Run-off triangles: formula sheet

Key formulas

Cumulative from incremental

C(i, j) = X(i, 0) + X(i, 1) + ... + X(i, j)

i is the origin period, j the development period, X the incremental amount. Cumulative values never fall for paid data.

Incremental from cumulative

X(i, j) = C(i, j) − C(i, j−1), with X(i, 0) = C(i, 0)

Differences along each row. Do not difference down a column.

Incurred claims

Incurred = Paid + Case estimates of outstanding claims

Excludes IBNR, which is the reserve for claims not yet reported plus allowance for further development.

Calendar period of a cell

Calendar period = origin period + development period

With both numbered from 0. Cells on the same diagonal share a calendar period.

Number of known cells in a run-off triangle

n(n + 1) ÷ 2

For n origin periods and n development periods. For n = 4 this gives 10 known cells.

Outstanding reserve (basic idea)

Reserve = Ultimate cost − Paid to date

The triangle gives the paid to date. Reserving methods estimate the ultimate.

Development factor (volume-weighted)

f(j) = Σ C(i, j+1) ÷ Σ C(i, j), summed over origin years i for which C(i, j+1) is known

C(i, j) is the cumulative claims for origin year i at development year j. Use the same origin years in the numerator and denominator.

Individual development ratio

d(i, j) = C(i, j+1) ÷ C(i, j)

A simple-average factor is the mean of these ratios. Use it only if the question asks for it.

Ultimate claims

Ultimate(i) = C(i, latest) × f(latest) × f(latest+1) × … × f(last) × tail factor

If no tail factor is given, assume the last factor completes development (tail factor = 1).

Outstanding reserve

Reserve(i) = Ultimate(i) − C(i, latest)

Valid when the triangle is of cumulative paid claims. Total reserve is the sum over all origin years.

Cumulative-to-ultimate factor

F(j) = f(j) × f(j+1) × … × f(last)

Calculate once for each development year, then apply to every origin year. It saves time.

Constant-price conversion

Constant-price payment = Actual payment × I(base year) ÷ I(payment year)

I is the claims inflation index. Apply it to incremental payments, by calendar year of payment.

Cumulative constant-price claims

C(i, j) = Σ (k = 0 to j) of constant-price incremental payment X(i, k)

Accumulate along each accident year i after the conversion, not before.

Development factor (volume weighted)

f(j) = Σ C(i, j+1) ÷ Σ C(i, j), summed over accident years with both values observed

Use the same accident years in numerator and denominator. Use constant-price values only.

Projection

Ĉ(i, j+1) = Ĉ(i, j) × f(j)

Future incremental payment = Ĉ(i, j+1) − Ĉ(i, j), in base-year prices.

Re-inflation

Future-price payment = Constant-price payment × (1 + e)^(t − base year)

For a constant future inflation rate e. If rates vary by year, multiply the yearly factors (1 + e1)(1 + e2)… up to year t.

Reserve

Reserve = Σ of re-inflated projected incremental payments

Discounting, if asked for, is a separate final step.

Ultimate claims for an accident year

Ultimate claims = Ultimate number of claims × Ultimate average cost per claim

Apply this to each accident year, then sum for the total.

Average cost per claim (triangle cell)

Average cost = Cumulative claim amount ÷ Cumulative number of claims

Use the same basis for both: incurred amounts with reported numbers, or paid amounts with the matching number of claims.

Development factor for claim numbers

f(j) = Σ N(i, j+1) ÷ Σ N(i, j), summed over accident years i with both values observed

N is the cumulative number of claims. Use only years that have both columns.

Development factor for average cost

g(j) = Σ A(i, j+1) ÷ Σ A(i, j), over years with both values observed

A is the cumulative average cost per claim. This ratio of column sums weights each year by the size of its average cost in the earlier column, so years with higher average costs count for more. If you want equal weights, use the simple average of the individual year-to-year ratios A(i, j+1) ÷ A(i, j) instead. State which one you use.

Projection to ultimate

Ultimate N(i) = latest N(i) × product of f(j) for the remaining development years; similarly for A(i) using g(j)

Multiply the latest diagonal value by all later factors.

Inflation-based average cost

Average cost for year i+1 = Average cost for year i × (1 + claim inflation rate)

Use when the question gives a claim inflation rate and asks you to adjust average costs between accident years.

Reserve

Reserve = Ultimate claims − Claims paid to date

If you use incurred amounts, the difference from incurred to date is the IBNR-type provision. Say which one you are giving.

Proportion developed

p = 1 ÷ (f₁ × f₂ × ... × fₖ), where f₁ ... fₖ are the remaining development factors

This is the proportion of ultimate claims expected to have emerged to date. For the latest development year, p = 1 ÷ (product of all remaining factors to ultimate).

BF reserve

Reserve = A × (1 − p)

A is the a priori ultimate claims for the origin year. (1 − p) is the proportion still to emerge.

BF ultimate

Ultimate = Claims to date + A × (1 − p)

Use the same basis for claims to date and for the pattern: both cumulative paid, or both cumulative incurred.

A priori ultimate

A = Premium × expected loss ratio

Other sources are allowed. State the source and any inflation or exposure adjustment.

Chain ladder ultimate (comparison)

Ultimate = Claims to date × f₁ × f₂ × ... × fₖ = Claims to date ÷ p

Chain ladder is fully driven by the data. BF replaces Claims to date ÷ p with a blend.

