What we claim, and how we know

The MRM class is defined for every VEV, is between 1 and 7, never decreases as the VEV rises, and changes class exactly at the limits in the table.

PRIIPs Read the proof of PR-01

The repetition count cutoff is the smallest integer C for which H times (C − 1) reaches the false positive exponent.

Entropy validation Read the proof of EN-01

The repetition count cutoff never rises when the claimed min-entropy rises.

Entropy validation Read the proof of EN-02

For every vehicle inside the family, the interpolated CO2 lies between the CO2 of the low vehicle and the CO2 of the high vehicle.

WLTP Read the proof of WL-01

Inside the family, a vehicle that demands more energy never gets a lower interpolated CO2, provided the high vehicle emits at least as much as the low one.

WLTP Read the proof of WL-02

For any non-negative module charges, the basic SCR is never above the plain sum of the charges and never below the largest single charge.

Solvency II Read the proof of SI-01

When only the quantum computer sets the date, the latest start is the years until it arrives less the years to keep the data secure and the years to migrate. Nothing later keeps the data safe, and the use is late exactly when those two add up to more than the years until it arrives.

PQC Read the proof of PQ-01

A use is at least medium risk under the EU roadmap exactly when it protects confidentiality or takes more than eight years to migrate. It is high risk exactly when it has a high impact and either protects confidentiality for ten years or more or takes more than eight years to migrate.

PQC Read the proof of PQ-02

A use that others wait for finishes no later than it would alone and no later than any of them can start, and no later finish does both. It starts strictly earlier than each of them, so a use that is late makes every use it waits for late too.

PQC Read the proof of PQ-03

For every number of days with data in a year, from 1 to 366, the rank that Table 1 of Appendix N gives for the 98th percentile is the smallest whole number that is at least 2 per cent of the days, and the rank that Table 1 of Appendix T gives for the 99th percentile is the smallest whole number that is at least 1 per cent of them. Every count of days has a rank, and more days never lower it.

AERMOD Read the proof of AE-01

In every stability class both spreads of the plume grow with the distance downwind, from 100 m to 10 km. At every distance the spread of a more stable class is smaller than that of the class before it, both across the wind and upward.

AERMOD Read the proof of AE-02

A day read as a monitor network keeps it is never above the same day read as a model takes it. It is below it by less than 0.2 for a mean of 24 hours and by less than 0.1 for the highest hour. The network cuts each hour off at a tenth and then cuts the mean of those off at a tenth again.

AERMOD Read the proof of AE-03

A daily value of PM10 that a monitor keeps in tenths is an exceedance of a level exactly when it is at least 5 below the next multiple of 10 above the level, so for the level of 150 it is an exceedance from 155.0 on. A day from 150.1 to 154.9 is above the level and is not counted, and every day that is counted is above the level.

AERMOD Read the proof of AE-04

A design value that is the mean of three values and is rounded half up as Appendices N and T say is at most the level exactly when the three add up to at most 1064 tenths for the 24-hour PM2.5 level of 35, which holds for values kept in whole tenths, and at most 2264 for the 1-hour SO2 level of 75 ppb. The annual mean is at most 9.0 exactly when the three annual means, which need not be whole tenths, add up to less than 27.15. For PM10 the design value is at most 1.0 exactly when the three yearly counts of days add up to at most 3.

AERMOD Read the proof of AE-05

For positive amounts, the development factor of a year is the mean of the ratios from that year to the next, weighted by the amounts of the first of the two. So it is not below the smallest of those ratios and not above the largest, and it is their common value when they are all alike.

Reserving and ESG Read the proof of RS-01

In a triangle of four origin years, what the chain ladder expects to be paid in the three calendar years to come adds up to the reserve, which is the ultimate less the latest amount of each origin year. The first year pays the next factor of every open origin year, and when no factor is below one, no payment is below zero.

Reserving and ESG Read the proof of RS-02

The variance that the formula of Mack gives to the last development year, which rests on a single ratio, is never below zero and never above either of the two estimates before it. It is the continuation of their decline when they do not rise, the earlier of the two when they rise, and nothing when the earlier is nothing.

Reserving and ESG Read the proof of RS-03

The ultimate forward rate of a year is the rate of the year before, or that rate plus 15 basis points, or that rate less 15 basis points. It rises exactly when the rate before the limit is at least 15 basis points above the rate of the year before, and falls exactly when it is at least 15 basis points below. It never ends farther from the rate before the limit than it began, and a higher rate before the limit never gives a lower ultimate rate.

Reserving and ESG Read the proof of RS-04

Beyond the first smoothing point, the weight of the last liquid forward rate, which is (1 − e^(−a×h)) over a×h, is above zero and not above one. It is not below 1 − a×h/2, and a×h times it is not above one.

Reserving and ESG Read the proof of RS-05

Beyond the first smoothing point, the forward rate of each horizon lies between the last liquid forward rate and the ultimate forward rate. Its distance from the first is at most a×h/2 of the distance between the two, and its distance from the second is at most 1/(a×h) of it.

Reserving and ESG Read the proof of RS-06

Beyond the first smoothing point, the zero rate of each horizon lies between the zero rate at the point and the forward rate of that horizon. It is nearer to the one that covers more years, which is the forward rate when the horizon is longer than the point and the zero rate at the point when it is shorter.

Reserving and ESG Read the proof of RS-07

The expected real rate is rounded to a whole number of five basis points, upwards when it is below the rounded rate of the year before and downwards otherwise. The rounded rate is less than five basis points from the unrounded one, a rate that is already a whole number of five basis points stays as it is, and a higher unrounded rate never gives a lower rounded one.

Reserving and ESG Read the proof of RS-08

When the hedges of a bucket offset every one of its risk factors in the same proportion, the quantity under the root of its capital is (1 − ratio)² × Q + R × ratio² × N, where Q is the form of the CVA weighted sensitivities through the correlation matrix, N is the sum of their squares and R is the hedging disallowance of 0.01. No ratio takes it below R × Q × N / (Q + R × N), the ratio Q / (Q + R × N) reaches that value, and so the capital of a bucket with a position is never zero.

