How Factor Selections Change Ultimate Losses

Why Factor Selections Matter

A chain ladder projection is sensitive to the development factors selected for each age. A change to one factor does not necessarily affect every accident year equally: an accident year that has already passed that development age is unaffected, while a less mature year may still be projected through it. The financial effect also depends on the cumulative loss amount being developed.

This tutorial quantifies those effects. It treats alternative selections as deterministic scenarios, not as probability distributions or confidence intervals. Unless stated otherwise, amounts are hypothetical, in thousands of dollars, and the examples assume no interest adjustment or mortality assumption. The selected factors are assumed appropriate for the cumulative loss measure being projected and for the accident years to which they are applied.

Definition: Let $C_{AY,a}$ be an accident year's observed cumulative loss at development age $a$, and let $f_{j,j+1}$ be the selected age-to-age factor from age $j$ to the next age. The age-to-ultimate multiplier at age $a$, denoted $M_a$, is the product of the selected factors from age $a$ through the final projected age. Under the deterministic chain ladder projection, the estimated ultimate is $U_{AY}=C_{AY,a}M_a$. If the unpaid amount is defined on the same cumulative basis, it is $R_{AY}=U_{AY}-C_{AY,a}$.

For example, if a year is observed at 12 months and factors remain to be applied at 12–24, 24–36, and 36 months to ultimate, its multiplier is the product of all three selected factors. If it is already observed at 24 months, only the latter two factors remain. Thus factor selections must be compared using the factors that actually apply at each year's maturity.

How a Factor Change Flows Through the Projection

Suppose one selected factor, $f_{k,k+1}$, changes by $\Delta f$, while all other selected factors remain fixed. An accident year is affected only if its projection still includes this factor. For an affected year observed at age $a\leq k$, separate the multiplier into the factors before, at, and after the changed age. Its change in ultimate is:

\Delta U_{AY} = C_{AY,a} \left(\prod_{j=a}^{k-1} f_{j,j+1}\right) \Delta f \left(\prod_{j=k+1}^{n-1} f_{j,j+1}\right).

Here $n$ is the final projected age, and an empty product is defined to equal 1. The formula follows by subtracting the original product from the revised product: every unchanged factor is common to both products, leaving only the difference $(f_{k,k+1}+\Delta f)-f_{k,k+1}=\Delta f$. This also shows why the change in unpaid amount is the same as the change in ultimate when the observed cumulative amount is held fixed.

A useful proportional form follows when the original factor is positive. For an affected accident year, with all other factors held fixed, the revised ultimate divided by the original ultimate is $(f_{k,k+1}+\Delta f)/f_{k,k+1}$. That percentage change is common to affected years, but their dollar changes need not be: the cumulative amounts and the factors before and after the changed factor can differ. A year already beyond age $k$ does not include the changed factor and has no direct change under this calculation.

Worked Example: Two Alternative Selections for One Accident Year

A hypothetical accident year has cumulative incurred losses of $800 at 12 months. The selected factors are 1.40 from 12–24 months and 1.15 from 24 months to ultimate. An alternative selection uses 1.45 and 1.18, respectively. Calculate the ultimate and unpaid amount under each selection, then quantify the change. The projection is deterministic; no interest, mortality, or probability distribution is assumed.

Original selection. The 12-month age-to-ultimate multiplier is the product of the two remaining factors:

M_{12}=1.40(1.15)=1.610.

The estimated ultimate is $800(1.610)=1{,}288$. The estimated unpaid amount on the incurred basis is $1{,}288-800=488$.

Alternative selection. Its multiplier is $1.45(1.18)=1.711$. The ultimate is $800(1.711)=1{,}368.8$, and the unpaid amount is $1{,}368.8-800=568.8$.

Change and check. The ultimate increases by $1{,}368.8-1{,}288=80.8$; the unpaid estimate also increases by $568.8-488=80.8$. This agrees with the difference in multipliers: $800(1.711-1.610)=80.8$. The method is to recompute the full remaining multiplier, rather than add the factor changes directly to the ultimate. The common mistake avoided here is adding $0.05+0.03$ as though factor changes were additive; the factors compound.

