How we built the five-component history

Units: all decomposition rates are annual continuous percentages; 1 basis point (bp) is 0.01 percentage points. Frozen model CSVs generally store 12 significant digits; new numerical selections retain 17 significant digits to preserve the parsed value, without implying additional source accuracy.

The charts begin with market prices and independent estimates of expectations. We construct comparable forward rates, estimate the expected-rate anchors, and calculate the compensation left between the two. We then divide UK real compensation between a foreign benchmark and a UK-relative component. This walkthrough explains each operation, the assumptions it needs and the statistical conclusions it supports.

The research checkpoint is 17 September 2026, using market data through 31 July. This explanation accompanies the existing estimates; it does not introduce a new fit. The results and sensitivity note preserves the detailed checkpoint.

1. The object we want to explain

A five-year, five-year forward rate, or 5y5y, is the interest rate implied by today's curve for the five-year interval beginning five years from now. Our target is its actual level, in per cent per year.

We use the following accounting identity:

The three compensation components in the five-part decomposition are amounts left in market pricing after subtracting the stated expectations. The broader market inflation-compensation input C includes expected inflation; subtracting π from C produces the fifth component. The site uses this label consistently; historical CSV field names containing “premium” retain their original spelling for reproducibility. Their economic interpretation also depends on the measurement and expectation assumptions explained below.

The sequence is important. We do not independently observe five components. We observe fitted market curves and survey responses, estimate expectation paths, and construct the remaining components from them.

2. Turn yield curves into comparable forward rates

We use zero-coupon rates: rates for a single payment at a specified maturity. A five-year coupon bond's yield averages payments over several dates, so it is not the five-year zero rate needed here.

All curve calculations use continuously compounded annual rates. If the five-year zero rate is z₅ and the ten-year zero rate is z₁₀, then:

The intuition is to remove the first five years' accumulation from the ten-year accumulation, leaving years 5–10. The same operation applies to nominal and real zero curves. Annual-effective quotes are converted with log(1 + rate). The Fed TIPSY05 and TIPSY10 columns, like the BoE zero curves, already quote continuous annual percentages. German inputs are clean real bond prices. Australian F16 inputs are quarterly-compounded real redemption yields, converted into normalised prices as set out in section 6a.

For Germany and Australia, bootstrapping means finding discount factors that price the observed bonds' coupon and principal payments. We calculate the five- and ten-year real zero rates from that curve, then the real forward. Both target maturities must lie within supported bond maturities. Interpolation between log discount factors is the baseline; linear-zero interpolation is a sensitivity.

This produces market-based real forwards. They retain differences in inflation baskets, bond liquidity, indexation lags, principal protection and market conventions. The global technical report records those constructions and pricing checks.

3. Translate the UK's RPI market into a CPIH-equivalent measure

UK index-linked gilts reference RPI. The research aims to compare their inflation compensation with broader consumer-price measures and foreign markets, so it reuses the project's previously constructed synthetic CPIH history. This is a modelled conversion of gilt pricing, not a directly traded CPIH curve. For independent downstream replication, the daily CPIH-equivalent history is a frozen input. Reconstructing its statistical and reform correction is a separate exercise. The numerical input pack supplies all three retained market-conversion histories.

The accounting is:

The statistical index adjustment

Using ONS inflation data, the inherited model first calculates the difference between log RPI and log CPIH inflation. It separates the published formula effect from the remaining housing, coverage and weighting differences, then forecasts those parts forward.

The central rule uses data with reference months at least two months before the valuation month and requires 60 observations. It anchors the non-formula component on the latest ten years, allows recent conditions measured over the latest twelve months to fade towards that anchor, and estimates annual persistence from a 12-month-lag regression. Persistence is shrunk towards 0.5 and bounded between 0.05 and 0.90. The formula anchor uses the latest three years, with pre-2010 observations excluded after that regime boundary.

These are statistical forecasts of an index difference. The lookback windows, shrinkage, persistence bounds and data lag are modelling choices. The retained ONS vintage includes historical backcasts, so the recursive date cutoff does not make this an available-at-the-time reconstruction.