SA-CVA Read the proof of SC-01

With the hedges of the previous statement, a ratio of hedging above zero and below 2 × Q / (Q + R × N) lowers the quantity under the root of the capital of a bucket, that ratio leaves it where it was with no hedge, and a ratio above it or below zero raises it. With the disallowance of 0.01 and a single factor, a hedge of more than 198 per cent of the exposure costs more capital than no hedge.

SA-CVA Read the proof of SC-02

In an interest rate bucket of six factors, the share that rests on the CVA sensitivity of each factor, which is the CVA weighted sensitivity times the matching entry of the correlation matrix times the net weighted sensitivities, and the share that rests on its hedge, which is the weighted hedge times R times itself less that entry, add up to the quantity under the root, whatever the sensitivities are. Divided by the capital, the shares add up to it.

SA-CVA Read the proof of SC-03

For three buckets that each have the same correlation of 0.5 with every other, or of 0.6, and whose sums of weighted sensitivities are cut to the range of their capital, the quantity under the root of the capital of a class is never below zero and never above the square of the sum of the bucket capitals. The capital of the class is then a real number, and the diversification it gives never lifts it above the plain sum of the buckets.

SA-CVA Read the proof of SC-04

For any five charges of the life risks that are not below zero, the quantity under the root of the life charge, which is the form of Table 6 of the Level 2 text, is at least 15/16 of the square of each charge and at most the square of their sum. So the life charge is never below the root of 15/16, which is 96.8 per cent, of its largest charge, and never above the sum of the charges. The floor is reached with a mortality charge of 4 and a longevity charge of 1, so that no higher floor can be proved.

ICS Read the proof of IC-01

With a mortality charge M above zero and a longevity charge L, and nothing else in the life risks, the quantity under the root of the life charge is M² + L² − M × L / 2. It is below M² for any L above zero and below M / 2, it equals M² at M / 2, and it is above M² beyond. It is lowest at L = M / 4, where it is 15/16 of M². A longevity charge of a quarter of the mortality charge therefore lowers the life charge by 3.2 per cent, and a larger one lowers the requirement too, as long as it is below half.

ICS Read the proof of IC-02

For charges that are not below zero, the quantity under the root of the market charge, in the row of Table 16 for an upward spread stress and in the row for a downward one, and the quantity under the root of the top level of Table 34, are each at least the sum of the squares of the charges and at most the square of their sum. So each aggregate is at least the root of the sum of the squares, which is what no correlation at all gives, and at most the sum of the charges. The charge of a category is real, and diversification never lifts it above the plain sum.

ICS Read the proof of IC-03

Before the operational charge, a life, catastrophe or market risk has the part of the insurance risk charge that is its share inside its category, c × (R × c) / A, times the weighted charge of the category over the aggregate. Non-life and credit take their own charge times that factor. The parts add up to the aggregate. The terms c × (R × c) of each table add up to the square of the charge of their category, and the terms of Table 34 add up to the square of the aggregate. Terms that add up to the square of a number add up to it when divided by it. A factor times shares that add up to a charge adds up to the factor times it.

ICS Read the proof of IC-04

The tax effect taken off the insurance risk charge lies between zero and 80 per cent of the notional effect, which is the requirement times the group effective tax rate, so at least a fifth of the requirement is left. The carry-back component is at most 85 per cent of the notional effect. The net deferred tax asset is at most 15 per cent of the requirement and is never held against a net liability. An asset can only lower what is used, and a sum of components between nothing and the cap is what is used. It holds for any notional effect up to the requirement and any three sub-groups, capacity, earnings and deferred taxes.

ICS Read the proof of IC-05

For any capital resources and any requirement, what counts and what is left out add up to what is held, for a group that is not mutual and for a mutual. For a group that is not mutual, Tier 1 Limited counts up to 15 per cent of the requirement, Tier 2 up to 50 per cent and the basket up to 15 per cent, and none of the Non-Paid-Up capital counts. For a mutual, Tier 1 Limited counts up to 30 per cent, Non-Paid-Up up to 10 per cent, and Tier 1 Limited and Tier 2 together up to 60 per cent. What Tier 1 Limited has over its limit moves to Tier 2. What is left out is not below zero and what counts is not above what is held.

ICS Read the proof of IC-06

For margins of the six risk classes that are not below zero, the quantity under the root of the margin of a product class is at least the sum of the squares of the margins and at most the square of their sum. So the margin is at least the root of the sum of squares, which is what independent risk classes give, and at most the sum of the six margins.

SIMM Read the proof of SM-01

The concentration factor is never below one and is one up to the threshold. Above it, its square times the threshold is the exposure, so the margin of a lone factor grows with the exposure to the power 3/2. It rises with the exposure, so more exposure never lowers a weighted sensitivity, and it depends only on the exposure over the threshold, so the unit of both does not matter.

SIMM Read the proof of SM-02

In a bucket, the parts that rest on the factors add up to K² + S × C, where K² is the quantity under the root of the bucket margin, S is the clipped sum of the weighted sensitivities and C is what the other buckets put in the cross term. It holds when the sum is inside the clip, above it or below it, or when K is zero, and for any number of factors, because each step of the sum keeps the invariant.

SIMM Read the proof of SM-03

The parts of a product class add up to its margin. In a risk class, the parts of the regular buckets add up to the quantity under the root over the root, which is the core margin, and the residual bucket gives its own margin. The parts of the six risk classes, each the margin times the weighted margin over the total, add up to the total, which is the Euler identity of the table between risk classes.

SIMM Read the proof of SM-04

For three buckets with the cross-bucket correlation of 3/10, any clipped sums within their K and any ratios of the concentration factors between 0 and 1, the quantity under the root of a risk class is at most the square of the sum of the bucket margins and at least 2/5 of the sum of their squares. So the cross-bucket aggregation never adds to the plain sum, the quantity is never below zero, and the floor in the engine is not reached.

SIMM Read the proof of SM-05

The clipped sum of a bucket is never beyond K and has the sign of the sum. It is the sum inside the clip and K with the sign of the sum outside it. It rises with the sum, and a book with every sign changed has the clipped sum with its sign changed, so that the margin does not depend on which side of the market the book is on.

