate of improvement of generations born between 1925 and It is not yet understood precisely why the members of the generation born about 1931 have been enjoying so much lower death rates throughout adult life than the preceding generation (GAD, Report 1995). While further research is needed to determine how long the cohort effect will last and to what extent it will influence the life expectancy of the entire population, it is undisputed that the cohort effect has increased the life expectancy level for individuals born in England & Wales. As a result, future generations will continue to experience improvements in life expectancy, although at a lower rate (The cohort effect is discussed in more details in section ). Mortality assumptions used for pension valuations and investment strategies vary widely and a significant number of schemes have underestimated the life expectancy of plan members. Professor Angus MacDonald, chairman of the Continuous Mortality Investigation (CMI), which carries out research into mortality and morbidity experience, argues that recent trends suggest that the medium cohort projection may now be understating likely future improvements (TPR 2006 Norgrove speech). However, Figure 1.2 and Figure 1.3, show that only 12% of FTSE 100 companies calculate the mortality rates of scheme members who are currently 60 years old using PA92 and 3% of FTSE 100 companies use PA92 table to project future mortality. 1 This means that most FTSE 100 pension schemes are using mortality assumptions that are underestimating the life expectancy of scheme members. Figure 1.2: Life expectancy assumptions reported in 2006 Males Retiring at age 60 on the accounting date (LCP 2007) 1 76 out of 93 out of the FTSE 100 companies with defined benefit schemes provided sufficient information for us to derive basic mortality statistics, specifically the life expectancy of a male aged 60 in the UK. January

9 Figure 1.3: Life expectancy assumptions reported in 2006 Males Retiring at age 60 in the future (LCP 2007) During the early years of the 21 st century, a large number of DB pension schemes turned pension fund surpluses into deficits. The main reasons for this were a crash in equity markets, falling interest rates and increased life expectancy. The average pension deficit for a FTSE 350 company in 2001 averaged 4.4million, while by 2004 the average had increased to 254 million (Gupta, 2006). The main risks that impact the liabilities of DB pension schemes are interest rate risk, inflation risk and longevity risk. Pension schemes are increasingly using a Liability Driven Investment (LDI) strategy as this allows the effective hedging of interest rate risk and inflation risk. Consequently, the biggest unhedged risk facing DB pension schemes is longevity risk. It is important to note that the effectiveness of interest rate and inflation hedging is based on the accuracy of the future cash flow forecasts, which, in turn are heavily influenced by the mortality assumptions employed by a pension scheme. Therefore, understanding longevity risk will also allow schemes to improve the effectiveness of hedging techniques used to mitigate interest rate and inflation risks while simultaneously reducing the overall risk in a scheme. Finally, the regulatory framework in the UK has become increasingly strict in recent years. The Pensions Act 2004, the Pensions Regulator and the Pension Protection Fund (PPF) were introduced to encourage pension schemes to reduce their risk exposure. For example, FRS17 states that sponsor companies are required to disclose the assets and liabilities of the company s pension scheme on the sponsor s balance sheet. Before any forecasts about longevity risk can be made, it is important to understand the different ways of modelling the risks. January

10 1.2 Deterministic vs. Stochastic Models Traditional pension scheme actuaries use deterministic mortality tables to calculate and value pension liabilities and to test the sensitivity of liabilities to changes in mortality rates by reasonably shifting the future mortality improvement rate up or down, or by adding one year of life expectancy to all ages in a scheme to observe the magnitude of the change. But these deterministic tables, such as the PA92 table, short (medium, long) cohort projection produced by the CMI can only provide scenario tests, which have three major drawbacks. First, the traditional approach cannot show the full distribution of possible future mortality rates. Second, the probability that one specific scenario happens cannot be estimated. Finally, only limited cases are covered by the scenario tests and the overall longevity risk is not accurately quantified. Therefore, to fully quantify longevity risk simply using deterministic mortality tables is not sufficient. Stochastic mortality models provide a more accurate representation of the full distribution of possible future mortality rates. This allows pension schemes to see not only the range of projected future mortality rates, but also the probability of occurrence for each scenario. These features provide the pension schemes with the information required to derive the Value at Longevity Risk (VaLR), which is the increase in the value of a scheme s liabilities that is a result from lower than expected future mortality rates. VaLR is the only way to incorporate longevity risk into a comprehensive liability risk analysis, which is the cornerstone of integrating longevity risk into the overall asset liability management of a portfolio. January

11 2.0 Explaining the Concepts This section defines the key terms necessary to understand longevity risk and the main variables used during the modelling phase. It then discusses mortality risk and its components and explains the three key population dynamics considered during the modelling phase. Finally, it explains and analyses the deterministic cohort projections produced by the Continuous Mortality Investigation (CMI) and illustrates the impact of how changes in these projections impact the value of pension scheme liabilities. 2.1 Defining Longevity Risk The randomness of mortality rates is recognised by pension schemes because it causes fluctuations in their liability values. This risk is defined as mortality risk. Mortality risk can be subdivided into three categories: pooling risk, catastrophic risk and longevity risk Pooling Risk Figure 2.1: Pooling Risk Pooling Risk mortality rates for 22 years old male in E&W mortality rate realised mortality rates projected mortality rates Source: The Human Mortality Database Pooling risk represents the random fluctuations of realised mortality rates around the expected trend line. Figure 2.1 shows that even though the mortality trend line is correctly projected, the actual mortality experience deviates from the projection because people randomly die earlier or later than expected. This risk is equivalent to white noise that can be eliminated by pooling a large number of participants into the pension scheme. This is tantamount to reducing random fluctuations January

12 by increasing the size of the scheme such that the total pension cash outflows should equal the projected amount (ceteris paribus). Therefore, pooling risk is smaller for large pension schemes, and vice versa Catastrophic Mortality Risk Catastrophic mortality risk is defined as sudden and unexpected spikes in mortality and can be a result of human made events such as wars, and/or natural disasters like earthquakes, tsunamis or pandemics. These events occur at random times and are of an unknown magnitude. Figure 2.2 shows the spike in mortality as a result of World War II and is widely considered the last known spike which was a direct result of human initiated events. Figure 2.2: Catastrophic Mortality Risk Catastrophic Mortality Risk mortality rates for 22 years old male in E &W mortality rate realised mortality rates projected mortality rates Source: The Human Mortality Database Catastrophic mortality risk is a greater concern for life insurance providers than pension schemes because in the event of a catastrophic event, pension schemes are only responsible for paying embedded death benefits which usually represent a small proportion of the overall liability in the scheme. January

13 2.1.3 Longevity Risk There are two types of longevity risk: systematic longevity risk and longevity basis risk. Systematic longevity risk refers to the portion of the risk that is non diversifiable and longevity basis risk is defined as the difference in mortality rates of the reference population against the actual mortality rates of the population analysed. From a modelling perspective, the longevity basis risk is implied by the difference between the mortality results from the calibration data against the mortality rates specific to the scheme. In Figure 2.3, systematic longevity risk is illustrated by comparing the projected mortality rates of the entire population (the solid red line) to the realized mortality rates of the whole population (the solid purple line). The deviation of the projected mortality trend from the realized mortality rates cannot be explained by random fluctuations. This means that the actual mortality rates are systematically lower than the projected mortality rates and this deviance cannot be minimized by increasing the size of the pool. Figure 2.3: Longevity Risk Longevity Risk mortality rate Calendar Year sector 1 sector 2 sector 3 sector 4 realised mortality rates of 65 years old male pensioners projected mortality rates of 65 years old male pensioners Source: The Human Mortality Database In the same graph, longevity basis risk is exemplified with four dotted lines representing different sectors (e.g. mining, manufacturing, IT and finance). This is important because people working in different sectors (or geographical areas, socio economic groups etc.) have their own mortality January