The reform calendar and historical discount

The model integrates the forecast gap over the portion of years 5–10 assumed to precede reform, then divides by the full five years. Before 4 September 2019 the reference assumes unreformed RPI throughout the horizon. Thereafter it assumes reform in February 2030, with a three-month indexation lag moving the payment-time gate to May 2030. An earlier 2025 reform date is a consultation-period sensitivity. The adjustment therefore depends on the valuation date and horizon; it is not a fixed deduction such as 80bp.

There is an additional complication: RPI market compensation could already have fallen because investors anticipated reform. Statistical subtraction alone can then overcorrect. The preferred version adds back a non-negative estimated discount during two selected episodes, May 2012–January 2013 and January 2019–November 2020. The path is bounded by the relevant estimated index gaps, is encouraged to change gradually between announcements, and is set to zero before each episode and from its terminal decision. These restrictions identify a conditional path; the model does not directly observe reform probabilities.

To distinguish reform repricing from other market moves, a robust regression fits weekly changes using a prior-date US inflation-market control and the UK nominal 5y5y. It uses 632 non-policy weeks during 2010–2024. Fitted control coefficients are about 0.125 on the US series and 0.215 on UK nominal rates. Leave-year-out prediction errors select the control regression’s ridge penalty, which limits unstable coefficients. Separate held-window tests select the discount smoothness, balancing false corrections when no discount is added against recovery of simulated discounts. The chosen smoothness parameter is 0.125. These are upstream correction parameters, not the weights of the global real benchmark.

The US control is FRED T5YIFR, using a strictly earlier observation for each London date. It is a different market series from the fitted TIPS real-zero curve used later in the global decomposition. Later training data and assumed episode boundaries make the correction retrospective. This is also a point where market-price dependence enters the constructed UK inflation history: independence from gilt prices applies to the survey expectation estimator, not to every step of the overall pipeline.

At 31 July 2026, years 5–10 lie wholly after the assumed reform gate. The inherited row therefore has zero statistical and policy adjustments: raw RPI, statistical-only and preferred CPIH-equivalent compensation all equal 3.003166% in continuous units. That equality follows from the reform mapping; it does not establish that all instrument differences disappear.

The integration audit records the inherited fields, control dependence and exact accounting identities.

The preferred history includes the reform adjustment. We also retain the statistical-only history and a reform specification using only the US market control. These are alternative constructions of the same UK market object, not three independent observations.

Once inflation compensation C is defined, the UK adjusted real-related forward is R = N − C, where N is the nominal forward. Thus a 10bp upward revision to the CPIH market estimate lowers the real-related forward by 10bp. This link is built into the arithmetic.

One further index assumption enters later: the expectation surveys generally measure CPI, while this market series is CPIH-equivalent. The reference calculation sets the expected CPIH-minus-CPI difference to zero. We examine what an unmeasured difference would do to levels and changes, rather than treating the two indices as identical observations.

4. Build expectation anchors that do not use gilt prices

The long UK history uses 84 rounds of HM Treasury's independent forecasts, from February 2005 to May 2026. For each round we retain the provider's average terminal-calendar-year short-rate and CPI forecasts. These are forecasts collected by HMT, not Treasury's own forecasts. The published “Independent” average is used directly, rather than recalculated across contributors. The target year is the publication year plus four. A missing individual short-rate forecast is not filled with zero or reweighted by this study; the precise contributor membership of each published mean is not reconstructed. The admitted panel omits May 2017 and November 2019, with no synthetic rounds inserted.

The terminal year is around four years ahead. To use it for years 5–10, we assume rates and inflation have substantially converged by then. This horizon mapping is an economic assumption. Historical short-rate definitions also vary across contributors; some include interbank rates. They are not a homogeneous direct survey of the average future Bank Rate over years 5–10.

We estimate expected nominal policy n and expected inflation π separately. Expected real policy is then:

Inflation forecasts quoted as annual percentage changes are converted to log rates before subtraction. Annualised overnight policy quotes are treated as continuous-accrual rate proxies; they are not treated as annual-effective investment returns. These conventions make the ledger internally consistent while leaving the survey-to-policy mapping explicit.

The key advantage is that government bond prices do not enter this expectation estimator. A rise in gilt yields therefore does not automatically raise the expected-rate state. Unless independent expectation inputs change, it is allocated to compensation; the real/inflation split also depends on how the market inflation measure moves.