SIMM Read the proof of SM-06

exp and ln stay within three units in the last place of the platform library over generated inputs.

Kernel

Changing any single bit of an input number changes the receipt, except that every not-a-number is written as the same one.

Kernel

Node, workerd, Chromium, Firefox and WebKit return identical receipts for every parity fixture.

Kernel

The cube root stays within two units in the last place of the platform library over generated inputs. Sine and cosine in degrees are exact at every quarter turn, never return a negative zero, and stay within a few units of 1e-16 of the platform at every whole degree.

Kernel

Risk factor shares add up to the capital charge in every correlation scenario, and each matches a numerical derivative of the charge.

FRTB-SA

The capital charge is positively homogeneous of degree one and unchanged when every position flips sign.

FRTB-SA

The risk weights, the decay parameter, the floor and the scenario rules reproduce every entry of the published correlation table and the worked example in the January 2019 text of MAR21.

FRTB-SA

A bucket whose quadratic form is negative is floored at zero, and the clipped fallback applies when the cross-bucket quantity under the root is negative.

FRTB-SA

The capital charge is never negative and equals the largest of the three scenario charges.

FRTB-SA

At every level of the Solvency II trace, the shares of the parts add up to the share of the whole.

Solvency II

Every module’s share of the basic SCR is at least zero and at most the module’s own charge.

Solvency II

The SCR is the basic SCR plus operational risk less the loss-absorbing adjustment, which is limited so the SCR cannot go below zero.

Solvency II

The VEV matches a case worked by hand, rises with negative skew and with fat tails, and tracks annualised volatility for normal returns.

PRIIPs

The parts of the Cornish-Fisher value at risk add up to the whole, and for a normal return they reduce to the normal value at risk less the drift. Returns are the logarithms of price ratios and add up to the logarithm of the whole move.

PRIIPs

The adaptive proportion cutoffs reproduce every entry of Table 2 in SP 800-90B. They match exact big integer arithmetic for whole bit claims of one to eight bits, except at an exact tie with the false positive rate, where rounding can put a cutoff one above.

Entropy validation

The most common value estimate reproduces the worked example in SP 800-90B section 6.3.1 and matches the closed form for a biased stream.

Entropy validation

Each adaptive proportion window counts its first sample as often as it occurs in the window, and the test fails at the first sample that brings a window to the cutoff.

Entropy validation

A health run reports what each test and the estimate report on their own, and puts its alarm at the earlier of the two failures.

Entropy validation

The repetition count test fails at the sample that completes a run of the cutoff length and never when every run is shorter, and it reports where the longest run ended.

Entropy validation

Energy demand over a cycle is the sum of force times distance over its one second periods, counting only positive force. It equals force times distance at a steady speed and rises with every road load coefficient and with test mass.

WLTP

The class 3b trace has 1,801 speeds in phases of 589, 433, 455 and 323 seconds, and its sums over each phase and over the whole cycle reproduce the checksums that the regulation prints in Table A1/13.

WLTP

An individual vehicle with the mass, tyres and drag of vehicle H has the road load of H. One with those of vehicle L has the refitted road load of L. In between, the first coefficient moves in proportion to test mass times rolling resistance and the second in proportion to drag area.

WLTP

The curve of vehicle L is refitted, over the reference speeds from 20 to 130 km/h, by least squares with the linear coefficient of vehicle H, and a curve that already has that coefficient is not moved.

WLTP

An individual vehicle that has the parameters of vehicle H or of vehicle L is given the CO2 of that vehicle, in every phase and over the cycle, and one between them is given a figure between theirs.

WLTP

The method is allowed when vehicle H emits at least 5 g/km more than vehicle L and no more than the lower of 30 g/km and 20 per cent of its own CO2. The line may be extended by 3 g/km at either end, and no further than the largest difference.

WLTP

The CO2 for the whole cycle weights the phases by their driven distance. The CO2 of the test vehicles is carried to two decimal places, and an individual vehicle is declared to the nearest whole g/km, for each phase and for the cycle.

WLTP

An individual vehicle takes the class value of its tyres from Table A4/1 for C1 tyres, which are 5.9, 7.1, 8.4, 9.8, 11.3 and 12.9 kg per tonne for classes A, B, C, E, F and G. There is no class D.

WLTP

Every key, ciphertext and signature size of ML-KEM, ML-DSA and SLH-DSA is derived from the formulas of FIPS 203, 204 and 205 and equals the size that the standard prints, in a table or in the sentence that gives it. That covers the three ML-KEM sets, the three ML-DSA sets and the six rows of Table 2 of FIPS 205. The hybrid key shares of TLS 1.3 are derived from RFC 10024 and equal the sizes it writes.

PQC

The dates are those of NIST IR 8547, the EU roadmap and the NCSC timeline, at every boundary they set, and the EU risk level follows the four rules of its section 5, including the difference between nine years of confidentiality and ten and between a migration of eight years and nine.

PQC

A use is exposed exactly when the years to keep it secure plus the years to migrate exceed the years until a quantum computer. It is late when its latest start is before the planning year, and a quantum computer that is further away never makes a use start earlier.

PQC

Every use that changes is listed once, in order of its latest start, key establishment first when two tie, and never before a use it waits for. A use never starts later than it could on its own, and the same inventory written in another order gives the same plan.

PQC

What a replacement adds on the wire is the sum of sizes that the standards give: both key shares for a hybrid exchange, the ciphertext for a wrapped key, and the signature with the public key for a signature. A use whose algorithm survives a quantum computer has nothing to send.

PQC

The receipt of a plan covers the dates, the thresholds, the sizes, the inventory and every figure of the plan. It changes when any input changes, and it does not change with a name or with the order in which the inventory is written.

PQC

The spread of the plume across the wind and upward is the open-country row of Table 4.5 of the Handbook on Atmospheric Diffusion for each class. The engine gives the values that the formulas give at 100 m, 1 km and 10 km, and refuses a distance outside 100 m to 10 km, which is the range the table holds for.

AERMOD

The rise of the plume follows equation 91 of the AERMOD formulation, in which a momentum term and a buoyancy term are added under a cube root. It starts from nothing, never falls and stops at the limit of the Handbook, which is equation 2.25 in the unstable and neutral classes and equation 2.19 in the stable ones. A gas no warmer than the air, a wind below calm and a stack outside the range are refused.

