Modeling for a Model Retirement
1. Separating the Return Environment from Sequence-of-Returns Risk
The question
A conventional retirement model generates paths of future asset returns, applies a spending rule to each path, and records outcomes such as ruin, terminal wealth or consumption shortfall. The proposed refinement is to separate two sources of uncertainty:
- the collection of returns experienced over retirement; and
- the order in which those returns occur.
For each simulated return path, we could hold its realized returns fixed and generate many permutations of exactly those returns. Variation across permutations would measure sequence-of-returns risk conditional on the return environment. Variation between different original return paths would measure uncertainty about the return environment itself.
The approach is attractive because two paths can have exactly the same arithmetic return, geometric return, volatility and empirical distribution, yet produce radically different retirement outcomes when withdrawals occur between investment returns.
Sequence risk in retirement
Suppose real wealth evolves according to \[W_{t+1}=(W_t-C_t)(1+R_{t+1}),\] where \(C_t\) is real spending and \(R_{t+1}\) is the real portfolio return. If there are no external cash flows, terminal wealth depends only on the product of the gross returns and is therefore invariant under their ordering. Withdrawals destroy that invariance because losses occurring while wealth and withdrawals are large have greater consequences than the same losses occurring later.
The retirement literature has long recognized that poor returns near retirement can have a disproportionate effect on sustainable withdrawals. Bengen’s historical analysis of inflation-adjusted withdrawals is the foundational practical example Bengen (1994). Pfau studies sequence risk over the complete accumulation and decumulation lifetime and emphasizes the exceptional sensitivity of outcomes around retirement Pfau (2015). Clare et al. provide a more explicit treatment of sequence risk and demonstrate the effect of rearranging a common collection of returns Clare et al. (2020).
The exact nested permutation experiment is nevertheless not part of the traditional standard toolkit. Conventional historical analysis, bootstrapping and Monte Carlo simulation usually vary the return environment and its ordering simultaneously. They show that sequence matters without cleanly decomposing sequence risk from variation in the returns available to be sequenced.
Recent working papers have moved closer to the exact counterfactual question. Varadi defines measures of the cost associated with a realized ordering relative to alternative orderings of the same returns Varadi (2026). Ko develops a mathematical framework for order-dependent wealth processes and permutation-based measures of sequence risk Ko (2026). These papers indicate that a formal permutation approach is a recent development rather than established actuarial practice.
The correct interpretation of the outer path
Fixing a simulated path and permuting its returns holds much more than its average return constant. It fixes every symmetric statistic of the realized collection, including:
- the arithmetic and geometric means;
- volatility and all empirical moments;
- skewness and tail thickness;
- the number, size and frequency of market crashes;
- the best and worst annual returns;
- the complete empirical return distribution.
The natural decomposition is therefore not merely
average-return risk versus sequence risk.
It is more accurately
return-environment risk versus ordering risk conditional on the return environment.
The return environment consists of the unordered multiset of returns. The sequence experiment asks what retirement outcomes would have occurred if that same economic material had arrived in another order.
A nested simulation
Let the outer simulation produce return environments \[R_i=(R_{i1},\ldots,R_{in}),\qquad i=1,\ldots,N.\] For each outer path \(i\), generate permutations \(\pi_{ij}\), \(j=1,\ldots,M\), and evaluate \[Y_{ij}=f\left(R_{i,\pi_{ij}(1)},\ldots,R_{i,\pi_{ij}(n)}\right),\] where \(Y\) could be terminal wealth, lifetime consumption, a shortfall measure or an indicator of ruin.
Define the conditional permutation mean and variance by \[\mu_i=P_\pi(Y_{ij}\mid R_i),\qquad v_i=\operatorname{Var}_\pi(Y_{ij}\mid R_i).\] The law of total variance gives \[\operatorname{Var}(Y)=\operatorname{Var}\!\left(P(Y\mid R)\right)+P\!\left(\operatorname{Var}(Y\mid R)\right).\] The two terms can be estimated by \[\widehat V_{\mathrm{environment}}=\operatorname{Var}_i(\mu_i),\qquad \widehat V_{\mathrm{order}}=P_i(v_i).\] A descriptive sequence-risk share is then \[s_{\mathrm{order}}=\frac{\widehat V_{\mathrm{order}}}{\widehat V_{\mathrm{environment}}+\widehat V_{\mathrm{order}}}.\] The ratio is not an invariant property of retirement risk. It depends on the scenario generator, spending rule, asset allocation, horizon and chosen outcome \(Y\). It nevertheless provides a clean decomposition within a specified model.
