## _wp1129

## Source details

**Canonical URL:** [_wp1129](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1129.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1129.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1129.pdf.json)

---

### Introduction — purpose and scope
- Purpose: develop a simple methodology for stress testing the funding ratio of defined benefit pension plans aimed at a non-actuarial audience while providing detailed discussion for practitioners.
- Structure: sections include stress testing context and supervision; actuarial factors and functions; pros and cons of liability and costing methods; a real DB plan case and a model plan for single-factor stress tests; conclusions and extensions.
- Key methodological objective: quantify sensitivity of the funding ratio to asset and liability shocks using a parsimonious template (“Model.xls”) and rescale model shock responses to the plan’s reported liabilities.

### Model plan baseline characteristics and core inputs
- Reported plan aggregates:
  - Total assets: LCU3,773 million.
  - Plan actuary liabilities (reported): LCU4,000 million.
  - Support ratio: 413 percent.
  - Funding ratio (reported): 94 percent.
- Valuation method used in the template: Projected Benefit Obligation – Constant Dollar (PBOcd) for active members; single life inflation indexed annuity for benefits.
- Key template assumptions (Appendix III / Input! cell mappings):
  - Entry age y = 20 (Input!D12).
  - Normal retirement age r = 55 (Input!D13).
  - Max age = 115 (Input!D14).
  - Accrual rate ܾ = 1.0% (Input!D22).
  - Labor productivity ݎ݌ = 1.0% (Input!D17).
  - Wage inflation ߨ௪ = 3.5% (Input!D18).
  - Annuity inflation ߨ௔ = 3.5% (Input!D21).
  - Level discount rate baseline for liabilities: 9 percent (template default); alternative yield curves allowed via Input!O11:O136.
- Duration statistics (model plan):
  - Total liability duration: 15 years.
  - Active members duration: 17 years.
  - Retired members duration: 10 years.

### Asset stress tests — concentration and market shocks (Table 5 results)
- Design: symmetrical up to ±25 percent flat shocks by issuer type and by vehicle; FX shock modeled as 25 percent local currency appreciation (unhedged FX).
- Baseline funding ratio: 94.33.
- Funding ratios by market shock (%) — Market shocks and resulting funding ratios (preserved exactly):
  - -25%: Govt. 89.74, Financial 89.99, Real 87.47, Bonds 82.12, Stocks 83.06, FX 88.10, TOT 70.75
  - -20%: Govt. 90.66, Financial 90.86, Real 88.84, Bonds 84.56, Stocks 85.31, FX 89.35, TOT 75.46
  - -15%: Govt. 91.57, Financial 91.73, Real 90.21, Bonds 87.00, Stocks 87.56, FX 90.59, TOT 80.18
  - -10%: Govt. 92.49, Financial 92.59, Real 91.58, Bonds 89.44, Stocks 89.82, FX 91.84, TOT 84.90
  - -5%: Govt. 93.41, Financial 93.46, Real 92.96, Bonds 91.89, Stocks 92.07, FX 93.08, TOT 89.61
  - 0%: Govt. 94.33, Financial 94.33, Real 94.33, Bonds 94.33, Stocks 94.33, FX 94.33, TOT 94.33
  - +5%: Govt. 95.25, Financial 95.20, Real 95.70, Bonds 96.77, Stocks 96.58, FX 95.57, TOT 99.05
  - +10%: Govt. 96.17, Financial 96.06, Real 97.07, Bonds 99.21, Stocks 98.84, FX 96.82, TOT 103.76
  - +15%: Govt. 97.08, Financial 96.93, Real 98.44, Bonds 101.65, Stocks 101.09, FX 98.07, TOT 108.48
  - +20%: Govt. 98.00, Financial 97.80, Real 99.82, Bonds 104.10, Stocks 103.35, FX 99.31, TOT 113.19
  - +25%: Govt. 98.92, Financial 98.67, Real 101.19, Bonds 106.54, Stocks 105.60, FX 100.56, TOT 117.91
- Notable asset-shock observation: an unhedged 25 percent FX shock reduces the funding ratio from 94 percent to 88 percent (FX column: 88.10 at -25%).
- Methodological note: flat percentage changes provide a rough test; duration-based tests are preferred for bonds but duration data were unavailable for this exercise.

### Liability stress tests — interest rates, yield-curve shifts, inflation, longevity, termination

- Interest rate (level rate) shocks — Table 6 (Model Plan – Level Interest Rates):
  - Baseline 9% → Funding Ratio 94.33
  - 8% → Funding Ratio 80.76
  - 7% → Funding Ratio 68.33
  - 6% → Funding Ratio 57.11
  - 5% → Funding Ratio 47.08
  - 4% → Funding Ratio 38.25
  - Observation: decreasing the interest rate assumption from 9 percent to 4 percent reduces the funding ratio from 94 to 38 percent, described as an average 11 percent funding ratio decline for every percentage point change in the interest rate.

- Market yield curve and parallel shifts — Table 7 (Model Plan – Shifts in the Yield Curve):
  - -150bps → FR 26.52
  - -100bps → FR 28.51
  - -50bps → FR 30.59
  - 0 → FR 32.70
  - +50bps → FR 34.86
  - +100bps → FR 37.04
  - +150bps → FR 39.24
  - Example: using a AAA government debt yield curve (end of 2009 US domestic debt curve in the example) the funding ratio is around 33 percent on a termination basis.

- Inflation shocks — Table 8 (Model Plan – Inflation Shocks) — sensitivity matrix (selected rows/columns preserved verbatim):
  - Key aggregated sensitivities:
    - A 100bps increase in the annuity inflation assumption ߨ௔ reduces the funding ratio from 94 percent to 85 percent.
    - A 100bps increase in the wage projection inflation assumption ߨ௪ reduces the funding ratio from 94 percent to 89 percent.
    - A 100bps increase in both ߨ௔ and ߨ௪ reduces the funding ratio from 94 percent to 80 percent.
  - Preserved matrix excerpts:
    - -150bps row: 118.44, 112.85, 107.36, 101.97, 96.70, 91.53, 86.49, -10.24
    - -100bps row: 115.45, 110.01, 104.66, 99.40, 94.26, 89.22, 84.31, -10.24
    - -50bps row: 112.49, 107.19, 101.97, 96.86, 91.84, 86.93, 82.14, -10.24
    - 0bps row: 109.56, 104.39, 99.31, 94.33, 89.44, 84.67, 80.00, -10.25
    - +50bps row: 106.66, 101.63, 96.68, 91.83, 87.07, 82.42, 77.87, -10.25
    - +100bps row: 103.78, 98.88, 94.07, 89.35, 84.72, 80.19, 75.77, -10.25
    - +150bps row: 100.94, 96.17, 91.49, 86.90, 82.39, 77.99, 73.69, -10.25
    - Final row: 5.28, 5.28, 5.28, 5.28, 5.28, 5.28, 5.28
  - Observation: funding ratio is more sensitive to the annuity inflation assumption than to wage inflation; elasticity with respect to annuity inflation is much larger.

- Longevity shocks — Table 9 (Model Plan):
  - Scenario: increasing number of improvement years between 0 and 70; normal retirement age r = 55.
  - Life expectancy at age 55 and FR by improvement years:
    - 0 improvement years: life expectancy 28.39 — FR 94.33
    - 30 improvement years: life expectancy 31.09 — FR 89.46
    - 40 improvement years: life expectancy 31.88 — FR 88.21
    - 50 improvement years: life expectancy 32.62 — FR 87.10
    - 60 improvement years: life expectancy 33.31 — FR 86.13
    - 70 improvement years: life expectancy 33.96 — FR 85.26
  - Example calculation: using US mortality improvement rates from a cited table projected T = 30 years for an individual aged 55 produces Δe_55 = 2.70.

- Termination rate shocks — Table 10 (Model Plan):
  - Decreases in termination rates and resulting FR:
    - -0%: FR = 94.33
    - -10%: FR = 92.18
    - -15%: FR = 91.11
    - -20%: FR = 90.04
    - -25%: FR = 88.97
    - -30%: FR = 87.91
  - Context: retention increases of at least 10 percent across the board motivate testing termination rate decreases between 10 and 30 percent.

### Actuarial costing and liability method findings
- Costing methods produce materially different liabilities and normal cost patterns:
  - ABO (Accrued Benefit Obligation): lowest individual actuarial liability; recognizes only past service.
  - RBO (Retirement Benefit Obligation): highest liability; fully recognizes accrued and accruable benefits.
  - PBOcd, PBOcp, EAOcd, EAOcp produce intermediate liabilities; EAOcd and EAOcp tend to front-load reserving for the sponsor; ABO and PBOcp tend to back-load reserving.
  - TER (Terminal funding method): reserves accrued liability only at normal retirement age; effectively zero reserves for active members under accrual accounting.
- Individual normal cost patterns:
  - Benefit allocation methods: normal costs increase over time, can be steep.
  - Cost allocation (entry age) methods: produce more stable or decreasing normal costs; an entry age cost allocation method prorated as a percentage of salary produces a stable cost pattern by definition.
- Aggregate vs individual valuation:
  - For stable population plans, aggregate costs can be stable even when individual cost patterns vary.
  - Individual (cohort) calculations are preferable when detailed precision is required; aggregate cohort approach acceptable for stress-testing when individual data unavailable.

### Key methodological simplifications and their directional impacts
- Simplifications tending to underestimate liabilities:
  - Exclude ancillary benefits (death, disability).
  - Ignore lump sum commutation.
  - Consider only single life annuities (no joint life).
- Simplifications tending to overestimate liabilities:
  - Assume all members join at entry age y and have immediate vesting.
  - Assume final salary pensions (versus career average).
  - Assume full indexation of pension rights and full longevity insurance irrespective of conditional indexing.
- Ambiguous-impact simplifications:
  - Assume all active members retire at normal retirement age r.
  - Assume pensionable salary equals total remuneration.
- Valuation perspective: template uses PBOcd (plan continuation / ongoing concern) rather than termination valuation; model results are rescaled so that percentage changes under shocks are applied to plan actuary liabilities for stress-test outputs.

### Policy recommendations, supervisory uses, and suggested refinements
- Supervisory role:
  - Use stress testing alongside risk-weighted solvency margins; stress testing complements solvency margins by quantifying assets needed under prescribed scenarios and by graduating supervisory responses.
  - Supervisors can require stress testing results to prioritize scrutiny of plans with poorer results (example jurisdiction: Denmark).
- Suggested refinements to improve stress-test realism and supervisory usefulness:
  - Asset side:
    - Identify interest rate risk at various maturities and credit risk of large exposures or sponsor-related exposures.
    - Incorporate duration information to conduct duration-based bond shocks.
    - Design liquidity shocks to capture forced asset sales at distressed prices for closed plans or plans with low support ratios.
    - Consider multi-asset factor shocks using estimated asset return correlations.
  - Liability side:
    - Improve granularity of age, wage and pension distributions.
    - Consider additional decrement factors (disability, labor-force entry/exit, voluntary unemployment).
    - Reflect gender and types of pensions (joint life, spouse benefits).
    - Consider extreme scenarios (temporary decreases in longevity due to catastrophe) and parametric plan reforms (changes in accrual rates, retirement ages).
    - When discounting liabilities using market yield curves, offset liability changes with changes in interest-rate sensitive asset portions (requires asset durations).
  - Expected cash flow analysis:
    - Present shocks in terms of impact on future cash flows to assess how long assets on a termination basis can meet liabilities; extend template to project asset cash flows assuming future returns, asset dispositions, and allocation rules.