AERMOD

The wind at the height of the stack is the wind at ten metres raised by the power that Table 4.6 gives for the class. It stops growing at 200 m and never falls below calm, which is 0.5 m/s.

AERMOD

The concentration at the ground is equation 4.1 of the Handbook with the plume and its image added, and it matches that equation worked independently, on the axis and off it. The plume carries the whole emission through every plane across the wind, which a numerical sum confirms. It is proportional to the emission, highest on the axis, equal on either side of it and nothing far to the side.

AERMOD

The curve of the figure lies on 400 distances at an equal ratio from 100 m to 10 km. Its maximum is the one that an independent search finds, in every class, and no point of the curve or neighbour of it is higher. In the classes C and D a taller stack, from 30 m to 120 m, gives a lower maximum when the wind at the stack is the same.

AERMOD

The illustrative weather is five years of hourly records made from a seed, 43,824 hours with the leap day of 2020. The same seed gives the same weather and another seed gives another. Every record is in the range of what it means, and stable classes come at night and unstable ones only by day, as the key of Table 4.1 says. It looks like weather, with a mean wind near 4 m/s, every class present, neutral the most common, one prevailing wind, and harder wind in winter and in the afternoon.

AERMOD

The calendar knows the leap years and names every day of every year, so that each day follows the one before it. It refuses a day that the year does not have. It puts 90, 91, 92 and 92 days in the quarters of an ordinary year, and one more in the first quarter of a leap year.

AERMOD

The rank tables are Table 1 of Appendices N and T, row by row. They cover every count of days from 1 to 366 once, and give the nearest rank above the percentile. The nth highest of a list agrees with sorting, and of equal values it takes the one that comes first. The annual mean is the mean of the quarterly means, so a quarter counts the same however many days it has.

AERMOD

A number is read as the decimal that prints for it, so that a value such as 1.005 is cut off or rounded by its digits and not by the binary number below them. Cutting off and rounding half up agree with whole-number arithmetic on every decimal with up to five places. A fraction is rounded half up exactly, with no drift on a tie.

AERMOD

The expected number of PM10 exceedances reproduces Examples 1, 2 and 3 of Appendix K, from the 2.36 of the first to the 2.19 of the third. A quarter or a year that lands on a tie rounds up, the sums agree with a whole-number formula on every quarter given, equation 3 reduces to equation 1 when each stratum has one sample, and the rate 1.05 rounds to 1.1.

AERMOD

The four forms carry the levels of 40 CFR Part 50. They are 35 and 9 micrograms per cubic metre for PM2.5, 150 micrograms per cubic metre with one expected exceedance a year for PM10, and 75 parts per billion for sulfur dioxide. A name that is not a form is refused.

AERMOD

The proofs about a day, an exceedance and a design value are about the formulas that the engine runs. A day read by a monitor is below the model by less than two tenths for a mean and one tenth for a highest hour, and never above it. A day of PM10 is an exceedance from 155. A mean of three values kept in tenths is rounded as the whole-number formula does.

AERMOD

Worked cases, each made by hand, show where a monitor and a model part. A day of 35.37 is 35.3 to a monitor and a design value of 35 that meets the standard, and a model reads it as over. The hour 35.49999 stays 35.4 for a monitor, and 35.5 gives 36. The annual form rounds to a tenth and the model rounds nothing. Sulfur dioxide becomes parts per billion and takes the highest hour of each day. The counts of three years are averaged and rounded to a tenth, and the peak hour keeps the weather it came in.

AERMOD

Over the five years the engine gives a row for every year and a window for every three. The day it takes in a year is the one that an independent sort puts at the rank of the table. A monitor never reads above a model, and by less than the cutting off at a tenth can take, and never counts more days of PM10. The value of the model is the mean of the years, the peak is the highest hour of the record, and moving the background moves a model by exactly that much. The defaults start where the figure starts, with a monitor reading 35 where the model reads just over it.

AERMOD

A receptor is placed in the plume at its along-wind and cross-wind distance in every hour. Straight downwind it gets the plume of the equation. It gets nothing upwind, nothing crosswind and nothing nearer than the spread formulas hold for, and a bearing or distance that is not valid is refused. The wind rose puts a steady wind from the west-southwest in the sector toward the east-northeast.

AERMOD

The receipt of a plume and the receipt of a design run are the same for the same input and different when any input moves. Running a design again changes nothing.

AERMOD

A run-off triangle has from two to forty origin years, and each has one amount fewer than the year before it. Every amount is a finite number, and every development year that has a successor adds up to more than zero, or no factor could be found. A zero or a negative amount is allowed where its column still adds up to more than zero. A message names the origin year or the development year, counting from one. The first origin year is held to the number of years, since no year comes before it, and a later one to the year before it.

Reserving and ESG

The amounts of each development year are added from left to right into running totals, and taken back out of them. The two are inverses of each other for whole amounts, keep the shape and check it. The amounts of Taylor and Ashe give the running totals that the vignette prints as the latest amounts.

Reserving and ESG

The development factors of the RAA triangle are the nine that the vignette prints, to the three decimals it prints, and the ones that an independent script gives, to the last digit but a few. A factor is the total of the next year over the total of this one, for the origin years that have both. It is a mean of the link ratios with the earlier amounts as weights, it lies between the smallest and the largest of them, and it agrees with exact fractions. The argument is checked before anything is done.

Reserving and ESG

On the RAA triangle the chain ladder keeps what was observed and has for the rest the amounts that the vignette prints, to the unit. The ultimate and the reserve of each origin year, and the totals of 213,122.23 and 52,135.23, are the ones that the vignette prints. On the triangle of Taylor and Ashe the reserve of each origin year is the one that the vignette prints, to the unit. The reserve is the ultimate less the latest amount and each total is the sum of its column. The oldest year has nothing to come, a triangle of two origin years is enough, and the ultimate of every origin year agrees with exact fractions to fourteen digits.