5. Estimate smooth paths from intermittent surveys

Surveys arrive occasionally; market prices move daily. We use a local-level state-space model to translate noisy survey observations into an expectation path. “State” means the underlying expectation we are trying to estimate.

The model has two equations:

Both disturbances are assumed Gaussian, with zero mean, and independent across time and of one another. The noise settings are modelling allowances, not measured survey standard errors. The variance of the underlying change grows with elapsed time, measured in months. The model does not impose a pull back towards 2% inflation or any other fixed long-run value.

We fix the survey-noise standard deviation at 50bp for nominal policy and 25bp for inflation, then estimate the underlying state's innovation variance q by maximising the survey prediction-error likelihood. In plain English, the fit asks how much genuine movement in expectations is needed to explain the sequence of survey answers once the assumed noise has been allowed for. A zero-variance solution is permitted.

The fitted Gaussian prediction-error likelihood is the sum, after the first observation, of one half times log(2πS) + error²/S, where S is predicted state variance plus measurement variance. Elapsed months equal actual days divided by 365.25/12. The search compares q = 0 with 161 log-spaced values from 0.0001 to 10,000 bp² per month, then refines a positive candidate by bounded minimisation in log q and compares it again with zero. The full specification records the refinement bounds and solver settings.

The first survey initializes the estimate at its observed value, with the assumed measurement variance. A Kalman filter updates the estimate whenever a subsequent survey arrives. It gives more weight to a new observation when the previous estimate is uncertain relative to the assumed survey noise. Between observations, the expected level is carried forward and its conditional uncertainty can grow. Daily market-date output does not create daily survey information.

This explains the flat-looking inflation line. With q = 0 and equal observation noise, the filtered inflation state becomes the expanding average of past log forecasts. The latest of 84 observations receives only 1/84 of the weight. Source inflation forecasts ranged from about 1.85% to 2.51%; their underlying observations were not constant. A retrospectively smoothed version uses later observations to revise earlier states and is constant in this zero-q case.

We therefore retain alternatives: 12.5bp inflation measurement noise, a fixed 5bp monthly state-change standard deviation, and the latest published forecast with no smoothing. Under 12.5bp noise the fitted q is about 3.09 bp² per month, so the estimate responds more to new observations. The fit diagnostics and historical model diagnostics contain the numerical estimates.

The main histories use filtered states but full-sample fitted parameters and retrospectively acquired data vintages. They are historical reconstructions. They should not be read as the precise estimates an investor could have produced at each past date. Fixed-parameter filtering is also tested separately to verify that later observations do not enter earlier state updates.

6. Construct the global real benchmark

For each foreign country, calculate:

The global bar is the equal-weight average of US, German and Australian real compensation, with the UK excluded. It is an average in rate units: no demeaning or standardisation removes its level. Equal weights and a UK loading of one are baseline choices, not regression estimates. We compare compensation in each currency’s own real-rate units; this is not the return on a portfolio hedged into sterling.

The UK-relative real component is then:

A negative value means the UK's total real compensation lies below this benchmark. It does not mean total UK real compensation is negative. Changing the foreign benchmark moves the global and UK-relative bars in opposite directions and leaves their sum unchanged.

Why there are separate long-history and survey versions

Comparable foreign neutral-rate surveys do not cover the entire historical period. We therefore preserve distinct versions rather than splice unlike expectation methods together.

The macro versions treat model estimates of neutral real rates—rates consistent with the economy operating around balance—as imperfect anchors for expected real policy in years 5–10. That convergence assumption provides the longer history. Published macro estimates are revised over time; they are not direct observations of historical investor expectations. For Australia, subtracting 2.5% is our declared mapping of the published nominal-neutral series, with model and conversion alternatives retained.

The recent foreign survey states use the same local-level approach. Survey medians, modal forecasts, means and horizons differ across countries, so the construction also assumes these can serve as comparable expected-policy anchors. The UK's HMT expectation method remains unchanged in both global versions. The phrase “survey comparison” here refers to the foreign benchmark; it does not replace UK HMT inflation with the recent UK MPS series.

There is a second distinction relevant to the macro story. Global investment demand can affect expected real rates as well as compensation. The first bar is the full UK expected real rate; it can contain worldwide influences. We export the foreign expected-real average separately, but do not add it again to the five components. The global compensation bar is therefore not a complete measure of every global influence on UK rates.

