Converting Claim Severities Between Price Bases

Why Convert Claim Amounts Between Price Bases?

A claim amount recorded in an earlier year reflects the prices prevailing at that time. To compare it directly with a claim from a later year, or to express it at a specified valuation date, the amounts must be put on the same price basis. The conversion uses a price index to adjust for the change in the general price level represented by that index.

The direction is determined by the target date. Converting an earlier claim to a later date means trending it forward; converting a later claim to an earlier date means deflating it backward. The same index ratio handles both operations. What changes is which index value is in the numerator.

Definition: Let $I_t$ be the value of a specified price index at date $t$, and let $C_s$ be a claim amount observed at date $s$. Under the assumption that the index adequately represents price change for the claim being converted, the amount expressed on the price basis of target date $T$ is $$ C_T=C_s\frac{I_T}{I_s}. $$ The ratio $I_T/I_s$ is the index ratio from the observation date to the target date. It is greater than 1 when the index has risen between those dates and less than 1 when it has fallen.

The amount $C_T$ is the observed claim adjusted to the target date’s price basis. It is not a new observation of what the claim would actually have cost, nor does the conversion allow for claim-specific changes in severity. It is a standardized comparison using the chosen index and the stated assumption.

Use dates that match the intended measurement. For example, a problem may specify the claim’s occurrence date, reporting date, or payment date as the date whose price basis is to be adjusted. Do not silently substitute one date for another. If the index is reported monthly or annually, use the date convention given in the problem; an unstated interpolation assumption should not be introduced.

Choosing the Direction of the Ratio

A reliable way to set up the calculation is to write the target index over the observation-date index. This makes the conversion direction explicit before any arithmetic is done.

Identify the observed amount and its date.
Record the claim amount $C_s$ and the date $s$ to which it applies.

Identify the target date.
The target $T$ is the date whose price basis is required, not necessarily the date of the latest claim in the data.

Form the target-to-observation index ratio.
Calculate $I_T/I_s$. If $T$ is later and the index has risen, this ratio increases the amount; if $T$ is earlier, it usually reduces it.

Multiply and check the direction.
Compute $C_s(I_T/I_s)$. Confirm that the size of the adjusted amount is consistent with the index movement.

The direction rule follows from the meaning of an index. If a fixed basket of prices costs $I_s$ index units at date $s$ and $I_T$ index units at date $T$, its relative price change is $I_T/I_s$. A claim proportional to that price level is adjusted by the same ratio. Reversing the dates gives $I_s/I_T$, the reciprocal conversion; multiplying by both ratios returns the original amount.

\frac{I_T}{I_s}\frac{I_s}{I_T}=1, \qquad C_T\frac{I_s}{I_T}=C_s.

Index values can be reported on different numerical scales. For example, an index with values 100 and 125 gives the same ratio as one rebased to 80 and 100, because both numerator and denominator are multiplied by the same constant. The index must, however, refer to the same consistent series at both dates. If the problem supplies separate series or a discontinuity, a link between them is needed before taking a ratio.

Worked Applications

Worked Example: Trending a Historical Claim Forward

A hypothetical property claim was recorded as $18,000 in 2020. The selected price index was 120 in 2020 and 150 in the 2023 valuation year. Under the assumption that this index represents the relevant price change, express the claim on the 2023 price basis. State the index ratio and check the result.

Method. Since the target date, 2023, is later than the observation date, 2020, use the target index in the numerator:

\frac{I_{2023}}{I_{2020}}=\frac{150}{120}=1.25.

The index has risen by a factor of 1.25. Apply that factor to the observed amount:

C_{2023}=\$18{,}000(1.25)=\$22{,}500.

Check. The adjusted amount is greater than $18,000, as expected when moving forward to a date with a higher index. Reversing the conversion gives $\$22{,}500(120/150)=\$18{,}000$.

Common mistake avoided. Using $120/150$ for a forward conversion would give $\$14{,}400$, incorrectly reducing the claim even though the index rose. The numerator must be the index at the target date.

Worked Example: Deflating a Claim Backward

A hypothetical equipment claim of $28,800 was recorded in 2024. The index was 144 in 2024 and 120 in the requested 2021 price basis. Convert the claim to 2021 dollars and verify it by trending forward again. Assume the supplied index values are comparable.

Method. The target date, 2021, is earlier than the observation date, 2024. The target index therefore goes in the numerator:

\frac{I_{2021}}{I_{2024}}=\frac{120}{144}=\frac{5}{6}.

Apply the ratio to the amount observed in 2024:

C_{2021}=\$28{,}800\left(\frac{5}{6}\right)=\$4{,}800(5)=\$24{,}000.

Check. Moving forward from 2021 to 2024 uses the reciprocal ratio $144/120=1.2$, so $\$24{,}000(1.2)=\$28{,}800$, the original observed amount.

Common mistake avoided. Multiplying by $144/120$ while converting backward would produce $34,560. That factor trends from 2021 to 2024, the opposite of the requested conversion.