Reserving and ESG

What the factors say will be paid in each calendar year to come is listed with the next year first. Each year holds the next development year of every origin year that is still open. Together the years pay out the whole reserve, on every triangle the generator made. A proof shows the same for four origin years.

Reserving and ESG

When every amount is scaled by a power of two, the factors stay the same and the ultimates, the reserves and the payments scale by that number exactly. When every origin year is a multiple of the first, the factors are those of the first. The triangle that was given is not changed, and the same triangle gives the same answer on every run.

Reserving and ESG

The standard error of Mack needs at least four origin years, because the variance of the last factor is extrapolated from the two before it. It needs every amount above zero, wherever that amount stands. The reason is one of two messages, the shape is looked at first, and a chain ladder that was made from other rows is refused.

Reserving and ESG

On the RAA triangle the standard error of each origin year is the one that the vignette prints, to the tenth, and that of the whole reserve is the 26,909.01 that it prints, to the hundredth. An independent script gives the variances, the errors and the total to thirteen digits, for this triangle and for the one of Taylor and Ashe. The variances, the squares of the errors and the square of the total agree with exact fractions. The oldest year has no error, and the smallest triangle that can be used has four origin years and a variance for each of three factors.

Reserving and ESG

The error of the whole reserve is at least that of any one origin year, because the years share their factors. Scaling every amount by a power of two scales every error by it. A triangle whose origin years are all multiples of the first has next to no error. The variance of the last factor is the geometric continuation of the two before it when they do not rise, held to no more than either, and nothing when the earlier one is nothing. What was given is not changed.

Reserving and ESG

A triangle copied from a sheet is read with tabs, spaces, commas or semicolons between the amounts, and with either kind of line end. Cells copied empty or as NA at the end of a line are dropped, and so are blank lines, but NA in the middle of a line is not. Signs, decimals, exponents and the minus sign of typography are read. A text that is not an amount is named with its line and shortened if it is long, and an empty text says that there is nothing to read. A triangle written out is read back as it was, and the two sample triangles come back as the page prints them.

Reserving and ESG

A run takes running totals, or the amounts of each development year, which it adds first, and gives the chain ladder and the formula of Mack. It says why when the formula cannot be applied and then gives no error. It refuses a triangle that is not one on either basis, and shares no array with its input. The receipt covers the amounts, the basis, the factors, what comes to be paid and every error, with nothing where there is no error. It is the same for the same amounts, and it changes with the last bit of any amount and with the basis.

Reserving and ESG

On the numerical illustration of Annex D of the technical documentation of October 2025, the Smith-Wilson fit finds the convergence parameter and the quasi-constant that the page prints. It gives the vector Q b to its three decimals and every one of the 242 numbers of Table 29 to the fifth decimal, and it prices each of the twenty swaps at par. The search for the parameter runs a few hundred times at most.

Reserving and ESG

The table of Annex D was made with the root of the gap, which is 0.12376048 to eight places, and the page prints it as 0.123760. The rule of the documentation asks for the lowest value of six decimals that brings the gap within one basis point. That value is 0.123761, one millionth above the root, and the printed 0.123760 misses the tolerance by a hair. The engine follows the rule, and the test keeps the difference in view.

Reserving and ESG

The rule in force is Smith-Wilson up to 29 January 2027 and the closed form from 30 January 2027, and it never goes back to the old one on a later day. A day is written year-month-day, and the calendar knows which centuries have a leap day. A day that is not on the calendar, or that is written another way, is refused, and the message says what it wants.

Reserving and ESG

The convergence parameter of the closed form is 11 per cent for every currency but the Swedish krona and 40 per cent for the krona. An undertaking that phases in follows Table 4 year by year from 2027 and ends where the plain value starts, stepping down by the same amount each year within each group. At 40 years past the first smoothing point every one of these values leaves at least 77.5 per cent of the weight on the ultimate forward rate, which is the floor of Article 77a, and 11 per cent is the lowest value of two decimals that does. Before 2027 there is nothing to phase in, and the message says so.

Reserving and ESG

The ultimate forward rate moves by 15 basis points when the rate before the limit is 15 or more away, and stays otherwise. At the boundary of 15 it moves, as the text puts it. Rates are read to a hundredth of a basis point, so a sum of doubles cannot decide a boundary. The rate changes by one step or none, never against the pull and never past it, and it does not fall when the rate before the limit rises. A proof shows the same for every rate.

Reserving and ESG

An inflation target falls in one of four buckets, with the edges where paragraph C.4.1 puts them, and a target at or below one per cent falls in the lowest. The expected real rate is rounded to five basis points, up when it is below the rounded rate of the year before and down when it is not, a negative rate included, and a rate already on the grid stays. A year of Annex C gives the euro rate 4.05 per cent from 4.20 when the rounded real rate falls to 2.00 per cent and the target is 2 per cent. It says when the rate does not move and when it rises.

Reserving and ESG

The function of Wilson is the same from either side, and is the form that the documentation writes in three ways. It is zero at zero, and on the diagonal it is alpha u less half of one minus the exponential of minus two alpha u. It does not overflow where a product of an exponential and a hyperbolic sine would. Its slope and its second derivative are those that the documentation gives, on both sides of the diagonal and across it.

Reserving and ESG

The fit refuses a setup that cannot be fitted and says which part, an alpha that is not above zero, and instruments that do not fix the curve. It gives the same fit twice for the same alpha and another for another. The price starts at one and the spot is the zero-year limit of the yield. Beyond the last date the forward intensity is the closed form, it is the slope of minus the logarithm of the price, and the yield is its average. The gap is the distance of the forward intensity at the convergence point from the ultimate one.

Reserving and ESG

The fit reprices what it was fitted to. Every par swap of a smooth curve is priced at par, for any curve the generator makes, and so is every zero coupon bond at its price, at the maturities it was fitted to and at the odd ones in between. A curve that already sits at the ultimate forward rate stays there. A curve of negative rates is fitted too, because the method needs the instruments to be independent and nothing more.

Reserving and ESG

The search takes the lower bound when that is enough. Otherwise it returns an alpha that meets the tolerance and, to the decimals asked for, no lower one does. It agrees with a scan of a coarser step for any curve the generator makes, and a looser tolerance gives a lower alpha and a tighter one a higher alpha. It looks at the convergence point of the last date, which is 40 years after the last liquid point and never before year 60. Options that make no sense, and a tolerance that nothing can meet, are refused.