Maturity Determines Which Years Respond

A selected factor can affect several accident years, but only if each year is still being projected through that development age. Consider a change to the 24–36 month factor. A 12-month year includes that factor and the subsequent factors; a 24-month year includes it and subsequent factors; a year already at 36 months does not. The older year can therefore be insensitive to a factor change that materially affects younger years.

This is a direct implication of the projection formula, not a claim that immature years are always more sensitive in dollars. A large mature year can have a substantial response to a factor that still applies to it, and a small immature year can have a modest dollar response. Always identify the remaining factors, then calculate the dollar effect using each year's cumulative amount.

Worked Example: Comparing Accident Years at Different Ages

A hypothetical portfolio has three accident years with observed cumulative incurred losses of $500 at 12 months, $900 at 24 months, and $1,200 at 36 months. Under the original selections, the factors are 1.40 for 12–24 months, 1.20 for 24–36 months, and 1.10 from 36 months to ultimate. An alternative selection changes the first two factors to 1.45 and 1.22, while keeping 1.10 unchanged. Calculate the ultimate and unpaid amount for each year under both selections. No interest, mortality, or probability distribution is assumed.

12-month year, original. Its multiplier is $1.40(1.20)(1.10)=1.848$. Ultimate is $500(1.848)=924$, so unpaid is $924-500=424$.

12-month year, alternative. Its multiplier is $1.45(1.22)(1.10)=1.9459$. Ultimate is $500(1.9459)=972.95$, so unpaid is $972.95-500=472.95$. The increase is $972.95-924=48.95$.

24-month year, original. Only the factors from 24 months onward apply: $1.20(1.10)=1.32$. Ultimate is $900(1.32)=1{,}188$, and unpaid is $1{,}188-900=288$.

24-month year, alternative. Its multiplier is $1.22(1.10)=1.342$. Ultimate is $900(1.342)=1{,}207.8$, and unpaid is $1{,}207.8-900=307.8$. The increase is $19.8$.

36-month year. It uses only the unchanged tail factor of 1.10 under either selection. Its ultimate is $1{,}200(1.10)=1{,}320$, and its unpaid amount is $1{,}320-1{,}200=120$. Its change is zero.

Portfolio check. Original total ultimate is $924+1{,}188+1{,}320=3{,}432$; alternative total ultimate is $972.95+1{,}207.8+1{,}320=3{,}500.75$. The total increase is $68.75$, which equals the sum of the year-level increases, $48.95+19.8+0=68.75$. Total unpaid increases by the same amount because total observed cumulative losses are unchanged. The method makes each year's maturity explicit; the common mistake avoided is applying the 12–24 factor again to the 24- and 36-month observations.

Changing One Factor Across a Portfolio

The sensitivity formula is especially useful when an alternative selection changes just one age-to-age factor. The following example holds all other selections fixed so that the affected accident years and the aggregate impact can be traced directly.

Worked Example: Aggregate Effect of a Revised 24–36 Month Factor

A hypothetical portfolio has cumulative incurred losses of $400 at 12 months, $700 at 24 months, and $1,000 at 36 months. The selected factors are 1.40 for 12–24 months, 1.20 for 24–36 months, and 1.10 from 36 months to ultimate. A revised selection raises only the 24–36 month factor from 1.20 to 1.25. Calculate each ultimate and unpaid estimate before and after the revision, and the total change. Assume a deterministic projection without interest, mortality, or a probability distribution.

12-month year. Before revision, the multiplier is $1.40(1.20)(1.10)=1.848$, giving ultimate $400(1.848)=739.2$ and unpaid $739.2-400=339.2$. After revision, the multiplier is $1.40(1.25)(1.10)=1.925$, giving ultimate $400(1.925)=770$ and unpaid $770-400=370$. The increase is $770-739.2=30.8$.