6a. Foreign curve construction

The US input is the Federal Reserve’s fitted TIPS zero curve: exact same-date TIPSY05 and TIPSY10, using 100 × (2 × TIPSY10 − TIPSY05) in basis points. The coupon-equivalent TIPS5F5 series is not substituted.

For Germany, the Finance Agency’s clean real prices are converted to dirty real prices by adding accrued interest. Annual contractual coupons and principal are dated using T+2 TARGET settlement, ACT/ACT ICMA accrual and ACT/ACT ISDA curve time. A sequential root solve finds each bond-maturity log discount factor so the curve prices that bond exactly. Earlier solved nodes stay fixed. Straight lines between log discount factors, including D(0) = 1, define the reference curve; linear zero rates are the alternative. There is no least-squares price fit, imposed monotonic discount constraint or extrapolated endpoint.

For Australia, F16 quarterly redemption yields become prices normalised to indexed principal. If y is the yield in per cent, c the quarterly coupon per 100, f the days to the next coupon and d that coupon period’s days, the normalised price is the sum of each cash flow discounted by (1 + y/400)^(−f/d − j). The cash-flow schedule includes the ex-interest convention. The reference uses quote-date settlement and actual days/365.25; settlement and day-count alternatives are retained. Sequential exact pricing yields log-discount nodes at retained maturities, using bonds only through the first maturity of at least ten years. Both target maturities must have observed support.

Germany’s three-month HICP-ex-tobacco indexation lag and the Australian contractual CPI indexation remain economic features of the instruments. The calculation uses real-normalised prices and cash flows; it does not reconstruct or remove every indexation lag or principal floor. No unobserved base-index history is imputed. German ISINs and Australian RBA TI identifiers, dated membership, root bounds, coupon schedules and calendar limitations are in the full specification.

Germany’s ten-year point therefore spans bonds about 6.7 and 19.7 years away. Linear-zero interpolation lowers its forward by 21.49 bp and the equal-weight global benchmark by 7.16 bp in the endpoint comparison. Accurate repricing of the observed bonds does not establish an accurate unobserved ten-year point.

7. A worked example: July 2026

The following calculation uses the long-history three-market macro specification shown in the article's main historical chart. All numbers are annual continuous percentages, rounded here to three decimals.

Across the retained inflation-belief paths, the July inflation-compensation level ranges from 0.646% to 1.030%. The point estimate below is the HMT reference, not an assumption-free observation.

The final five-part sum is:

The recent survey-based foreign benchmark instead gives 1.934% global and −0.370% UK-relative real compensation on that date. The UK inputs and total are identical. Keeping these versions labelled avoids confusing a change in foreign assumptions with a change in the observed gilt rate. Macro monthly data · Survey monthly data.

The general algebra is equally simple. Let T = R − (n − π) be total UK real compensation, G the foreign benchmark, L = T − G the UK-relative part, and I = C − π inflation compensation. Then:

Exact addition is a necessary implementation check. It does not independently validate the expectation anchors or identify the economic cause of each component.

8. Date matching, gaps and the recent inflation crosscheck

Market inputs are joined on the same actual valuation date before making monthly charts. We take the last common quote no more than seven calendar days before month end. We do not join different countries simply because their observations share a month label. Unsupported months remain gaps; market prices are not carried across missing cross-country dates. For a three-market month to count, UK nominal and CPIH inputs, UK expectation states, and the US, German and Australian real curves and chosen anchors must all be present on that actual date. The two-market history uses the same rule for its fixed US/Australian membership. The common calendar is the intersection of these observed dates, not an assumed universal business-day calendar. Missing countries are not reweighted. This still leaves differences between intraday market fixing times.

Expectation inputs have their own timing. UK HMT updates use release dates, with seven older month-end proxies: 31 May, 31 August and 30 November 2010; 28 February, 31 May, 31 August and 30 November 2011. The foreign survey histories use a mixture of submission or fieldwork deadlines and reference-month proxies; these are not uniformly public-release dates. The macro histories use retrospectively revised estimates dated to their economic quarters, carried forward within a 270-day age limit. Their historical dating does not establish what was known at that time. These conventions are separate from the source-vintage download date.