Reserving and ESG

A curve lists 150 years, each with an intensity, an annual rate and a forward, and a spot. It tends to the ultimate forward rate from below and passes it by no more than the tolerance at the convergence point. It keeps the rate at 20 years where the swaps put it. A fit whose price falls to zero or below, which very high swap rates can cause, has no rate for that year, so it is refused with the first such year named, and a curve that is given never has a gap.

Reserving and ESG

The bootstrap of Annex D gives the zero rates that the swaps imply, as an independent Newton-Raphson on equation D.5.9 gives them. It prices every swap at par, for any curve and any choice of maturities, and has one rate for each year to the last. It agrees with the Smith-Wilson fit at every maturity the swaps cover, is flat up to the first maturity, and keeps the forward constant between two maturities that are not a year apart. Zero coupon rates give back the rate at each maturity, with the forward that D.9.3 gives. What cannot be bootstrapped is refused, and the message says why.

Reserving and ESG

The weight of the last liquid forward rate is one at the start and falls to zero, whatever alpha is. It is one less the exponential of minus a h, over a h, the same in alpha and in the horizon, and it keeps its digits for a small product, to four units in the last place, where a difference of two numbers close to one would lose them. It is continuous where the series gives way to the closed form. The forward rate is a weighted mean of the last liquid forward rate and the ultimate one for every horizon, the first at once and the second in the end. An alpha or a horizon that is not above zero is refused.

Reserving and ESG

The extrapolation from the first smoothing point gives the zero rates, the last liquid forward rate and the ultimate forward rate that an independent script gives, to thirteen digits. That holds for a point at 20, 15, 10, 5 and 1 years, for other values of the ultimate rate and of alpha, and for forwards weighted by notionals in the order of the maturities from the point. The zero rates up to the point are those of the bootstrap.

Reserving and ESG

The forward just before the point is the only leg, with the whole weight, when the point is the longest maturity, and the first leg runs from zero when the point is the first maturity. With gaps between the maturities the forward before the point is the constant forward of the bootstrap. The curve is continuous at the point, and a point at the last year has nothing after it.

Reserving and ESG

A curve run builds the Smith-Wilson curve before 30 January 2027 and the closed form from that day, with the alpha of the day, and it says that the date decided. It takes the rule that is asked for whatever the date, and says that it was a choice. It uses the alpha of the krona and the phasing in when they apply, gives the same rates under both rules where the swaps fix the curve and different ones after, fits zero coupon rates too, and uses the notionals that it is given. A date that is not a date is refused under either rule, and so is a point that is not a maturity.

Reserving and ESG

A payment at the end of year t is discounted by one over one plus the rate of that year, to the power t. Payments add, nothing is worth nothing, and the value is linear in the payments and falls at a higher rate when no payment is negative. A payment after the last year of the curve, and a payment that is not a number, are refused.

Reserving and ESG

The receipt of a curve covers every input, the rule that was applied and how it was chosen, alpha and the rate of every year. It is the same for the same inputs, it does not depend on the type of the arrays, and it changes when any input changes, even by the last bit.

Reserving and ESG

After the point the zero rate lies between the rate at the point and the forward for every horizon. The forward runs from the last liquid forward rate to the ultimate one, between the two the whole way and faster for a higher alpha, and a curve that already sits at the ultimate rate stays there. What does not fit is refused in a fixed order, and the message says what.

Reserving and ESG

The rule is as the July 2020 text of the Basel Committee prints it. The multiplier is 1, the hedging disallowance is 0.01 and the risk-weighted assets are 12.5 times the capital. The seven specified currencies, and every risk weight and every correlation of interest rates, FX and counterparty credit spreads, are those of Tables 3 to 7 and paragraphs MAR50.54 to MAR50.65. Every matrix is symmetric, with a diagonal of one and entries from zero to one.

SA-CVA

A specified currency and the reporting currency have six interest rate factors, and any other currency has two. The weighted sensitivity of a line is its sensitivity times the weight of its factor, which is the weight that the tables give for each factor of each risk, and for each sector of credit spread, for investment grade and for the rest. A bucket of one factor and no hedge has a capital equal to its weighted sensitivity, which is the weight times the change that one shift makes, over the size of the shift.

SA-CVA

The correlation of two factors in a bucket is the entry of the table for a specified currency. It is 0.4 for the two factors of any other currency and for the volatilities of rates. For two credit spreads it is the product of three factors, one for the tenors, one for the names and one for the qualities. The two sectors of bucket 1 make one bucket, with their own weights and one correlation between their names.

SA-CVA

For one factor with an exposure A and a hedge of L times it, the capital of the bucket is A times the square root of (1 - L)^2 + R L^2, with R the hedging disallowance. A perfect hedge leaves A times the square root of R, not zero. The capital is lowest at L = 1 / (1 + R), and is that of no hedge again at 2 / (1 + R). A hedge of the same sign as the CVA offsets it, one of the opposite sign is more exposure, and a hedge alone is exposure of its own. The net weighted sensitivity of a factor enters the correlation and its hedge enters its own R term only. A share of the hedges scales every hedge and nothing else, and the capital at each share is listed in order.

SA-CVA

The capital of a class puts its buckets together at 0.5 for every pair of currencies, 0.6 for every pair of FX buckets and the entries of Table 6 for credit, times the multiplier. The sum of the weighted sensitivities of a bucket enters at no more than its capital and at no less than minus it. A one sided bucket is held to its capital, a bucket whose sum is smaller is left alone, and a bucket of one factor is never held. The class says which of its buckets were held, and the quantity under its root is that of the book with its hedges.

SA-CVA

The capital is the sum of the capital of the three delta classes and of the two vega classes, and the risk-weighted assets are 12.5 times it. The book without its hedges is worked class by class, and the difference is the benefit of the hedges. A class that the book does not name has no buckets and no capital, and a book of zero sensitivities has none. Each factor in the result carries what the book said, its weight and its weighted sensitivities, under a name that a person reads.