14 trends. The difference between sector specific mortality rates and the mortality rates of the whole population can be significant enough to create what is called longevity basis risk. In theory, if the general population data is used as the base for mortality projection, then pension schemes can correct the longevity basis risk by pooling a variety of sectors into the scheme. However, in practice, most pension schemes are heavily weighted toward one sector of the population. So longevity basis risk is greater for small schemes concentrating in a particular sector and/or location if general population data is used. Therefore, with respect to DB pension schemes pooling risk and catastrophic mortality risk are less of a risk than longevity risk. Without effective management of a scheme s assets and liabilities, longevity risk has the potential to significantly weaken the funding level of a DB pension scheme. 2.2 Population dynamics: age, period and cohort effects There are three main factors that influence the historical mortality trend; namely age effect, period effect and cohort effect. In order to forecast future mortality rates with any accuracy, the stochastic models should include these three effects Age Effect Age effect describes the relationship between age and mortality rates. The probability of dying increases as people become older and these mortality trends exhibit exponential behaviour. Figure 2.4 shows that between the ages 20 and 38, mortality rates increase slowly from to , a 68% increase in 18 years. However, for the following 20 years (ages 39 59) mortality rates increase by over 600% to a level of The most dramatic increases in mortality rates occur over the age of 60. For Example, an 85 year old male has a mortality rate of But in the same year, the probability of death for a male 20 years younger is eight times less. January

15 Figure 2.4: Age Effect Age Effect mortality rates for male aged 20 to 100 in E &W in mortality rate Age realised mortality rates for male aged 20 to 100 in year 2003 Source: The Human Mortality Database Period Effect The period effect illustrates the impact that time specific events and changes have in the number of deaths experienced in a given year. The time period can span one year or multiple years. The period effect includes contemporary factors, such as the standard of living of the population, availability of health services, and critical weather conditions (Olivieri, 2007). Figure 2.5 shows a decreasing trend in realized mortality rates for 65 year old males between 1920 and 2003 which is a result of increased medical knowledge and improvements in technology. Figure 2.5: Period Effect Period Effect mortality rates for male aged 65 in 1920 to 2003 in E &W mortality rate realised mortality rates for 65 years old male Source: The Human Mortality Database January

16 2.2.3 Cohort Effect The cohort effect describes how being born in a specific year of era can influence the rate of mortality for a cohort. (LifeMetrics, JPMorgan). The cohort effect cites historical factors such as changes in generational eating habits, improvements in healthcare and change in smoking habits (Olivieri, 2007). As noted earlier in this paper, individuals born in the UK between 1925 and 1945 experienced a higher improvement in life expectancy than people born before and after this generation. Figure 2.6 and Figure 2.7 show the cohort effect for males and females in England and Wales, and analyse the mortality improvements for eight age groups between 1975 and Figure 2.6: Cohort Effect for England and Wales males by age group and year Cohort Effect male mortality improvement in E&W annual mortality improvement 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% age group calendar year Source: Office of National Statistics January

17 Figure 2.7: Cohort Effect for England and Wales females by age group and year Cohort Effect female mortality improvement in E&W 5.0% annual mortality improvement 4.0% 3.0% 2.0% 1.0% 0.0% age group calendar year Source: Office of National Statistics The arrows in both graphs highlight the observation that both males and females between the ages of 40 to 49 in 1975 have a higher mortality improvement than any other age group in the same year. This phenomenon continues as this group moves into the age groups in 1980, the age groups in 1985 and the age group in This 1936 cohort effect is more significant for males. It can be seen that year old males in 1975 experienced an approximately 3% lower mortality rate on average than year old males in 1974 (i.e. people born 1 year earlier) and when this cohort reached years in 2000, their average annual mortality improvement rate is over 4.5%. The exact reason for this sudden jump in the mortality improvement rate is not clear. According to Willets (2004), it is highly likely that the cohort effect has been caused by a number of different factors in combination, including the prevalence of smoking; the effect of World War II on the and previous generations; the diet in post war Britain; NHS and welfare reform, and the three baby booms during the period from 1920 to mid 1960 (Willets, 2004). 2.3 Deterministic Projections This section discusses how deterministic projections are used by pension schemes to value a scheme s liabilities using CMI s deterministic cohort projections and cohort projections with underpins. Underpins represent the minimum improvement factors applied to mortality rates. January

18 2.3.1 Explaining CMI s Deterministic Cohort Projections Similar to the cohort effect discussed above, the CMI has observed that the birth cohort from 1910 to 1942 in the UK has experienced a much faster rate of improvement than people born outside this period (CMI, 2002). The CMI has produced a set of mortality projections to include this 1926 cohort effect. Figures demonstrate a set of mortality projections published by the CMI in These graphs show the projections of future mortality improvements ranging from 0% 0.6% (light green) to larger than 4.2% (dark purple). The data used in the analysis is from 1984 to From 2000 onwards the improvement factors are based on CMI projections. Figure 2.8: 92 Series Mortality Improvement Projection Source: CMI Working Paper 1. Figure 2.8 shows that the 1926 cohort effect is significant from 1984 to Compared with other generations, individuals aged 61 in 1987 show a much smaller mortality rate. However, the 92 series projection from 2000 onwards does not capture the 1926 cohort effect, and as a result the CMI revised its deterministic mortality projections. January

19 Figure 2.9, Figure 2.10 and Figure 2.11 illustrate the revised set of mortality projections, namely Short Cohort (SC), Medium Cohort (MC) and Long Cohort (LC) projections. To make the comparison meaningful, the same historical dataset is used, but from 2000 onwards the improvement factors are replaced by the Short, Medium and Long Cohort projections. The revised projections are closer to actual data than the 92 Series. However, these projections represent different assumptions / predictions on how this cohort behaves in the future Short Cohort Projection As shown by Figure 2.9, the Short Cohort (SC) projection assumes that the 1926 cohort effect will continue for ten years after 2000 but with a diminishing impact by age band, which is shown graphically by the reduced width. We observe that the age band with the higher than normal mortality improvement rate has narrowed since 2000 and finally disappears by the end of the cohort period in After that, the mortality improvement rate is assumed to be the same as that projected by the original 92 Series. Figure 2.9: Short Cohort Projection Source: CMI Working Paper 1. January

20 2.3.3 Medium Cohort Projection The Medium Cohort projection in Figure 2.10 shows a similar trend as the Short Cohort projection, but the important difference is that the annual reduction of the cohort effect by age is smaller in the Medium Cohort projection. This means that the cohort effect is predicted to last until 2020 as opposed to 2010, as is the case in the Short Cohort projection. Figure 2.10: Medium Cohort Projection Source: CMI Working Paper Long Cohort Projection The Long Cohort Projection expands the cohort effect until 2040 and under this projection people are assumed to live much longer than assumed by other projections discussed previously. January

21 Figure 2.11: Long Cohort Projection Source: CMI Working Paper 1. In 2007, the CMI released a library of mortality projections. Within this library, one type of projection imposes minimum improvement factors on cohort projections. This type of projection has slowly gained popularity in pension fund valuation as the minimum improvement factors have a significant impact on people s projected life expectancy, especially the elderly. Mortality improvement factors projected in 92 Series and Cohort projections for ages 90 to 110 are less than 1% and zero for those above 110 year olds. The minimum improvement factors are applied by taking the maximum difference between the projected improvement rate and the minimum value in the mortality table. For example, if we apply a 1% minimum improvement factor to the Short Cohort projection, then values that are less than a 1% mortality improvement are replaced by the minimum 1% level; the produced table is called Short Cohort projection with 1% underpin. A higher mortality improvement rate indicates a lower mortality rate and longer life expectancy. Therefore, using different mortality projections impacts the shape and timing of pension benefit January

22 cash flows that reflects on the duration and value of liabilities. The next section discusses the impact of these deterministic mortality assumptions on pension liability valuation. 2.4 Implication of Deterministic Mortality Projections The purpose of mortality projections for pension funds is to determine the value of their liabilities. Various deterministic mortality projections represent different assumptions on future mortality, and in turn produce different cash flow structures, present value and duration of the liabilities of a pension scheme. This section demonstrates what the implications of the deterministic mortality projections discussed are to the liabilities of pension funds by applying these projections to a sample pension fund Future Cash Flows Figure 2.12 shows the difference between cashflows generated by different deterministic mortality projections, including the original 92 Series projection, cohort projections and cohort projections with 1% and 2% underpin. Figure 2.12: Cashflows under ten mortality projection assumptions Projected Pension Fund Cashflow Under 10 different deterministic mortality assumption pension payout ( million) sc 2017 mc 2027 lc 2037 sc 1% 2047 sc2% mc 1% 2077 mc 2% 2087 lc 1% 2097 lc 2% Source: Redington Partners, LLP: Proprietary Research January

