Risk Management and Insurance Review

Measuring Longevity Risk: An Application to the Royal Canadian Mounted Police Pension Plan


  • M. Martin Boyer,

  • Joanna Mejza,

  • Lars Stentoft

  • This research is financially supported by CIRANO and by the Social Science and Humanities Research Council of Canada. This article was subject to double-blind peer review.


An employer that sets up a defined benefit pension plan promises to periodically pay a certain sum to each participant starting at some future date and continuing until death. Although both the future beneficiary and the employer can be asked to finance the plan throughout the beneficiary's career, any shortcoming of funds in the future is often the employer's responsibility. It is therefore essential for the employer to be able to predict with a high degree of confidence the total amount that will be required to cover its future pension obligations. Applying mortality forecasting models to the case of the Royal Canadian Mounted Police pension plan, we illustrate the importance of mortality forecasting to value a pension fund's actuarial liabilities. As future survival rates are uncertain, pensioners may live longer than expected. We find that such longevity risk represents approximately 2.8 percent of the total liability ascribable to retired pensioners (as measured by the relative value at risk at the 95th percentile) and 2.5 percent of the total liabilities ascribable to current regular contributors. Longevity risk compounds the model risk associated with not knowing what is the true mortality model, and we estimate that model risk represents approximately 3.2 percent of total liabilities. The compounded longevity risk therefore represents almost 6 percent of the pension plan's total liabilities.


Although we would all like for our life to last as long and be as healthy as possible, we would also like to maintain a level of consumption that satisfies us. Consequently, we would prefer not to run out of cash once we are old and gray. Lacking financial resources is, however, one of the pernicious effects stemming from longevity risk, which we define as the risk that a population lives longer than anticipated. Obviously, longevity risk greatly affects the profitability of institutions that offer lifetime pensions, but longevity risk also consistently affects all levels of society, from individuals to organizations and governments.

One element that highlights the importance of longevity risk is the systematic historical underestimation of the survival rates at older ages (see, e.g., Oeppen and Vaupel, 2002; Turner, 2006). For example, Turner (2006) reports that in the past 100 years, life expectancy has systematically been underestimated by about 18 months for each decade. Clearly, the decrease in the mortality rate has important consequences on the viability of pension funds since underestimating life expectancy after retirement will lead to insufficient accumulations before retirement. The importance of longevity risk and some organizations’ exposure to it has recently been recognized by the international accounting standards board, which has begun to tackle the problem of how to disclose longevity risk in financial statements of publicly traded corporations (see Fujisawa and Li, 2010). Another important particularity of aggregate longevity risk is its systematic and undiversifiable nature (see Milevsky et al., 2006).

In this article, we consider one Canadian defined benefit pension plan and stress the impact of adequate mortality forecasting on actuarial liabilities.1 Any shortcoming of funds in an employer-sponsored pension plan is often the employer's responsibility. It is thus essential for the employer to be able to predict with a high degree of confidence the total amount that will be required to cover its obligations to the future retirees. To do so, several assumptions must be made, such as the rate of return on the assets before they are liquidated, the amount of the periodic payments, and the number of years employees are expected to live after retirement. All these factors have sizable impacts on the actual cost of the pension plan and on its solvency ratio.

To be specific, we implement the two-factor stochastic mortality model of Cairns et al. (2006) (the CBD model hereafter) using the Canadian mortality tables, and measure the longevity risk associated with issuing immediate and/or deferred annuities for the pension plan of the Royal Canadian Mounted Police (RCMP, 2008). The present value of the expected future benefits (i.e., the plan's actuarial liability) is also calculated using the projected mortality rates as forecasted by the Office of the Chief Actuary (OCA hereafter) of Canada and these are compared to the results obtained with the CBD model. Given the CBD results, we then run Monte Carlo simulations to obtain the plan's value at risk (VaR) to emphasize the uncertainty about the future. Our approach allows us to show that the longevity VaR varies depending on the age group of the participants as well as on the type of the annuity (immediate vs. deferred), and we estimate that to compensate for longevity risk alone, pension funds should increase their reserves by 6–9 percent, depending on the pension plan's maturity, composition of beneficiaries, structure and level of certainty needed.

The next section presents the key features of the CBD mortality model and gives a broad outlook of Canadian mortality trends. In the section titled “An Application to the RCMP Pension Plan,” we apply the CBD model to the RCMP pension plan. It reveals how the risk embedded in future mortality rates affects the current liabilities of pension plans. As we will see the OCA's projected mortality decreases are quite different from the variation produced by the CBD model. The CBD model reveals an actuarial liability significantly superior to the funds’ reported liability based on the OCA approach. The “Conclusions and Further Discussion” section concludes.