SA-CVA

Positions that name the same line are added before anything else, in the CVA and in the hedge. Positions that differ in the risk, the bucket, the factor or the name are not netted. The capital does not depend on the order of the positions, to the last digit when the amounts are whole. The factors of a bucket are listed in the order of the table, the buckets of a class by currency or by sector, and the credit lines by name and then tenor.

SA-CVA

A book that the rule cannot take is refused, and the message names the position, counting from one. A book has a reporting currency of three capital letters and from one to 200 positions, and every sensitivity is a finite number of at most 1,000,000,000,000 in size. A factor is one that the risk has in that bucket, and the message lists the ones it has. A credit spread has a sector of the table, an entity, a quality and a group, and the positions of one entity agree on all three. The checks of a position run in a fixed order, and so do the positions.

SA-CVA

A second program that shares no code with the engine works the capital in exact fractions and takes each square root once. For 26 books that use every risk, with a bucket held on each side and one that is not held, the capital of every class, with the hedges and without, and the capital of the book agree with the engine to a relative error under 3e-15. The saved answers are what the program says now, so that nobody edited them by hand.

SA-CVA

MAR50 has no rule for where the capital comes from, and the product gives one, the Euler allocation. The part of a bucket is what the capital of its class changes by along the bucket, and the factors of a bucket share its part, the CVA and the hedge of each apart. The parts add up to the capital for every book of the second program. A hedge that offsets takes a negative part, a bucket that the others more than offset takes a negative part, and a bucket with no capital takes none. A minus zero never comes out.

SA-CVA

Over generated books of up to 14 positions, the capital doubles when every sensitivity doubles and halves when they halve, to the last digit. It is the same with every sign reversed, and with the positions in another order when the amounts are whole. The capital of a class is a real number and not above the sum of its buckets. The capital of the book is delta plus vega, not below its largest class, and the risk-weighted assets are 12.5 times it. The shares add up to the capital, and the benefit of the hedges is never more than the capital without them.

SA-CVA

The receipt of a calculation holds the rule, the book as the calculation took it, which is netted and in a fixed order, and what came out. It is the same for the same book in another order when its amounts are whole, and different when any input moves by one unit. It lists the lines by risk before the bucket, and never carries a minus zero, which would give another hash.

SA-CVA

The tables and rules are as the IAIS Level 2 text of 5 December 2024 prints them. Table 6 has the five life risks, symmetric, and only mortality and longevity offset each other, at minus 25 per cent. The four catastrophe scenarios are independent. Table 16 has seven rows, and each direction of the spread stress has the six by six table left when the other is taken out. Table 34 has life and non-life independent and every other pair at 25 per cent. Table 33 has the factors of operational risk, with growth above 20 per cent. The tax rules are those of L2-339 and L2-348 to L2-353, and the limits on resources those of L2-122, L2-127 and L2-129.

ICS

The life charge is the root of the form of Table 6, and a longevity charge under half the mortality charge lowers it below what mortality alone sets, which is what the negative entry means. The catastrophe charge is the root of the sum of the squares of the four scenarios. The market charge is read from Table 16 in the row of the larger of the two spread stresses, and the smaller has no effect. When the stresses are equal and not zero the text gives no direction, so the engine takes the larger of the two results and says that there was a tie. With no spread charge there is none.

ICS

The operational charge of a line is the larger of its premium charge and its liability charge, plus the growth charge, with the charge on life business without risk added for the group, at the factors of Table 33. Premium counts as growth only above 120 per cent of the year before, at the factor of its line and on top of the larger charge, and at exactly 120 per cent or after a fall there is none. The engine says which of premium and liabilities decides, and takes the premium when they are equal. A group with no business has none. The charge is added after the aggregation and is never diversified.

ICS

The five categories are aggregated with Table 34, so life and non-life, which are independent, add in quadrature, and five equal charges give the root of 9.5 times the square. The tax effect comes off the insurance risk charge, and the charge for the non-insurance entities is added after it with no tax effect of its own. A group with nothing has a requirement of zero. The undiversified sum is every charge and the operational charge, and two diversification figures, between the categories and within them, explain the rest. The sample group has the charges that the second program confirms.

ICS

The group effective tax rate weights the rate of each entity by the earnings that it made in the previous three years. The example that the IAIS prints, three entities at 30, 25 and 20 per cent, gives 23.03 per cent. Each loss is set to zero in its own year and not against the sum of the three years. The rate of one entity is its own, and the unit of the earnings does not change the result. When no entity made a profit there is nothing to weight the rates by, and the engine refuses, as it does for an entity that is not one.

ICS

Under the default rule the tax effect is 80 per cent of the notional effect, which is the requirement times the group effective tax rate. A supervisor can limit it by four components. Component a is the smaller of the share by liabilities of the notional effect and the carry-back capacity of each tax sub-group, summed, less 15 per cent. Component b is half of five years of earnings times the rate, and zero for a projected loss. Component c is a net deferred tax liability and d a net asset held to 15 per cent of the requirement, and a group has one of them. The sum a plus b plus c less d is held between zero and 80 per cent of the notional effect. The IAIS example gives 2,005.

ICS

Tier 1 Limited counts up to 10 per cent of the requirement, or 15 per cent when the part above 10 has a loss absorbency mechanism. What is above the limit moves to Tier 2, which counts up to 50 per cent. The Tier 2 basket is half of the pension fund asset, all of the deferred tax asset and a tenth of the software intangibles, held to 15 per cent. A mutual counts Tier 1 Limited up to 30 per cent, Non-Paid-Up up to 10 per cent, and Tier 1 Limited with Tier 2 up to 60 per cent. The limits are shares of the requirement after tax and with the non-insurance charge, so they rise and fall with it, and what does not count is exactly what is above them. The text defines no ICS ratio.

ICS

A charge below zero is refused with a message that names it and says that a charge is the fall in capital resources under its stress, so a stress that raises them has a charge of zero. A charge that is not finite, or is above 1,000,000,000,000 in size, is refused by name. A wrong exposure, non-insurance charge or tax rule is refused with a message that says what is wrong. The checks run in a fixed order, charges first, then exposures, the non-insurance charge, the tax rule and last the resources. A group that gives no resources gets no result for them, which is not resources of zero.

