Liability valuation on several bases
Projected benefit cash flows are discounted on funding, accounting, guarantee and buyout bases, so the board sees how the same promises look under each purpose and which assumptions drive the differences.
GRA · Institutional Funds Nexus
Spatial financeWhere capital meets hazard, exposure and transition
Actuarial mathematics for pension funds
Actuarial mathematics turns a promise to pay into a number a board can manage: the present value of benefits, the contributions needed to fund them and their sensitivity to rates, inflation and survival. For institutional funds it is the measure of the liability side, and every assumption it rests on is a judgement that must be stated.
01Contents
Actuarial mathematics for pension funds in five parts.
02Application
What actuarial mathematics does here.
Actuarial mathematics applies probability, statistics and financial theory to measure and price contingent risks, particularly in insurance and pensions. It determines reserves and premiums for uncertain future obligations. In capital markets and investment and society and human development, actuarial methods determine whether risks can be insured and at what cost. They help assess accumulation of correlated losses across portfolios and regions. The canonical description is on the STEM Commons.
Mortality and longevity projection
Base tables are adjusted to scheme experience and combined with improvement models, and the liability is revalued under alternative improvement paths to show longevity sensitivity.
Funding plan and recovery path
Contributions, expected returns and benefit outgo are projected forward to show how a deficit is closed over the recovery period, and how the path changes under stressed returns or a weaker sponsor.
03Monitor
Tail fit.
The screen a team would use to follow this method: its measures, states and decisions. Names are examples and values are illustrative.
Tail fit
Institutional funds · Actuarial mathematics for pension · Run R-7350
Shape parameter uncertainGRA · Institutional Funds Nexus
Return level curve
Method card
- Method
- Actuarial mathematics
- Approach
- peaks over threshold
- Threshold
- 98th percentile
- Label
- MODELLED
Parameters
generalised Pareto fit
| Shape ξ | 0.29 | −0.04 | 0.47 | heavy tail if above zero |
| Scale σ | 3.0 | 1.1 | 5.1 | units of the variable |
| Exceedances | 68 | declustered |
04Data
What the method consumes.
Datasets and records used by actuarial mathematics on this platform, with the role each plays and the label its values carry. Sources are listed in the source register.
| Data | Role | Label |
|---|---|---|
| Scheme membership and benefit data | Projected benefit cash flows | REPORTED |
| National and actuarial mortality tables | Base mortality and improvement | Published statistics |
| Government bond and swap curves | Discount rates and inflation expectations | OBSERVED |
| Scheme mortality experience studies | Adjustment of tables to the membership | ESTIMATED |
| Disclosed actuarial assumptions | Comparison across funds | DECLARED |
05Loss model
Loss exceedance, average annual loss and probable maximum loss.
The loss a portfolio in the footprint can expect per year and at the 1 in 100 and 1 in 250 levels that insurers and lenders price, from the same three blocks every catastrophe model uses. Teaching model with stated illustrative values, not a prediction.
Controls
- Typical annual hazard intensity (location)
- 1.00 intensity index
- Spread of extremes (scale)
- 0.80 intensity index
- Intensity at which half the value is lost
- 4.00 intensity index
- Dispersion of vulnerability
- 0.55
- Share of value in the hazard footprint
- 40%
- Protection standard (return period)
- 50 years
Readouts
- Average annual loss
- 0.51% of value
- Loss at 1 in 100 years
- 24.5% of value
- Loss at 1 in 250 years
- 28.4% of value
- Intensity at 1 in 100 years
- 4.68 intensity index
Numbers, methods and sources
| Return period (years) | Intensity (intensity index) | Loss (% of value) |
|---|---|---|
| 1 | 0.09 | 0.00 |
| 2 | 1.27 | 0.00 |
| 4 | 1.91 | 0.00 |
| 7 | 2.47 | 0.00 |
| 13 | 3.00 | 0.00 |
| 24 | 3.51 | 0.00 |
| 44 | 4.02 | 0.00 |
| 82 | 4.53 | 23.55 |
| 154 | 5.03 | 26.44 |
| 287 | 5.53 | 28.87 |
| 536 | 6.03 | 30.88 |
| 1,000 | 6.53 | 32.53 |
Methods
- Gumbel extreme-value hazard Gumbel (1958), Statistics of Extremes, Columbia University Press
- Lognormal fragility (vulnerability) FEMA (2003), HAZUS-MH Technical Manual
- Exceedance-probability curve and average annual loss Grossi and Kunreuther (2005), Catastrophe Modeling, Springer