23 Figure 2.12 shows that for the first 10 years all projection methods show similar results. However, the cash flows are diverging at an increasing rate. Different projection assumptions result in the presumed cash flows reaching their peaks at different years with the largest cash outflows occurring in later years for models that include aggressive mortality improvement assumptions. This is because under mortality assumptions with higher improvement factors, all people tend to live longer and therefore the period with the greatest number of pensioners emerges in later years. As a result, the peak is much lower and earlier than for pessimistic mortality projections, such as 92 Series. Other projections such as Short and Medium Cohort compare with Long Cohort, while other more extreme projections such as Long Cohort with 1% Underpin and Long Cohort with 2% Underpin show the highest cash flows in later years. As well as having different peak times, the rate that cash flows decrease also varies according to the underlying mortality improvement assumption. Not surprisingly, cohort projections with a 2% underpin show the slowest decreasing cash flow rate because under the 2% underpin projection there are more pensioners in the scheme for each calendar year than under any other projection assumption. Finally we observe that, under most mortality assumptions, the cash flow projections approach zero by 2087 which is the time when a 20 years old member reaches his 100 th birthday. However, for those mortality assumptions with underpins, especially those with 2% underpin, pension payouts are still relatively high and do not fall until 2101 when the mortality table ends Present Value and Duration The selection of a mortality projection method has implications when determining the present value (PV) of a scheme s liabilities as well as their duration because the present value and the duration are directly linked to the cash flow dynamics. This is important to highlight because of the significant impact that PV and duration have when establishing both an investment strategy and a hedging strategy. Table 2.1 and Table 2.2 show the present value and duration of the cashflows under the ten mortality projections. The projected benefits are linked to the RPI (Retail Prices Index) for active and deferred January

24 participants and to the LPI (Limited Price Index) for retirees. The table shows that changing the mortality projections assumptions can have significant effects on PV and duration. With respect to total liabilities, the difference in PV between the smallest mortality improvement projection ( 92 Series) and the largest improvement (Long Cohort projection with 2% underpin) is 12.8% ( 988m to 1,115m). Other 2% underpin projections and Long Cohort with 1% Underpin also show almost 10% increases in the value of the liabilities; while the Short Cohort, Medium Cohort with 1% underpin projections increased the value of the liabilities less than 5%. When analysing the effect of different mortality assumptions on sub groups of scheme members, it can be seen that these assumptions have a larger impact on active and deferred members than pensioners. This is because active and deferred members have a higher life expectancy compared to pensioners and the mortality improvement factors can impact these younger members for a longer period of time. It is the cumulative impact of these factors that contributes to the significant changes in cash flow projections. Duration measures the sensitivity of the value of pension liability to changes in interest rate. The larger the duration, the more sensitive the liability value is to fluctuations in interest rates. Therefore, by using more conservative mortality assumptions (i.e. mortality assumptions with larger improvement factors), the liabilities are increasingly exposed to interest and inflation risk. The analysis of duration yields similar results to those in PV. For example, in the Long Cohort with a 2% underpin the total duration increased 13.2% compared to a duration of 17 years in the 92 series projection. January

25 Table 2.1: PV and duration of liabilities under various deterministic mortality projections Source: Redington Partners LLP, Proprietary Research Table 2.2: Change in PV and duration of liabilities under various deterministic mortality projections Source: Redington Partners, LLP: Proprietary Research As demonstrated by Table 2.1 and Table 2.2, uncertainties around future mortality experiences have a significant impact on pension benefit cash flow structure, the present value of future benefit and duration of pension liabilities. The following two factors are important to note when considering the results of the tables: (i) The longevity risk will increase the value of the liabilities, which in turn will negatively affect the funding position of the pension fund. The sponsoring company will then be under increased pressure from The Pensions Regulator (TPR) and the Pension Protection Fund (PPF) to eliminate the deficit. January

26 (ii) It is commonly understood that interest rate and inflation risk are irrelevant to longevity risk and by applying LDI solutions the exposure to changes in interest rates and inflation can be hedged. However, as demonstrated in Figure 2.12, Table 2.1 and 2.2, using conservative mortality assumptions will result in underestimating the present value, duration, and cash flow of the liabilities, resulting in LDI solutions only hedging a portion of the scheme s liabilities. Thus the effectiveness of hedging depends on the accuracy of mortality projection and the level of longevity risk. January

27 3.0 Introduction to Eight Stochastic Mortality Models According to the FSA s regulatory guidance for actuaries: Where there is a considerable range of possible outcomes, the FSA expects firms to use stochastic techniques to evaluate these risks. In time, for example, longevity risk, where this constitutes a significant risk for the firm, may fall into this category. Redington studied eight stochastic mortality models from three main families: the Lee Carter family, the Cairns, Blake and Dowd (CBD) family and the Currie family. The eight models studied were developed with different underlying assumptions and theories that will be discussed later in this section, but all attempt to stochastically quantify longevity risk. The models studied in this paper were selected because they are the most widely used and accepted today. Additionally, JPMorgan published its q forward mortality bond designed for pension funds to transfer longevity risk, and identified these eight leading models for potential counterparties of the instrument to use to help understand and price the risk. Finally, the market expectation of future mortality rates and the price of instruments to hedge longevity risk can be derived by understanding the underlying model used in creating the instrument. The Lee Carter model was developed by Professors Ronald Lee and Lawrence Carter. This model has become the leading statistical model of mortality forecasting in the demographic literature in the United States (Deaton and Paxson, 2004). Lee and Carter originally calibrated their model to use United States mortality data from Girosi and King (2007) note that the model is now being applied to all cause and cause specific mortality data from many countries and time periods, and all well beyond the application for which it was designed (Girosi and King, 2007). However, the original Lee Carter model has thus far failed to fit the Australian and UK mortality data. Therefore, Steven Haberman and Arthur Renshaw subsequently designed the Lee Carter Extension model which laid the groundwork for the development of a wider class of generalized, parametric, non linear models. The Lee Carter extension model allows for the modeling and extrapolation of age specific cohort effects as well as the age specific period effects, which is beyond the scope of the original Lee Carter model. January

28 The APC (Age Period Cohort) model and P Splines model were developed by Dr. Iain Currie at Heriot Watt University. The main design feature of the APC model is that it ensures the smoothness of data. The P Splines model was also designed to ensure the smoothness of data and adjusts for large jumps guaranteeing a smoother projected mortality surfaces (Andrew et al, 2007). The Cairns, Blake and Dowd (CBD) model was developed by three Professors in the UK; Professor David Blake from Cass Business School, Professor Andrew Cairns from Heriot Watt University and Professor Kevin Dowd from Nottingham University Business School. The CBD model was developed for and tested using mortality data from males living in England and Wales and has yet to be tested with data from any other countries. However, the model has already been taken up widely by actuaries in Germany and is currently being investigated by the CMI (Pension Institute, 2007). The Three Factor CBD Extension and Four Factor CBD models are still under review by the designers, who are determining how the models should be calibrated and used for forecasting purposes. In early 2008 they will release a discussion paper detailing findings of their current research. Table 3.1 below shows the underlying formula for the eight models. Note that for all models the () i β x () i κ t functions reflect age effects; the functions reflect period effects and the functions () i γ c reflect cohort effects, with c= t x, where t reflects time and x reflects age. Table 3.1: Formula for the stochastic mortality models MODEL FORMULA (1) (2) (2) Lee Carter logm(, t x) = βx + βx κt (1) (2) (2) (3) (3) Lee Carter Extension logm(, t x) = βx + βx κt + β γt x APC (1) (2) (3) P Splines logm(, t x) = βx + κt + γt x ay logm(, t x) = θ ijbij ( x,) t Two Factor CBD (1) (2) logitq(, t x) = κ + κ ( x x) Three Factor CBD (1) (2) (3) logitq(, t x) = + ( x x) + i, j t t κ κ γ t t t x Three Factor CBD Extension (1) (2) (3) 2 2 (4) logitq(, t x) = κ + κ ( x x) + κ (( x x) δ ) + γ t t t x x t x Four Factor CBD (1) (2) (3) logitq(, t x) = + ( x x) + ( x x) κ κ γ t t t x c January