24-month year. Before revision, the multiplier is $1.20(1.10)=1.32$, giving ultimate $700(1.32)=924$ and unpaid $924-700=224$. After revision, the multiplier is $1.25(1.10)=1.375$, giving ultimate $700(1.375)=962.5$ and unpaid $962.5-700=262.5$. The increase is $962.5-924=38.5$.

36-month year. It does not use the changed factor. Its ultimate remains $1{,}000(1.10)=1{,}100$, and its unpaid amount remains $1{,}100-1{,}000=100$.

Portfolio totals and check. Before revision, total ultimate is $739.2+924+1{,}100=2{,}763.2$; after revision, it is $770+962.5+1{,}100=2{,}832.5$. The increase is $69.3$, equal to $30.8+38.5$. Total unpaid rises from $339.2+224+100=663.2$ to $370+262.5+100=732.5$, also an increase of $69.3$. The common mistake avoided is changing the ultimate for the 36-month year even though its projection no longer includes the revised 24–36 month factor.

Interpreting the Results and Avoiding Traps

A factor sensitivity calculation answers a conditional question: how would the projection change if the selected factor changed and the observed cumulative losses and all other selections stayed fixed? It does not establish that either selection is preferable. Selection judgment still depends on the loss experience and assumptions supporting the factor, which are separate from the arithmetic of translating that selection into ultimates.

Keep the loss basis consistent. If $C_{AY,a}$ is incurred loss, subtract that incurred amount to calculate an unpaid amount on the incurred basis. If the projection is instead based on paid cumulative losses, the corresponding unpaid estimate is ultimate less paid-to-date. Do not subtract paid losses from an incurred ultimate, or treat a change in estimated unpaid amount as a change in losses already observed.

A frequent exam trap is to apply a changed factor to every year in the triangle. First compare each year’s maturity with the changed factor’s age. Another is to interpret an increase of 0.05 in a factor as an increase of five dollars per hundred of ultimate. The change acts within a product of factors, so its dollar effect must be calculated using the cumulative amount and the other factors in the multiplier. Finally, report both the accident-year effects and their aggregate: a small change to one factor can affect several projected years, while an older year may be untouched.

Key takeaway: Recompute each accident year's remaining age-to-ultimate multiplier under the alternative selection, multiply by its observed cumulative amount, and subtract the same observed amount to obtain unpaid. A factor change affects only years whose projections still include that factor.

Check Your Understanding

For each hypothetical deterministic projection, calculate the requested effect. No interest, mortality, or probability distribution is assumed.

A year has cumulative losses of $250 at 12 months. Its selected factors are 1.30 for 12–24 months and 1.10 from 24 months to ultimate. An alternative uses 1.32 and 1.12. Find the ultimate and unpaid amount under each selection and the increase in each. (Answer: Original ultimate $357.50 and unpaid $107.50; alternative ultimate $369.60 and unpaid $119.60; each increases by $12.10.)
A year has cumulative losses of $640 at 24 months. The 24–36 month factor is revised from 1.18 to 1.20, while the 36-month-to-ultimate factor remains 1.08. Find both ultimates and unpaid estimates. (Answer: Original ultimate $815.616 and unpaid $175.616; revised ultimate $829.44 and unpaid $189.44; increase $13.824.)
A year has cumulative losses of $950 at 36 months, with a tail factor revised from 1.07 to 1.05. Calculate the ultimate and unpaid amount under both selections, and state the change. (Answer: Original ultimate $1,016.50 and unpaid $66.50; revised ultimate $997.50 and unpaid $47.50; each decreases by $19.00.)
A portfolio has cumulative losses of $300 at 12 months, $500 at 24 months, and $800 at 36 months. The factors are 1.40 for 12–24 months, 1.20 for 24–36 months, and 1.10 from 36 months to ultimate. Only the 24–36 factor changes to 1.25. Calculate the change in each ultimate and the aggregate ultimate. (Answer: The 12-month ultimate increases by $23.10, the 24-month ultimate by $27.50, and the 36-month ultimate is unchanged; aggregate ultimate increases by $50.60.)

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