Worked Example: Bringing Claims From Different Dates to One Basis

An analyst wants to compare three hypothetical claims on the 2022 price basis. Claim A was $10,000 in 2020, Claim B was $11,000 in 2021, and Claim C was $12,100 in 2022. The price index values were 100 in 2020, 110 in 2021, and 121 in 2022. Convert each claim to the 2022 basis and find their total on that basis. Assume the same index applies to all three claims.

Method. Use the target index, 121, over each claim’s own observation-date index. Claim C is already on the target basis, so its ratio is 1.

C_{A,2022}=\$10{,}000\frac{121}{100}=\$12{,}100, \qquad C_{B,2022}=\$11{,}000\frac{121}{110}=\$12{,}100,

C_{C,2022}=\$12{,}100\frac{121}{121}=\$12{,}100.

The total expressed on the 2022 basis is

\$12{,}100+\$12{,}100+\$12{,}100=\$36{,}300.

Check. The index ratios from 2020 and 2021 are $1.21$ and $1.10$, respectively, so both earlier amounts increase to $12,100. The 2022 amount stays unchanged. The total is therefore $36,300 on the common basis.

Common mistake avoided. Adding the original amounts gives $33,100, but that sum combines amounts from three different price bases. It is not the requested total on the 2022 basis.

Worked Example: Using Compounded Annual Changes in a Conversion

A hypothetical claim of $5,000 is observed in 2021. The index is 100 in 2021, and its annual increases are 10% in 2022 and 10% in 2023. Convert the claim to the 2023 price basis. Assume each stated increase applies successively to the index level at the start of that year.

Method. Build the index ratio by compounding the annual factors. After the 2022 increase, the index is $100(1.10)=110$. After the 2023 increase, it is $110(1.10)=121$. Thus the cumulative forward ratio is $121/100=1.21$, not 1.20.

C_{2023}=\$5{,}000(1.10)(1.10) =\$5{,}000(1.21)=\$6{,}050.

Check. Deflating $6,050 by the reciprocal cumulative ratio gives $\$6{,}050/1.21=\$5{,}000$. The amount rises because the target is later and the index has increased.

Common mistake avoided. Adding the two annual increases and using a 1.20 factor ignores compounding. The second 10% increase applies to the already increased index, so the correct factor is $1.10^2=1.21$.

Assumptions and Practical Checks

The index-ratio conversion is exact as a calculation once the index and dates are specified, but its interpretation depends on the modeling assumptions. The index is being used as a proxy for the price change relevant to the claims. If claim costs change differently from the selected index, the converted amounts remain index-adjusted comparisons rather than exact reconstructions of each claim at another date.

Before accepting a result, check the following:

Both dates are clear. Identify the date attached to the observed claim and the date requested for the converted amount.
The index series is consistent. Use values from the same series, or use an explicitly supplied link when the series changes.
The ratio points in the right direction. Write target index over observation-date index, even when the target is earlier.
The price movement agrees with the amount. A higher target index gives a higher adjusted amount; a lower target index gives a lower adjusted amount, all else equal.
Rounding is delayed. Keep the index ratio unrounded during intermediate calculations when possible, then round the final money amount to the nearest cent if dollars and cents are requested.

A frequent exam trap is to think that “deflating” requires subtracting an inflation percentage. It does not. Deflation is multiplication by the relevant index ratio, which may be the reciprocal of a forward factor. Another trap is to use the newest available index automatically. The correct numerator is the index at the requested target date, whether that date is earlier or later than the observation date.

Key takeaway: To convert an amount observed at date $s$ to the price basis of date $T$, multiply by $I_T/I_s$. Forward conversions commonly increase amounts when prices have risen; backward conversions commonly reduce them. Always choose the ratio from the dates, not from the words “inflation” or “deflation” alone.

Check Your Understanding

For each problem, use the stated index values and assume the index is appropriate for the claim. Give dollar answers to the nearest cent where needed.

A hypothetical claim of $9,600 is recorded in 2021, when the index is 120. The target date is 2023, when the index is 150. What is the claim on the 2023 price basis? (Answer: $12,000.00)
A hypothetical claim of $18,900 is recorded in 2024, when the index is 135. Express it on the 2022 price basis, when the index is 120. (Answer: $16,800.00)
A hypothetical $4,000 claim is recorded in 2020. The index is 100 in 2020, 105 in 2021, and 115.5 in 2022. Convert the claim to the 2022 basis using the index values directly. (Answer: $4,620.00)
Two hypothetical claims are $8,000 at index 100 and $10,500 at index 125. Convert both to a target basis with index 150, then find their combined amount on that basis. (Answer: $24,600.00)
A hypothetical claim of $14,000 is observed at index 140. The requested basis has index 100. A calculation incorrectly uses the factor $140/100$. What is the correct converted amount, and what amount would the incorrect factor produce? (Answer: Correct $10,000.00; incorrect $19,600.00)

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