### Selected actuarial tables and exact inputs (Appendix II / III highlights)
- Mortality samples (USA male annuitant mortality rates, Table A1):
  - Age 0: 0.002080
  - Age 20: 0.000499
  - Age 50: 0.002994
  - Age 80: 0.050643
  - Age 115: 1.000000
- Early retirement probabilities (Table A3):
  - Age 55–59: 0.051 (each for 55–59)
  - Age 60: 0.202
  - Age 65: 1.000
- Discount rate (yield curve samples, Table A6):
  - Period 0: rate 0.00000, (1 + s_0) 1.00000, discount factor 1.00000
  - Period 1: rate 0.00466, (1 + s_0) 1.00466, discount factor 0.99536
  - Period 10: rate 0.03021, (1 + s_0) 1.04349, discount factor 0.74261
  - Period 30: rate 0.03929, (1 + s_0) 1.05255, discount factor 0.31469
  - Period 35: discount factor 0.25954

*Source: _wp1129 (IMF working paper excerpt; Model.xls template description, stress-test results, and appendices as provided).*

### References .............................................................................................................

### _wp1129 - References

### Document structure and inventory
- Tables listed (1–10) with topics including: regulatory provisions, risk based solvency requirements, accrued benefits of main actuarial cost methods, actuarial liabilities as a share of RBO, and a series of Model Plan – Stress Tests (Asset Concentration Risk; Level Interest Rates; Shifts in the Yield Curve; Inflation Shocks; Longevity Shocks; Termination Shocks).
- Figures listed (1–7) covering straight life annuities (expected nominal and real cash flows; cash flows per surviving annuitant), individual normal costs, individual actuarial liabilities, actuarial liabilities as a share of RBO, active and retired member distributions, and impact of longevity improvements on survival probabilities.
- Box 1: Continuation and Termination Valuation Methods.
- Appendices I–III: Basic Pension Concepts; General Actuarial Assumptions; The Accompanying Template (Model.xls) and Assumptions.
- Appendix Tables A1–A8: mortality, termination, retirement, wage, discount rate assumptions, and actuarial factors used in the model plan.

### Key terminology and acronyms (Glossary)
- ABO Accrued Benefit Obligation
- AL Actuarial Liability
- DB Defined Benefit
- DC Defined Contribution
- EAOcd Entry Age Obligation – Constant Dollar
- EAOcp Entry Age Obligation – Constant Percent
- EET Exempt-Exempt-Taxed
- EU European Union
- IFRS International Financial Reporting Standards
- IOPRP Institutions for Occupational Retirement Provision
- NC Normal Cost
- OECD Organization for Economic Cooperation and Development
- PBOcd Projected Benefit Obligation – Constant Dollar
- PBOcp Projected Benefit Obligation – Constant Percent
- RBO Retirement Benefit Obligation
- TEE Taxed-Exempt-Exempt
- TER Terminal Funding Method
- VaR Value at Risk

### Introduction — purpose and scope
- Purpose: develop a simple methodology for stress testing the funding ratio of defined benefit pension plans aimed at a non-actuarial audience while providing detailed discussion for practitioners.
- Structure:
  - Section II: stress testing for pension plans in an international context and risk based supervision.
  - Section III: actuarial factors and functions for estimating actuarial liabilities of DB plans.
  - Section IV: pros and cons of alternative pension liability calculation methods and costing methods.
  - Section V: a real DB plan case and construction of a model plan for single-factor stress tests.
  - Section VI: conclusions and discussion of possible extensions.
- Supporting material: Appendix I for basic pension terminology and Appendix III for mechanics of the accompanying template “Model.xls”.

### What jurisdictions do in the area of stress testing — regulatory context
- Stress testing is an element of risk based supervision alongside regulatory requirements and quantitative techniques for risk assessment.
- Regulatory tools:
  - Investment rules: restrict investable universe, improve asset quality/mix, reduce volatility, increase liquidity; may be quantitative limits or “prudent person” rules.
  - Benefit regulation: limit types and amounts of insurance to beneficiaries to prevent tax avoidance and ensure adequate benefits.
- Typical privately sponsored occupational DB plan benefits covered/guaranteed: retirement benefits, vested benefits, disability benefits, death benefits.

### Retirement, disability, death, and vesting rules (key features)
- Retirement benefits:
  - Eligibility upon retirement subject to age and, sometimes, minimum service requirements.
  - Normal retirement age often defined as earliest age to retire without benefit reduction; early retirement with reduced benefits is common.
  - Benefit formulas typically: accrual factor × pensionable salary × years of credited service (example: “2 percent of the average of the final five years’ base salary times the number of years of service”).
- Disability benefits:
  - Often take the form of life annuities with variable eligibility (may commence at early retirement age, normal retirement age, or soon after accident).
  - Alternatives include waiver of contributions with accrual of additional retirement benefits.
  - Eligibility usually includes minimum age and/or service requirements.
- Death benefits:
  - Forms include lump sums (multiples of salary) or life annuities to surviving spouses; eligibility often coincides with vesting.
  - Values vary considerably across plans and jurisdictions.
- Vesting:
  - Vesting schedules may be immediate, after x years, or graded; regulations often require full vesting after a short period such as two or five years.

### Valuation rules — assets and liabilities
- Asset valuation:
  - Can be marked to market or amortized cost; market-based methods may require methodologies for illiquid assets.
  - Valuation requirements may be regulated or follow professional standards such as IFRS.
  - Amortized cost smoothing reduces volatility; regulations may define amortization periods and smoothing techniques.
- Liability valuation:
  - Requires actuarial assumptions (mortality, termination, retirement, inflation, discount rate, wage increases) and membership/rule data.
  - Multiple actuarial valuation methods exist for balance sheet valuation and funding policy; valuation and costing methods can be numerous.
- Consistency:
  - Use of consistent bases for assets and liabilities enhances usefulness (e.g., both market values with market interest rates, or both with smoothing).
- Valuation group basis:
  - Closed group valuations consider only members at valuation date (typical for private plans).
  - Open group valuations assume infinite life and project demographics (typical for social security).
- Valuation perspective:
  - Going concern valuations assume plan continues and members accrue future benefits.
  - Termination valuations assume plan is closed at valuation date and obligations are settled (used for liquidation or insolvency scenarios).
  - Jurisdictions vary in required valuation types and frequency (example: some require both types every three years; Canada requires both; US restricts termination valuations for funding purposes).

### Solvency rules and international diversity
- Objective: ensure pension plans have sufficient assets to meet liabilities.
- Diversity stems from lack of international regulatory standards on assumptions and methodologies.
- Examples of divergent practices:
  - Liability cash flows based on current salaries vs. salaries projected to normal retirement age.
  - Discount rate choices vary: market discount rate (e.g., yield on government bonds) used in Belgium, Canada, and Japan; fixed discount rate used in Finland, Ireland, Germany, Norway, Portugal, and Spain; rate equal to future expected return on plan assets (United Kingdom).
- Impact: Different provisions produce different liability levels and different sensitivities under identical stress test scenarios.

* _wp1129 - References

### 3.75 percent. Minimum vested rights

### _wp1129 - 3.75 percent. Minimum vested rights

### Country examples: discount rates and liability valuation practices
- Belgium
  - Minimum vested rights are calculated on the basis of current salaries with an interest rate of 6 percent and specific mortality tables (MR 88-90 table for males and the FR 88-90 table for females).
  - Belgian prudential legislation: the discount rate for the calculation of the technical provisions has to be chosen in a prudent manner and taking into account: (i) the return on covering assets as well as future returns and/or (ii) the return on bonds of a Member State or on other high-quality bonds.
- Canada
  - Plan termination liability (current unit credit).
  - Interest rate of x percent per annum for 10 years and y percent per annum thereafter.
  - The rate “x” is equal to the annualized market yield on seven-year Government of Canada benchmark bonds plus 90 basis points.
  - The rate “y” is a more complicated blend of market yields on such seven-year bonds and on long term Government of Canada benchmark bonds, again plus 90 basis points.
  - Lower interest rates apply when the plan provides indexation of pensions; the formulas are specified in the CIA Standards of Practice.
  - Note: The information for Canada in this table applies to defined benefit pension plans regulated at the federal level. Provincially regulated plans may have different requirements, particularly for the maximum allowable amortization period.
- Finland
  - Accrued benefits calculated under current unit credit method.
  - 3.5 percent-3.8 percent depending on the plan.
- Germany
  - The technical provisions are the present value of the future liabilities minus the present value of the future premiums.
  - The maximum discount rate for Pensionskassen and Pensionsfonds (if the latter offer insurance-like guarantees) is currently 2.25 percent for new schemes.
  - Pensionsfonds can use market interest rates on a best estimate basis if they offer no insurance-like guarantees.
- Ireland
  - Plan termination liability (current unit credit), including mandatory revaluation of benefits with 4 percent cap, until retirement.
  - (a) a pre-retirement discount rate of 7.50 percent; (b) a long term post-retirement discount rate of 4.50 percent; (c) a pre-retirement price inflation rate of 2.00 percent; and (d) a post-retirement long term rate of price inflation of 2.00 percent.
- Japan
  - Plan termination liability (current unit credit).
  - 80-120 percent of 10-year government bonds issued during the previous 5 years.
- Netherlands
  - Accrued benefits calculated under current unit credit method.
  - Discount rate for the valuation of liabilities is based on swap rates.
  - Smoothing is allowed for determining contributions.
- Norway
  - Accrued benefits calculated under current unit credit method.
  - 4 percent discount rate until 1993.
  - For contributions due after 1 January 2004 and pension funds established after 1993 the maximum rate is 3 percent, 2.75 percent for new contracts after 2006.
- Portugal
  - Accrued benefits calculated under current unit credit method.
  - If indexing of pensions is contractually guaranteed, then an allowance for the effect of future indexing must be included in the calculation of the accrued liabilities.
  - 4.50 percent.
- Spain
  - Projected Benefit Obligation (including salaries at retirement - projected unit credit method).
  - 4 percent discount rate.
  - Inflation assumption of 1.5-2.0 percent.
- Switzerland
  - Accrued benefits calculated under current unit credit method.
- United Kingdom
  - Accrued benefits must be calculated on a prudent basis.
  - The discount rate in the UK can broadly be described by the following equation: discount rate = risk free rate + risk premium.
  - A proxy such as a government bond yield is typically used for the spread over the risk free rate is assumed, typically based on: the time horizon of liabilities; the potential for additional investment return; and a prudence adjustment, based on the employer’s covenant.
- United States (Single Employer Plans)
  - Accrued benefits calculated using projected unit credit method for financial reporting. Other methods are allowed for funding purposes.
  - For financial reporting: modified yield curve (three segments) based on a two-year average of top three levels of high-grade corporate bonds of appropriate duration.
  - For funding methods: IRS determined rate plus allowed spread for inflation.

