Phantoms Never Die: Living with Unreliable Mortality Data ANDREW - PowerPoint PPT Presentation

Phantoms Never Die: Living with Unreliable Mortality Data ANDREW CAIRNS, DAVID BLAKE, AND KEVIN DOWD Tuesday, December 10, 2013 Thought-Leading Sponsor 0255848-00001-00 Prudential Financial Inc. headquartered in the United States is not affiliated with Prudential plc in the United Kingdom

1 Phantoms Never Die Living with Unreliable Mortality Data Andrew Cairns Heriot-Watt University David Blake City University London Kevin Dowd Durham University Supported by Prudential Retirement

2 Journey of Discovery • December 2012: ONS revises population estimates • Investigation commissioned by Prudential Retirement • Process of discovery ⇒ – there is a need for fundamental reviews of all official mortality data and how people interpret this data – stochastic mortality modelling potentially flawed until this review is complete.

3 Potential Errors in Revised Population Estimates Estimated Error (%), phi(t,x) 10 5 90 0 80 −5 Age, x 70 −10 60 −15 50 −20 −25 40 1970 1980 1990 2000 2010 Year, t Source data: ONS EW males deaths and revised population estimates.

4 Plan 1. Background and motivation 2. Data issues: deaths, population, exposures 3. Graphical diagnostics and signature plots 4. Model-based analysis of historical population data 5. Impact of exposure errors on future mortality projections 6. Conclusions and next steps

5 1: Background and Motivation • England and Wales data • D ( t, x ) : Deaths counts and age at death are considered to be accurate • P ( t + 1 2 , x ) : Mid-year population estimated by Office for National Statistics (ONS) – No ID card system with 100% coverage – Censuses every 10 years: . . . . . . , 1981, 1991, 2001, 2011 – Censuses ⇒ ’under-enumeration’ – Further difficulties between censuses ∗ imperfect migration records ∗ attribution of deaths to specific cohorts

6 Revisions After the 2011 Census ONS: December 2012 ⇒ • Results of 2011 census finalised • At some ages: 2011 census not consistent with post-censal estimates ( 2001 → 2011 ) • Mid-year population estimates for 2002 to 2010 revised • Significant revisions at some ages

7 Impact of Population Revisions on Mortality Rates EW Males Mortality Rates in 2010 EW Males Mortality Rates in 2010 Ratio of revised rates to old rates 1.2 0.500 Old rates 1.1 Revised Rates 0.200 1.0 0.050 0.9 0.020 0.005 0.8 60 70 80 90 100 0 20 40 60 80 100 Figure 1:

8 Why Does it Matter? Potential impact on • Population mortality forecasts • Forecasts of sub-population mortality • Calibration of multi-population models • Calculation of annuity liabilities and Value-at-Risk • Assessed levels of uncertainty in the above • Buyout pricing • Assessment of basis risk in longevity hedges • Assessment of hedges and hedging instruments

9 Types of Impact: Base Table; Central Trend; Future Uncertainty M7 − CBD 0.500 Age 94 Revised Mortality Age 84 0.100 Death Rate (log scale) Original Mortality Age 74 0.020 Age 64 Age 54 0.005 0.001 1990 2000 2010 2020 2030 2040 2050 2060 Year

10 Questions/Issues to Address: • What immediate impact on forecasts resulting from the 2012 revisions? • Be aware that 2012 revision not a one off. • Be aware that issues are relevant to all countries. • Can we further clean the exposures data? • Impact of potential further anomalies on forecasts? • How do we allow for future revisions?

11 2: Data Issues: Deaths, Population, Exposures • Data: deaths, population, births – where can errors occur? • Themes: – facts – conjectures – consequences of facts and conjectures

12 2.1: Deaths • Published death counts, D ( t, x ) – deaths in calendar year t – age x last birthday at date of death • Regarded as accurate, BUT ... – potential errors in recorded age at death

13 2.2: Population Estimates, Exposures, Death Rates Death rate m ( t, x ) = D ( t, x ) E ( t, x ) • E ( t, x ) = ’exposure’ in year t (central exposed to risk) = average value of P ( s, x ) from t to t + 1 – P ( s, x ) = population at exact time s aged x last birthday • England & Wales ⇒ only P ( t + 1 2 , x ) reported • Common assumption: E ( t, x ) = P ( t + 1 2 , x ) – e.g. ONS reported death rates: m ( t, x ) = D ( t, x ) /P ( t + 1 2 , x )

14 2.3: Where Can Errors in E ( t, x ) Occur? • Known errors: Inaccurate P ( t + 1 2 , x ) – no ID card system – infrequent censuses, under-enumeration – migration etc. – mis-reported age at census • Lesser known errors: – inaccurate shift from census date to mid-year – assumption that P ( t + 1 2 , x ) ≈ E ( t, x )

15 Where Can Errors in E ( t, x ) Occur? ✲ Four sources of error: ✲ ✲ Census PPPPP P q Mid-year Exposures ✲ Population ✏ ✶ ✏✏✏ Migration ✲ Errors that can be mitigated using CBD Exposures Methodology