Mortality Models and Trends

There exists a vast literature on mortality modeling (see Gaille, 2010, for a recent survey). Commonly used, the Lee–Carter model (see Lee and Carter, 1992) is a simple one-factor model that offers a relatively good fit over a full range of ages. Li and Chan (2007) even call the Lee–Carter model the “gold standard of mortality trend fitting and forecasting.” The main advantage of the Lee–Carter model is its simplicity since only one factor is needed to identify the time series properties of longevity.2

The simplicity of the Lee–Carter model, however, is also its main disadvantage.3 Cairns et al. (2006) propose a more sophisticated model that uses a second factor to capture nonhomogenous improvements in mortality rates (see also Renshaw and Haberman, 2006; Cairns et al., 2009). In essence, the CBD model introduces a factor that distinguishes the different age group dynamics (see Li and Chan, 2005, as well as the Appendix for more details). Letting inline image be the probability that a person aged x in year t dies within 1 year, the CBD model posits that the mortality rate is represented by a two-factor model where the first parameter, inline image, affects all ages by the same amount whereas the second parameter, inline image, affects higher ages much more than lower ages. The particular specification used is given by

display math

We can adjust our results for parameter uncertainty by using a simulation approach. We are then able to compute different trajectories to obtain a distribution of mortality improvements that we then use to calculate the estimated and projected mortality table for a given population. Because we have a distribution of a retired population's mortality, we are able to calculate the present value of a portfolio of annuities paid to this population, as well as the portfolio's VaR.

It is not trivial to forecast longevity and mortality in the long run. Figure 1 depicts life expectancies at age 65 for both Canadian men and women, and Figure 2 shows the male population survival curves (the probability that a newborn will survive to a given age) for three different years.4 From Figure 1, we see that life expectancy at age 65 increased faster for men than for women over the period 1985 to 2006. The bar chart shows that the gap between the two sexes has been narrowing since its peak in 1980. As mentioned in Li et al. (2011), mortality rate reductions have accelerated since the 1970s. Figure 2 shows that survival curves have been shifting to the right, a process known as “rectangularization” because of the eponymous geometric form that survival curves are trending toward.

Figure 1.

Life Expectancy at 65 Years for Canadian Males and Females

Figure 2.

Anticipated Survival Rates for Men by Age for Cohorts Born in 1921, 1965, and 2006

For pension plans, the important forecast is the life expectancy at age 65. Individuals that reached age 65 in 2000 were born in 1935 and accumulated 35 years of pensionable benefits starting in 1965. This means that these individuals’ pension fund was initially calculated using mortality forecasts for 2000 made in 1965 that underestimated real life expectancy at retirement by 1.5 years (see Oeppen and Vaupel, 2002, for more on the dynamics of longevity). And even though mortality tables were updated regularly, life expectancy at 65 remained underestimated for most of the period leading to 2000 (see Waldron, 2005). Consequently, the amount of wealth accumulated in the pension plan became insufficient compared to the pension plan's actuarial liability. We can therefore be reasonably certain that there is a risk—the pension plan's longevity risk—that pensioners will live longer than expected and thus expose pension funds to a liability that is difficult to forecast and costly to manage.

An Application to the RCMP Pension Plan

We want to apply the CBD two-factor model to the case of the RCMP pension plan to get an estimate of its longevity risk.5 We will therefore compare the pension plan's published actuarial liability using the longevity forecast made by the OCA of Canada with its liability obtained from fitting and forecasting the CBD model to the recent Canadian mortality experience. We will stress the difference between the OCA and the CBD approaches to future mortality improvements by calculating the ratio of the number of survivors predicted using the CBD model to the number of survivors predicted by the OCA model. As we will see, the CBD model using the entire Canadian male population predicts higher survival rates for all ages and categories than the pension-plan-specific model used by the OCA. The 1.5 percent difference between the two models’ predictions may be only due to the different populations used to calibrate the mortality forecasts.6

The RCMP's workforce is divided into two categories: Regular and civilian members. Regular members account for the majority of the personnel. Of the 15,505 pensioners, 94 percent are men and 88 percent are former regular members. In terms of contributors, there are 15,990 men who represent 75 percent of the total number of contributors that number 21,212. Since the RCMP is male dominated, longevity risk is mostly dependent on the evolution of male mortality rates. Finally, 84 percent of contributors are regular members (17,862) and 16 percent are civilian members (3,350). Table 1 presents the RCMP's pension plan's balance sheet in millions of dollars for 2005 and 2008. For both years, the plan's total actuarial liability has been assigned to the corresponding type of member (regular or civilian, retired or contributing, disabled or surviving).

