Reserving and ESG

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.

5 checks, decided by Z3 5.3.3.0 (WebAssembly) and rerun from the emitted file by Z3 5.3.3 (native) and cvc5 1.4.2 (native, nl-cov). 5 broken variants are refuted. proofs/smt2/RS-04.smt2

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.

4 checks, decided by Z3 5.3.3.0 (WebAssembly) and rerun from the emitted file by Z3 5.3.3 (native) and cvc5 1.4.2 (native, nl-cov). 3 broken variants are refuted. proofs/smt2/RS-05.smt2

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.

4 checks, decided by Z3 5.3.3.0 (WebAssembly) and rerun from the emitted file by Z3 5.3.3 (native) and cvc5 1.4.2 (native, nl-cov). 3 broken variants are refuted. proofs/smt2/RS-06.smt2

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.

4 checks, decided by Z3 5.3.3.0 (WebAssembly) and rerun from the emitted file by Z3 5.3.3 (native) and cvc5 1.4.2 (native, nl-cov). 3 broken variants are refuted. proofs/smt2/RS-07.smt2

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.

6 checks, decided by Z3 5.3.3.0 (WebAssembly) and rerun from the emitted file by Z3 5.3.3 (native) and cvc5 1.4.2 (native, nl-cov). 4 broken variants are refuted. proofs/smt2/RS-08.smt2

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/reserving.test.ts

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.

tests/kernel/curve.test.ts

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.

tests/kernel/curve.test.ts

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.

tests/kernel/curve.test.ts

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.

tests/kernel/curve.test.ts

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.

tests/kernel/curve.test.ts

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.

tests/kernel/curve.test.ts

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.

tests/kernel/curve-fit.test.ts

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.

tests/kernel/curve-fit.test.ts

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.

tests/kernel/curve-fit.test.ts

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.

tests/kernel/curve-fit.test.ts

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.

tests/kernel/curve-fit.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts

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.

tests/kernel/curve-rules.test.ts