The three-market gaps in April 2013–March 2014 arise from German ten-year maturity support. December 2014 fails the common-date rule: the last common quote is 23 December, eight days before month end. The two-market gap in August 2009–July 2010, and the June 2020 gap in both histories, arise from missing Australian five-year support. The gap audit supplies month-by-month evidence. A US-only fill would change benchmark membership.

A more direct UK inflation check

The Bank's Market Participants Survey (MPS) asks for a distribution of average annual CPI inflation between five and ten years ahead. We use 27 admitted releases from March 2023 to June 2026, as a separate recent comparison.

For each release, we assign each probability bin its midpoint and calculate the probability-weighted log inflation rate. Open tails need finite endpoints: the reference uses −1% and 5%, with alternatives of −2% and 6%, and −5% and 10%. All 27 releases use the same seven printed bins: ≤1%; 1.01–1.40%; 1.41–1.80%; 1.81–2.20%; 2.21–2.60%; 2.61–3.00%; and >3%. Each closed bin uses its two-endpoint midpoint. The reference open-tail midpoints are 0% and 4%. Divide each published probability by the sum of the seven probabilities to normalise publication rounding. Treating an average annual inflation outcome this way also assumes a mapping to the compounded horizon return.

This recent comparison uses the latest published distribution, carried forward until the next release. It does not fit the heavily smoothed HMT inflation state to those observations or invent earlier MPS forecasts. The 19 June distribution is carried through July; it is 42 days old at 31 July. The last-published MPS path admits an observation only for 180 days after publication. The arithmetic annual mean, about 2.25% at both the first and last releases, is a different statistic from the expected-log rate used in the decomposition. Survey source audit.

For a simple conversion example, 100 × log(1 + 2.25/100) = 2.2250609% in continuous units, 2.4939 bp below 2.25%. MPS instead averages the log of each bin midpoint, weighted by its normalised probability; that generally differs from taking the log of the arithmetic mean.

Across the 27-release overlap, MPS expected-log inflation minus the latest published raw HMT log forecast averages +12.8 bp, ranges from −6.2 to +25.9 bp, and widens from 9.6 to 25.3 bp between endpoints. The gap is not stable. It mixes horizons, respondent populations and aggregation statistics, so it is not a separately identified horizon correction. The releases draw on 14 distinct HMT rounds; fieldwork-start and publication-date matching select the same rounds. Matched observations and summary.

We make inflation-trend comparisons using complete-quarter daily averages. A quarter must cover at least 80% of the reference dates, with at least 30 reference observations; the reported comparison quarters have full coverage. Multi-quarter windows give each quarter equal weight. Thus the article's 2019-to-July contribution chart uses daily endpoints, while its inflation-trend ranges use quarter averages. These answer different questions.

9. What changing an assumption does to the bars

The components are linked. A sensitivity must update every affected part of the identity.

The broader global study retains 41 scenarios for membership, weighting, UK loading, curve interpolation and expectation mappings. The later inflation study isolates four HMT expectation methods plus three MPS tail conventions, crossed with three market conversions: 21 UK cases, embedded in each of three existing foreign benchmarks to give 63 five-part cases. The recent market-conversion alternatives coincide after 2020, so the nine MPS combinations contain only three distinct numerical paths.

These ranges answer: does the conclusion survive this declared menu of alternatives? They are not confidence intervals or probabilities assigned to the alternative methods. In the inflation exercise, nominal policy expectations and the foreign benchmark are held fixed to isolate the UK inflation allocation; it is not a joint refit of every assumption.

A constant unmeasured CPIH/CPI or horizon gap shifts the level of inflation compensation but cancels from its change. A changing gap changes the measured trend one for one. We therefore also test an illustrative ±25bp change in that gap between comparison windows. The 2023–26 MPS compensation increase of about 25–26bp only just survives the adverse version of that stress. This quantifies how much changing measurement would be needed to explain the movement; it is not a probability statement about that measurement error.

10. Which statistical inferences have we drawn?

The evidence falls into five categories.

The local-level likelihood supports a very slow-moving UK inflation state conditional on 25bp survey noise. Reducing that assumed noise changes the inferred amount of genuine movement. The likelihood-support diagnostics and state-error bands do not encompass uncertainty in the horizon mapping, CPIH conversion, curve measurement or economic interpretation. A q = 0 boundary estimate is not a hypothesis test proving perfectly anchored expectations.