Table 1. Balance Sheet (Millions of Canadian Dollars) of the Pension Plan of the Royal Canadian Mounted Police as of March 31, 2008 and March 31, 2005, and Actuarial Liability by Type of Member
 As of March 31, 2008As of March 31, 2005
Actuarial value of assets14,823  12, 284  
   % of Actuarial  % of Actuarial
   Liability  Liability
Actuarial liability14,191 100%11,303 100%
Regular members 13,06892.1% 10,41692.2%
of which: Contributors 5,73740.4% 4,99644. 2%
Retired 6,39345.0% 4,82342.7%
Disability 6234.4% 3653.2%
Surviving 3152.2% 2322.0%
Civilian members 1,1237.9% 8877.8%
of which: Contributors 6274.4% 5424.8%
Retired 4112.9% 2822.5%
Disability 630.4% 430.4%
Surviving 220.2% 200.2%
Administrative expenses110  79  
Actuarial surplus522  902  

The plan's total actuarial liability as of March 31, 2008, as calculated by the actuaries of the RCMP's pension plan using the OCA model parameters, amounts to Canadian $14,191 million. The present value of pensions paid to current retired regular and civilian pensioners represents 48 percent of this total inline image. Another 45 percent includes the present value of future payable benefits to regular and civilian contributors that are not yet in payment inline image. The remainder of the actuarial liability is ascribable to the pensions offered to pensioners with a disability and to surviving dependants. We also note from the balance sheet that pension promises made to regular members (contributors and retired pensioners) represent 85 percent of the pension plan's total actuarial liability. In comparison, on the date of the previous actuarial audit on March 31, 2005, pension promises made to regular members (contributors and retired pensioners) accounted for 87 percent of the pension plan's total actuarial liability. Given the steady importance of regular members in the plan's total liability, we concentrate our analysis on this subgroup.

Retired Pensioners

The actuarial report does not contain all the information we need to evaluate the total actuarial liability. For instance, the report does not give us the exact decomposition by age of the plan's covered population so that we are forced to use the median age for each quinquennial age group in our calculations in lieu of the plan members’ actual age.7 We will also assume that a pensioner's annual payment is equal to the average annual pension of his age group. Finally, we will ignore the liability ascribable to retired regular members younger than 45 and older than 84 since they represent only 2 percent of the pensioners. The impact of these assumptions on the liability of the RCMP's pension plan is shown in Table 2.8

Table 2. RCMP's Pension Plan's Liability as Calculated Using the CBD or the OCA Mortality Model, by Age Group and Sex
   CBDOCA% Retired
Age RangeMedianCBD/OCA($ Millions)($ Millions)Model
Panel A: Liability for Retired Men by Age Group, Calculated at the Median Age
80– 84821.018619.5019.140.317
For all men 1.01506,009.075,920.2697.82
Panel B: Liability for Retired Women by Age Group, Calculated at the Median Age
For all women 1.0060133.68132.882.176

Using median age and average annual pension benefits, and ignoring pensioners younger than 45 and older than 84 years old, we applied the OCA's forecast to future mortality to find a total liability ascribable to retired regular members of (5,920.26 + 132.88) $6,053 million, thus a 5 percent difference with the value $6,393 million reported in the pension plan's balance sheet. Hopefully the use of median age and income should only have a small and economically marginal impact on the pension plan's measure of longevity risk. As we can see, the OCA model assigns a lower total liability by age group than the CBD model. This is true for both men (in Panel A) and for women (in Panel B). Our calculations result in a total liability for regular members of (6,009 + 134) $6,143 million according to the CBD model and of (5,920 + 133) $6,053 million according to the model used by the OCA. Table 2 also highlights the fact that men account for close to 98 percent of the retired regular members’ actuarial liabilities.

Compared to the OCA model, the CBD model predicts higher reductions in mortality rates on average (hence the CBD/OCA ratio is always greater than one). The discrepancy between the two models is lower for intermediate ages: 1.35 percent for males in the 65–69 age group versus 1.84 percent for males aged 45–49 and 1.86 percent for males aged 80–84. But because it is for intermediate ages that we find the majority of the total actuarial liabilities, the difference between the CBD-calculated total liability and the OCA-calculated total liability is only 1.50 percent.9 The difference between the CBD and the OCA approaches is much lower for women (the CBD-to-OCA ratio ranges from 0.57 percent to 1.06 percent). One possible reason why the gap is larger for men could be that their longevity is currently increasing much faster than the women's, which induces a higher level of uncertainty for the men.

Another interesting feature of the CBD-to-OCA ratios is its U-shape that appears much more pronounced for men than for women. Although this impression is not false, it does not take into consideration the difference in the CBD-to-OCA ratio levels. When we normalize each ratio by its respective overall CBD-to-OCA ratio, the difference between the two errors is much less as we see in Figure 3.

Figure 3.