- Levee effect Di Baldassarre and others (2013), Socio-hydrology, HESS 17
- Uncertainty fan (parameter Monte Carlo) Spiegelhalter, Pearson and Short (2011), Visualising uncertainty about the future, Science 333
- One-at-a-time sensitivity (tornado) Saltelli and others (2008), Global Sensitivity Analysis: The Primer, Wiley
Data
The model runs on the teaching values set by the controls. To calibrate it for a place, read it against these admitted layers and series, which this page’s map can draw:
- Rivers Natural Earth
- Urban extents Natural Earth
- Cities Natural Earth
- Earthquakes M4.5+, past week (USGS) U.S. Geological Survey · register usgs.earthquake.summary-feeds
- Rain rate, last half hour (IMERG) NASA GPM IMERG Early · register nasa.gibs.context-live-layers
- River discharge forecast, 30 days with range (GloFAS via Open-Meteo) Open-Meteo.com (CC BY 4.0), Copernicus GloFAS · register open-meteo.public-pages
- Built-up and settlement extent, 2010 (HBASE, Landsat) CIESIN/Columbia, NASA SEDAC HBASE · register sedac.context-urban-layers
Across the six pillars, led by Nexus Consortium
- Experiments Labs
- Built in Institutional Funds Nexus in Foundry
- Tested at Institutional Funds Nexus in Universe
- Codified in Finance, Insurance and the Economy in Standards
- Published in Institutional Funds Nexus in Reports
- Taught in Faculty of Finance and Economics in Academy
06Validation
How results are checked.
Assumptions are tested through experience studies and actual-versus-expected analysis. Independent actuarial review checks methods and results.
| Check | How it is done |
|---|---|
| Actual versus expected analysis | Deaths, retirements and withdrawals are compared with what the assumptions implied, and persistent gaps prompt a change of assumption |
| Analysis of surplus | The change in funding between valuations is decomposed into returns, rates, experience and assumption changes, and the residual is reported |
| Sensitivity reporting | Liabilities are restated under shifts in discount rate, inflation and longevity so each assumption’s weight is visible |
| Independent actuarial review | Methods and assumptions are reviewed by an actuary not involved in the valuation, under professional standards |
Historical experience may not reflect emerging or correlated risks. Long-tail liabilities are sensitive to assumptions about future trends.
07Exposure, loss and capital
What actuarial mathematics for pension funds means for exposure, loss and capital.
The exposure, loss, scenario and capital views a risk officer, supervisor or ministry of finance would ask for, with the frameworks they report against. Structures follow supervisory and actuarial practice. Values are illustrative: never prices, ratings or advice.
Exposure and loss at return periods
How losses from actuarial mathematics for pension funds grow with severity, gross and net of reinsurance, against capital.
| Return period | Annual probability | Gross loss | Net loss | Net, capital = 100 |
|---|---|---|---|---|
| 1 in 10 | 0.10 | 14 | 10 | 10 |
| 1 in 50 | 0.02 | 33 | 21 | 21 |
| 1 in 100 | 0.01 | 46 | 27 | 27 |
| 1 in 250 | 0.004 | 68 | 39 | 39 |
| Average annual loss | 3.1 | 2.2 | 2.2 |
NGFS climate scenarios · Network for Greening the Financial SystemIFRS S2 Climate-related Disclosures · ISSBTCFD recommendations · Financial Stability Board TCFDPrinciples for the effective management of climate-related financial risks · Basel Committee on Banking Supervision
08Connections
Topics that rely on this method.
09Stewardship
Who maintains the method.
Actuarial and Quantitative Risk stewards actuarial mathematics for the Consortium and reviews how it is applied across platforms.
10Connected
Where this page leads across the Consortium.
11Questions
Questions and answers.
What are the limits of actuarial mathematics in this field?
Historical experience may not reflect emerging or correlated risks. Long-tail liabilities are sensitive to assumptions about future trends.
How are results checked?
Assumptions are tested through experience studies and actual-versus-expected analysis. Independent actuarial review checks methods and results.
Are the values on this page real?
The data names, roles and checks are real descriptions of practice. The canvas shows a design of the method’s use, not a run; no result is reported here.
Where else is this method applied?
The canonical method page lists every platform that applies it.
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