Conditional ruin probability
For retirement planning, the most revealing object may be the conditional probability of ruin over random orderings: \[q_i=P_\pi(\tau\leq n\mid R_i),\] where \(\tau\) is the time at which assets are exhausted.
The value \(q_i\) distinguishes three fundamentally different environments:
- \(q_i\approx0\): the environment is robustly adequate; nearly every ordering succeeds;
- \(q_i\approx1\): the environment is fundamentally inadequate; rearranging returns does not rescue it;
- \(0<q_i < 1\): the environment contains adequate aggregate investment performance, but success depends materially on sequencing.
The middle category represents sequence risk in its purest form. The first and third categories might have identical average returns, yet very different robustness to ordering.
A continuous shortfall measure is often preferable to the ruin indicator. Ruin treats a path that misses the final payment by \(1\) as equivalent to one that exhausts assets twenty years early. Alternatives include discounted unmet spending, minimum funded ratio, duration of shortfall and utility loss.
The permutation distribution is counterfactual
Randomly permuting annual returns destroys any serial structure in the original path. It removes or alters:
- serial correlation;
- volatility clustering;
- persistence of economic regimes;
- valuation-dependent expected returns;
- relationships between inflation, interest rates and asset returns;
- correlations between labour income, pension income and markets.
An unrestricted permutation therefore answers a precise counterfactual:
Given exactly this collection of annual outcomes, what would retirement have looked like under a uniformly random ordering?
It does not necessarily describe the distribution generated by a realistic economic process. Uniformity over permutations is itself a probability model, although it is conditional and deliberately artificial.
Several restricted rearrangements can provide complementary information:
- annual permutations measure maximum sensitivity to ordering;
- block permutations preserve some short-term dependence;
- regime-block permutations keep complete recessionary or inflationary episodes intact;
- cyclic shifts preserve almost all adjacent relationships while changing calendar placement;
- path reversal gives a simple comparison between early and late adversity;
- targeted optimization identifies the best and worst possible orderings.
The comparison among these experiments can show whether retirement vulnerability arises from the broad placement of adverse regimes or from fine annual ordering.
Alternatively, deliberately destroying auto-correlation in the data gives a measure of the importance of that correlation.
Practical conclusion
The permutation approach is useful and conceptually clean. It isolates a source of risk that ordinary Monte Carlo simulation mixes together with uncertainty about the overall return environment. Its strongest use is diagnostic rather than predictive: it reveals whether a given collection of returns is intrinsically inadequate or merely vulnerable to ordering.
A serious retirement model should report both unconditional outcomes and conditional order sensitivity. The nested analysis could include:
- the distribution of \(q_i\);
- the sequence-risk variance share;
- best, worst and median permutations within each environment;
- sensitivity to annual, block and regime rearrangements;
- characteristics of environments having intermediate \(q_i\);
- the effect of adaptive spending or asset-allocation rules on \(q_i\).
The resulting analysis would distinguish bad economic material from bad timing.
2. Red and Green Scenario Maps Without a Trusted Probability Model
The question
An alternative way to think about retirement planning is to begin with a set of possible futures rather than a probability distribution. The futures could be displayed conceptually as a two-dimensional grid, although a realistic scenario space would be much higher-dimensional. Each scenario is classified as acceptable or unacceptable: green if the retirement strategy achieves its objectives and red if it fails.
The first goal is to understand the geometry of success and failure without asserting that a reliable probability distribution exists. A secondary temptation is to calculate the fraction of the scenario region that is green. That calculation appears non-probabilistic, but it implicitly treats the grid or sampling design as a probability measure.
The questions are what this framework is called, how it relates to robust risk measurement, and whether it is genuinely useful.
The safe set
Let \(u\) denote a retirement strategy and \(z\in\mathcal Z\) a possible future. The scenario \(z\) might specify returns, inflation, longevity, care costs, taxation and spending shocks. Let \(G(u,z)\) be a performance measure and \(c\) the minimum acceptable level.
The safe set is \[S_u=\{z\in\mathcal Z:G(u,z)\geq c\}.\] The green region is \(S_u\), while the red region is \(\mathcal Z\setminus S_u\).