CBD-to-OCA Ratio Normalized by the Overall Ratio for Each Sex for Different Age Groups

The CBD-to-OCA ratio increases as pensioners reach the normal retirement age of 65, suggesting that the OCA methodology underestimates the financial risk associated with an aging Canadian population. What is surprising is that the CBD model overshoots the OCA model at younger ages. The reason may be that the CBD does a better job at forecasting longevity for higher age tranches than it does for lower age tranches.10 Another possibility is that the CBD model incorporates a compounding effect of longevity risk measurements: for higher ages the stochastic component of the CBD model overtakes the OCA, whereas the implicit compounding of risk before normal retirement age in the CBD model increases the risk of surviving to a higher age. The CBD and the OCA model are most similar at the normal retirement age where the two types of risk reach a saddle point.

The evaluation in Table 2 is accomplished using the most probable trajectory of mortality rates. Longevity risk is defined as a departure from this trajectory. The annual report of the RCMP's pension plan gives no information regarding the OCA's parameter uncertainty with respect to mortality improvements. In contrast, the CBD model is constructed to give us a distribution of future mortality improvements.

To demonstrate how sensitive the present value of future payments is to forecasted mortality rates, we calculate the relative and nominal VaR at the 95th and 99th levels for the CBD model. This risk measure is defined as the maximum probable loss (with 95 percent or 99 percent certainty) in the case of an adverse deviation in the evolution of mortality rates. All values are presented in Table 3, with the relative VaR on the left (relative to the total liability for each age group and sex) and the nominal VaR on the right. We simulated 100,000 trajectories to calculate the VaRs.

Table 3. Relative (in %) and Nominal (in $ Millions) VaR at the 95th and 99th percentile Levels for the CBD Mortality Forecast Model of the RCMP's Pension Plan's Liability, by Age Group and Sex (Regular Retired Members Only)
 RelativeRelative Value at Risk
 Value at Risk (%)($ Millions)
For the whole plan2.874.350.761.12172.34262.571.011.49

For both men and women, we see a positive correlation between the relative longevity risk and the age of the retired pensioners. The relative VaR increases at younger ages to reach a peak for the age bracket 70–74. In dollar terms, the VaR is remarkably superior for members for whom the actuarial liability is highest, that is, for members aged 50–59. Finally, because longevity risk is mainly driven by the male population in this pension plan, the male population's VaRs are orders of magnitude greater than those of the female population.

If we believe the CBD model over the OCA model, our results suggest that the total liability ascribable to the RCMP's retired regular male members, who represent 94 percent of all retired members, should be close to (1.015 × 1.0287 − 1) 4.545 percent higher than what it is at the moment.11 This represents a total of over $275 million for the $6.393 million of liability ascribable to regular retired pensioners.


To determine actuarial liability with respect to contributors we must determine the amount of future payments based on current annual earnings. According to the RCMP's annual report, the liability ascribable to all regular contributors is $5.737 million. As for retired pensioners, we do not have access to the exact decomposition of the insured population contributors. We therefore use the median age of each group, the median year of pensionable service and the average pensionable income. We also assume for simplicity that a member who has less than 20 years of service receives a deferred annuity payable at age 60 whereas a member with more than 25 years of service collects an immediate annuity. A member that retires after 20–24 years of service is given an annual allowance. All benefits are adjusted for the cost of living after retirement.12 In Table 4, we present total liabilities ascribable to all age categories of contributors (men and women, and age) depending on whether their annuity payment is deferred or immediate.

Table 4. CBD Model Versus OCA Model Mortality Rate Ratio, Calculation of the RCMP's Pension Plan's Liability, and Percentage of Total Contributors’ Population, by Age Group, Sex, and Type of Pension (Deferred in Panels A and C and Immediate in Panels B and D)
AgeRatioCBD LiabilityOCA Liability% Retired
RangeMedianCBD/OCA($ Millions)($ Millions)CBD Model
Panel A: Male Contributors: Deferred Annuity Payments
Panel B: Male Contributors: Immediate Annuity Payments
For all men 1.02075,125.025,021.0887.397
Panel C: Female Contributors: Deferred Annuity Payments
– 25251.01861.951.910.033
Panel D: Female Contributors: Immediate Annuity Payments
For all women 1.0086739.04732.7412.603

Assuming that we constrain the mortality forecast model to be that of the OCA, it is interesting to note that the set of assumptions we make to calculate the total liability of regular contributors gives us a value of (5021.08 + 732.74) $5,754 million. Comparing this approximate total liability of regular contributors to the actual liability of $5,737 million ascribable to the regular contributors in the RCMP's annual report, we note that the difference is less than 0.5 percent. It therefore seems that, on average, our median age and other assumptions make very little difference in calculating the total liability ascribable to regular contributors.

Similar to the results we had in Table 2 for pensioners, we observe in Table 4 larger CBD/OCA ratios for men than for women, but we do not observe the same concave pattern. The reason why the CBD/OCA ratios are larger in Table 4 than in Table 2 is that the amount of time between the calculation date and the payment date is much longer for contributors. Consequently, dissimilarities across the mortality models’ forecast are accentuated and compounded.