29 The next section of this paper explains each model in detail, defining the variables and the assumptions used in each model. 3.1 Lee Carter model Lee Carter model is one of the most popular models for forecasting mortality rates in the United States. The model is a simple bilinear model with variables X (age) and T (calendar year). The model is defined as: (1) (2) (2) logm(, t x) βx βx κt ε, = + + x t Where: mt (, x) : central death rate at age x in year t (1) β x : average age specific pattern of mortality (by constrains) (2) β x : pattern of deviations from the age of profile as the κ t varies (2) κ t : a time sires index of general mortality level ε xt, : the residual term at age x and time t Since the parameterization in the Lee Carter model is not unique it is necessary to establish restrictions on the model which are defined as: n x= 1 [ β ] (2) 2 x = 1 T t= 1 κ (2) t = 0 Given the imposed restrictions and when we have n T observations of mt (, x), then the estimators of (1) β x ˆ T lnm (1) t=1 β x = T For X=1,2...n x,t are given by the means: While this function can sometimes be used as a constant, the model designers argue that this is best used as a variable and that is how it was used when we built and tested the model. January

30 (2) β x (2) κ t To estimate the series of and in the model, we use the above constrains and minimize the mean squared error. This is equivalent to solving the following equation system: = 0 (1) (2) 2 (ln mxt, βx βx κt) t (1) (2) 2 (ln m β β κ ) κ = 0 t xt, x x t t (1) (2) 2 (ln m β β κ ) b = 0 x xt, x x t x (1) Where m are from the observations of historical data and are defined as above. ln x,t β x After solving these equations, the necessary parameters are established to forecast future mortality rates with: κ = κ + θ + ε t t 1 t Note that κ t is nothing else that a standard univariate time series model ARIMA (0,1,0). Where θ 2 equates to the mean of [ln( m ) ln( )] and ε is white noise and mean of xt, mxt, 1 σ = (ln( mxt, ) ln( mxt, 1) θ ) 2 for t=1 to n. β (2) (2) x = βx 1 + mu β 1 (2) Note that x follows a linear increasing function where: mu = mean( bt bt ). 3.2 Lee Carter Extension The original Lee Carter model captures the cohort effect through the error term and as a result the cohort effect is considered independent of the age and period effects. In an attempt to address the assumption that age and period effects are independent, Renshaw and Haberman added an independent cohort effect factor to the Lee Carter model, creating the Lee Carter Extension model. The formula is given by: (1) (2) (2) (3) (3) logm(, t x) = βx + βx κt + βx γt x + εx, t January

31 From a parameterization point of view, the Lee Carter Extension model follows a similar process as the Lee Carter model and the only difference is two more parameters have been added to evaluate (3) β x (3) γ t x the cohort effect through and. The constrains are defined as: β (2) = x x 1, (2) κ = t t 0, (3) β = x 1, x (3), t x γ xt = 0 For parameter estimation, the Lee Carter Extension model also incorporates ˆ(1) ln( mt (, x)) / T, β x = t and all other parameters are estimated by minimizing the mean square error with constrains. To increase the accuracy of the model results, the more parameters used in building the models will minimize the error when fitting the model to the historical observations. For the Lee Carter and Lee Carter Extension models this means that including the cohort effect should improve the accuracy of fit. However, recent mortality experience suggests that the cohort effect is decreasing each year and becomes inconsequential as individuals move further into retirement. This means that future model selection in order to predict longevity risk will be highly dependent on the future projections of the cohort effect. 3.3 APC Model and P Splines The APC model is a period, age and cohort model similar to the models in the Lee Carter family, while the P Splines model is a non parametric model and differs significantly from the other seven models discussed. P Splines model provides results of a smoothed surface which fits the historical data and then projects to age or to cohort and assumes that the only uncertainty in forecasts is parameter and model uncertainty. As a result, historical volatility in mortality rates will not affect the parametric estimation of the model. It would therefore, be very difficult to extrapolate meaningful results from this model. January

32 3.4 Two Factor CBD Model The CBD model 2 is a discrete time stochastic model, designed to fit UK mortality data, and adopted for pricing mortality linked financial instruments. The general form of the CBD model is as follows: e qtx %(, ) = 1 ptx %(, ) = 1 + e Where: At ( + 1) = At ( ) + μ + cz( t+ 1) Here, qtx ˆ(, ) year t, whilst A1( t+ 1) + A2( t+ 1)( x x) A1( t+ 1) + A2( t+ 1)( x x) is defined as the projected one year mortality rate for the group of people aged x in ptx ˆ(, ) is defined as the projected one year survival probability for the group of people aged x in year t. At () is a two dimensional stochastic process, where A() t 1 and A () t 2 are assumed to be measurable at time t. The values are estimated using Maximum Likelihood Estimator (MLE) by transforming the ungraduated mortality rates from logitq t x A t A t x x ε Moreover: (, ) = 1() + 2()( ) + xt, qtx (, ) μ is a 2 x 1 vector representing the mean value of first order difference of 1 and 2. c is a 2 x 2 matrix, estimated by satisfying the condition that V = cc where V is covariance matrix for series 1 and 2. Zt () is a two dimensional random variable following a standard normal distribution. into cz() t represents the random volatility of mortality improvement. The Two Factor CBD model assumes that the age and period effects are different in nature and that both affect future mortality rates. The A() t 1 applies to all ages for that given year, on the other hand part of the model measures the period effect that A ()( t x x) 2 measures mortality dynamics from the age specific period effect and the (x x) parameter ensures the age specific period effect is diminishing as age increases. With respect to the age effect, the Two Factor CBD model assumes mortality rates of different ages for a specific year follow a functional relationship, and that the mortality rates are increasing smoothly and exponentially as age increases. 2 More details about the CBD model are described in Cairns, Black, Dowd (2006), A two factor model for stochastic mortality with parameter uncertainty: theory and calibration January

33 3.5 Three Factor CBD Model The Three Factor CBD model builds a parameter on top of the two factor model in order to capture the cohort effect. The general form of three factor CBD model is as follows: e qtx %(, ) = 1 ptx %(, ) = 1 + e Where: At ( + 1) = At ( ) + μ + cz( t+ 1) And: % γ ( ) t x= γt x+ φ1+ φ2 t x x A1( t+ 1) + A2( t+ 1)( x x) + γ ( t x) A1( t+ 1) + A2( t+ 1)( x x) + γ ( t x) In addition to the parameter matrix At (), the Three Factor CBD model includes an extra factor, γ t x which represents the cohort effect observed in historical England & Wales mortality data. The model effectively decomposes the effects of mortality rates into three parts: age effect, period effect and cohort effect. Currently there is no consensus about the relationship between cohort factor and period factor, and this relationship is difficult to test. In this research, we consider the correlation between the two factors is not significant because period effect, by definition, affects mortality rates for all age groups in a given year while the cohort effect can affect mortality rates for a specific group of people who share the same cohort at birth. As a result, it is assumed that the cohort effect is independent of period effect. In the process of forecasting, the future cohort factors % t xare estimated using the formula above, where φ1, φ2 are obtained by fitting historical data. γ 3.6 Three factor CBD Model Extension The model designers also proposed an extension to the Three Factor CBD model, which adds ( x c x) as a decay factor to reduce the impact of cohort effect as age increases, where x c is the estimated parameter to fit the historical effect and forecast future mortality rates. This additional parameter has been added to model the belief that mortality improvements due to cohort effect will diminish as the group grows old. The general form of the Three Factor CBD model extension is shown below: e qtx %(, ) = 1 ptx %(, ) = 1 + e A1( t+ 1) + A2( t+ 1)( x x) + γ ( t x)( xc x) A1( t+ 1) + A2( t+ 1)( x x) + γ ( t x)( xc x) January

34 The parameters are defined in the same manner as for the Three Factor CBD model. However, the design of diminishing cohort effect and the method to estimate a proper by the model designers. x c, is still being reviewed 3.7 Four factor CBD Model The Four Factor CBD model adds a quadratic component to capture a non linear relationship between cohort effect and the logit function of mortality rate, logitq(, t x ) follows: e qtx %(, ) = 1 ptx %(, ) = 1 + e 2 2 A1( t+ 1) + A2( t+ 1)( x x) + A3( t x)(( x x) ˆ σ ) + γ( t x) 2 2 A1( t+ 1) + A2( t+ 1)( x x) + A3( t x)(( x x) ˆ σ ) + γ( t x) 3. The general form is as David Blake, one of the CBD model designers, maintains that this model is a work in progress as the model design team has yet to determine how the Three Factor CBD Extension model and the Four Factor CBD model should be calibrated and used for forecasting purposes. In January 2008, they will release several discussion papers on their current research. As a result, these two models were not tested. 3 x logitq(, t x) = log( ) 1 x January