The construction needs:
- a set of plausible futures \(\mathcal Z\);
- a model connecting strategies and futures to outcomes;
- an adequacy criterion;
- no initial probability distribution on \(\mathcal Z\).
The binary classification is an example of satisficing: the strategy must meet a threshold rather than maximize an expected objective.
Robust Decision Making and scenario discovery
The closest established framework is Robust Decision Making, or RDM, developed principally at RAND by Lempert, Popper and Bankes Lempert, Popper, and Bankes (2003). RDM is part of the broader field of decision-making under deep uncertainty.
Deep uncertainty arises when analysts or decision-makers cannot agree on one or more of:
- the correct model;
- the probability distributions governing important variables;
- the appropriate valuation of outcomes;
- the decisions that will be available in the future.
RDM treats models as generators of possible cases rather than trusted predictors. A typical process is:
- specify candidate strategies;
- generate a large and deliberately diverse ensemble of plausible futures;
- evaluate every strategy in every future;
- classify outcomes using agreed acceptability criteria;
- identify the combinations of assumptions under which strategies fail;
- design more robust or adaptive strategies;
- repeat the analysis.
The red-green classification step is called scenario discovery. Bryant and Lempert formalized an influential computer-assisted approach in which statistical learning algorithms identify regions of the input space strongly associated with policy success or failure Bryant and Lempert (2010).
A common algorithm is the Patient Rule Induction Method, or PRIM. PRIM searches for relatively simple boxes in the scenario space containing a high concentration of cases of interest. A retirement application might identify a vulnerability region such as:
failure is concentrated where first-decade real returns are low, essential-cost inflation is high and retirement lasts more than 35 years.
Scenario-discovery regions are commonly judged by:
- density: the fraction of cases inside the identified region that are failures;
- coverage: the fraction of all failures captured by the region;
- interpretability: whether the region can be explained using a manageable number of conditions.
The central output is not a probability forecast. It is an intelligible description of the conditions under which the strategy is vulnerable.
Why the proportion of green scenarios is not probability-free
Suppose a finite design contains scenarios \(z_1,\ldots,z_N\), and define \[r(u)=\frac{1}{N}\sum_{i=1}^N1_{\{z_i\in S_u\}}.\] The quantity \(r(u)\) is the empirical measure of the safe set under the scenario-generation design. Calling the scenarios merely “possible” does not remove the weighting induced by the design.
The problem is especially clear under reparameterization. Uniform spacing in annual return is not uniform spacing in continuously compounded return. Uniform spacing in mortality improvement is not uniform spacing in life expectancy. Refining the grid selectively in one region changes the fraction green even though the underlying safe set has not changed.
Consequently,
80% of scenarios are green
has no intrinsic meaning unless the scenario sampling measure has a defensible interpretation.
RDM usually treats scenario frequencies as exploratory design weights rather than estimates of real-world probabilities. It emphasizes whether vulnerabilities persist under alternative designs and whether a strategy can be improved, rather than presenting the raw proportion of successful scenarios as a probability of success.
Measure-free comparisons
Several useful comparisons do not require a probability measure.
Safe-set inclusion
If \[S_u\supseteq S_v,\] then strategy \(u\) succeeds whenever strategy \(v\) succeeds and perhaps in additional futures. Strategy \(u\) therefore dominates \(v\) by safe-set inclusion.
Safe-set inclusion is genuinely measure-free, but it produces only a partial ordering. In realistic cases, different strategies will usually succeed in different regions.
Worst-case performance
Classical robust optimization chooses \[\max_u\min_{z\in\mathcal Z}G(u,z).\] The criterion is clean but can be excessively conservative. If \(\mathcal Z\) contains remotely plausible catastrophes, every strategy may be evaluated according to an extreme and largely uninformative case.
Minimax regret
Let \[G^*(z)=\max_vG(v,z).\] The regret of strategy \(u\) in scenario \(z\) is \[L(u,z)=G^*(z)-G(u,z),\] and minimax regret selects \[\min_u\max_{z\in\mathcal Z}L(u,z).\] Minimax regret does not demand excellent absolute performance in every future. It limits how badly the selected strategy performs relative to the strategy that would have been chosen with hindsight.
Distance to failure
Choose a metric \(d\) on the scenario space. For a green scenario \(z\), define its robustness margin by \[m_u(z)=\inf\{d(z,z'):z'\notin S_u\}.\] A large margin means that a substantial perturbation is required to cause failure. The measure distinguishes barely successful scenarios from comfortably successful ones.