### Solvency rules, margins, and stress testing
- Solvency rules typically require assets to exceed liabilities by a solvency margin; in some cases the level of assurance is increased by requiring assets to exceed liabilities by a solvency margin calculated as:
  - a simple percentage of liabilities, or
  - risk weighted, or
  - stress test related.
- Risk weighted solvency margins
  - Require the definition of asset and liability weights.
  - Weights are attached to various proxies for the asset and liability risks and often just added to one another to determine the total required solvency margin, or sometimes combined using a non linear formula.
  - Example: The Netherlands requires a solvency margin of 5 percent of technical provisions (following the EU IORP Directive) and the standard model defines the various solvency buffers associated with each type of risk (interest risk, equity and real estate risk, currency risk, commodity risk, credit risk, and insurance risk), as well as how such factors are combined to derive the overall solvency margin.
- Stress testing complements risk weighted solvency margins
  - Risk weighted margins are affected by model risk and might not provide a reliable solvency estimate needed to withstand adverse conditions.
  - Stress testing requires plans to calculate the additional amount of assets needed to meet obligations under prescribed stress scenario(s).
  - Stress testing generally uses the standard or accepted internal model to conduct calculations under different scenarios of increasing but still plausible severity.
  - Supervisors use stress testing results to graduate policy and supervisory responses with more intense scrutiny devoted to plans with poorer results.
  - Example jurisdiction with such requirements: Denmark.

### Risk monitoring techniques used by supervisors
- Techniques include:
  - Sensitivity testing using different actuarial factors.
  - Analysis of sources of earnings (investment gains/losses and actuarial gains/losses).
  - Roll-forward calculations to project financial position from a valuation date to future dates.
  - Value at Risk (VaR) calculations.
  - Duration and maturity gap analysis.
  - Deterministic and stochastic stress testing.
- Valuations with different actuarial factors
  - Objective: identify inappropriate assumptions and provide sensitivity testing across alternative macro or financial scenarios.
- Analysis of sources of earnings
  - Compare actual experience to previous assumptions and recalculate valuations to identify and quantify sources of gains and losses.
- Roll-forward calculation
  - Projects plan financial position forward using valuation outputs; usefulness declines with projection horizon due to increasing projection error.
- Maturity gap analysis
  - Compare future cash flows of assets and liabilities to identify period-by-period maturity mismatches; requires detailed information on maturity of fixed income investments.
- Duration and convexity analysis
  - Duration compares sensitivity of liabilities and interest rate sensitive assets to interest rate changes; convexity measures second-order change in duration for a small change in rates.
  - Typical asset-liability management requires convexity of assets be larger than convexity of liabilities.
- Key rate duration and convexity analysis
  - Key rate (partial) durations measure local sensitivity to portion of yield curve; matching partial durations can protect against nonparallel shifts in the yield curve.
- Value at Risk (VaR) analysis
  - Measures expected dollar loss from adverse market movements with a specified probability (example confidence interval: 97.5 percent) over a particular period (example: one day or more).
- Stress testing analysis
  - Deterministic stress testing: scenarios defined a priori without reference to likelihood.
  - Stochastic stress testing: scenarios randomly generated to produce distribution of results based on assumed distributions of underlying assumptions.

### Actuarial cost factors and decrement assumptions
- The calculation of DB pension liabilities relies on assumptions for various actuarial cost factors.
- Key actuarial cost factors include decrement assumptions, related survival probabilities, salary assumptions, inflation, and discount rate assumptions.
- Decrement assumptions
  - Describe how plan members’ liabilities are affected by various risks/contingencies (e.g., mortality, termination, disability, retirement).
  - Rates of decrement (given by available tables) are assumed for each contingency or risk.
  - Notation used: Ԣݍ
௫
ሺ௞ሻ denotes the rate at which members of age x transit from one status to another within the period between x and x + 1; the prime indicates rates (as opposed to probabilities) and (k) indicates the type of contingency (mortality (m), termination (t), and retirement (r)).
- Mortality assumptions are a core component of decrement assumptions (section continues into mortality discussion in the source).

*Source: Adapted from Yermo and Severinson (2010).*

### 33.      Mortality rates are one of the most important actuarial assumptions used in

### _wp1129 - 33.      Mortality rates are one of the most important actuarial assumptions used in

### Mortality rates: role and characteristics
- Mortality rates are one of the most important actuarial assumptions used in calculating the liabilities of DB plans because they affect all active and retired members since entry age and over a long period of time.
- Mortality is typically indicated with Ԣݍ
௫
ሺ௠ሻ
 and ; for solvency it is important that deaths be at least equal to their expected number according to the mortality assumption.
- Mortality eliminates the pension benefit obligation (it is “good” for the plan) but may trigger other benefits:
  - If the deceased had not vested, accumulated contributions are typically returned to survivors as a cash lump sum.
  - If benefits were vested, death benefits often take the form of immediate or deferred annuities with or without a cash lump sum.
- Mortality assumptions depend on age, gender, and occupational status:
  - Assumed mortality rates increase with age until the maximum assumed age when the rate equals 1.
  - Females typically die at much lower rates than males of the same age.
  - Hazardous occupations (mining, army, police, et cetera) have much higher mortality rates than average.
- Jurisdictional constraints and basis risk:
  - In many jurisdictions tables are mandated (e.g., unisex or population tables).
  - The basis risk between mandated tables and the covered population can result in large unrecorded liabilities requiring extra provisioning by the actuary.
- Two key types of mortality tables:
  - Static (period) tables — give today mortality rates of individuals of different ages x.
  - Dynamic (cohort) tables — give mortality rates of individuals of different ages x, n years from now.
- Example: Table A1 reports the USA 1996 male static annuitant mortality rates.

### Termination assumptions
- Termination rates are another key actuarial assumption, affecting only active members between entry age and retirement age.
- Termination rates are typically indicated with Ԣݍ
௫,௬
ሺ௧ሻ
 and specify the rate at which active members of age x who entered at age y leave the plan (without retiring) within x to x + 1.
- Impact depends on vesting status:
  - For non-vested members, termination eliminates the pension benefit obligation (typically only a cash lump sum return of accumulated contributions).
  - For vested members, benefits take the form of a deferred annuity with or without a lump-sum option.
- Termination assumptions depend on age, length of service, gender, and occupation:
  - Unlike mortality rates, termination rates decrease over time until zero when active members retire (or can retire).
  - For a given length of service, termination rates typically decrease with age.
  - For a given age, termination rates typically decrease with length of service.
  - Distinction: ultimate termination rates (age dependent only) vs. select termination rates (depend on entry age and length of service).
- Example: Table A2 provides an example for members aged 20 to 64 with a select period of five years.
  - Note: the table assumes a vesting period of 10 years, normal retirement age of 64 and early retirement age of 55; termination rates for people aged at least 55 and with at least 10 years of service are assumed equal to zero.

### Retirement assumptions
- Retirement decrement rates apply when the plan provides for early or delayed retirement and affect only active members between earliest and latest permissible retirement ages.
- Retirement rates are typically indicated with Ԣݍ
௫
ሺ௥ሻ
 and specify the rate at which active members of age x retire within x to x + 1.
- Generally, retirement rates increase until the maximum allowed retirement age, when they equal 1 (everybody assumed to retire at that age).
- Influencing factors:
  - Whether reduction for early retirement is actuarially fair affects early retirement behavior.
    - If reduction is less than actuarially fair, members are encouraged to retire early and early retirement rates are higher.
    - If more than actuarially fair, members are encouraged to retire at the normal retirement age and early retirement rates are lower.
  - Timing of social security benefits often causes a large increase in retirement rates in that year.
  - Gender, social expectations, and prevailing economic conditions also influence retirement rates.
- Common observed shapes:
  - Spike at the earliest effective retirement age and a hump near the age associated with social security eligibility.
- Examples and notes:
  - Some plans provide unreduced early retirement benefits if other criteria are met (e.g., age 50 with 30 years of service, or age plus years of service equal to 85).

### Other decrement assumptions
- Plans may include other decrements (e.g., disability rates, death and continuance of disability) which affect funding and costing calculations.
- The paper does not consider other decrement assumptions in detail since they complicate calculations and are unlikely to materially affect FSAP stress-test outcomes.

### Contingent, composite survival probabilities
- Composite survival probabilities derive from the decrement assumptions and give the probability that an individual will survive in the plan from one period to the next.
- Single decrement environment:
  - Rate of decrement equals the probability of decrement (e.g., retirees exposed only to death).
  - Notation: Ԣݍ
௫
ሺ௞ሻ
 is the rate of decrement for cause k at age x and ݍ
௫
ሺ௞ሻ
 the probability; then Ԣݍ
௫
ሺ௞ሻ
ݍൌ
௫
ሺ௞ሻ
.
- Multiple decrement environment:
  - The rate of decrement is higher than the probability of decrement; probabilities are smaller than rates because of competing risks.
  - Under uniform distribution of rates over the period, decrement probabilities can be approximated with equation (1) as presented in the source:
    - ݍ
௫
ሺ
௞
ሻ
ݍൎ
௫
ᇱ
ሺ
௞
ሻ
ෑቀ1െ
ଵ
ଶ
ݍ
௫
ᇱ
ሺ
௝
ሻ
ቁ
௝ஷ௞
- Composite survival probability for an individual aged x exposed to mortality, termination and early retirement contingencies is given by equation (2):
  - ݌
௫
ሺ்ሻ
ൌ1െቀݍ
௫
ሺ௠ሻ
ݍ൅
௫
ሺ௧ሻ
ݍ൅
௫
ሺ௥ሻ
ቁ
  - where ݍ
௫
ሺ
௞
ሻ
 is defined in equation (1) and (T) indicates multiple decrement.
- Multi-period survival:
  - Given time independence of probabilities, for age x the probability of surviving the next n periods is given by equation (3):
    - ݌
௡
௫
ሺ்ሻ
݌ൌෑ
௫ା௦
ሺ்ሻ
௡ିଵ
௦ୀ଴
- Note: the accompanying spreadsheet uses two decrement factors (mortality and termination) to derive composite survival probabilities for active members.

