Prudential Financial, Inc. · NYSE:PRU
Prudential Financial — Technical Presentation (December 2013)
65 slides from Prudential Financial's technical presentation, Dec 10, 2013
Slide 1: Phantoms Never Die: Living with Unreliable Mortality Data
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
Slide 2: Phantoms Never Die: Living with Unreliable Mortality Data
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
Slide 3: Journey of Discovery
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 potenti
Slide 4: Potential Errors in Revised Population Estimates
3 Potential Errors in Revised Population Estimates Estimated Error (%), phi(t,x) 10 90 5 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.
Slide 5: Plan
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
Slide 6: 1: Background and Motivation
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 + 21 , x): Mid-year population estimated by Office for National Statistics (ONS) – No ID card system with 100% coverage – Censuses every 10 years: . . . . . . , 1981
Slide 7: Revisions After the 2011 Census
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
Slide 8: Impact of Population Revisions on Mortality Rates
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 60 70 80 90 100 0.8 0 20 40 60 80 100 Figure 1:
Slide 9: Why Does it Matter?
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
Slide 10: Types of Impact: Base Table; Central Trend; Future Uncertainty
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) Age 74 Original Mortality 0.020 Age 64 Age 54 0.005 0.001 1990 2000 2010 2020 2030 2040 2050 2060 Year
Slide 11: Questions/Issues to Address
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?
Slide 12: 2: Data Issues: Deaths, Population, Exposures
11 2: Data Issues: Deaths, Population, Exposures • Data: deaths, population, births – where can errors occur? • Themes: – facts – conjectures – consequences of facts and conjectures
Slide 13: 2.1: Deaths
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
Slide 14: 2.2: Population Estimates, Exposures, Death Rates
13 2.2: Population Estimates, Exposures, Death Rates D(t, x) Death rate m(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 + 12 , x) report
Slide 15: 2.3: Where Can Errors in E(t, x) Occur?
14 2.3: Where Can Errors in E(t, x) Occur? • Known errors: Inaccurate P (t + 12 , 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 + 12 , x) ≈ E(
Slide 16: Where Can Errors in E(t, x) Occur?
15 Where Can Errors in E(t, x) Occur? - Four sources of error: - - Census PP PP P q P Mid-year - Exposures Population 1 Migration - Errors that can be mitigated using CBD Exposures Methodology
Slide 17: 2.3.1: Propagation of General Errors Through Time
16 2.3.1: Propagation of General Errors Through Time • Errors follow cohorts • Phantoms never die
Slide 18: Phantoms Never Die
0 500 1000 1500 Census 0 True + error Estimate 1 2 Phantoms Never Die 3 4 True Population 5 Year 7 6 Never Die Phantoms 8 9 10 Post−censal Estimate 10 New Census Estimate 17
Slide 19: Factual Consequence: Backfilling (ONS Methodology)
18 Factual Consquence: Backfilling (ONS Methodology) 1500 True + error Inter−censal Estimate Inter−censal 1000 Adjustments New Census Estimate Revised 500 True Population 0 0 1 2 3 4 5 6 7 8 9 10 10 Year
Slide 20: 2.3.2: Census to Mid-year Shift
19 2.3.2: Census to Mid-year Shift Census Mid−year x+1 A A P(Tc, x) B P(Tc + ωc, x) B x C C P(Tc, x − 1) D D x−1 Time ONS 2001 assumption: birthdays spread evenly throughout the year Conjecture: – Different methodology used in earlier censuses and in 2011
Slide 21: Can We Improve on This Assumption?