35 4.0 Methodology In our research we tested four models: Lee Carter, Lee Carter Extension, Two Factor CBD and Three Factor CBD. The models were calibrated using two different data samples in order to analyse the sensitivity of each model to different types of mortality data. The data used in testing the four models was provided by the Human Mortality Database 4 (HMD), and included: e x The life expectancy ( ) for the total population of England and Wales for ages 20 to 100 from 1920 to 2003 (from England & Wales Life tables); D tx, The number of deaths ( ) experienced in the total population for England and Wales for ages 20 to 100 from 1920 to 2003 (from England & Wales deaths); E tx, The exposure ( ) for the total population of England and Wales for the ages of 20 to 100 from 1920 to 2003 (from England & Wales Exposure to risk by year of death / period). Following the HMD methods protocol 5, the population size is derived on January 1 st by averaging the two adjacent mid year estimates for each age and as result the death rates calculated from the mortality data may differ slightly from those published by Office of National Statistics (Philipov, 2005). From the number of deaths( ) and the corresponding exposures ( E ), the mortality rates ( q ) are calculated by applying the formula x represents the age. D tx, tx, x D q =, where t represents the time period and 1 exp( tx, E ) tx, x In the first test the four models were calibrated with population mortality data for ages 20 to 100 from the period 1920 to As shown in Figure 4.1 mortality rates were very volatile during this period and the data also includes the effect of World War II. We carried out stochastic projections from 1961 to 2003 for ages from 20 to 100. The projections were then compared against actual mortality experience during that period to test the projection accuracy of each model. 4 Data in the HMD is supplied by the Office of National Statistics (ONS). 5 Please see the HMD website ( for further information January

36 Figure 4.1: Mortality Rates for Age 65 in Period Mortality Rates for Age 65 in Period mortality rate Source: Data provided by the HMD In the second test the four models were calibrated for the same ages as the first test but for the period from 1963 to Unlike mortality rates in the first test, as shown in Figure 4.2, the mortality rates during this 20 years period were relatively smooth and show improvement over time. Stochastic projections from 1984 to 2003 were carried out for ages 20 to 100 and like the first test the 20 years projections were compared with the actual data during 1984 to Figure 4.2: Mortality Rates for Age 65 in Period mortality rate Mortality Rates for Age 65 in Period Source: Data provided by the HMD After calibrating the models against the and periods, 1000 simulations for ages were performed for each year between and respectively. January

37 4.1 Evaluation Criteria The objective of these tests was to compare the actual mortality experience in the forecasting period with the projection results for both best estimated mortality rates and the range of possible variations. The Mean Square Error (MSE), sign test and outliers analysis were all calculated to aid in examining the forecasting accuracy of the models. Additionally we performed more detailed analysis for mortality rate and life expectancy for key ages (25, 35, 45, 55, 65, 75, and 85) MSE Test The MSE calculates the squared difference between the realised and projected values and it is one of the key metrics used to rank a models forecasting accuracy. It is defined as: MSE = T ( qˆ 1 20 xi, qxi, ) i= x= T X Where T is the total number of projected years and X is the total number of age groups in the q ˆx, i q xi, analysis, and are respectively the projected and actual mortality for age x at time i. The MSE is measured by averaging all of the squared residuals between realized mortality rates and the projected mortality rate (the expected value from 1000 simulation), and therefore a smaller MSE value implies better projection accuracy. The MSE is the sum of all of the error values for each individual in the simulation. Because the mortality rates for older ages have higher magnitude than the mortality rates for younger ages, a 1% estimation error for the mortality rate of a person aged 20 will be given less weight in the calculation of MSE than a 1% estimation error for the mortality rate of someone aged Sign Test The sign test is used to test the hypothesis that the model residuals are unbiased. Mathematically, if m is the number of positive residuals, then m should follow a binomial distribution with parameters being the number of residuals (n) and 0.5. Then, for large sample sizes, the normal approximation can be used such that: n n m~ N(, ) 2 4 January

38 n ( m ) With test statistic: 2 n 4 For an accurate model, the projections should not systematically over or under estimate future mortality rates. Thus, the number of positive residuals should not significantly differ from the number of negative residuals. Therefore, with a sign test, the closer the value is to zero the greater the accuracy of a model s forecasting capability Outliers Test The outliers test is used to study the capability of a model to project confidence intervals which can cover most of the realized mortality rates expected. If the realized mortality rates follow the same distribution as the one proposed by the model, then we do not expect to have more outliers than those predicted by the confidence level. In this research, where the confidence interval is between 5% and 95%, the expected number of outliers for every 100 projections is 10, and the 10 outliers should scatter randomly among the 100 projections. If the number of outliers is more than 10 or the outliers are not randomly distributed, this can be interpreted to mean the model has failed to project a proper confidence interval and further analysis is required. As a result, a model that displays accurate forecasting abilities should have a small MSE value, have unbiased residuals, and the number of outliers should correspond with the confidence level of the test and be randomly distributed. January

39 5.0 Discussion of Results Two steps are required in the selection process of a valid model which can be used in further longevity research. The first is a subjective review of the philosophical background and mathematical theories behind the models which we discussed in Chapter 3; this analysis identified four models that should be built and tested. The second step requires mortality data to be used to test the forecasting accuracy of the four models. The aim of the following chapter is to provide qualitative and quantitative measures? That will allow us to better understand the models and to use them effectively in future research and applications. 5.1 MSE Test and Sign Test After calibrating the four models against the and periods, we performed 1000 simulations for each age between 20 and 100 and for each year between and , respectively. The overall fitting results for the Lee Carter, the Lee Carter Extension, the Two Factor CBD and Three Factor CBD are shown in Table 5.1. Table 5.1: Overall Fitting Statistics Analysis of the MSE in the 40 year data set reveals that the Lee Carter model provides the best projection accuracy in each age group and time horizon and both the Two Factor and Three Factor CBD models provide poor mortality predictions. In simulations for the 20 year data, the MSE criterion reveals that CBD models provide more accurate predictions than the Lee Carter family, while the Three Factor CBD model is the overall best model in projection accuracy in the 20 year test. The MSE test reveals that the forecasting accuracy of the Two Factor and Three Factor CBD models is more sensitive to the sample data used to calculate projections than the Lee Carter and Lee Carter Extension models. The two factor CBD model has two period related factors and gives more flexibility in mortality dynamics due to period effect. The Lee Carter model, however, only has one factor to capture the January

40 period effect and this factor is diluted by the more significant age effect. Thus, the Two Factor CBD models are more sensitive to the choice of sample mortality data. In this case, the inclusion of the period including WWII largely affected the forecasting accuracy of CBD models. This effect is more significant for Three Factor CBD model, as it has one extra cohort factor which is more sensitive to the selection of data. The CBD models can produce a better projection if the data sample can be selected by the modeller. When it is unclear how large the sample should be, the Lee Carter model is the best choice. The sign test results for all four models (with the exception of the Three Factor CBD model) are positive, indicating that the death rates are over estimated compared with the observations, meaning that life expectancies are consistently under estimated. As the projection results are biased, it is necessary to closely evaluate the results from the outlier test to further explore the properties of residuals to help determine which models demonstrate the best predictive capabilities. 5.2 Outlier Test In this section, the results from the outlier test are studied in order to see whether these models can successfully project a reasonable confidence interval of future mortality rates. This is done in two steps. First, the total number of outliers for each model for both the 40 and 20 year sample tests are compared to provide an overall picture of the reasonableness of the range of mortality projections produced by these stochastic models. Secondly, fan chart analysis for mortality rates and life expectancies for key ages (i.e. 45, 65 and 85 years old) is carried out to see the main statistics (mean, 50% percentile, 5% percentile, 95% percentile) of the projected mortality rates. By including in the fan charts the realized mortality rates during the same period of time, the accuracy of projection and appropriateness of projected confidence interval can be observed easily Compare Number of Outliers The outlier test in the 40 year sample projected a mortality table that is a 43x81 matrix, which means the test is projecting a total of 3483 future mortality rates. In the 20 year sample, the projected mortality table is a 20x81 matrix, which means the test is projecting a total of 1620 future mortality rates. With the projected confidence interval at 90%, 348 outliers are expected in the 40 year sample test and 162 in the 20 year sample test and these outliers should be randomly distributed in the projected tables. Table 5.2 and Table 5.3 show that for both projection periods, January