The metric is a modelling choice. Scaling, parameterization and dependence assumptions all matter, but the choice is explicit rather than hidden in an arbitrary count of grid cells.
Info-gap decision theory
Info-gap decision theory is a related explicitly non-probabilistic framework developed by Ben-Haim Ben-Haim (2006). Begin with a nominal model \(z_0\) and nested uncertainty sets \[\mathcal Z(\alpha)=\{z:d(z,z_0)\leq\alpha\}.\] For strategy \(u\), define the robustness radius \[\alpha^*(u)=\sup\{\alpha:G(u,z)\geq c\text{ for every }z\in\mathcal Z(\alpha)\}.\] The value \(\alpha^*(u)\) is the greatest departure from the nominal assumptions that the strategy can tolerate while still meeting the required outcome.
Info-gap analysis makes the trade-off between aspiration and robustness explicit. A higher spending target may be feasible under the nominal model but have a small robustness radius; a more modest target may survive a much wider range of deviations.
The method is useful but depends strongly on:
- the nominal model;
- the distance or uncertainty measure;
- the shape and nesting of the uncertainty sets;
- the chosen adequacy threshold.
Info-gap theory does not make uncertainty disappear. It replaces a probability model with a geometric specification of uncertainty.
Relationship to robust risk measures
The phrase robust risk measure usually has a different mathematical meaning. Coherent and convex risk measures were developed to assign a monetary capital requirement to a random financial position Artzner et al. (1999), Föllmer and Schied (2002). A robust representation often takes the form \[\rho(X)=\sup_{Q\in\mathcal Q}\{Q(-X)-\alpha(Q)\},\] where \(\mathcal Q\) is a family of probability measures and \(\alpha\) penalizes less plausible measures.
For a coherent risk measure, the penalty often reduces to an admissibility condition, giving \[\rho(X)=\sup_{Q\in\mathcal Q}Q(-X).\] This framework does not dispense with probability. It recognizes ambiguity by considering a family of probabilities rather than a single probability measure.
Applied to the retirement safe set, imprecise probability or distributional robustness might report \[\underline p_u=\inf_{Q\in\mathcal Q}Q(S_u),\qquad \overline p_u=\sup_{Q\in\mathcal Q}Q(S_u).\] The result is a range of success probabilities across an ambiguity set.
The red-green scenario map is logically prior to that calculation. It first constructs \[z\longmapsto1_{\{z\in S_u\}},\] and studies the geometry of the classification. Probabilities or families of probabilities may then be laid over the map.
The terminology can therefore be organized as follows:
| Framework | Principal object |
|---|---|
| Scenario analysis | Outcomes under selected futures |
| Exploratory modelling | A large ensemble used to investigate assumptions |
| Scenario discovery | Regions associated with success or failure |
| Robust Decision Making | Strategies that perform adequately across diverse futures |
| Satisficing | Meeting an adequacy threshold |
| Info-gap theory | Radius of uncertainty tolerated before failure |
| Robust optimization | Worst-case performance over an uncertainty set |
| Minimax regret | Worst opportunity loss relative to hindsight |
| Distributionally robust optimization | Optimization over a family of probability distributions |
| Imprecise probability | Lower and upper probabilities |
| Robust risk measurement | Risk evaluated over multiple probability measures |
| Viability analysis | States from which acceptable operation can be sustained |
| Stress testing | Performance under selected adverse cases |
Application to retirement planning
A retirement scenario space could include:
- long-run asset returns;
- sequence of returns;
- interest rates;
- general and category-specific inflation;
- asset-return and inflation dependence;
- longevity and mortality improvement;
- health and long-term-care states;
- taxation and public-pension rules;
- essential and discretionary spending;
- housing shocks;
- willingness and ability to adapt spending;
- annuitization or other contingent decisions.
For each strategy, the model would classify whether essential consumption and other core objectives are maintained.
The most valuable analysis would not ask only how much of the scenario space is green. It would ask:
- Where is the failure boundary?
- Which assumptions most strongly distinguish red from green?
- Are failures shallow or severe?
- Does one strategy contain another strategy’s safe set?
- Which modest policy adaptations turn large red regions green?
- Are conclusions stable under alternative scenario designs and probability weightings?