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 •
Slide 22: Quarterly Births
21 Quarterly Births 250 Quarterly Births (’000s) 200 150 100 50 WW1 Spanish Flu 0 1910 1912 1914 1916 1918 1920 1922 1924 Time, t
Slide 23: Percentage Difference Between ONS and CBD Methodology
22 Percentage Difference Between ONS Methodology and Cohort Births/Deaths Methodology 10 1919 Cohort +9.2% CBD Benchmark ONS relative to CBD Percentage Difference 5 1941 0 1947 −5 1920 −6.3% −10 1920 1940 1960 1980 2000 Mid−year Birth Cohort
Slide 24: 2.3.3: Proposal to Improve Estimates of Exposures
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 + 12 , x) • CBD Exposures Methodology: 1 E(t − x, 0) Assume E(t, x) = P (t + , x) × 2 P (t + 12 − x, 0) • E(t − x, 0)/P (t + 12 − x, 0) = Convexity Adjustment Ratio • CAR
Slide 25: CBD Exposures Methodology: Convexity Adjustment Ratio
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
Slide 26: 2.3.4: High Age Methodology
25 2.3.4: High Age Methodology • ONS reports 1 – P (t + 2 , 90+) only – D(t, x) for x = 90, 91, 92, . . . • P (t + 12 , 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+
Slide 27: 2.4: Facts and Conjectures, Consequences, Anomalies
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
Slide 28: 3.1: Graphical Diagnostic 1
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.
Slide 29: Cohort Death Rates: 1907 to 1911 birth cohorts
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.
Slide 30: Signature Plot: Emergence of Phantoms
29 Signature Plot: Emergence of Phantoms Cohort Death Rates: 1917 to 1921 birth cohorts 0.15 0.20 1921 1920 1919 1918 1917 0.10 Phantoms emerging 0.05 65 70 75 80 85 90
Slide 31: 3.2: Graphical Diagnostic
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, x0 ) = log m(t, x0 + t) ( ) − 12 log m(t, x0 + t − 1) + log m(t, x0 + t +
Slide 32: Concavity: 1924 Cohort
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.
Slide 33: Concavity: 1920 Cohort
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
Slide 34: Concavity: 1947 Cohort
33 1947 Cohort Log Death Rates: Deviation Between 1947 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 Dosts mostly below 0 ⇒ cause for concern
Slide 35: Same Data in 2-Dimensions: Heat Map
34 Same Data in 2-Dimensions: Heat Map Concavity of log m(t,x) 0.4 90 0.2 80 Age, x 70 0.0 60 −0.2 50 −0.4 1970 1980 1990 2000 2010 Year, t Sampling variation ⇒ more extremes < 50 and > 90
Slide 36: Heat Map: by Age and Calendar Year
35 Heat Map: by Age and Calendar Year Identifiable non-random patterns Signatures: • Diagonals ⇒ issues with a cohort • Horizontals ⇒ anomalies in reported age at death ??? • Age at death errors are more plausible than systematic age-dependent errors in exposures.
Slide 37: 3.3: Graphical Diagnostic
36 3.3: Graphical Diagnostic 3 Hypothesis: Changes in cohort population sizes should match pattern of reported deaths • Underlying data: – mid-year population, P (t + 12 , x) – deaths in one calendar year, D(t, x) ˆ + 1 , x) = P (t + 1 , x) − P (t + 3 , x + 1) • Define d(t 2 2 2 ˆ + 1 , x) by cohort
Slide 38: Standard Graphical Diagnostic 3: 1924 Cohort, Deaths Curve
37 Standard Graphical Diagnostic 3: 1924 Cohort, Deaths Curve 1924 Cohort 10000 d−hat(t,x) D(t,x) D(t+1,x) D(t,x+1) D(t+1,x+1) 5000 Deaths 0 Age: 50 60 70 80 −5000 1970 1980 1990 2000 2010
Slide 39: Signature Plot: Backfilling the 1919 Cohort by ONS
38 Signature Plot: Backfilling the 1919 Cohort by ONS 1919 Cohort Age 90 10000 d−hat(t,x) D(t,x) D(t+1,x) D(t,x+1) D(t+1,x+1) 5000 Deaths Parallel shift 0 Age: 60 70 80 90 −5000 1970 1980 1990 2000 2010
Slide 40: Possible Explanation: Census to Mid-year Pop Error