### Salary assumptions and the wage function
- Future salary assumptions are key because benefits accrual can depend on future salaries and estimations can span many years.
- Three key components for future salary assumptions:
  - Merit increase assumptions — reflect career progression and increased member contributions.
  - Productivity improvement assumptions — sector-dependent; 1 percent annual increases in productivity is common.
  - Inflation assumptions — typically the most important for future salaries.
- Example merit scale (from Table A4):
  - Merit increases of about 4.5 percent for the first year for a worker entering at age 20, peaking at 5.4 percent at age 35 and declining to zero by age 64.
- Wage function definitions:
  - Future salaries for a member of age x who joined at age y with beginning-of-period wage ݓ
௬,௬
 are calculated as per equation (4):
    - ݓ
௫,௬
ݓൌ
௬,௬
ݏ݉
௫,௬
ݏ݉
௬,௬
ሾሺ
ߨ1൅
ሻሺ
ݎ݌1൅
ሻሿ
ሺ௫ି௬ሻ
    - where ݏ݉
௫,௬
 is the cumulative merit increase at age x, ݏ݉
௬,௬
ൌ1 for service before age y, ߨ is the inflation assumption, and pr is productivity improvement.
  - The wage function ݂ݓ
௫,௬
 gives wage ݓ
௫,௬
 as a multiple of entry wage ݓ
௬,௬
 and is shown in equation (5):
    - ݂ݓ
௫,௬
ൌ
ݓ
௫,௬
ݓ
௬,௬
ൌ
ݏ݉
௫,௬
ݏ݉
௬,௬
ሾሺ
ߨ1൅
ሻሺ
ݎ݌1൅
ሻሿ
ሺ௫ି௬ሻ
  - Table A5 reports select and ultimate end-of-period wage function assumptions for the merit scale and productivity assumptions together with a 1 percent annual inflation assumption.
  - Salary at age x can be expressed as a function of salary at age z via equation (6):
    - ݂ݓ
௫,௭
ൌ
݂ݓ
௫,௬
݂ݓ
௭,௬
ൌ
ݏ݉
௫,௬
ݏ݉
௭,௬
ሾሺ
ߨ1൅
ሻሺ
ݎ݌1൅
ሻሿ
ሺ௫ି௭ሻ
  - Cumulative salary of an individual of age x who joined at age y is given by equation (7):
    - ܹ
௫,௬
ݓൌ෍
௦,௬
௫ିଵ
௦ୀ௬

### Discount rate assumptions
- Discount rates, together with survival probabilities, are among the most critical assumptions in valuing DB liabilities because expected future cash flows can be very distant (e.g., the last pension payment for a new entrant could be more than 80 years in the future).
- Debate on appropriate discount rate:
  - Traditional actuarial view: discount at expected long-term returns on a pension fund’s assets.
  - Financial economists (and some actuaries): discount at market rates (risk-free government bond yields or high-grade corporate bond yields).
  - Debate is being superseded by requirements for consistent (market) valuation of assets and liabilities and the introduction of countercyclical/dynamic solvency buffers.
- General formulation used in the template:
  - The rate ߥ
௡
 that discounts the cash flow expected at the beginning of period n to time zero is given by equation (8):
    - ߥ
௡
ൌෑ
ሺ
1൅݂
௦
ሻ
ି௦
௡ିଵ
௦ୀ଴
    - where ݂
௦
 is the forward interest rate assumed for the sth year.
  - Forward rates can be extracted from the government debt yield to maturity curve ݉ݐݕ
௦
 via equation (9):
    - ሺ
1൅݂
௦
ሻ
ൌ
ሺ
݉ݐݕ1൅
௦
ሻ
௦
ሺ
݉ݐݕ1൅
௦ିଵ
ሻ
௦ିଵ
  - Example: Table A6 provides a discount rate function derived from the yield curve where yield-to-maturity rates above period 30 are assumed constant.
- Notes:
  - There is a debate regarding discount rates for financial reporting versus funding standards; regulations typically require market rates for financial reporting and allow more stable rates for funding standards.
  - Use of forward rates rather than spot rates implies an underestimation of the discount rate curve because yield rates typically contain an illiquidity premium at longer maturities; the template ignores this difference.

### Actuarial liabilities and methods (introductory)
- Actuarial liabilities are estimated for each category of plan member; most important categories are retirees and active members (others can include active members with past service benefits, active but disabled members, and terminated members with deferred benefits).
- For simplicity the discussion focuses on retired and active members.
- Actuarial liability for a retired member is calculated using annuity functions: project year-by-year contingent cash flows and discount them with an annuity function corresponding to the form of pension provided.
- DB plans provide a variety of annuities to insure risks like longevity and inflation, can cover spouses and beneficiaries, can be for a period or for life, and sometimes provide participation in investment and mortality experience.
- The paper will use contingent survival probabilities and discount rate functions developed in section III and limit discussion to standard types of annuities; begins with straight life annuities.

*Source: IMF working paper excerpt (section text provided).*

### 55.      A straight life nominal annuity insures individuals against longevity and investment

### _wp1129 - 55.      A straight life nominal annuity insures individuals against longevity and investment

### Straight life nominal annuity (section 55)
- Insures individuals against longevity and investment risks only.
- Provides periodical benefit payments fixed in nominal terms B starting with the normal retirement age r = 55 until death.
- Present value of expected cash flows is given by equation (10): the sum over s of (ܽܤሷ௥) times the contingent survival probability ݌௦௥ሺ௠ሻ and the discount factor ߥ௦ (as defined in equation (8)).
- Key symbols retained from source: ܽܤሷ௥ (present value of a straight life annuity due), ݌௦௥ሺ௠ሻ (contingent survival probability for an individual retired at r), ߥ௦ (discount rate for period s).

### Straight life real annuity (section 56–57)
- Insures also against inflation risk by increasing periodic benefit payments after the initial payment of B at inflation rate ߨ.
- Present value of expected cash flows is given by equation (11): sum over s of ( (1 + ߨ)^{1} ) times ݌௦௥ሺ௠ሻ and ߥ௦ (formula as in source).
- Comparative findings (section 57):
  - Initial periodic payments of real annuities are much lower than nominal annuities that have the same present value.
  - Example/calculation assumptions stated in source:
    - r = 55
    - ߨ = 0.035
    - B = 1 for nominal annuity
    - B = 0.6246 for real annuity with same present value
    - Mortality rates defined in Table A1
    - Discount rate defined in Table A6
  - The areas under the expected cash flow curves equal the present values given by equations (10) and (11); because present values are equal, initial payments from the real annuity must be lower than nominal payments.

### Joint life annuities (sections 58–59)
- Provide longevity insurance to more than one beneficiary.
- Present value of a nominal joint life annuity due that pays B to two individuals of ages x and z when both are alive and pays ܤߙ when either beneficiary is dead is given by equation (12) (explicit summation formula in source).
- Common variation: annuity due that pays full pension when both beneficiaries, or when only the primary beneficiary (the retiree), are alive, but pays only a fraction ߙ of the pension to the surviving spouse. Present value for this variant is given by equation (13) (explicit summation formula in source).
- Note from source: other modifications exist but are beyond the scope of this presentation.

### Individual actuarial liabilities for active members (section B, sections 60–64)
- Calculation depends on the plan benefit formula (benefit function) and the actuarial cost method used.
- Accrued benefit function (section 61):
  - Denote ܾ௫,௬ as benefits accrued between age x and x + 1 for a member entered at age y (accrual factor).
  - Accrued benefit function ܤ௫,௬ is the sum of past accrued benefits: equation (14).
- Three common accrued benefit function forms among DB plans:
  - Flat dollar function (section 62):
    - Benefits fixed in dollar levels, depend only on length of working career.
    - For attained age x who joined at age y: ܤ௫ = (x–y) ܾ (equation (15) form in source).
  - Career average salary function (section 63):
    - Benefits based on average compensation over the whole working career; accrual factor typically fixed as a percentage of pensionable salary.
    - Accrued benefit function expressed as sum over years of accruals; see equation (16) and accompanying notation ݓ௫,௬ for average salary.
  - Final average salary function (section 64):
    - Benefits based on average compensation over a period defined by plan rules (e.g., 1, 3, 5, or 10 years prior to retirement).
    - Example: if average is calculated over last five years, accrued benefit function given by equation (17) using ݓ௫,௬:ହ for the final five years average salary.
    - Numerical example in source: with a 2 percent accrual factor, an average salary during the final 5 years of US$100,000, and 40 years of service, the retired member would receive a pension worth US$80,000.

### Actuarial cost methods (sections 65–74)
- Purpose: define how expected costs of a pension plan are allocated over the period of active membership and generate an annual normal cost ܥܰ௫,௬ (present value of benefits earned allocated to that year).
- Formula for the annual normal cost (equation (18)):
  - ܥܰ௫,௬ = ܾ௫,௬כ × sum over s of composite survival probability ݌௥ି௫௫ሺ்ሻ times discount factor ߥ௥ି௫ and ܽሷ௥ (present value at normal retirement age r of the annuity provided by the plan).
  - Notation: ܾ௫,௬כ is the benefit accrual allocated to the period between attained age x and x + 1 under the chosen cost method.
- Actuarial liability for an individual aged x who joined at age y (equation (19)):
  - Sum of normal costs up to attained age equals actuarial liability; defined using accrued benefit value ܤ௫,௬כ and the summation formula shown in source.
- Two broad categories of cost methods (section 68):
  - Benefit allocation methods: allocate total benefits to years of service, then value annual allocations.
  - Cost allocation methods: value total cost of benefits, then allocate cost among years of service.
- Benefit allocation methods (sections 69–74):
  - Unit credit method (section 70):
    - Considers only past service.
    - Accrual function ܾ௫,௬ determined by plan's benefit formula (often flat dollar).
    - Actuarial liability called accrued benefit obligation (ABO): present value of benefits accrued up to attained age x (equation (20)).
  - Projected unit credit method (sections 71–73):
    - Considers future service until the normal retirement age r.
    - Accrued benefit function ܤ௥,௬ is sum of accruals between entry age y and normal retirement age r.
    - Produces retirement benefit obligation (RBO): present value of benefits that will be accrued at retirement evaluated with plan formula (equation (21)).
    - Source notes RBO is generally not used in practice because it forces reserving the present value of all possible benefits at entry age; regulators allow proration to produce projected benefit obligation (PBO).
  - Proration approaches (section 73–74):
    - Purpose: allocate projected benefits equally to all years of service to reduce front-loaded reserves of RBO.
    - Two common proration methods:
      - Constant dollar pro rata method:
        - Accrual factor ܾ௫,௬ constant and defined as fixed share of retirement benefits: ܾ௫,௬ = ܤ௥,௬ / (r–y).
        - Accrued benefits ܤ௫,௬ = (x–y) × ( (r–y)^{-1} × ܤ௥,௬ ); actuarial liability expressions as in equation (22).
      - Constant percent pro rata method:
        - Accrual factor ܾ௫,௬ is a fraction of accrued benefits at retirement proportional to individual salary.
        - Accrued benefits and liabilities follow forms given in equation (23).

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1129.pdf*

### 75.      Cost allocation methods are fundamentally different from, and more complex than,

### _wp1129 - 75.      Cost allocation methods are fundamentally different from, and more complex than,

### Overview: cost allocation versus benefit allocation
- Cost allocation methods allocate the total costs of the benefits to the various years of service, rather than assigning units of benefits to specific years of service.
- Their derivation is more complex than benefit allocation methods.
- Benefit allocation methods assign units of benefits to service years; cost allocation methods take the total cost of the RBO liability and allocate that cost across service years.