Adaptive strategies are particularly important. A fixed withdrawal rule might fail in a large region, while a rule that temporarily reduces discretionary spending, changes asset allocation or purchases an annuity at a funded-ratio trigger may substantially enlarge the safe set.
Practical conclusion
The red-green framework is useful, especially when probability estimates are unreliable or contentious. It makes modelling assumptions visible and changes the central question from
What is the probability that the plan succeeds?
to
Under what circumstances does the plan fail, how severe is the failure, and what can we do about it?
The framework does not eliminate subjective judgement. The scenario set, adequacy threshold, geometry and sampling design remain choices. Its advantage is that these choices are exposed to inspection.
A strong retirement analysis would combine:
- safe and failure sets;
- scenario-discovery descriptions;
- margins to failure;
- worst-case or regret measures;
- probabilities under several alternative models;
- adaptive actions that can be taken as uncertainty resolves.
We should first understand the geometry of retirement success and failure and only then decide how much probability we are willing to place over that geometry.
3. Real Returns, Inflation and the Evolution of Retirement Spending
The question
Retirement models often express assets and spending in real terms. Nominal portfolio returns are deflated by an inflation index, and withdrawals are held constant in real terms. That representation appears to remove inflation from the model.
The concern is that retirement spending is not genuinely constant. The household consumption basket changes with age, health, housing, family circumstances and preferences. Different components of spending experience different inflation rates, and some important costs, such as long-term care, are contingent rather than smoothly inflation-linked.
The questions are when folding inflation into real returns is legitimate and how a more realistic model should represent retirement consumption.
Real returns are a change of numéraire
Let nominal wealth be \(W_t\), nominal return \(R_{t+1}\), nominal spending \(C_{t+1}\), and a price index \(I_t\). Nominal wealth evolves as \[W_{t+1}=W_t(1+R_{t+1})-C_{t+1}.\] Define real wealth by \(w_t=W_t/I_t\), and let \[1+\pi_{t+1}=\frac{I_{t+1}}{I_t}.\] Then \[w_{t+1}=w_t\frac{1+R_{t+1}}{1+\pi_{t+1}}-\frac{C_{t+1}}{I_{t+1}}.\] Define the real portfolio return by \[1+r_{t+1}=\frac{1+R_{t+1}}{1+\pi_{t+1}}.\] The real-wealth equation becomes \[w_{t+1}=w_t(1+r_{t+1})-c_{t+1},\] where \(c_{t+1}=C_{t+1}/I_{t+1}\).
Holding \(c_t=c\) constant is therefore mathematically legitimate when nominal spending satisfies \[C_t=cI_t.\] The transformation is exact. It does not approximate inflation or require low inflation.
The substantive issue is the choice of \(I_t\). “Real wealth” has meaning only relative to a specified numéraire. Constant spending in CPI units means that the household purchases a stream whose cost tracks CPI. It does not necessarily mean that the household maintains its actual standard of living.
Prices and quantities must be separated
Let retirement consumption contain categories \(k=1,\ldots,m\). Nominal expenditure is \[C_t=\sum_{k=1}^mp_{k,t}q_{k,t},\] where \(p_{k,t}\) is the category price and \(q_{k,t}\) the quantity or service consumed.
Changes in expenditure arise from two distinct sources:
- prices change;
- the consumption bundle changes.
The second source includes:
- voluntary changes in preferences;
- reduced travel or recreation with age;
- substitution toward cheaper goods;
- housing changes and mortgage expiry;
- disability or reduced physical capacity;
- death of a spouse;
- family support or gifts;
- new health and care requirements;
- forced spending reductions following poor investment outcomes.
A broad consumer price index is designed to measure changes in the price of a representative basket. It does not model the path of an individual household’s quantities, and its representative weights may differ substantially from those of a particular retiree.
Kalwij, Alessie and Knoef show that retirees experience different inflation rates because household expenditure weights differ and category prices evolve differently Kalwij et al. (2018). The relevant issue is therefore not only whether CPI is forecast accurately, but whether CPI is the correct liability index for the household.
Constant real spending as a benchmark
The conventional safe-withdrawal model specifies an initial withdrawal and increases its nominal amount with general inflation. Bengen’s historical analysis is the classic example Bengen (1994). In real units, the rule is a constant withdrawal.