16 2.3.1: Propagation of General Errors Through Time • Errors follow cohorts • Phantoms never die

Phantoms Never Die 0 500 1000 1500 True + error Census 0 Estimate 1 True Population 2 3 4 5 Year Never Die Phantoms 6 7 8 9 10 Post−censal Estimate 10 New Census Estimate 17

18 Factual Consquence: Backfilling (ONS Methodology) 1500 True + error Inter−censal Inter−censal Estimate 1000 Adjustments New Census Estimate Revised 500 True Population 0 0 1 2 3 4 5 6 7 8 9 10 10 Year

19 2.3.2: Census to Mid-year Shift Census Mid−year x+1 A A B P ( T c , x ) P ( T c + ω c , x ) B x C C D P ( T c , x − 1 ) D x−1 Time ONS 2001 assumption: birthdays spread evenly throughout the year Conjecture: – Different methodology used in earlier censuses and in 2011

20 Can We Improve on This Assumption? The Cohort Births/Deaths (CBD) Exposures Methodology Underlying hypothesis: • At any point in time t , pattern of birthdays at t will reflect – actual pattern of births x years earlier – deaths (impact at high ages) – migration and birth patterns of immigrants • Irregular pattern of births can lead to errors in census → mid-year shift

21 Quarterly Births 250 Quarterly Births (’000s) 200 150 100 50 Spanish Flu WW1 0 1910 1912 1914 1916 1918 1920 1922 1924 Time, t

22 Percentage Difference Between ONS Methodology and Cohort Births/Deaths Methodology 10 1919 Cohort CBD Benchmark +9.2% ONS relative to CBD Percentage Difference 5 1941 0 1947 −5 1920 −6.3% −10 1920 1940 1960 1980 2000 Mid−year Birth Cohort

23 2.3.3: Proposal to Improve Estimates of Exposures • Death rate m ( t, x ) = D ( t, x ) /E ( t, x ) • Current assumption: E ( t, x ) = P ( t + 1 2 , x ) • CBD Exposures Methodology: Assume E ( t, x ) = P ( t + 1 E ( t − x, 0) 2 , x ) × P ( t + 1 2 − x, 0) • E ( t − x, 0) /P ( t + 1 2 − x, 0) = Convexity Adjustment Ratio • CAR based on monthly pattern of births over t − x − 1 to t − x + 1

24 CBD Exposures Methodology: Convexity Adjustment Ratio CBD Convexity Adjustment Ratio 1.04 1919 1.02 1946 1.00 0.98 1947 1920 0.96 1910 1920 1930 1940 1950 Cohort Year of Birth, t−x

25 2.3.4: High Age Methodology • ONS reports – P ( t + 1 2 , 90+) only – D ( t, x ) for x = 90 , 91 , 92 , . . . • P ( t + 1 2 , x ) for x = 90 , 91 , . . . derived using the Kannisto-Thatcher Method (extinct cohorts) • Conjecture: Potential for inconsistencies at the boundary between ages 89 and 90+

26 2.4: Facts and Conjectures ⇒ Consequences, Anomalies Statistically, how significant are these anomalies? • Graphical diagnostics – hypothesis ⇒ plot should exhibit specific characteristics • Signature plots – what if it does not? • Model-based analysis

27 3: Graphical Diagnostics and Signature Plots 3.1: Graphical Diagnostic 1 Hypothesis: Crude death rates by age for successive cohorts should look similar. ⇒ Plot crude death rates against age.

28 Cohort Death Rates: 1907 to 1911 birth cohorts 0.50 1911 1910 1909 1908 1907 0.20 0.10 0.05 70 75 80 85 90 95 100 Cohort death rates by age for 1907 to 1911 cohorts. ONS revised EW males data up to 2011.

29 Signature Plot: Emergence of Phantoms Cohort Death Rates: 1917 to 1921 birth cohorts 0.20 1921 1920 0.15 1919 1918 1917 0.10 Phantoms emerging 0.05 65 70 75 80 85 90

30 3.2: Graphical Diagnostic 2 Hypothesis: Underlying log death rates are approximately linear ⇒ Plot concavity of log death rates: the difference between log of one death rate and the average of its immediate neighbours: C ( t, x 0 ) = log m ( t, x 0 + t ) ( ) − 1 log m ( t, x 0 + t − 1) + log m ( t, x 0 + t + 1) 2 If log death rates are linear then this should be close to 0.

31 1924 Cohort Log Death Rates: Deviation Between 1924 Cohort and the Average of its Nearest Neighbours 0.2 0.1 Concavity 0.0 −0.1 −0.2 1960 1970 1980 1990 2000 2010 Year, t Dots are randomly above and below 0.

32 1920 Cohort Log Death Rates: Deviation Between 1920 Cohort and the Average of its Nearest Neighbours 0.2 0.1 Concavity 0.0 −0.1 CBD adjustment for Emerging phantoms in uneven births 1919 cohort using Cohort Adjustment Ratio −0.2 1960 1970 1980 1990 2000 2010 Year, t Signature plot: births pattern ⇒ true E ( t, x ) < P ( t + 1 2 , x )