39 Possible Explanation: Census → Mid-year Pop Error 1919 cohort (stylized) Census −> Mid−year Census −> Mid−year Estimate 1500 Estimate Post−censal Estimate 1000 500 True Population 0 1991 1993 1995 1997 1999 2001 Year
Slide 41: Factual Consequence: Backfilling (ONS Methodology)
40 Factual Consquence: Backfilling (ONS Methodology) 1919 cohort (stylized) Census −> Mid−year Estimate 1500 Inter−censal Estimate Inter−censal 1000 Adjustments Revised 500 True Population 0 1991 1993 1995 1997 1999 2001 Year
Slide 42: 1918, 1919 and 1920 Cohorts, Deaths Curves
41 1918, 1919 and 1920 Cohorts, Deaths Curves 1918 Cohort 1919 Cohort 1920 Cohort Age 90 Age 90 Age 90 10000 10000 10000 d−hat(t,x) D(t,x) D(t+1,x) D(t,x+1) D(t+1,x+1) 5000 5000 5000 Deaths Deaths Deaths 0 0 0 Age: 60 70 80 90 Age: 60 70 80 90 Age: 60 70 80 90 −5000 −5000 −5000 1970 1980 1990 2000 2
Slide 43: Signature Plot: Backfilling the 1947 Cohort
42 Signature Plot: Backfilling the 1947 Cohort 1947 Cohort 10000 d−hat(t,x) D(t,x) D(t+1,x) D(t,x+1) D(t+1,x+1) 5000 Deaths 0 Age: 30 40 50 60 −5000 1970 1980 1990 2000 2010 Again consistent with ONS versus CBD methodologies
Slide 44: 3.4: Summary
43 3.4: Summary • Errors remain in the ONS population data • Combination of three graphical diagnostics highlight known anomalies (e.g.1919) and some unexpected discoveries (e.g. 1920, 1947 cohorts; age 89/90) • Anomalies characterised by cohort and by age • CBD Exposures Methodology can be used to
Slide 45: 4: Model-Based Analysis of Historical Population Data
44 4: Model-Based Analysis of Historical Population Data 4.1: Proposed Solution: Bayesian Adjustment of Exposures Bayesian prior hypotheses: A: Death counts are accurate B: Exposures are subject to errors – errors following cohorts are correlated through time C: Within each calendar year: – curve of
Slide 46: 4.2: Results: Assume E(t, x) = P(t+1/2, x)
45 4.2: Results: Assume E(t, x) = P (t + 12 , x) Mid-year Population Mean Smoothing Adjustment, phi(t,x) 0.10 Age 90/91 90 0.05 1919 0.00 80 1886 −0.05 Age, x 70 −0.10 1900 60 −0.15 50 −0.20 1920 1947 −0.25 40 1970 1980 1990 2000 2010 Year, t
Slide 47: Exposures Adjusted Using CBD Convexity Adjustment Ratio
46 Exposures, E(t, x), Adjusted Using CBD Convexity Adjustment Ratio Mean Smoothing Adjustment, phi(t,x) 0.10 Age 90/91 90 0.05 1919 0.00 80 1886 −0.05 Age, x 70 −0.10 1900 60 −0.15 50 −0.20 1920 1947 −0.25 40 1970 1980 1990 2000 2010 Year, t E(t, x) = P (t + 12 , x)× Convexity Adjustment Ratio
Slide 48: 4.3: Results
47 4.3: Results 1 • Results confirm conclusions based on graphical diagnostics (e.g. problems with 1919, 1947 cohorts; age 89/90 boundary) • Bayesian approach allows us to quantify rigorously the size of the error
Slide 49: Results
48 Results 2 • CBD Exposures Methodology: – convexity adjustment for E(t, x) ̸= P (t + 21 , x) explains 1920 anomaly – CBD dampens other anomalies (e.g. 1947 cohort) • Other anomalies remain but we have some explanations – 1919 cohort explained by 2001 census + backfilling – age 89/90 ⇒ issues with
Slide 50: Mortality Fan Chart
49 5: Impact of Exposure Errors on Future Mortality Projections Mortality Fan Chart 0.50 Age 94 0.20 1919 Death Rate (log scale) Age 84 Cohort 0.10 1919 1947 Age 74 0.05 1947 0.02 1947 0.01 1990 2000 2010 2020 2030 2040 2050 Year Red: unadjusted exposures. Blue: adjusted exposures
Slide 51: 5.2: Impact on Annuity Valuation: 2% Interest Rate
50 5.2: Impact on Annuity Valuation: 2% Interest Rate 0.06 0.04 0.02 1919 1947 Difference −0.02 Proportional Difference in 2% Annuity Values −0.06 1920 1930 1940 1950 Cohort Year of Birth Significant impact on some annuity values
Slide 52: Why are 1919 annuity values so different?