Foreign real-compensation changes are positively correlated in the recent survey sample: about 0.58–0.82 across country pairs, using 29 common monthly changes. That supports describing co-movement. The baseline foreign weights are still chosen equal weights; those correlations do not identify a unique global shock, and no causal significance claim follows from them.

The principal inflation-trend findings are also descriptive robustness findings. The 2013–15 to 2017–19 increase is 34–49bp across the declared long-history menu; the 2019 Q4 to 2026 Q2 increase is 38–67bp. These are neither p-value-based rejection statements nor estimated Brexit effects. The direct MPS comparison supports little net change in mean expected inflation alongside higher market compensation over 2023–26. It also shows that compensation fell during 2025 and partly rebounded in 2026.

The checks underpinning these statements include 63 five-part cases and 114,675 daily rows, 132 quarter-window comparisons, nine targeted inflation-checkpoint tests and two byte-identical executions of its 21 generated files. The reported accounting discrepancy is below 0.000000003bp after CSV rounding. That tiny number measures arithmetic consistency; it is not an error bar on risk compensation. Internal AI numerical checks.

11. What the decomposition lets us say

The method gives us a transparent conditional macro account: how much of the forward rate is allocated to expected real rates, expected inflation and compensation under a stated set of anchors; how much UK real compensation is shared with the chosen foreign benchmark; and how the inflation allocation behaves when the expectation assumptions change.

It also tells us how to frame the debate. A larger global real component is consistent with more expensive global capital, changes in duration supply or a higher price of risk. Inflation compensation can rise without a higher mean inflation forecast. A small UK-relative real component can coexist with a large total borrowing cost. Those distinctions are useful before selecting a causal explanation.

Two definitions govern the interpretation. UK-relative real compensation is a comparison with a benchmark, not a direct fiscal-credibility meter. UK inflation compensation is the full UK market-minus-expectation residual, including international inflation-risk influences; we have not separately removed a global inflation component. A strictly local inflation-risk series would require an additional identification step.

The earlier nominal common-factor, persistence, fiscal and QE exercises are separate research stages. Their fitted coefficients or significance diagnostics do not determine these five bars; the final series use the construction described here.

The completed work supports discussion of these competing explanations. It does not turn the five bars into five independently observed causes, and it does not make a monetary-policy prescription automatic.

Sources, results and further detail

• Results and inflation sensitivity checkpoint: comparison windows, ranges and downloads.
• Global construction and longer histories, with the full global technical report.
• RPI-to-CPIH integration audit: inherited versions, controls, timing and accounting.
• Survey source audit, methodology review and internal AI validation.
• Current evidence ledger, global evidence ledger and the chart book.

Appendix: the full downstream specification

The following rules describe the frozen September checkpoint. They support an independent implementation using the published numerical inputs.

R  = N - I                     adjusted UK real-related forward
er = n - pi                    expected UK real-rate proxy
PR = R - er                    total UK real compensation
PI = I - pi                    UK inflation compensation
G  = sum_c w_c*(R_c - er_c)     foreign real-compensation benchmark
L  = PR - beta*G               UK-relative real compensation
N  = er + pi + beta*G + L + PI

f = [log D(5) - log D(10)] / 5 = 2*z10 - z5

P_norm = sum_j CF_j * (1+y/400)^(-(f/d+j)), j=0,...,n
P_cash = P_norm * K_next * (1+p/100)^(-f/d) / 100

y_i = x_i + epsilon_i,  epsilon_i ~ N(0,R)
x_i = x_(i-1) + eta_i,  eta_i ~ N(0,q*Delta_i)
Delta_i = elapsed calendar days / (365.25/12)

P_minus = P_previous + q*Delta
S = P_minus + R
K = P_minus/S
m = m_previous + K*(y-m_previous)
P = (1-K)*P_minus
negative_log_likelihood = 0.5*sum_{i=2..n}[log(2*pi*S_i)+(y_i-m_previous)^2/S_i]

expected_log_bp = 10000*sum_j p_j*log1p(m_j/100)
arithmetic_mean_pct = sum_j p_j*m_j