41 the numbers of outliers is greater than expected. The exception is the two factor CBD model, which produced significantly less outliers than expected in the 40 year test. Table 5.2: projection: Analysis of outliers by year Table 5.3: projection: Analysis of outliers by year January

42 Figure 5.1: Lee Carter model in 40 years sample test Figure 5.3: Two factor CBD model in 40 years sample test Figure 5.2: Lee Carter extension 40 years sample test Figure 5.4: Three factor CBD model 40 years sample test January

43 Figure 5.5: Lee Carter 20 years sample test Figure 5.7: Two factor CBD model 20 years sample test Figure 5.6: Lee Carter extension 20 years sample test Figure5.8: Three factor CBD model 20 years sample test January

44 The test results are shown in Table and Figure The tables summarize key statistical results while the graphs show a more thorough view of the test results. In the graphs, the green area represents observations of realized mortality rate within the projected confidence interval; the orange area indicates outliers below the 5% confidence band which represents the left hand tail of the distribution (i.e. underestimate mortality rates); the red area indicates outliers above the 95% confidence band, which represents the right hand tail of the distribution (i.e. overestimate mortality rates). In the 40 years sample test, both the Lee Carter model and the Lee Carter Extension produce more outliers than expected, with most occurring in the year age group and very few in the 70 to 80 year age group. The detailed results reveal that almost all of the outliers occur in the red region which means the Lee Carter models failed to capture the properties of mortality rates, systematically over estimated the mortality rates and did not properly capture the realized mortality scenarios for older ages. In the 20 years sample test, the Lee Carter model produced similar results with most of the left hand tail outliers concentrated in the year old bands and right hand tail outliers concentrated in the age groups. In addition, unlike the Lee Carter model, the Lee Carter Extension also failed to predict the mortality rates for the age group for those 80 to 100 years old. The CBD models are more sensitive to variations in the sample data, particularly the three factor CBD Extension model. For example, in the 40 years sample where data from World War II is included, the estimation and projection of model parameters are disrupted and both CBD models that were tested cannot produce reasonable projections. This is illustrated by the results in the 40 years sample test, with the Three Factor CBD model producing 2,188 outliers in total and the model only produces reasonable projections for the age group The Two Factor CBD model only produced 31 outliers in the first test with the 40 year data. This is much less than expected, because the projected confidence intervals were much wider as a result of the volatility in the underlying sample input data. However, the performance of both CBD models tested improved significantly when more stable mortality data from the 20 years sample was used. Most of the outliers for both the two and three factors CBD model are concentrated in the age groups, where the mortality rates for very January

45 young ages are underestimated and the mortality rates for medium ages are overestimated with only a limited number of outliers represented in older ages. These systematic mistakes in projections can be explained. In the 20 years sample test, all of the four models under estimated the mortality rates for the very young. This is because they often assumed a smooth exponential increase in mortality rates as age increases, and failed to capture the hump effect that appeared in the mortality rates for young persons in their 20s to 30s. The hump effect refers to the increased mortality that is observed in young adult males. Meanwhile, all of the models observed a red zone in the medium age group in the second test. This is the cohort effect. This effect did not appear in the first test because the mortality improvements due to cohort effect only start to affect the mortality rates when the cohort group born in the 1930s reached their 50s. While the Lee Carter Extension and Three Factor CBD models account for the cohort effect by model design, both failed to fully capture the extent of mortality improvements. The Three Factor CBD model did, however provide better predictions than the Lee Carter Extension model. Finally, the Lee Carter models consistently over estimated the mortality rates for the 80 to 100 age band, and were unable to cover the realized mortality experience in their projected confidence interval; in contrast, both CBD models successfully achieved this Fan Chart Analysis A fan chart is another tool used to determine the accuracy of model projections. It charts the model results within the given confidence intervals and shows different degrees of uncertainty of future mortality rates. Figures A.1 A.48 in Appendix show mortality rates and life expectancy fan charts for individuals 45, 65 and 85 years old and graphically show the mean, 50% percentile, 5% percentile, 95% percentile and realized mortality rates. For the 40 years test, the observed mortality rates for age 45 and 65 are in the range of the 90% confidence interval from all models except the CBD Extension model. However, to analyse the fan chart for an 85 year old, the only model to cover a 90% confidence interval is the CBD model. The rest of the models tested overestimate the mortality rate with all observation rates being lower than the lower boundary. January

46 The 20 year data test provides similar results but the Two Factor CBD Extension model exhibits the best performance, indicating that it is the model most sensitive to the input data used in the testing process. The simulations reveal that each model has its own strength and weaknesses. The Lee Carter models provide more accurate projections of future mortality rates for younger ages but systematically over estimate the mortality rates and fail to project a reasonable confidence interval for older ages. In contrast, CBD models provide accurate projections of future mortality rates and project a confidence interval which covers most realised mortality rates for old ages. But their performance is comparatively weak when used to forecast mortality rates for younger ages. Consequently, if the sample data can be chosen, the Two Factor CBD model is the preferred choice because the model can better quantify the underlying volatility of future mortality improvements, which is longevity risk. January

47 6.0 Longevity Risk Analysis based on Two Factor CBD Model This chapter demonstrates how to use the Two Factor CBD to quantify longevity risk and derive the Value at Longevity Risk (VaLR). This analysis will allow pension schemes to incorporate longevity risk analysis into the overall asset and liability management of the scheme and aid in implementing effective longevity risk hedging. 6.1 Stochastic Cash Flow & Value at Longevity Risk (VaLR) In this example we used the actuarial cash flow from a sample pension scheme and the mortality assumption used by the scheme actuary to derive the undiscounted cash flow before considering survival probability. The CBD model was calibrated with historical data for ages from 20 to 100 in the period of to generate stochastic simulations of future mortality rates simulations were performed on the sample data and 1000 simulated mortality tables were calculated. Finally the pension liabilities were valued and cash flow structures were estimated with these mortality tables which produced a distribution of the expected present value of future pension benefit and the possible cash flow structures. The key statistics from the stochastic simulation are summarized in the Table 6.1 below: Table 6.1: Key Pension Metrics based on Two Factor CBD Model Key Statistics CBD Model ( '000,000) Mean 1,002 Standard Deviation 32 95% Percentile 1,054 5% Percentile 954 VaLR 52 VaLR as % of Total Liability 5.17% The results show that the projected present value of future pension liabilities is about 1 billion on average with a possible range of billion to billion and standard deviation of 32 million. The annual VaLR95 is defined as the maximum increase of pension liabilities in one year s January

48 time solely due to mortality improvements with a 95% confidence level. The annual VaLR for this sample pension scheme is equivalent to 5.17% of total pension liability. 6.2 Stochastic Cash Flow & Deterministic Cash Flow A major drawback of the traditional deterministic projection is the inability to determine the probability of a scenario occurring. This problem can be solved by combining the stochastic cash flow structures with the deterministic cash flow structures. Table 6.1 shows that the CBD model projected a confidence interval of possible future cash flow structures which covers most of the cash flows estimated with deterministic mortality assumptions. The Short Cohort projection is close to the expected value of CBD simulations. The long cohort projection is near the 95% upper boundary of CBD simulations and the Medium Cohort projection does not vary significantly from the Short Cohort projection. The expected present value of future pension benefits under Long Cohort assumption is slightly higher than the 95% boundary value produced by the CBD simulations and the mean of CBD simulations is a value that is between the Short Cohort projection and Medium Cohort projection. The Two Factor CBD model can produce an estimation of the value at longevity risk (VaLR) and the confidence interval of expected present value, while deterministic projections can only provide a mean value. January

49 Figure 6.1: Cash Flow Structures (Two Factor CBD Model vs. CMI Deterministic Assumptions) Annual Cashflow Millions Cash Flow Structures Two Factor CBD Model vs. CMI Deterministic Assumptions Medium Cohort Short Cohort Long Cohort Mean 95% 5% Source: Redington Partners LLP, Proprietary Research Table 6.2: Comparison of Present Value derived from Two Factor CBD model and Cohort Projections Present Value Two Factor CBD( 000,000) Short cohort ( 000,000) Medium cohort ( 000,000) Long cohort ( 000,000) Mean 1, ,020 1,067 Standard Deviation 32 n/a n/a n/a 95% 1,054 n/a n/a n/a 5% 954 n/a n/a n/a VaLR 52 n/a n/a n/a As % of Total Liability 5.17% n/a n/a n/a Source: Redington Partners LLP, Proprietary Research January