The difference between the CBD/OCA ratio results of Table 2 and of Table 4 are even more important for deferred annuities (in Panels A and C of Table 4) than for immediate annuities (Panels B and D). For contributors younger than 25 years, the dissimilarity between the two models is as great as 6.49 percent for men and 1.86 percent for women. Even for contributors of the same age cohort, we see that the CBD/OCA ratio is greater in the case where contributors have to defer their annuity payment than if they are allowed to receive it immediately. The discrepancy between the CBD model and the OCA model diminishes with age for both deferred and immediate annuities. Overall, the actuarial liability is 2.07 percent higher for male contributors under the CBD model than the OCA model. For women, the CBD model values their actuarial liability only 0.86 percent higher than the OCA model.

Table 5 presents our relative and nominal VaR calculations for both immediate (i) and deferred annuities (d), for both male and female contributors. As we mentioned earlier, there is less precision in any model when looking further in the future, so that future mortality rates are naturally harder to predict for younger cohorts. This is illustrated by a decreasing relative VaR with age. The nominal VaR is of course greater for the age bracket and contract type where contributors are mostly concentrated and more numerous.

Table 5. Relative (in %) and Nominal (in $ Millions) VaR at the 95th and 99th percentile Levels for the CBD Mortality Forecast Model of the RCMP's Pension Plan's Liability, by Age Group, Type of Pension, and Sex (Regular Contributors Only)
 RelativeRelative Value at Risk
 Value at Risk (%)($ Millions)
− 25 (d)
25–29 (d)
30–34 (d)5.858.621.682.475.5258.1380.7061.038
35–39 (d)5.438.061.542.2711.93717.7101.3221.946
40–44 (d)4.957.401.402.0616.82025.1151.2201.797
40–44 (i)1.932.870.560.837.07710.5520.6991.029
45–49 (d)4.406.611.241.836.75610.1460.3180.469
45–49 (i)
50–54 (d)3.795.721.081.591.8862.8460.1170.172
50–54 (i)2.383.580.691.0140.81061.5390.7861.159
55 + (d)3.395.120.971.430.4490.6790.0300.044
55 + (i)2.503.780.731.0720.62231.1800.1710.252

We note a higher risk associated with deferred pensions compared to immediate pensions, a higher risk that is compounded by the age of the cohort. In the worst case scenario, say for the group of men 40–44 with deferred pensions, the longevity risk (at the 95 percent level) represents 4.95 percent of the CBD model's expected cost. We also found in Table 4 that the expected cost of this group's pension liability is 4.20 percent higher using the CBD model than the OCA approach (with the caveat of the additional necessary assumptions we needed to make to obtain this number). If we believe the CBD model to be correct, the total liability of the RCMP's pension plan ascribable to 40- to 44-year-old men with deferred pensions should be more than (1 + 0.0495) × (1 + 0.0420) − 1 = 9.36 percent higher. To be sure at 99 percent, the ascribable liability should be (1 + 0.0740) × (1 + 0.0420) − 1 = 11.91 percent higher.

Bringing Everything Together

We bring all of our calculations together in Table 6 to evaluate the compounded longevity risk of each type of regular member (contributing and retired) by age group and sex. We call compounded longevity risk the compounding of model risk, represented by the CBD-to-OCA ratio, and of longevity risk, calculated as the 95th percentile VaR using the CBD model projection of future mortality. The column labeled “Weight” refers to the liability importance of that category in the pension plan.13 Panel A presents the case of current retired pensioners, Panel B presents the case of contributors who are allowed a deferred pension, and Panel C presents the case of contributors who are allowed an immediate pension.

Table 6. CBD Model Versus OCA Model Mortality Rate Ratio and Calculation of the RCMP's Pension Plan's Liability, and percentage of Total Members by Age Group and Sex
Age RangeRisk(95%)Long. RiskTotalRisk(95%)Long. RiskTotal
Panel A: Retired Regular Members
Panel B: Regular Member Contributors: Deferred Annuity Payments
Panel C: Regular Member Contributors: Immediate Annuity Payments

In Table 7, we calculate the averages by type of member and by sex. This means that for the entire pension plan of the RCMP we can calculate the weighted model risk as (1.0338 × 92.73 percent + 1.0082 × 7.27 percent − 1 = ) 3.2 percent for these members who represent 85 percent of the total liabilities of the RCMP's pension plan. Weighted longevity risk (at the 95th percentile level) represents 2.5 percent of the best estimate of the cost of the pension plan given by the CBD model for longevity. The compounded longevity risk of the RCMP's pension plan is therefore almost 6 percent of total liabilities ascribable to retired and contributing regular members. In dollars, 6 percent of the total liabilities calculated according to the OCA model represent $720 million. This means that if the RCMP's pension plan was to recognize model risk and longevity risk in its balance sheet, and price it accordingly, the plan's actuarial surplus would completely disappear… not to mention that we calculated the longevity and model risks for less than 85 percent of the plan's total liabilities.