### Main variants of cost allocation methods
- Two main variants:
  - Constant dollar method (entry age cost allocation — EAOcd): allocates a constant share of the cost of the RBO liability.
  - Constant percent method (entry age cost allocation — EAOcp): allocates a fixed percent of the employee’s salary to each year of past service starting from the entry age y until the attained age x.
- Both methods use the present value at entry age y of a temporary annuity.
  - A temporary annuity: an annuity provided for a fixed period of time or until death, whichever comes first.
  - Two types needed:
    - Constant benefit temporary annuity (for constant dollar method).
    - Salary-based temporary annuity (for constant percent method).
- The annuities are evaluated at entry age; hence these are called entry age cost allocation methods.

### Mathematical characterizations (as given)
- Present value at entry age y of a constant benefit temporary annuity: present value at entry age y of a temporary annuity that pays one unit of benefit (B = 1) until the normal retirement age r. (Equation (24) in source.)
- Present value at entry age y of a constant salary-based temporary annuity: present value at entry age y of a temporary annuity that pays multiples ݓ
௦
ݓ
௬
⁄
 of the entry age salary until normal retirement age r. (Equation (25) in source.)
- EAOcd accrual function ܾ
௫,௬
:
  - ܾ
௫,௬
 is a fraction of the final RBO liability where the fraction is the ratio of:
    - the present value at entry age y of one unit of benefits (B = 1) paid at the attained age x through a temporary annuity, over
    - the present value at entry age y of a temporary annuity paying one unit of benefits (B = 1) between entry age y and normal retirement age r.
  - (Equations and summation expressions provided in source; see equation (26).)
- EAOcp accrual function ܾ
௫,௬
:
  - ܾ
௫,௬
 is a fraction of the final RBO liability where the fraction is the ratio of:
    - the present value at entry age y of salary multiples ݓ
௫
ݓ
௬
⁄
 paid at attained age x through a temporary annuity, over
    - the present value at entry age y of a temporary annuity paying salary multiples ݓ
௫
ݓ
௬
⁄
 between entry age y and normal retirement age r.
  - (Equations and summation expressions provided in source; see equation (27).)

### Comparing benefit and cost allocation methods — key findings
- Individual normal costs:
  - Individual cost methods amortize the RBO obligation at entry age (ܱܤܴ
௥,௬
) over the member’s working life.
  - Choice of method produces different patterns of normal costs:
    - Benefit allocation methods: normal costs increase over time and can become as steep as the accrual function.
    - Cost allocation methods: can produce normal costs that are more stable over time and sometimes decreasing.
  - An entry age cost allocation method prorated as a percentage of salary (݌ܱܿܣܧ
௫,௬
) produces by definition a stable cost pattern over time.
- Figures and tables referenced:
  - Figure 3: Individual Normal Costs – Percentage of Salary. Calculations assume:
    - y = 20, r = 55, pr = 0.01, ߨ = 0.035, b = 0.01, five year average final salary formula, nominal straight life annuity, mortality rates defined in Table A1, termination rates defined in Table A2, merit scale defined in Table A4, and discount rate defined in Table A6.
  - Table 3: Accrued Benefits of Main Actuarial Cost Methods — lists ABO, PBOcd, PBOcp, EAOcd, EAOcp, TER, RBO representations as in source.
- Actuarial liabilities and method extremes:
  - Different cost methods produce very different individual actuarial liabilities.
  - Lowest value produced by ABO method (does not recognize expected future benefit accrual).
  - Highest value produced by RBO method (fully recognizes accrued and accruable benefits).
  - All prorated methods produce intermediate values.
  - Terminal funding method (TER): reserves accrued liability only at normal retirement age; not allowed by accrual accounting such as IFRS and amounts to zero reserves for active members (a pay-as-you-go plan).
  - Figure 4: Individual Actuarial Liabilities – Various Cost Methods (same calculation assumptions as Figure 3).
- Actuarial liabilities as share of RBO:
  - ABO and PBOcp methods tend to backload reserving for the sponsor.
  - EAOcd and EAOcp methods tend to front-load reserving for the sponsor.
  - PBOcd is intermediate between RBO and TER; produces an individual actuarial liability linearly increasing over time as a share of RBO.
  - Figure 5 and Table 4 report these shares formally (calculation assumptions same as Figure 3).

### Aggregate versus individual values
- Two ways to calculate aggregate liabilities:
  - Individual method: calculate individual liabilities and sum across members.
  - Aggregate method: calculate values on an aggregate basis by averaging across individuals or cohorts.
- For a plan with a stable population (stable average age and average salary of active members), aggregate cost will be stable over time even if individual cost patterns differ.
- Choice of cost method for plans with stable population is ultimately dictated by:
  - the value of liability it produces,
  - sponsor philosophy regarding prefunding,
  - type of benefit formula,
  - tax restrictions,
  - regulatory requirements,
  - professional standards.
- An aggregate (cohort) approach is acceptable if individual data is unavailable or valuation purpose does not require much precision (e.g., stress test in an FSAP). Individual calculations are preferable.

### Stress test methodology — model and real plan characteristics (sections 86–88)
- Purpose: describe a possible stress test methodology implemented with accompanying template “Model.xls” using a real DB plan; simplifications were made for operational usability.
- Real pension plan characteristics (selected items from source):
  - Benefits: retirement, death, and disability benefits with various withdrawal options in form of single gender specific and/or joint straight life real inflation indexed annuities or cash lump sums.
  - Retirement benefits: largest liability; based on final 5 year average salary formula with accrual rate of 1 percent. Entry age varies from a minimum of y = 20 to normal retirement age r = 55. Early retirement allowed starting at x = 45. Early retirement reductions less than actuarially fair. No information on delayed retirement accruals.
  - Assets and liabilities: total assets amount to LCU3,773 million. Plan actuary uses PBOcd actuarial method for active members’ individual actuarial liabilities. Latest actuarial report shows liabilities amounting to LCU4,000 million.
  - Solvency regulation: no minimum solvency margin required. Rules require assets in excess of liabilities to be considered “funded”. Reported solvency ratio is 94 percent.
  - Actuarial cost factors: plan actuary uses male mortality and termination rates defined in Table A8. Inflation, salary increase and discount rate assumptions in Table A7. Other decrement information unavailable.
  - Plan membership distributions (Figure 6):
    - Active workers: density distribution with two peaks around ages 42 and 52.
    - Wage remuneration of active workers: density distribution with two peaks around ages 40 and 52.
    - Retired workers and pensions: large concentration soon after normal retirement age of 55.
    - Active distributions include individuals active beyond normal retirement age; retired distributions include individuals retired before normal retirement age. Actual data in Table A8.
- Model pension plan constructed from collected information (selected items):
  - Benefits: only retirement benefits considered, in the form of single life inflation indexed annuity.
  - Retirement benefits: final salary formula with constant accrual rate. Effective entry and retirement ages set at y = 20 and r = 55 for all members.
  - Assets and liabilities: use real plan investment portfolio data. Use aggregate (annual cohorts) PBOcd method to calculate active member liabilities. Model liabilities are rescaled to coincide with liabilities calculated by the plan actuary: change in model plan liability from a shock is applied to real plan actuarial liability to compute change in funding ratio.
  - Actuarial cost factors: use male mortality for all members and termination rates in Table A8. Wage projections use actual merit scale, inflation, and productivity assumptions in Table A7.
  - Plan membership: use real plan distributions from Figure 6. Cutoff at r = 55 means early retirees ignored and late retirees assumed to retire at normal retirement age; all members assumed to have full career service.
- Note on simplifications: Appendix III contains a detailed description of the template; Tables A7 and A8 summarize actuarial cost factors used.

*Source: _wp1129 (sections 75–88).*

### 89.      Already in this very simple plan we had to make several simplifications, which do not

### _wp1129 - 89.      Already in this very simple plan we had to make several simplifications, which do not

### Simplifications in the template and their directional impact on liability estimation
- Simplifications that imply an underestimation of true liabilities:
  - Only consider pension benefits and disregard ancillary benefits like death and disability.
  - Ignore lump sum commutation (footnote: in the US the discount rate used to calculate lump sum commutation is linked to the Fed rate and established year by year by the IRS; such disconnection can result in very large unfunded liabilities).
  - Consider only single life annuities and disregard joint life annuities, which underestimates the annuity factor for active members and undervalues retired members’ liabilities.
- Simplifications that imply an overestimation of true liabilities:
  - Assume all plan members join at entry age y with immediate vesting (in reality, some enter later and may not have immediate vesting).
  - Consider only final salary pensions, which overestimates liabilities relative to career average salary pensions.
  - Consider full indexation of pension rights to inflation and full longevity insurance; where indexation is conditional on plan performance, the model overestimates liabilities.
- Simplifications with ambiguous impact:
  - Assume active members all retire at the normal retirement age r; depending on actuarial fairness of early retirement provisions, estimated liabilities may be higher or lower than plan liabilities.
  - Assume pensionable salary equals total remuneration; this overestimates liabilities when pensionable salary is lower than total remuneration and underestimates when it is higher.
- Valuation approach:
  - The template uses a PBOcd valuation method (plan continuation / ongoing concern) rather than a plan termination valuation; this affects liabilities for active members and may differ from local regulatory requirements.

### Validation and rescaling for stress testing
- The template’s simplifications are considered immaterial for stress testing purposes because:
  - The interest is in rates of change of liabilities under shocks rather than absolute liability levels.
  - Model liabilities will be rescaled to coincide with true liabilities; the model rate of change from shocks is then applied to true liabilities to obtain changes in the funding ratio.
- Assumption on timing of payments (footnote): calculations assume all benefit payments are made annually at the beginning of the year; on average they will be made one-half year later.

### Box 1 — Continuation and Termination Valuation Methods (summary)
- Plan continuation methods (ongoing concern): assume plan terminated to new entrants only; current members continue accruing benefits under the closed plan.
- Plan termination methods: assume plan terminated and members do not accrue more benefits.
- For retirees: continuation and termination liabilities are the same.
- For active members:
  - Continuation liability equals accrued benefits at normal retirement age r prorated by service, multiplied by contingent composite survival probability until retirement, annuitization factor at retirement, and discounted to valuation date.
  - Termination liability equals accrued benefits at valuation, multiplied by contingent simple survival probability until retirement, annuitization factor at retirement, and discounted to valuation date.
- Note: For termination liabilities no future benefit accrual is assumed; in this case PBOcd collapses to ABO. Actuarial assumptions (including choice of discount rate) typically differ across methods.