50 6.3 Managing Longevity Risk This section discusses how longevity risk contributes to the overall Value at Risk (VaR) of a pension scheme Risk Components in DB Pension Fund Figure 6.2 shows that the risk components in most DB pension funds change after executing Redington recommendations and LDI hedging strategies. Figure 6.2: Risk Components in DB Pension Funds Source: Redington Partners LLP, Proprietary Research In the first step, Redington offers strategies of reduction in equity, interest rate and inflation risk. However, as these risks are hedged, the proportion of longevity risk in the pension fund liability increases significantly. The second step of Redington s ALM is to help the pension scheme to reduce longevity exposure. This can be accomplished by first quantifying the longevity risk within the scheme. Redington s current stochastic mortality modelling research has laid the groundwork for future research to determine how to calculate the VaLR first based on general population and then to be able to quantify the scheme specific VaLR. We will then be able to compare the VaLR with the January

51 longevity products available in the market and derive the best way of managing longevity risk. After longevity risk has been hedged, the final step is to add alternative assets to the portfolio to increase the expected return of the scheme s portfolio without significantly increasing the scheme s risk VaR Road Map The VaR Road Map (Figure 6.3 Table 1) is a powerful tool to help the trustees develop a risk budget. For example, it plots the VaR of a portfolio relative to a scheme s equity holdings and relative to the percentage of the portfolio that is hedged against changes in interest rates and inflation. The VaR road map reveals that as a scheme moves from equities into cash, the overall VaR of the scheme is, not surprisingly, reduced. Less intuitively, perhaps, if the equity allocation of a portfolio is left unchanged and the portfolio s real yield risk is hedged using swaps, the VaR of the portfolio will fall very significantly, allowing the scheme to retain the expected return from equities while simultaneously reducing the overall risk. This is a key concept. It is important to note that once the interest rate and inflation risks have been immunized, moving portfolio assets from equities to cash or cash line assets yielding a spread over Libor significantly reduces VaR. One possible way Redington has identified to incorporate longevity risk into VaR to assist those responsible for managing pension fund liabilities is demonstrated in the VaR Road Map. Figure 6.3: Comparison of VaR Road Map Source: Redington Partners LLP, Proprietary Research January

52 Figure 6.3 shows three VaR Road Maps. A traditional VaR Road Map (Figure 6.3 Table 1: No Longevity Risk ) is described above. However, this VaR Road Map does not consider the impact of longevity risk on the value of the sample scheme s liabilities. In order to understand the true VaR of the pension scheme, it is necessary to account for the longevity risk in the portfolio. The stochastic mortality modelling described in this paper will allow a pension scheme to quantify the Value at Longevity Risk (VaLR) thus incorporating the VaLR into the overall VaR. Figure 6.3 Table 3 (Unhedged Longevity Risk) demonstrates how the VaR of this sample pension scheme changes when longevity risk is included when calculating the scheme s VaR. In this example it is 19.3% 6, based on the assumption that longevity risk for this sample pension fund is 5.92% and longevity risk is independent from all other risks facing the pension fund. When the equity holding in the sample pension fund is reduced to zero and 20% of the portfolios exposure to interest rate and inflation risks is hedged using swaps, the VaR with and without considering longevity risk is 15.3% and 14.1% respectively. In this scenario longevity risk contributes 8.4% to the total VaR of the scheme. When the sample portfolio s equity holdings are reduced to zero and 100% of the interest rate and inflation risks are hedged, the VaR without considering longevity risk is 2.1% and the VaLR is 6.3%, which represents a 200.7% increase in the schemes total VaR. Understanding how longevity risk impacts the overall VaR is crucial for longevity risk management and the overall Asset Liability Management framework used by the scheme because all management and investment strategies are dictated by the risk composition of the pension fund. Figure 6.4 graphically shows how longevity risk affects the overall VaR and how VaLR exponentially contributes more to the overall VaR as equity holdings are reduced and as the percentage of interest rate and inflation risks that are hedged increases. 6 Based on the assumption of independence between longevity risk and other risks, the VaR with longevity risk added in is calculated using the formula: VaR = 2 2 a + b, where a = VaR without considering longevity risk and b=valr. For example, the VaR = 19.3% in Table 3 of figure 6.4 is calculated in the following way: 2 2 (18.35%) + (5.92%) = 19.3% VaR without considering longevity risk = 18.35%, VaLR = 5.92%. January

53 Figure 6.4: Increase in VaR due to longevity risk Source: Redington Partners LLP, Proprietary Research Finally, Figure 6.3 Table 2 shows the VaR Road Map at a 95% confidence level with 83% of the longevity risk hedged. Redington is currently researching the products available to hedge longevity risk including the JPMorgan q forward and the Credit Suisse longevity swap. The VaR of a portfolio that has hedged longevity risk is slightly higher than the VaR of a portfolio that has not hedged longevity risk. This is due to the fact that new risks like credit risk, counterparty risk and basis risk are likely to be introduced into the portfolio when a new transaction to hedge longevity risk is executed. While there is a slight increase in the portfolio s VaR when the longevity risk is hedged, any such increase is reduced if longevity risk is not quantified and managed The Efficient Frontier Figure 6.5 shows the effect of how longevity products such as mortality swaps and longevity bonds impact the efficient frontier of the sample pension scheme. The dark red line represents the current efficient frontier, whilst the purple to the light blue represents the possible efficient frontiers after adopting longevity hedging products. January

54 Figure 6.5: Change in Efficient frontier by availability of longevity risk hedging products Efficient Frontier Expected Return % 2% 4% 6% 8% 10% 12% 14% 16% 0.01 Volatility (% p.a.) The two red circles on the graph highlight the fact that the more volatile the assets and liabilities of a scheme are, the less effective longevity hedging will be. This would suggest that a pension scheme should first hedge risks like equity risk, inflation risk, and interest rate risk before hedging longevity risks. However, as indicated by the large circle at the bottom of the graph, the difference between the expected return for small changes in volatility is significant. In this instance, by hedging longevity risks in a portfolio, the expected return of the portfolio can increase whilst maintaining the same level of volatility LE01 LE01 measures the variation in the present value of pension liabilities following a one year increase in life expectancy of a particular age and salary group. A LE01 table (e.g. Table 6.5) can therefore be used to assist pension schemes to identify which groups of members benefits are most affected by longevity risk. January

55 Table 6.3 to Table 6.5 show the three steps to deriving LE01 table. The first step is to estimate changes in present value of future benefits for a single member in a particular age and salary group by adding one year in life expectancy. This is done by parallel shifting the future mortality rate for the corresponding cohort so that the life expectancy of this person is increased by one year, i.e. multiply the row by a fixed number. The new mortality table is then used to calculate the present value of benefit. The difference between the new present value and the original present value is the sensitivity of the member s benefits to a 1 year rise in life expectancy for individual members in different age and salary groups. This is done for every age and salary group and the results are shown in Table 6.3. As can be seen by the colours and numbers (the darker the colour, the higher the sensitivity to adding 1 year to life expectancy), older members earning higher salary are most sensitive to mortality rates. This can be attributable to the fact that, all other factors being equal, members with pensions, who are living longer, receive more benefit, compared to members with smaller pensions. From the age perspective, based on the actuarial assumption that no one lives longer than 120 years, a one year increase in life expectancy for a person who is currently 110 years old means that his probability of death in future years drop more significantly than for a member currently aged 20. For a generic pension scheme, in order to discover the group of members benefits which is most vulnerable to a 1 year increase in life expectancy, we need the age and salary structures of the pension scheme. In this paper, for demonstration purposes, we assume the age and salary structure for this generic pension scheme follow Weibull distribution. Table 6.4 shows the distribution of members in each age/salary combinations. With results from Table 6.3 and Table 6.4, the LE01 table for the generic pension scheme can be derived by combining the sensitivity of individual members with the age and salary distribution of the pension scheme. The results (shown in Table 6.5) show that members aged between 45 to 75 with a salary between 45,000 and 105,000 are most heavily affected by a 1 year increase in life expectancy. This LE01 analysis helps pension schemes better understand the longevity risk they face and also provides them with a tool to identify the group of members benefits which is most risky in terms of longevity risk, and therefore which part of risk should be removed first. Most importantly, this analysis is scheme specific; schemes with different age and salary distribution result in a different LE01 table. January