Table 7. CBD Model Versus OCA Model Mortality Rate Ratio and Calculation of the RCMP's Pension Plan's Liability by Member Category and Sex
Panel A: Total by Type
Age RangeRisk(95%)Long. RiskTotalRisk(95%)Long. RiskTotal
Panel B: Total by Sex
Age RangeRisk(95%)Long. RiskTotalRisk(95%)Long. RiskTotal

It is true that our estimations are not precise since we do not have access to all the information that is available to the actuaries of the RCMP's pension plan. We nevertheless obtain that our estimation error represented only 1 percent of total liabilities.14 Also, we used a 95th percentile level for our VaR calculations, which means that on average once every 20 years exceptional contributions could become necessary. Using a 99th percentile level instead, the longevity and model risk rises to over 7.25 percent of total liabilities ascribable to retired and contributing regular members, which amounts to $890 million.

Conclusions and Further Discussion

This article presents an assessment of the implicit cost of compounded longevity risk for a pension plan using the case of the RCMP as the backdrop. Risk is defined in our context as the possibility that the pension plan will run out of funds before the last pensioner has died, not because of a misappropriation of funds or bad investment, but because the pensioners’ life expectancy was underestimated. This risk is becoming more and more important in value as the general population is living longer upon reaching normal retirement age and the financial impact of misestimating future mortality is sizable for pension plan sponsors, life-annuity providers, and the economy in general (see Olshansky et al., 2009). It is thus imperative for a pension fund to quantify correctly not only future longevity improvements, but also the risk associated with misestimating future mortality rates for the segment of the population that is retired. Failure to do so will lead to higher payouts than anticipated, which will result in the sponsoring company suffering major losses.15 Ultimately, unforecasted longevity improvements could cause financial distress for the pension plan's sponsor, or delay the retirement date for workers.

We implement the CBD two-factor stochastic mortality model of Cairns et al. (2006) using the Canadian mortality tables. We then apply the CBD model to calculate the potential risk associated with an increase in longevity for the pension plan of the RCMP. The present value of the expected future benefits (i.e., the plan's actuarial liability) is also calculated using the projected mortality rates as forecasted by the OCA of Canada and these are compared to the actuarial liability obtained using the CBD model. We calculate the necessary reserves that would ensure that, with 95 percent or 99 percent certainty, the plan will not require an extraordinary contribution because the number of deaths was overestimated. Our estimates show that to compensate for longevity risk alone pension funds should increase their reserves by 6–9 percent, depending on the pension plan's maturity, composition of beneficiaries, and structure and level of certainty needed.

If our projections from the CBD model turn out to be correct, the economic impact of compounded longevity risk is not trivial. However, our model may even be too optimistic in the sense that stochastic mortality trends obtained with the CBD model may be overestimated. The reason is that both models assume implicitly that longevity improvements will be continuous, even in the far future. If, on the other hand, future mortality trends vary by leaps and bounds rather than continuously, our forecast overestimates future mortality, which means that current plan sponsors are underreserving for future longevity risk shocks. There are no clear ways of knowing ex ante whether the CBD model is adequate and superior to any other. For instance, by comparing the assumptions made by the OCA of Canada with those of the CBD model, we can clearly ascertain that there is a significant divergence in the present value of future benefits. These differences can eventually imply several hundreds of millions of dollars in either additional contributions or surplus. Both cases can imply negative consequences for the plan's promoter.

This article contains results on the magnitude of longevity and mortality model risk for a particular pension plan and a particular mortality model. It provides a comparison between commonly used actuarial forecast of mortality and the increasingly important stochastic mortality models. This issue is clearly of interest to other defined benefit pension plan sponsors and public policymakers. In particular, there exists some evidence that governments are systematically underestimating the longevity exposure they face at a national level in terms of social security pensions, etc. (see Shaw, 2007, for evidence of this for the United Kingdom). Now if a government's chief actuary is underestimating life expectancy at the level of the national population, there is a good chance that the same will be done at the level of the population in a public sector pension plan. This question is important and requires one to conduct a large-scale empirical analysis to provide further evidence from other countries.

  1. 1

    See Ambachtsheer (2008) for an analysis of the general issues surrounding the pension debate in Canada and Antolin (2007) for the case in the Organisation for Economic Co-operation and Development (OECD).

  2. 2

    A second advantage of using a one-factor model is that it makes the pricing of derivative instruments less complex since only one parameter needs to be forecasted. The added complexity associated with a second factor will become apparent below and is even more evident when one tries to price financial derivative instruments that have more than one cash flow, such as swaps (see Boyer et al., 2012, for more details on the pricing of survival swaps, and Boyer and Stentoft, 2013, for the pricing of survivor options).