Loss ratio ultimate

Ultimate = Earned premium × Expected loss ratio

The expected loss ratio is an a priori assumption. It does not come from the triangle.

Loss ratio reserve

Reserve = Ultimate − Claims paid to date

Use incurred instead of paid if you want the IBNR only. State which basis you use.

Chain ladder ultimate

Ultimate = Latest cumulative claims × CDF

CDF is the product of the remaining development factors, from the current development year to ultimate.

Proportion developed

p = 1 ÷ CDF

The share of ultimate claims expected to have emerged so far.

Bornhuetter-Ferguson ultimate

Ultimate = Claims to date + A priori ultimate × (1 − 1/CDF)

The a priori ultimate is usually premium × expected loss ratio.

BF as a weighted average

BF ultimate = p × Chain ladder ultimate + (1 − p) × Loss ratio ultimate

Here p = 1/CDF. Valid when the same a priori ultimate and CDF are used in all three methods.

Age-to-age (development) factor

f(j) = Σ C(i, j+1) ÷ Σ C(i, j), summed over accident years i with both values observed

C(i, j) is cumulative claims for accident year i at development year j. This is the volume-weighted chain ladder factor. A simple average of individual ratios is an alternative.

Projected ultimate with tail

Ultimate(i) = C(i, latest) × f(latest) × ... × f(last) × tail factor

The tail factor is 1 only if you believe claims are fully developed at the end of the triangle.

Reserve

Reserve(i) = Ultimate(i) − Paid to date(i)

Ultimate − Paid to date gives the total outstanding reserve, which is case reserves plus IBNR. With incurred data, Ultimate − Incurred to date gives IBNR in the wide sense, covering both IBNR and IBNER.

Implied ultimate loss ratio

Ultimate loss ratio(i) = Ultimate(i) ÷ Premium(i)

Use as a reasonableness check across accident years. Premiums should be on a consistent basis (earned, same gross or net treatment).

Paid-to-incurred ratio

Paid ÷ Incurred at each development year

A rising ratio at the same age can indicate faster settlement or weaker case estimates, because incurred is then lower. A falling ratio can indicate slower settlement or stronger case estimates. Check other data, such as claim closure rates and average case estimates, to tell the causes apart.

Quick revision

• A triangle shows claims by origin year and development year, and only the top-left part is known.
• Chain ladder factors are ratios of column totals, using only the years where both columns are observed.
• Ultimate = latest cumulative claims × product of the remaining development factors.
• Reserve = ultimate − claims paid to date (for a paid triangle).
• Chain ladder assumes future development follows the same pattern as past development.
• For inflation adjustment, deflate, project, then reinflate at the assumed future rates.
• Average cost per claim projects the number of claims and the average cost separately, then multiplies them.
• BF ultimate = claims to date + a priori ultimate × (1 − proportion developed).
• BF is more stable than chain ladder on immature years because it leans on the a priori estimate.
• The loss ratio method sets ultimate = premium × expected loss ratio and ignores actual experience.
• Check that the triangle is consistent across years: look for changes in claim handling, mix of business or reserving practice.
• State your assumptions clearly in every answer, including inflation and the treatment of the tail.

Common mistakes

• Mixing up origin period and calendar period. Fix: Rows are when the claim arose. Calendar period is origin plus development, shown by the diagonals.
• Differencing down the columns instead of along the rows. Fix: Incremental and cumulative conversion is always within one origin row, across development periods.
• Using incremental figures to calculate factors Fix: Always read the heading. Convert to cumulative before calculating any ratio.
• Including the wrong origin years in a factor Fix: Only sum the earlier column for the origin years where the later column is known. The numerator and denominator must cover the same years.
• Adjusting cumulative figures instead of incremental payments. Fix: Convert incremental payments by their payment year, then accumulate. A cumulative figure mixes payments from different years.
• Using the accident year instead of the calendar year for the index. Fix: The payment year is accident year plus development year. Inflation factors are constant along diagonals, not along rows.
• Mixing bases, for example dividing paid amounts by reported claim numbers. Fix: Write the basis beside each triangle before dividing. Match incurred with reported and paid with the matching settled or paid number, or state the assumption you are making.
• Projecting total claims with chain ladder and then multiplying by something, or multiplying two cumulative triangles together by mistake. Fix: Keep two separate projections: numbers and average cost. Multiply only at the end, using the ultimate values.
• Using p as the proportion still to emerge Fix: p is the proportion already developed. The reserve uses (1 − p). Check that your reserve is smaller than A for any year with some development.
• Multiplying the a priori ultimate by the development factor Fix: Apply factors only to claims to date in chain ladder. In BF, use them only to get p.

Exam tips

• Write the headings on every triangle you draw: origin basis, development period, paid or incurred, incremental or cumulative.
• Always check one row total after converting. It takes seconds and catches most arithmetic slips.
• For comment questions, give a point on each of paid, incurred and claim numbers: speed, objectivity and what can distort them.
• Name the assumptions the triangle needs, such as stable settlement pattern and no inflation change, because later method questions build on them.
• In computer-based questions, keep the triangle in a clear grid with blank cells for future periods so formulas do not pick up zeros.
• Write down each factor as a fraction first, for example 4,950 ÷ 3,300, so you can earn method marks even if the arithmetic slips.
• Always list the assumptions of the method. Written questions often award marks for them.
• Be ready to comment on limitations: the latest origin years rely on little data, and inflation, large claims and changes in settlement speed distort factors.