The assumption is useful because it:
- creates a transparent and reproducible liability;
- isolates investment, sequence and longevity risk;
- provides a common benchmark across strategies;
- avoids embedding debatable assumptions about declining consumption;
- may be reasonably conservative for discretionary spending.
Constant real spending should nevertheless be interpreted as a benchmark liability, not as a literal forecast of household behaviour.
Empirical spending patterns
Empirical research generally finds that real household spending tends to decline during much of retirement, although the pattern varies materially across households. Blanchett describes a “retirement spending smile”: real spending declines through much of retirement and may rise at advanced ages as healthcare or care costs become more important Blanchett2014a?.
Several cautions are essential.
First, an observed decline in spending does not establish that retirees prefer lower consumption. Spending may decline because resources are inadequate. A model that treats forced retrenchment as a harmless age effect risks understating the cost of an unsuccessful retirement strategy.
Second, cross-sectional age patterns need not equal longitudinal household paths. Cohort effects, mortality selection, household composition and wealth differences can make older households appear to spend differently even when a particular household would not follow the estimated curve.
Third, average spending conceals enormous heterogeneity. Retirees differ in health, housing tenure, family structure, wealth and desired lifestyle. A deterministic “retirement smile” should not replace constant spending as another universal rule.
Fourth, rising healthcare expenditure does not necessarily cause total household spending to rise. Reduced transport, travel and leisure expenditure can dominate increases in healthcare. Conversely, long-term care can produce very large individual tail costs even when average healthcare expenditure remains manageable.
Household-specific inflation
Let \(w_{k,t-1}\) be the share of the household’s expenditure devoted to category \(k\) immediately before period \(t\), and let \(\pi_{k,t}\) be category inflation. A household inflation approximation is \[\pi_t^H\approx\sum_{k=1}^mw_{k,t-1}\pi_{k,t}.\] The weights evolve as quantities, prices and household circumstances change. Household inflation is therefore path-dependent.
A retiree who owns a mortgage-free home, drives little and spends heavily on private healthcare has a different inflation exposure from a renter who travels frequently and relies on public healthcare. Neither exposure is captured perfectly by a national CPI.
A personal model can begin with actual spending records. Categories need not be excessively granular. A manageable division might be:
- essential recurring consumption;
- discretionary consumption;
- housing and maintenance;
- healthcare;
- long-term care;
- large irregular expenditures;
- transfers, gifts and bequests.
A component model of retirement spending
A useful representation is \[C_t=C_t^{E}+C_t^{D}+C_t^{H}+C_t^{M}+C_t^{L}+C_t^{X},\] where the superscripts denote essential, discretionary, housing, medical, long-term-care and exceptional expenditure.
Each component could be modelled as \[C_{k,t}=C_{k,0}A_k(t)\prod_{s=1}^t(1+\pi_{k,s})H_{k,t}F_{k,t}.\] The factors have distinct interpretations:
- \(A_k(t)\) is an expected age or lifecycle profile;
- \(\pi_{k,t}\) is category-specific inflation;
- \(H_{k,t}\) represents health, household or other state changes;
- \(F_{k,t}\) represents adjustment to financial circumstances.
Essential expenditure might have a nearly flat quantity profile and little financial flexibility. Discretionary spending might decline voluntarily with age and respond strongly to funded status. Housing could contain modest regular maintenance plus occasional large shocks. Long-term care should be state-dependent and potentially severe rather than represented solely through a higher deterministic inflation rate.
The model should distinguish desired spending from actual spending. Let \(D_t\) be desired consumption and \(C_t\leq D_t\) actual affordable consumption. The shortfall \[L_t=D_t-C_t\] is an economically meaningful retirement loss. Without that distinction, spending cuts mechanically improve portfolio survival and can be misclassified as success.
Joint modelling of returns and inflation
If all quantities are expressed relative to a single index, nominal returns and that index can be converted into real returns. However, the underlying simulation should usually model nominal asset returns and inflation jointly.
Inflation shocks affect:
- nominal bonds;
- inflation-linked bonds;
- cash interest rates;
- equity profits and valuations;
- taxes;
- public pensions;
- annuity income;
- wages or consulting income;
- several expenditure categories.
Subtracting an independently simulated inflation series from independently simulated nominal returns destroys economically important dependence. The problem is especially serious for sequence risk because an inflation shock early in retirement can simultaneously raise withdrawals and reduce the real value of nominal assets.