  3. 3

    Empirically, the reliance on a unique factor leads to problems of underfitting, especially at younger ages (see Gaille and Sherris, 2011; Ouellet, 2011). There are many other variants to the Lee–Carter model that are highly parameterized and usually include a single parameter that fits mortality across time (see, e.g., Tuljapurkar et al., 2000). Wilmoth (1993), Tabeau et al. (2001), and Barugola and Maccheroni (2007) apply the Lee–Carter model to individual causes of death (see also Hanewald, 2011). Lamb (2011) shows that Lee–Carter models that use socioeconomic-specific information have promising applications. Gaille and Sherris (2010) show how different causes of death have different age patterns by country and different time trends, so that an approach that allows for time dependency and long-run trends between the parameters is necessary.

  4. 4

    The data source for Figures 1 and 2 is the Human Mortality Database (2010). For more details on mortality trends in Canada, see Ouellet (2011).

  5. 5

    An important reason for choosing this particular pension plan is that the actuarial reports for the RCMP are publicly available.

  6. 6

    We used the entire Canadian population in the calibration of the CBD model whereas the OCA uses the population that has access to a pension plan. It is well known that mortality dynamics are not the same for these two populations. The reader interested in this “basis risk problem” is invited to consult Li and Hardy (2011), Cairns et al. (2011), and Zhou et al. (2011).

  7. 7

    The information provided is the only one that is required to be made public, presumably because of the risk of identification of individuals in the pension plan.

  8. 8

    It is important to recall that the total reported liabilities of the RCMP's pension plan ascribable to retired regular members total $6,393 million (see Table 1). This value was calculated by the pension plan's actuaries using the real distribution of age and benefits of each retired regular member and by using the OCA's forecast of future mortality.

  9. 9

    Interestingly, the difference between the CBD total liability and the OCA total liability ($89 million) is much less than the difference between the median age liability we calculated using the OCA method and the reported actual age liability ($340 million).

  10. 10

    We are indebted to an anonymous referee for pointing that out.

  11. 11

    For women, it has to be (1.006 × 1.0076 − 1) 1.365 percent higher.

  12. 12

    In line with the information contained in the annual report of the RCMP's pension plan, we also assume that the annual pension upon retirement corresponds to 2 percent of the average pensionable income multiplied by the number of credited years of service under the plan. Starting at age 65, the annual pension amount is reduced by a certain percentage due to the integration with the indexed Canadian Pension Plan (CPP) of with the Régie des rentes du Québec (RRQ). See the appendix for more details.

  13. 13

    We calculate weighted risk, for both model and longevity risks, asinline image, where n is one of the N categories of regular members (retired vs. contributors, men vs. women, younger vs. older, immediate vs. deferred), inline image is the total liability of the category's member as a percentage of total liability, and inline image is the risk value of members of category n.

  14. 14

    The total liability ascribable to retired regular pensioners and current regular member contributors, of all ages and both sexes, is $12,007 million using the OCA projections and our assumptions regarding age and salary distribution. The balance sheet of the RCMP gives a total liability for these two groups of individuals of 12, $130 million. The difference of 1 percent between our approximation of the OCA approach and the exact OCA number is therefore marginal.

  15. 15

    A similar issue afflicts reverse mortgage providers (see Wang et al., 2008).


In this appendix, we provide an overview of the two-factor stochastic mortality model of Cairns et al. (2006) and explain how parameter uncertainty can be accommodated in the model. We then review the economic and demographic hypothesis made in order to calculate plan liabilities. Finally, we explain how we take into account the integration of the RCMP's pension plan with the CPP/QPP.

Summary of the CBD Mortality Model

Let inline image represent the probability that a person aged x in calendar year t dies within 1 year that follows:

display math

Instead of modeling inline image directly, we can model inline image so that we can estimate inline image and inline image for all (t) using an ordinary least squares regression technique. Given the estimated parameters we run a simulation to obtain the forecasted values of inline image and use those values to calculate inline image as:

display math

This enables us to obtain the evolution of mortality rates through time as inline image

Since the initial parameters of our model are subject to some degree of uncertainty due to limited information, we will define the innovations in the parameter values as

display math
display math
display math

Given these dynamics, we can adjust our results for parameter uncertainty. For example, based on m observations, Cairns et al. (2006) suggest to simulate V from the inverse Wishart distribution (i.e., inline image) and μ from a multivariate normal distribution (i.e., inline image) and use these values for the rest of the sample path. We are then able to compute different trajectories that mortality improvements can take.