ICS

The text has no rule for where the capital comes from. The product gives one, the Euler allocation through three levels. A risk takes its share inside its category, the category takes its share of Table 34, and the part of the risk is the product of the two. The parts add up to the insurance risk charge for every case of the second program. The operational charge is a part of its own, and non-life and credit have a line each. Each part is what the insurance risk charge changes by when its charge is scaled, which a numerical derivative confirms for eleven charges. A longevity charge under a quarter of the mortality charge takes a negative part.

ICS

The text gives a spread charge for an up stress and for a down stress, and does not say which row of Table 16 the larger of the two uses. The engine reads the row of the direction that gave the larger. So a down stress that grows past the up stress turns the row, and the market charge can then fall, because equity is correlated at 75 per cent with the spread under the up stress and at nothing under the down stress. Only when the two are equal and not zero does the engine take the larger of the two results, and the result says so.

ICS

A second program that shares no code with the engine works every case in exact fractions and takes each square root once, to 60 digits. For 50 cases that use both directions of the spread and a tie, every outcome of the tax rule, both lines of operational risk on both of their binds and every limit on capital resources, every figure agrees with the engine to a relative error under 3e-15, and the amounts left out, which are differences of sums, to 1e-14. The saved answers are what the program says now, so that nobody edited them by hand.

ICS

Over generated groups, the requirement doubles when every amount doubles, to the last digit, through the tax rule and the limits. The tax effect never takes the requirement below a fifth of the insurance risk charge, and the parts add up to that charge. A larger charge never gives a smaller requirement, except in a pair that offsets and in the spread. Each category charge lies between its largest part and the sum of its parts. The group tax rate lies between the lowest and the highest rate of the entities that made a profit. The tax effect never falls when earnings, the deferred tax liability or carry-back capacity rise, and never rises with the asset.

ICS

The receipt of a calculation holds the rule, every number that went in and every number that came out, in a fixed order. It carries the tax rule with its sub-groups, and no resources as nothing. It is the same for the same input, and different when any input moves by one unit or any choice is turned over. It never carries a minus zero, which would give another hash.

ICS

The tables and weights are those of the sketch calibration, which is illustrative and is not the calibration of ISDA. Every correlation table is symmetric with ones on its diagonal and no entry above one or below zero. The vertices of a rate fall by a tenth for each vertex between them to a floor of a fifth. Credit, equity and commodity buckets have the structure of the methodology with numbers of our own. Every bucket has a weight, a threshold and a label.

SIMM

Each risk class is worked by hand in the tests. A rate has its weight times its size. Two vertices that offset in part give the root of the form and keep their sum inside the range of K, and two with one sign are cut to K with the side named. Sub-curves correlate at 0.95. Currencies, issuers, names and commodities take the correlation of their rule, and the residual bucket of credit and of equity is added after the root.

SIMM

The six risk classes of a product class are put together with the table, and the four product classes are added, so that one factor in two of them does not net. An empty risk class or product class has a margin of zero. The part of a factor that nets to exactly zero comes from the others that it correlates with. Amounts of ten digits and amounts of cents both work.

SIMM

Concentration is judged on the sum for a currency or an issuer and not on each vertex. Exposure at the threshold is not concentrated, and above it the factor is the root of exposure over threshold. For rates the ratio of the factors of two currencies lowers their correlation, and for the other classes the ratio of two names lowers theirs inside a bucket. A book of large names costs about a third more than its plain margin, which sets every factor to one.

SIMM

A line that breaks a rule is refused, the first one, with a message that names it by its place and says what to do. Refused are a product class or risk type that is not one of those read, an amount that is not finite or is above 1,000,000,000,000, a name over 40 characters, a rate with no tenor or sub-curve, a bucket that does not exist, a trade in two product classes, a name in two buckets, no lines and more than 400.

SIMM

The part of each factor is its size times what one more unit of it costs, which is its coefficient. The parts of the factors add up to the margin for every portfolio of the tests and every case of the second program, and the same parts give those of trades, risk classes and product classes. A hedge has a negative part. With concentration the coefficient holds the factors where they are, so for one large name it is two thirds of the whole, and a difference quotient confirms it elsewhere.

SIMM

Each trade has the part of the margin that rests on it, in the order the trades first appear, and the lines with no trade count as one. Next to the part is what the margin falls by without the trade, and the two are not the same. A line of the calculation currency in FX has no part and is counted. A portfolio of one trade can be taken down to nothing, and nothing has a margin of zero.

SIMM

A second program in Python that shares no code with the engine works 92 cases in decimals of 60 digits and finds the part of each factor by a difference quotient. The cases take both sides of the cut of S, the residual buckets, every risk class and product class, concentration in each risk class and every rule of correlation. Every figure agrees with the engine to 3e-14, and the parts to 1e-9. The saved answers are what the program says now.

SIMM

Over generated portfolios the margin is zero or more, finite, and no more than the margin with no diversification, and the parts of the factors, the trades and the risk classes add up to it to a part in a hundred billion. It does not change when a line of nothing is added. It changes by less than a part in a trillion when every sign changes, when the order of the lines changes or when a line is split in two, and a book split among its product classes has the sum of their margins to the same part. It scales with the amounts when nothing is above its threshold.

SIMM

Two portfolios are compared on their net factors, and the result gives the margin of each side, the difference and the classes that have a margin on either side. A line is valued at what ours falls by when that line alone is made like theirs, which is checked by running ours again, and the one that closes the most comes first. Each break is named as only ours, only theirs, a bucket, a sign, a scale of a hundred or ten thousand, or an amount. A difference of half a cent or less is not listed, and the result says so when the margins still differ.

SIMM

The receipt of a margin holds the calibration, the kind, the factors as the calculation took them and the figures that came out. It is the same for a copy of the lines and for a factor whose lines are written as one or as two, and another for any change to a line that the calculation reads. The trades are listed as they first appear, so the same lines in another order can give another receipt. It does not read what it does not use, and it never carries a minus zero. The receipt of a reconciliation holds both calculations and every difference, and is another when a figure on either side moves by one unit or the sides change places.

SIMM