### D. Stress Testing the Funding Position – Asset Shocks (model plan baseline)
- Model plan baseline characteristics:
  - Support ratio: 413 percent.
  - Funding ratio: 94 percent.
  - Observations: concentration risk in key asset classes; actuary assumptions may be optimistic (e.g., 9 percent discount rate, static mortality table, outdated termination rates); unhedged FX risk and rising inflation expectations.
- Asset stress test design:
  - Assumed up to a 25 percent symmetrical shock in asset values by issuer type (government, financial sector, real sector) and by vehicle (bonds, stocks, foreign assets / FX).
  - FX risk unhedged: a 25 percent local currency appreciation (or decrease in FX denominated assets by 25 percent) reduces funding ratio from 94 percent to 88 percent.
- Table 5. Model Plan – Stress Tests (Asset Concentration Risk) — Funding ratios by market shock (%)  
  - Market shocks and resulting funding ratios:
    - -25%: Govt. 89.74, Financial 89.99, Real 87.47, Bonds 82.12, Stocks 83.06, FX 88.10, TOT 70.75
    - -20%: Govt. 90.66, Financial 90.86, Real 88.84, Bonds 84.56, Stocks 85.31, FX 89.35, TOT 75.46
    - -15%: Govt. 91.57, Financial 91.73, Real 90.21, Bonds 87.00, Stocks 87.56, FX 90.59, TOT 80.18
    - -10%: Govt. 92.49, Financial 92.59, Real 91.58, Bonds 89.44, Stocks 89.82, FX 91.84, TOT 84.90
    - -5%: Govt. 93.41, Financial 93.46, Real 92.96, Bonds 91.89, Stocks 92.07, FX 93.08, TOT 89.61
    - 0%: Govt. 94.33, Financial 94.33, Real 94.33, Bonds 94.33, Stocks 94.33, FX 94.33, TOT 94.33
    - +5%: Govt. 95.25, Financial 95.20, Real 95.70, Bonds 96.77, Stocks 96.58, FX 95.57, TOT 99.05
    - +10%: Govt. 96.17, Financial 96.06, Real 97.07, Bonds 99.21, Stocks 98.84, FX 96.82, TOT 103.76
    - +15%: Govt. 97.08, Financial 96.93, Real 98.44, Bonds 101.65, Stocks 101.09, FX 98.07, TOT 108.48
    - +20%: Govt. 98.00, Financial 97.80, Real 99.82, Bonds 104.10, Stocks 103.35, FX 99.31, TOT 113.19
    - +25%: Govt. 98.92, Financial 98.67, Real 101.19, Bonds 106.54, Stocks 105.60, FX 100.56, TOT 117.91
- Methodological note (footnote): flat percentage changes are a rough test; duration-based tests are more appropriate for bonds and mortgages, but duration information was not available for this exercise.

### E. Stress Testing the Funding Position – Liability Shocks (overview)
- Liability shocks considered: interest rate, inflation, longevity, and termination rate shocks.
- Critical issue: liability valuation is highly sensitive to actuarial assumptions; mismatches can create unfunded liabilities that develop slowly and may become difficult for the sponsor to remedy.

Interest rate shock
- Two shock types:
  - Substitute the level 9 percent interest rate used for discounting with more reasonable level rates.
  - Derive market discount rates from a AAA government debt yield curve and then shock that curve.
- Duration information:
  - Total liability duration: 15 years.
  - Active members duration: 17 years.
  - Retired members duration: 10 years.
  - Active members expected to be more sensitive to interest rate changes due to higher duration.
- Impact of level interest rate changes (Table 6. Model Plan – Stress Tests (Level Interest Rates)):
  - Baseline 9% → Funding Ratio (FR) 94.33
  - 8% → FR 80.76
  - 7% → FR 68.33
  - 6% → FR 57.11
  - 5% → FR 47.08
  - 4% → FR 38.25
  - Observation: if the interest rate assumption decreases from 9 percent to 4 percent, the funding ratio decreases from 94 to 38 percent, or an average 11 percent for every percentage point change in the interest rate.
- AAA market rates and yield-curve shifts:
  - Using a AAA government debt yield curve (end of 2009 US domestic debt curve in the example), the funding ratio is around 33 percent.
  - Parallel shifts of the yield curve up to ±150 basis points produce plan termination funding ratios between 26 and 39 percent.
- Table 7. Model Plan – Stress Tests (Shifts in the Yield Curve) — FR by shock:
  - -150bps → FR 26.52
  - -100bps → FR 28.51
  - -50bps → FR 30.59
  - 0 → FR 32.70
  - +50bps → FR 34.86
  - +100bps → FR 37.04
  - +150bps → FR 39.24

Inflation shock
- Channels:
  - Inflation affects annuitization factor and wage projections (notations: ߨ1൅௔ and ߨ1൅௪ in the equations referenced).
  - Changes in annuity inflation assumption ߨ௔ increase both the annuitization factor and the value of accrued benefits.
- Sensitivity (summary from Table 8. Model Plan – Stress Tests (Inflation Shocks)):
  - A 100bps increase in the inflation assumption for annuity valuation ߨ௔ reduces the funding ratio from 94 percent to 85 percent.
  - A 100bps increase in the inflation assumption for wage projections ߨ௪ reduces the funding ratio from 94 percent to 89 percent.
  - A 100bps increase in both assumptions reduces the funding ratio from 94 percent to 80 percent.
- Table 8. Model Plan – Stress Tests (Inflation Shocks) — matrix of funding ratios (selected rows/columns preserved exactly as presented):
  - Row labels and columns preserved verbatim:  
    - -150bps row: 118.44, 112.85, 107.36, 101.97, 96.70, 91.53, 86.49, -10.24
    - -100bps row: 115.45, 110.01, 104.66, 99.40, 94.26, 89.22, 84.31, -10.24
    - -50bps row: 112.49, 107.19, 101.97, 96.86, 91.84, 86.93, 82.14, -10.24
    - 0bps row: 109.56, 104.39, 99.31, 94.33, 89.44, 84.67, 80.00, -10.25
    - +50bps row: 106.66, 101.63, 96.68, 91.83, 87.07, 82.42, 77.87, -10.25
    - +100bps row: 103.78, 98.88, 94.07, 89.35, 84.72, 80.19, 75.77, -10.25
    - +150bps row: 100.94, 96.17, 91.49, 86.90, 82.39, 77.99, 73.69, -10.25
    - Final row: 5.28, 5.28, 5.28, 5.28, 5.28, 5.28, 5.28
- Observation: funding ratio is more sensitive to inflation assumptions used for annuity valuation than for wage projection inflation assumptions; elasticity with respect to annuity inflation assumption is much larger.

Longevity shock
- Modeling approach:
  - Project period mortality rates over a number of improvement years t by multiplying the static table rates by (1 + r_t) for each cohort x, where r_t represents annual multiplicative improvements in longevity.
  - Equation (28): q_x,t = q_x * (1 + r)^t (notation preserved as in source).
- Example:
  - Using US mortality improvement rates from the 1994 Group Annuity Reserving Table projected for T = 30 years for an individual aged 55, the change in life expectancy at age 55 is Δe_55 = 2.70.
- Note: mortality improvement rates should be discussed with the local actuary and amended as needed.

*Italic: Source — _wp1129 (excerpt: paragraphs 89–99, Box 1, and sections D–E) — original PDF content supplied.*

### 100.      Longevity improvements translate into higher in life expectancy at different

### Longevity improvements translate into higher in life expectancy at different

### Longevity shocks — Model Plan (Table 9)
- Scenario: Increasing number of improvement years between 0 and 70.
- Plan normal retirement age: 55 (݁ହହ).
- Life expectancy at plan normal retirement age of 55:
  - 0 improvement years: 28.39
  - 30 improvement years: 31.09
  - 40 improvement years: 31.88
  - 50 improvement years: 32.62
  - 60 improvement years: 33.31
  - 70 improvement years: 33.96
- Funding ratio (FR) at corresponding improvement years:
  - 0 improvement years: 94.33
  - 30 improvement years: 89.46
  - 40 improvement years: 88.21
  - 50 improvement years: 87.10
  - 60 improvement years: 86.13
  - 70 improvement years: 85.26

### Termination rate shock — Model Plan (Table 10)
- Context: Retention rates reportedly increased by at least 10 percent across the board; analysis investigates decreases in termination rates between 10 and 30 percent.
- Termination rate decrease scenarios and resulting Funding Ratio (FR):
  - -0%: FR = 94.33
  - -10%: FR = 92.18
  - -15%: FR = 91.11
  - -20%: FR = 90.04
  - -25%: FR = 88.97
  - -30%: FR = 87.91

### Conclusions and other considerations (Sections 102–103)
- Main outcomes:
  - The paper described the basic mechanics of DB plan liability valuation and how to conduct simple stress tests of the solvency ratio, using the accompanying Excel template.
  - The accompanying template “Model.xls” uses a last salary DB formula and a projected benefit obligation constant dollar (PBOcd) actuarial method to value liabilities.
  - The template is not a substitute for a proper actuarial evaluation, but the simplifications introduced do not affect its usefulness to evaluate sensitivity of the funding ratio.
  - The stress test methodology is parsimonious and aimed at quantifying key risk exposures by non actuarial analysts; it is not a substitute for a full actuarial evaluation.
- Suggested extensions and refinements:
  - Refinements on the asset side:
    - Identify sources of risk stemming from interest rate shocks at various maturities, and credit risk shocks of large exposure, or of the sponsor (ability to pay contributions), to improve connection of stress tests with macro scenarios.
  - Refinements on the liability side:
    - (i) Improving the granularity of the age, wage and pension distributions.
    - (ii) Considering additional “decrement factors” beyond the mortality tables such as the distribution for entry into and exit from the labor force (retirement, disability, voluntary unemployment, et cetera).
    - (iii) Reflecting gender and types of pensions in the calculations.
    - (iv) Considering the possibility of decreases in longevity due to health, famine or natural catastrophe events.
    - Considering tests to assess the impact of plan changes (parametric reforms such as changes in accrual rate, retirement ages, any actuarial assumption, et cetera).
  - Liquidity shocks:
    - Asset shocks should include tests on the portion of assets that might need to be used to cover short term liabilities; these shocks can be more severe, affect closed plans or plans with very low support ratios, and may force sales at distressed prices.
  - Multi asset shocks:
    - Consider multi asset (factor) shocks requiring estimation of asset classes return correlations.
  - Asset-liability correlations:
    - When liabilities are discounted using a market yield curve, offset changes in liabilities with changes in the value of the portion of assets which are interest rate sensitive; this requires knowing durations of these assets.
  - Expected cash flow analysis:
    - Present shocks in terms of impact on future cash flows to see until when assets (on a termination basis) are enough to meet liabilities; the template already produces expected liability cash flows and could be extended to project asset cash flows by assuming future rates of returns on assets, dispositions of assets, and allocation of future cash flows to different types of assets.