Economic, Demographic, and Other Assumptions

We will use the OCA's economic and demographic hypotheses to calculate the present value of the pension plans’ liability. This allows us to focus exclusively on the mortality improvement differences between the CBD and OCA models. Table A1 presents the nominal rates and indexing adjustments used to calculate the actuarial value of pensions as presented in the OCA reports and those used to calculate the total expected liabilities of the RCMP pension plan. Table A2 shows the initial and ultimate plan year mortality reductions (percent) according to the OCA method and according to the CBD model applied to the Canadian experience.

Table A1. Main Inputs (Interest Rates and Indexing Adjustments) Used to Calculate the Actuarial Value of Pensions as Presented in the OCA Reports
YearInflation (%)Real Rate (%)Nominal Rate (%)Indexing (%)
Table A2. Initial and Ultimate Plan Year Mortality Reductions (%) for Men and Women According to the OCA Model and the CBD Model

For the CBD model, we fitted a least squares line through the natural logarithms of the projected central death rates. The average annual rates of improvement were then obtained by taking the complement of the exponential of the slope. To be consistent with OCA assumptions, the CBD model was calibrated using data from 1989 to 2006. The OCA anticipates that the difference between mortality rates for men and women will diminish in Canada over the next two decades. This means that Canadian men should experience, in the next two decades, higher mortality improvement rates than Canadian women. Mortality improvements should eventually be the same starting in 2029. In the OCA model, mortality reductions between 2009 and 2029 are calculated using linear interpolation so that mortality improvement rates are constantly diminishing over these 20 years. Yearly mortality rate improvements are then assumed to remain constant thereafter. In contrast, the CBD model assumes, by construction, that mortality rate improvements will be constant over the entire horizon, and therein lies one of the main differences between the OCA and the CBD models apart from the mere evolution of mortality rates through time.

Although our calculations uses the same mortality improvements as those of the overall population, we nevertheless use mortality rates that are plan specific and lower than the rates of the general population. Table A3 presents the assumed mortality rates for 2009 of individuals aged between 30 and 110 as they are reported in the actuarial report of the RCMP's pension plans.

Table A3. Assumed Mortality Rates for 2009 of Individuals Aged Between 30 and 110: Male and Female Regular Members of the Royal Canadian Mounted Police (RCMP)
 Regular Members

Mortality rates are given only for successive 10-year intervals of age. We used a cubic spline interpolation to find the mortality rates for intermediate ages. We then transformed mortality rates into central death rates and retro-engineered the values to the plan year by applying the following relation inline image, where inline image corresponds to the improvement in mortality for a person aged x at time t. Finally, starting in 2008, we applied mortality reductions to the central death rates for the remainder of the time horizon using both the OCA and the CBD models.

Integration with CPP/QPP

CPP payments increase as a function of the annual pensionable income multiplied by the number of years of CPP pensionable service. In our evaluation, all contributors will reach age 65 in 2012 or later. Consequently, in all cases, the appropriate reduction factor will be 0,625 percent as we see in Table A4, Panel A. Furthermore, as presented in Table A4, Panel B, in the case of an annual allowance, payments are further reduced by 5 percent for each year by which pensionable service is less than 25 or by which the age at retirement is less than 60 (the lesser of the two).

Table A4. Canada Pension Plan/Quebec Pension Plan Contribution to the RCMP's Pension Payments and an Example of the Liability Reduction
Panel A: Coordination Percentage by Year
 Calendar Years
Coordination percentage0.685%0.670%0.655%0.640%0.625$
Panel B: Example of the Impact That CPP/QPP Integration Has on the Pension Plan's Liability
Age at departure42 years old
Average salary$91,343
Years of pensionable service22
Pension that would have been payable starting at age 602% × 22 × $91,343 = $40,191
Reduction5% × (25 − 22) × $40,191 = $6,029
Annual allowance payable from age 42$40,191 − $6,029 = $34,162
CPP/QPP integration reduction0,625% × 22 × $42,460 = $5,838
CPP/QPP integration reduction$34,162 − $5,838 = $28,324

Taking into account the integration of the RCMP's pension plan with the CPP/QPP is important. The example in Panel B of Table A4 shows that, omitting any pension increases due to indexing, a regular male member that retires in 2008 at age 42 with 22 years of pensionable service receives $34,162 in annual allowances until he reaches the age of 65 at which point he receives $28,324 from the RCMP pension plan, and $5,838 from the CPP/QPP. This means that the liability for this member in the RCMP's pension plan is not the present value of the full amount until the member's death, but rather the present value of $34,162 until the member's 65th birthday, plus the present value of $28,324 until the member's death.


  • M. Martin Boyer is the CEFA Professor of Finance and Insurance, and CIRANO Fellow at HEC Montréal; e-mail: martin.boyer@hec.ca.

  • Joanna Mejza was a graduate student in Financial Engineering at HEC Montréal.

  • Lars Stentoft is an Associate Professor of Finance and member of CIRPÉE at HEC Montréal, and Visiting Professor of Finance and member of CREATES at the Copenhagen Business School.