When several expenditure indices matter, there is no single real-return series that removes all inflation. We can choose one base numéraire and retain relative price processes for the remaining categories. For example, use general CPI as the numéraire and model excess medical inflation by \[1+\widetilde\pi_{M,t}=\frac{1+\pi_{M,t}}{1+\pi_{\mathrm{CPI},t}}.\] Medical expenditure in CPI-real units then continues to grow with \(\widetilde\pi_{M,t}\).
Essential and discretionary objectives
A binary ruin criterion is often too crude once spending is decomposed. A household can remain solvent only by making unacceptable cuts.
A more informative hierarchy might be:
- essential consumption must be maintained;
- housing and care obligations must be met;
- discretionary spending may be adjusted within specified limits;
- legacy or terminal wealth is desirable but subordinate.
The retirement model could classify a scenario as:
- green: all essential and target discretionary spending is maintained;
- amber: essential spending is maintained but discretionary spending is reduced;
- red: essential spending, care or housing requirements cannot be met.
A continuous score could measure discounted consumption shortfall: \[L=\sum_{t=1}^n v_t\left[\lambda_E(D_t^E-C_t^E)_++\lambda_D(D_t^D-C_t^D)_+\right],\] with \(\lambda_E\gg\lambda_D\).
The approach avoids treating every euro of expenditure as equally necessary.
Interaction with robust scenario analysis
The component-spending model fits naturally into the red-green scenario framework. Scenario dimensions could include:
- general inflation;
- excess essential-cost inflation;
- medical and care-cost inflation;
- planned age-related spending decline;
- housing shocks;
- health transitions;
- longevity;
- financial flexibility;
- return-inflation dependence.
The safe set could be defined by maintenance of essential consumption rather than by a generic portfolio-survival indicator.
Three models would provide a useful progression:
- constant CPI-linked real spending;
- component spending with expected age profiles and category inflation;
- component spending with stochastic shocks and adaptive financial responses.
The difference between models 1 and 2 measures the effect of predictable basket evolution. The difference between models 2 and 3 measures the effects of contingent expenditure, household state and spending flexibility.
The models should not be collapsed prematurely into a single estimated success probability. Their disagreement is itself information about model risk.
Practical conclusion
Folding inflation into real asset returns is an exact and useful transformation when assets and liabilities are deflated by the same index. Constant real spending is therefore internally coherent.
The simplification becomes questionable when the chosen index does not represent the household’s evolving consumption liability. Retirement spending changes because both prices and quantities change, and different components have different dynamics and flexibility.
The most useful practical approach is not to discard constant real spending but to retain it as a benchmark and add progressively richer liability models. We should distinguish:
- general inflation from household-specific inflation;
- price changes from changes in quantities;
- desired consumption from forced spending reductions;
- essential from discretionary expenditure;
- gradual age profiles from contingent care and housing shocks;
- nominal return risk from return-inflation dependence.
Real wealth is not an absolute economic quantity. It is wealth measured in units of a specified liability. For retirement planning, the correct numéraire is ultimately the evolving cost of the standard of living we intend to protect.
4. An Integrated Retirement-Modelling Framework
The three lines of analysis fit together naturally.
First, generate broad economic and household scenarios, including nominal asset returns, inflation components, longevity, health states and spending needs.
Second, map each strategy and scenario into retirement outcomes. Outcomes should include more than terminal wealth: essential consumption, discretionary shortfall, ruin timing, care adequacy and legacy may all matter.
Third, examine sequence risk conditionally. For selected economic environments, rearrange returns using annual permutations, blocks, regime shifts or cyclic rotations. The exercise separates inadequate economic material from harmful timing.
Fourth, classify scenarios into green, amber and red sets. Use scenario discovery to identify the assumptions and combinations of events associated with failure.
Fifth, compare strategies through several lenses:
- safe-set inclusion;
- distance to failure;
- minimax regret;
- worst-case performance over carefully specified uncertainty sets;
- conditional sequence sensitivity;
- probability of success under several competing probability models.
Sixth, search for adaptive strategies. Withdrawal reductions, annuitization, asset reallocations and housing decisions should be represented as contingent rules rather than fixed decisions made once at retirement.
The integrated framework does not reject probability. It refuses to allow a single estimated probability model to conceal the geometry of retirement risk. Probability becomes one layer of the analysis rather than the foundation of every conclusion.