*Source: IMF working paper content (excerpt provided).*

### Appendix I. Basic Pension Concepts

### Appendix I. Basic Pension Concepts

### Pension plans
- A pension plan is a legal contract having an explicit retirement objective. The contract may be part of a broader employment contract, it may be defined in the plan rules or documents by the plan sponsor, or it may be required by law.
- Parameters of the pension plan (such as contribution rates, eventual guarantees, retirement age, types of benefits) may be mandated by law, or statute, or defined in the plan rules or documents by the employer, or defined in special laws or regulations. These parameters are often pre-requisites for the plan to be able to obtain special tax treatment.
- Pension plans may offer additional benefits such as disability, sickness, and survivor benefits.
- Plans can be public or private:
  - Public plans: general government administers the plan and its assets, and pays pension benefits. Social security and similar schemes provide minimum benefits at retirement (with or without longevity insurance) for the population at large (or at least the formal sector).
  - Private plans: an entity other than general government administers the assets during the accumulation phase, and/or administers the payment of pension benefits. Typically managed by the employer acting as the plan sponsor, a pension entity or a private sector provider; they may complement or substitute for social security schemes and can include plans for public sector or special categories of workers.
- Plans can be occupational or individual:
  - Occupational plans: participation linked to an employment relationship; may be established by a single employer or a group of employers, sometimes with labor associations.
  - Individual/personal plans: participation is not linked to employment; various financial industry products capture voluntary savings while offering tax exemption.
- Plans can be mandatory or voluntary:
  - Mandatory plans: established by law or require mandatory participation of workers (e.g., traditional social security schemes, private sector mandatory plans).
  - Private pension plans can be voluntary for the employer but mandatory for the employee once the employer decides to sponsor one.
- Plans by promise type: defined benefit (DB), defined contribution (DC), or hybrid:
  - DB plans: insure longevity risk; typically provide a life annuity based on pensionable salary, an accrual factor, and years of contributions/work under the plan. Investment and longevity risks (and sometimes inflation risks) are typically borne by the sponsor.
  - DC plans: do not insure longevity risk; typically provide a cash balance at retirement based on contributions and investment returns. Investment and longevity risks are typically borne by the worker.
  - Hybrid plans: mixture of DB and DC mechanisms; in some jurisdictions (i.e., the US) these are considered DB plans.
- Plans can be funded or unfunded:
  - DC plans are by definition fully funded.
  - For DB plans and DC plans with guarantees, actuarial valuation of assets and liabilities (present value of future benefits) determines funding status. If assets > liabilities (assets < liabilities), DB plans are considered fully funded (partially funded).
  - Many public plans have few or no assets and are pay-as-you-go (PAYG) financed; some OECD countries have partial pre-funding or replaced public plans by private pension plans. In Asia and many African countries, public plans (often called provident funds) provide retirement benefits with little or no longevity insurance.

### Pension funds
- Definition: pools of savings accumulated during working life; cumulative sum of employer and employee contributions and investment income, net of cumulative benefits and administration expenses paid.
- Autonomous pension funds:
  - Legally separated from the plan sponsor (special purpose legal entity or separate account managed by a financial institution).
  - Support personal pension plans by definition.
  - Plan/fund members have a legal or beneficial right or contractual claim against assets held in the autonomous pension fund, providing the highest degree of protection from sponsor bankruptcy (especially with an independent custodian).
- Non-autonomous pension funds:
  - Not legally separated from the plan sponsor; stay on sponsor's balance sheet (e.g., reserve in sponsor's balance sheet as in German Direktzsusage system) or held in legally separated vehicles but remain property of the sponsor.
  - Plan members have no legal claim on pension fund assets; provide lowest degree of protection from sponsor bankruptcy.
- Insured pension funds:
  - Sold by insurance companies; bought by sponsors on behalf of workers or directly by individuals.
  - Exclude cases where insurer acts as plan administrator or asset manager of an autonomous fund.
  - Provide protection from sponsor bankruptcy but expose beneficiary to insurer bankruptcy since assets are segregated from sponsor but merged with insurer assets.
- Claim structures:
  - Collective pension funds pool assets of pension plans of different sponsors.
  - Group pension funds pool assets of unconnected individuals and/or companies in the same pension plan.
  - Individual pension funds do not pool assets of multiple sponsors or beneficiaries; typically individual accounts invested in units of pooled investment funds (examples include occupational mandatory funds in Latin America and Eastern Europe, and 401(k) plans in the United States).
- Open vs closed funds:
  - Closed funds restrict membership to a specific group (company workforce, professional association, industry group).
  - Open funds do not restrict membership; found in Latin America and Eastern Europe where funds compete for market share.

### Pension entities and governance
- Four types of entities typically involved: the pension plan manager, the pension fund manager, the custodian, and external auditors.
- Pension plan manager:
  - Often a special-purpose legal entity (trust, foundation, corporate entity) that owns and may control the pension fund on behalf of plan/fund members.
  - Functions: collect contributions, maintain records, manage assets, pay benefits; may hire service providers for many services. In some jurisdictions, collection of contributions in mandatory DC plans is centralized and executed by the tax authority.
- Pension fund manager:
  - Often an external manager hired by plan managers to manage all or part of fund assets, especially for complex mandates (alternative investments, currency hedging).
  - Hiring external managers can provide benchmarking, shared mandates, and higher risk-adjusted expected returns.
- Custodian:
  - Processes trade settlement payments, reconciles transactions, holds custody records for each manager, reports to the plan manager, is subject to internal audits, and contributes to valuation of assets and liabilities.
  - Best practice discourages plan manager providing custodial services.
  - In sophisticated jurisdictions, daily valuation of assets and liabilities may be performed by the pension plan manager based on: (i) information from the custodian bank for completed transactions of the previous business day; (ii) accounting record of fund liabilities and transactions with its assets for the previous business day; (iii) information on the assets’ market prices on the previous business day; and (iv) determination of fair value for assets and liabilities without a market price, using applicable methods reviewed by supervisory authorities.
  - Custodian can be used by the supervisor as a whistle blower to signal serious breaches of contribution or investment regulations.
- External auditors:
  - Review financial statements, IT systems, internal controls, reconciliation of values with the custodian and accounting, and actuarial assumptions used by the plan manager.
  - Regulators typically: (i) allow the supervisor to call auditors for clarifications without need for approval of the pension plan’s board or management; (ii) grant access to auditors’ working papers; and (iii) require auditors to report serious breaches of regulation and prudential guidelines directly to the supervisor, especially in suspected money-laundering cases.
- Pension firms can be public or private depending on whether they are subject to public or private law.

### Tax incentives and tax treatment of pension savings
- Principle: appropriate tax treatment is when savings are taxed only once. Two alternative ways to achieve single taxation:
  - TEE (Taxed-Exempt-Exempt): contributions are from income already subjected to income tax; investment income and distribution of plan benefits are exempt from income tax.
  - EET (Exempt-Exempt-Taxed): contributions and investment income are exempt from income tax; plan benefits are liable to income tax.
- Expenditure tax regimes (TEE, EET) vs comprehensive income tax regimes (TTE, ETT):
  - Expenditure tax regimes: post-tax rate of return expected to equal in present value terms the pre-tax rate of return; consumption taxed at same rate now and in the future; maintain tax neutrality of consumption over time; avoid double taxation of savings and encourage accumulation of contractual savings for retirement.
  - Comprehensive income tax regimes treat income equally regardless of source and maintain neutrality between consumption and saving.
  - EET or TEE are usually preferred to either TTE or ETT (or TTT) and are prevalent as best practice in several advanced economies.
- Choice between TEE and EET:
  - Usually dictated by fiscal considerations.
  - TEE and EET are generally not equivalent: taxation will be lower in EET than in TEE owing to tax deferral.
  - Introduction of an expenditure tax regime has static and dynamic impacts on Government tax revenues. Static analysis requires information on: (i) amount of contributions to retirement agencies and life insurance companies by employers and employees; (ii) tax treatment of such contributions; (iii) average investment return of different retirement funds and life insurance companies; (iv) tax treatment of such returns; and (v) estimates of the present value of future streams of benefits disbursed to retirees and their tax treatments.
  - Dynamic impact: deferment (EET) means more pre-tax income is available for current investment and accumulation, likely translating into higher economic growth. If the economy is expected to grow at a substantial rate, and contributions are indexed to inflation, an EET scheme may provide a higher present value tax revenue income than a TEE scheme.
  - Credibility: the EET scheme is more credible than the TEE alternative because the TEE scheme entails uncertainty as to whether Government would, in the future, tax benefits as well.

*Source: _wp1129 - Appendix I. Basic Pension Concepts*

### Appendix II. General Actuarial Assumptions

### Appendix II. General Actuarial Assumptions

### USA Male Annuitant Mortality Rates (Table A1)
- Table A1 provides x and q′_x (t887 – 1996) for ages 0 through 115.
- Selected exact q′_x values from the table:
  - Age 0: 0.002080
  - Age 10: 0.000350
  - Age 20: 0.000499
  - Age 30: 0.000694
  - Age 40: 0.000953
  - Age 50: 0.002994
  - Age 60: 0.006428
  - Age 70: 0.018891
  - Age 80: 0.050643
  - Age 90: 0.112208
  - Age 95: 0.162179
  - Age 100: 0.225806
  - Age 105: 0.346177
  - Age 110: 0.584004
  - Age 115: 1.000000

### Select and Ultimate Termination Rates (Table A2)
- Table A2 provides select and ultimate termination rates q′_x,s for various select durations and entry ages (columns include indices such as ,ଶ଴, ,ଷ଴, ,ସ଴, ,ହ଴, ,ହହ, and ,଺଴ as presented).
- Selected exact rates (examples shown in table):
  - Age 20: 0.246913
  - Age 25: 0.163588 and 0.214564 (two column values shown)
  - Age 30: 0.107226, 0.107226, 0.172689
  - Age 35: 0.071678, 0.071678, 0.071678, 0.129676
  - Age 40: 0.051830 (repeated across several select columns) and 0.095359
  - Ages 55–59 show many 0.000000 entries in earlier select columns and nonzero values in later columns, e.g. Age 55: 0.034924 and 0.052843 (in later columns)
  - Age 60: 0.000000 (in many columns), 0.026117, 0.056150 (in later columns)
  - Age 64: 0.000000 (in many columns), 0.011945, 0.012856 (in later columns)

### Early Retirement Rates (Table A3)
- Table A3 lists early retirement probabilities q′_x (early) for ages 55–65:
  - Age 55: 0.051
  - Age 56: 0.051
  - Age 57: 0.051
  - Age 58: 0.051
  - Age 59: 0.051
  - Age 60: 0.202
  - Age 61: 0.303
  - Age 62: 0.405
  - Age 63: 0.304
  - Age 64: 0.304
  - Age 65: 1.000

### Cumulative Wage Merit Scale – Multiples of Entry Age 20 (Table A4)
- Table A4 gives cumulative wage scales (multiples of entry age 20) for ages 20–64.
- Selected exact scale multiples:
  - Age 20: 1.0000
  - Age 25: 1.2369
  - Age 30: 1.4930
  - Age 35: 1.7582 (presented as 351.7582 in the source layout — preserved numeric fragment in table context)
  - Age 40: 2.4780 (listed for age 50 column alignment)
  - Age 50: 2.4780
  - Age 55: 2.6440
  - Age 60: 2.7523
  - Age 64: 2.7908