56 Table 6.3: Sensitivity to 1 year rise in life expectancy for individual member Table 6.4: Age and salary distribution of pension Table 6.5: LE01 Table January

57 7.0 Conclusions Solutions to mitigate equity, interest rate and inflation risks can currently be found in products available in the capital markets. This paper has demonstrated that the first step towards understanding longevity risk is creating superior models to analyze, forecast and price longevity risk. A stochastic model produces a full picture of the impact of mortality improvements to the scheme s cash flows and identifies how these can affect the funding position or current hedging strategy of a scheme. Additionally, stochastic models can be used to conduct extreme case scenario analysis and to fully integrate longevity risk within the overall risk framework of the scheme. The CMI recently introduced two deterministic projection methods in 1990 and 1999 and both significantly underestimated future mortality rates. As a result, the mortality and longevity risk in life associated financial product are difficult to evaluate. Therefore, the stochastic projection model is required to support risk monitoring and decision making. Our research indicates that all models cannot correctly project the future mortality rates and can overestimate the mortality rate (underestimate life expectancy). However, the difference is in the volatilities projection for people over 65. Both CBD models tested have the capability of projecting a reasonable confidence interval which covers most of the mortality observations for those individuals over 65 of age. Both Lee Carter models tested are unable to accurately forecast these projections. These findings suggest that the Two Factor Cairns, Blake and Dowd (CBD) model is more relevant for pension funds risk management. While further research is needed to fully understand how longevity risk can be included in an overall Liability Driven Investment strategy, it is clear that stochastic modelling is the best way to capture and quantify longevity risk. As briefly described in Chapter 6, our future research will focus on how to integrate longevity modelling into an overall Asset Liability Management framework. January

58 References: Andrew J.G. Cairns, David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, Alen Ong, and Igor Balevich, Pension Institute Discussion paper PI 0701, March Barr, N. and Diamond, P. (2006) The Economics of Pensions, Oxford Review of Economic Policy. 22, 1, pp Barr, N. (2006a) Pensions: Overview of the Issues, Oxford Review of Economic Policy. 22, 1, pp Blake, D., Cairns, A.J.G., Dowd, K., and MacMinn, R. (2006) Longevity Bonds: Financial Engineering, Valuation and Hedging. Journal of Risk and Insurance 73, CMI, An interim basis for adjusting the 92 Series mortality projections for cohort effects. Working paper 1. December Cummins, J.D. (2004): Securitization of life insurance assets and liabilities. Preprint, The Wharton School, University of Pennsylvania. Deaton, A., and Paxson, C., (2004) Mortality, Income, and Income Inequality Over Time in the Britain and the United States. National Bureau of Economic Research Technical Report 8534, Cambridge. Girosi, F.; King, G., Understanding the Lee Carter Mortality Forecasting Method, Harvard University, (2007) Government Actuary's Department (1995). National population projections 1992 based. H.M.S.O., London, UK Latest research reveals man could live up to 12 years longer by 2050 than currently predicted, Pensions Institute Press Release, Cass Business School, 26 November January

59 Human Mortality Database [cited 5 December 2007] Lane Clark and Peacock (LCP) Accounting For Pensions 2007 UK and International Report LifeMetrics Glossary, JPMorgan Olivieri, A., The Longevity Risk in Pension Products, Cass Business School Lecture Notes. June 2007 Pensions Regulator, 2007: [cited 30 November 2007] Extract from a speech by david Norgrove, Chairman of the Pensions Regulator, delivered at the UK Pensions and Investment summit in Brighton on Monday 25th September, Published in September Renshaw A.E. and Haberman S. (2006), A cohort based Extension to the Lee Carter model for mortality reduction factors. Insurance: Mathematics and Economics Volume 38, Issue 3. Vrinda Gupta, Pensionrisk: Do employees care? Pensions 25 October (Original: Watson Wyatt Pension Risk Indicators database) journals.com/pm/journal/v12/n1/full/ a.html Willets, R. C. (2004), The cohort effect: insights and explanations. British Actuarial Journal, Volume 10, Number 4, 2004, pp (45). Publisher: Faculty of Actuaries and Institute of Actuaries. January

60 Appendix 1. Comparison of projected and realized mortality rates during the period for the four models tested. (All graphs were generated using proprietary Redington Partners pension scheme analysis) Figure A.1: Lee Carter 45 year old total population mortality rate in England & Wales Figure A.2: Lee Carter Extension 45 year old total population mortality rate in England & Wales Figure A.3: Two Factor CBD 45 year old total population mortality rate in England & Wales Figure A.4: Three Factor CBD 45 year old total population mortality rate in England & Wales January

61 Figure A.5: Lee Carter 65 year old total population mortality rate in England & Wales Figure A.6: Lee Carter Extension 65 year old total population mortality rate in England & Wales Figure A.7: Two Factor CBD 65 year old total population mortality rate in England & Wales Figure A.8: Three Factor CBD 65 year old total population mortality rate in England & Wales January

62 Figure A.9: Lee Carter 85 year old total population mortality rate in England & Wales Figure A.10: Lee Carter Ext. 85 year old total population mortality rate in England & Wales Figure A.11: Two Factor CBD 85 year old total population mortality rate in England & Wales Figure A.12: Three Factor CBD 85 year old total population mortality rate in England & Wales January

63 2. Comparison of projected and realized mortality rates during period for four models Figure A.13: Lee Carter 45 year old total population mortality rate in England & Wales Figure A.14: Lee Carter Ext. 45 year old total population mortality rate in England & Wales Figure A.15: Two Factor CBD 45 year old total population mortality rate in England & Wales Figure A.16: Three Factor CBD 45 year old total population mortality rate in England & Wales January

64 Figure A.17: Lee Carter 65 year old total population mortality rate in England & Wales Figure A.18: Lee Carter Ext. 65 year old total population mortality rate in England & Wales Figure A.19: Two Factor CBD 65 year old total population mortality rate in England & Wales Figure A.20: Three Factor CBD 65 year old total population mortality rate in England & Wales January

65 Figure A.21: Lee Carter 85 year old total population mortality rate in England & Wales Figure A.22: Lee Carter Ext. 85 year old total population mortality rate in England & Wales Figure A.23: Two Factor CBD 85 year old total population mortality rate in England & Wales Figure A.24: Three Factor CBD 85 year old total population mortality rate in England & Wales January

66 3. Life expectancy fan chart during period for four models Figure A.25: Lee Carter 45 year old total population mortality rate in England & Wales Figure A.26: Lee Carter Ext. 45 year old total population mortality rate in England & Wales Figure A.27: Two Factor CBD 45 year old total population mortality rate in England & Wales Figure A.28: Three Factor CBD 45 year old total population mortality rate in England & Wales January

67 Figure A.29: Lee Carter 65 year old total population mortality rate in England & Wales Figure A.30: Lee Carter Ext. 65 year old total population mortality rate in England & Wales Figure A.31: Two Factor CBD 65 year old total population mortality rate in England & Wales Figure A.32: Three Factor CBD 65 year old total population mortality rate in England & Wales January

68 Figure A.33: Lee Carter 85 year old total population mortality rate in England & Wales Figure A.34: Lee Carter Ext. 85 year old total population mortality rate in England & Wales Figure A.35: Two Factor CBD 85 year old total population mortality rate in England & Wales Figure A.36: Three Factor CBD 85 year old total population mortality rate in England & Wales January

69 4. Life expectancy fan chart during period for four models Figure A.37: Lee Carter 45 year old total population mortality rate in England & Wales Figure A.38: Lee Carter Ext. 45 year old total population mortality rate in England & Wales Figure A.39: Two Factor CBD 45 year old total population mortality rate in England & Wales Figure A.40: Three Factor CBD 45 year old total population mortality rate in England & Wales January

70 Figure A.41: Lee Carter 65 year old total population mortality rate in England & Wales Figure A.42: Lee Carter Ext. 65 year old total population mortality rate in England & Wales Figure A.43: Two Factor CBD 65 year old total population mortality rate in England & Wales Figure A.44: Three Factor CBD 65 year old total population mortality rate in England & Wales January

71 Figure A.45: Lee Carter 85 year old total population mortality rate in England & Wales Figure A.46: Lee Carter Ext. 85 year old total population mortality rate in England & Wales Figure A.47: Two Factor CBD 85 year old total population mortality rate in England & Wales Figure A.48: Three Factor CBD 85 year old total population mortality rate in England & Wales January