51 Why are 1919 annuity values so different? M9: 1919 Birth Cohort Conditional Survivorship from End 2010 0.50 Survival Probability (log scale) Mean 8−year survival 0.20 probability: 0.102 0.0912 0.05 Unadjusted exposures Adjusted Exposures Fewer survivors 0.01 92 94 96 98 100 102 104 Age
Slide 53: Annuity Valuation: 2% Interest Rate
52 Annuity Valuation: 2% Interest Rate a2% x Age Unadjusted Adjusted Percentage x Cohort Mean St.Dev. Mean Difference 63 1948 17.30 2.2% 17.23 +0.4% 64 1947 16.90 2.3% 16.67 +1.4% 65 1946 16.04 2.4% 16.08 −0.3% 92 1919 3.06 4.2% 2.93 +4.4%
Slide 54: 5.4: Future Mortality Conclusions
53 5.4: Future Mortality Conclusions Adjustments to exposures ⇒ • Forecasts and fitted cohort effects look more plausible • Limited impact on forecasts by age • Significant impact on some cohorts • Potentially significant impact on annuity portfolio valuation and buyout pricing
Slide 55: 6: Conclusions and Next Steps
54 6: Conclusions and Next Steps • Exposure errors – 2011 census revisions drew attention to potential for errors more widely in population data – errors remain in ONS population data – graphical diagnostics ⇒ powerful model-free toolkit for identifying anomalies (signature plots) – Bayesian framewo
Slide 56: CBD Exposures Methodology Explains Bigger Errors
55 • CBD Exposures Methdology can be used to explain many of the bigger errors – census → mid-year population estimates – mid-year population estimates → exposures – some big errors can be attributed to change in ONS methodology in 2001 – other errors have been identified but only partially explaine
Slide 57: Use of CBD Exposures Methodology to Mitigate Errors in Exposures
56 Use of CBD Exposures Methodology to Mitigate Errors in Exposures - Four sources of error: - - Census PP PP P q P Mid-year - Exposures Population 1 Migration - Errors that can be mitigated using CBD Exposures Methodology
Slide 58: Conclusions (continued)
57 Conclusions (continued) • Potential small biases in reported age at death • Anomaly at ages 89/90 might be due to interface with Kannisto-Thatcher high age methodology
Slide 59: Implications
58 Implications • Errors can make a big difference in annuity valuation • Financially: post WW-2 cohorts (especially 1947) need special consideration. • Sources of errors and variants will apply to other countries e.g. – Netherlands: good registration data, but still needs careful handling of data –
Slide 60: US Males, Ages 40-85, 1971-2010: Similar Issues
59 US Males, Ages 40-85, 1971-2010: Similar Issues Mean Smoothing Adjustment, phi(t,x) 0.04 80 0.02 70 0.00 Age, x −0.02 60 −0.04 −0.06 50 −0.08 40 −0.10 1980 1990 2000 2010 Source data: Human Mortality Database Year, t
Slide 61: Next Steps
60 Next Steps • Need for fundamental reviews of all official mortality data and how users interpret this data • Stochastic mortality modelling – potentially flawed until this review is complete – need to stop assuming exposures = mid-year population
Slide 62: Further Engagement with Stakeholders
61 • Further engagement with a range of stakeholders: – government and supra-national agencies – professional bodies – specialist longevity modelling consultancies – private spector holders of longevity risk (pension plans, insurers, reinsurers) • Better communication between government agencies – e
Slide 63: Methodology and Software Next Steps
62 • Census → mid-year population methodology needs to be revisited • Revisit high-age population methodology • Software for tackling errors in population data – module for graphical diagnostics – model-based quantification of errors – (revised) high-age methodology – output: updated population and
Slide 64: Further Numerical and Modelling Work
63 • Further numerical and modelling work – analysis of data from other countries (US, NL, Germany, ...) – further work on England and Wales ages 90+ ∗ Kannisto-Thatcher methodology – model future revisions to data – what if no further adjustments to ONS 2012 revisions? – impact on multi-population
Slide 65: Opinions expressed are those of the individual authors
The opinions expressed are those of the individual authors. Institutional Investor LLC is not responsible for the accuracy, completeness, or timeliness of the informa- tion contained in the articles herein. If you require expert advice, you should seek the services of a competent professional. No st
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