### Select and Ultimate Wage Function Assumptions (Table A5)
- Table A5 provides select and ultimate wage function multipliers across multiple select durations and entry cohorts (columns given for different select indices).
- Representative exact values from the table:
  - Age 20: 1.0000
  - Age 25: 1.5440
  - Age 30: 2.3263 (and column value 1.0000 for a later select column)
  - Age 35: 3.4198 (and later select column 1.4700)
  - Age 40: 4.9054, 2.1086, 1.0000 (three column values)
  - Age 50: 9.3744, 4.0297, 1.9111, 1.0000 (four column values)
  - Age 60: 16.2242, 6.9741, 3.3074, 1.7307, 1.0000 (five column values)
  - Age 64: 19.6444, 8.4443, 4.0047, 2.0955, 1.2108

### Discount Rate Assumptions (Table A6)
- Table A6 lists period y_0, nominal/annual rates (1 + s_0), discount factors v_t (present value factors) and related columns for periods 0 through 35.
- Selected exact period entries:
  - Period 0: rate 0.00000, (1 + s_0) 1.00000, discount factor 1.00000
  - Period 1: rate 0.00466, (1 + s_0) 1.00466, discount factor 0.99536
  - Period 2: rate 0.00938, (1 + s_0) 1.01413, discount factor 0.98150
  - Period 5: rate 0.01977, (1 + s_0) 1.03202, discount factor 0.90674
  - Period 10: rate 0.03021, (1 + s_0) 1.04349, discount factor 0.74261
  - Period 15: rate 0.03248, (1 + s_0) 1.03886, discount factor 0.61915
  - Period 20: rate 0.03475, (1 + s_0) 1.04342, discount factor 0.50502
  - Period 25: rate 0.03702, (1 + s_0) 1.04798, discount factor 0.40302
  - Period 30: rate 0.03929, (1 + s_0) 1.05255, discount factor 0.31469
  - Periods 31–35: rate 0.03929 repeated; discount factors:
    - Period 31: 0.30279
    - Period 32: 0.29135
    - Period 33: 0.28033
    - Period 34: 0.26973
    - Period 35: 0.25954

*Source: Appendix II. General Actuarial Assumptions (tables A1–A6) from the provided IMF content unit.*

### Appendix III. The Accompanying Template (Model.xls) and Assumptions

### Appendix III. The Accompanying Template (Model.xls) and Assumptions

### Basic model assumptions and inputs
- Entry age: 20 (Input!D12).
- Normal retirement age: 55 (Input!D13).
- Max age: 115 (Input!D14).
- Accrual rate ܾ: 1.0% (Input!D22).
- Labor productivity ݎ݌: 1.0% (Input!D17).
- Wage inflation ߨ
௪: 3.5% (Input!D18).
- Annuity inflation ߨ
௔: 3.5% (Input!D21).
- Effective entry and retirement ages set at ݕ = 20 and ݎ = 55.
- Retirement benefit form: single life inflation indexed annuity (inflation assumption ߨ
௔ = 0.035 in Input!D20).
- Benefit formula: final salary with constant accrual rate (ܾ = 0.01 in Input!D11).

### Decrement, salary, and discounting assumptions
- Mortality rates Ԣݍ
௫
ሺ௠ሻ rendered in Input!I11:I136 are derived from the 1996 US male annuitant table (t887).
- Termination rates Ԣݍ
௫
ሺ௧ሻ rendered in Input!J11:J136 are assumed between age 20 and 54.
- Mortality probabilities ݍ
௫
ሺ௠ሻ and termination probabilities ݍ
௫
ሺ௧ሻ are calculated using equation (1) and rendered in Input!K11:K136 and Input!L11:L136, respectively.
- Composite survival probabilities ݌
௫
ሺ்ሻ are calculated using equation (2) and rendered in Input!M11:M136.
- Wage growth composition:
  - 3 percent average merit increase (Input!T11:T136).
  - 3.5 percent inflation (ߨ
௪ = 0.035 in Input!D18).
  - 1 percent productivity improvements (ݎ݌ = 0.01 in Input!D17).
- Discounting:
  - Level interest rate assumed for liabilities: 9 percent.
  - Template allows alternative discount curves via yield-to-maturity rates Input!O11:O136 using equations (8) and (9).

### Data inputs for distributions
- For each age cohort:
  - Number of workers: Input!V11:V136.
  - Number of retirees: Input!W11:W136.
  - Cohort wage remuneration: Input!X11:X136.
  - Cohort retirement benefit paid: Input!Y11:Y136.
- Aggregates rendered:
  - Total number of workers: Input!V6.
  - Total retirees: Input!W6.
  - Total remuneration: Input!X6.
  - Total retirement benefits: Input!Y6.
  - Density distributions: Input!Z11:Z136 to Input!AC11:AC136.

### Actuarial liabilities — retired cohorts (sheet AL-R)
- Step 1: For each retired cohort אݔ
ሾ
,ݎ∞
ሻ (row 'AL-R'!F8:CH8) compute present value of $1 real life annuity using equation (11):
  - ܽ
ሷ
௫
గ = Σ ( ( (ߨ1൅
௔
) ௦ ) ݌
௦
௫
ሺ௠ሻ ߥ
௦ )
  - Inflation term structure ( (ߨ1൅
௔
) ௦ ) rendered in 'AL-R'!B11:B136.
  - Discount factor curve ߥ
௦ derived using equation (8) and rendered in 'AL-R'!C11:C136.
  - Conditional probabilities of survival ݌
௦
௫
ሺ௠ሻ derived using equation (3) and rendered in 'AL-R'!F11:CH136.
  - Present value ሷܽ
௫
గ rendered in row 'AL-R'!F7:CH7.
- Step 2: Aggregate actuarial liabilities for all retired cohorts using equation (30):
  - ܣܮ
ሺ
ܴ
ሻ = Σ ( ( (ܴܰ) (ܤܴ) (ܴ݀ܰ) (݀ܤܴ) ) ܽሷ
௫
గ )
  - Aggregate rendered in 'AL-R'!D1.

### Actuarial liabilities — active cohorts (sheets Brx and AL-A)
- Method: Projected unit credit, constant dollar (PBOcd).
- Step 1 (Brx):
  - Project wages until retirement for each active cohort in matrix Brx!F11:CH136 using formula (31):
    - ݂ݓ
௦,௫ = ݂ݓ
௦,௬ / ݂ݓ
௫,௬ = ݏ݉
௦,௬ / ݏ݉
௫,௬ ( (ߨ1൅
௪
) (ݎ݌1൅) )  (as rendered in text).
  - Calculate total accrued benefits at retirement for each cohort in row Brx!F1:CH1 using plan final salary benefit formula and distributions (equation (32)):
    - ܤ
௥,௫ ܾ = ( (ݕെݎ) ) ݂ݓ
௥,௫ ( (ܣܰ) (ܣܹ) (ܣ݀ܰ) (ܣܹ݀) )  rendered as described.
- Step 2 (AL-A):
  - Aggregate cohort liabilities using equation (33):
    - ܣܮ
ሺ
ܣ
ሻ = Σ ( ( (ݕെݔ) (ݕെݎ) ܤ
௥,௫ ݌ቀ
௥ି௫
௫
ሺ்ሻ ߥ
௥ି௫ ሷܽ
௥
గ ) )
  - ݌
௥ି௫
௫
ሺ்ሻ is conditional composite probability of survival between age x and age r rendered in 'AL-A'!E11:BM136 and 'AL-A'!E3:BM3.
  - ߥ
௥ି௫ rendered in 'AL-A'!C11:C136 and row 'AL-A'!F4:CH4.
  - ሷܽ
௥
గ is present value of a real life annuity at normal retirement age rendered in cell 'AL-R'!F7.
- Total plan liabilities (sheet AL-TOT): sum of actuarial liabilities for active and retired members.

### Key spreadsheet cell mappings (Table A7)
- ݕ = 20 — Input!D12 — Entry Age.
- ݎ = 55 — Input!D13 — Normal Retirement Age.
- ݔ
௠௔௫ = 115 — Input!D14 — Max Age.
- ݎ݌ = 1.0% — Input!D17 — Labor Productivity.
- ߨ
௪ = 3.5% — Input!D18 — Inflation (wages).
- ߨ
௔ = 3.5% — Input!D21 — Inflation (annuities).
- ܾ = 1.0% — Input!D22 — Accrual Rate.

### Select rows from Table A8 (actuarial factor series and cohort data)
- Ages and associated series (excerpted rows as rendered):
  - Age 0: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.002080; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.09; ݉ݐݕ
௦
ݏ݉
௫ = 0.02.
  - Age 20: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.000499; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.246913; ݉ݐݕ
௦
ݏ݉
௫ = 0.09; ݎ
௫ = 1; ܣܰ
௫ = 0.018.
  - Age 23: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.000566; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.193147; ݉ݐݕ
௦
ݏ݉
௫ = 0.09; ݎ
௫ = 1.1396974; ܣܰ
௫ = 0.013; ܣܹ
௫ = 1; ܤܴ
௫ = 13,051.2.
  - Age 30: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.000694; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.107226; ݉ݐݕ
௦
ݏ݉
௫ = 0.09; ݎ
௫ = 1.4929901; ܣܰ
௫ = 0.005; ܣܹ
௫ = 20; ܤܴ
௫ = 238,983.9.
  - Age 37: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.000749; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.062155; ݉ݐݕ
௦
ݏ݉
௫ = 0.09; ݎ
௫ = 1.8645042; ܣܰ
௫ = 0.006; ܣܹ
௫ = 41; ܤܴ
௫ = 882,063.2.
  - Age 45: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.001752; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.042618; ݉ݐݕ
௦
ݏ݉
௫ = 0.09; ݎ
௫ = 2.265375; ܣܰ
௫ = 0.014; ܣܹ
௫ = 44; ܤܴ
௫ = 1,025,525.9; additional cell value 4,129.0 shown.
  - Age 50: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.002994; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.038670; ݉ݐݕ
௦
ݏ݉
௫ = 0.09; ݎ
௫ = 2.477958; ܣܰ
௫ = 0.019; ܣܹ
௫ = 52; ܤܴ
௫ = 1,293,359.4; additional cell value 151.1.
  - Age 55: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.004534; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.09; ݉ݐݕ
௦
ݏ݉
௫ = 0.018; ܣܹ
௫ = 8; ܣܰ
௫ = 0.09; additional values: 225,327.3 and 38,941.3.
  - Ages 92–115: sample series end values include:
    - Age 92: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.131017; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.09; ݉ݐݕ
௦
ݏ݉
௫ = 0.003.
    - Age 100: ݍ
ᇱ
௫
ሺ
௠
ሻ = 0.225806; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.09; ݉ݐݕ
௦
ݏ݉
௫ = 0.000.
    - Age 115: ݍ
ᇱ
௫
ሺ
௠
ሻ = 1.000000; ݍ
ᇱ
௫
ሺ
௧
ሻ = 0.09; ݉ݐݕ
௦
ݏ݉
௫ = 0.000.

*Source: Appendix III. The Accompanying Template (Model.xls) and Assumptions (from _wp1129 - Appendix III. The Accompanying Template (Model.xls) and Assumptions).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1129.pdf_