Inflation

Study guides: Exam VEE-ECON, Exam P-1, Exam FM-2, Exam FAM, Exam ASTAM, Exam 5, Exam CP-351

Inflation in an insurance context is growth in the underlying loss random variable over time, $X' = (1+r)X$ . Many deductibles and benefit limits are stated in fixed dollars that do not rise with inflation, so inflation does not scale the insurer's cost proportionally. A fixed deductible leverages it, raising the insurer's expected payment by more than $r$ ; a fixed limit damps it, raising the capped payment by less than $r$ .

$X' = (1 + r)\,X$

$E[(X' - d)_+] = (1+r)\,E\!\left[\left(X - \tfrac{d}{1+r}\right)_+\right]$

The second identity is the whole mechanism: inflating losses against a fixed deductible $d$ is equivalent to holding losses fixed and lowering the deductible to $d/(1+r)$ . A lower deductible means more claims pierce it and each pierces it by more.
P-21-05's car policy shows both. At $10\%$ inflation a year, expected losses rise $46\%$ from year 1 to year 5. Under a fixed $500$ deductible, expected claim payments rise $54\%$ ( $650$ to $998$ ); add a fixed maximum claim payment of $12{,}500$ and they rise only $34\%$ ( $610$ to $819$ ).
The losses above a fixed limit take the leverage instead. With a positive severity trend, basic limits trend $\le$ total limits trend $\le$ excess losses trend (Werner & Modlin), and their example dampens a $10\%$ total-limits severity trend to $3.5\%$ in basic-limits losses. Deductibles have the same leveraging effect, with the censoring below the deductible rather than above the limit.
In ratemaking, monetary inflation is one of the factors that drive loss trends, alongside increasing medical costs and advancements in safety technology, and the actuary estimates loss trends to project the historical losses into the future.
In reserving, an increase in the inflation rate is one of the changes in the economic environment an actuary has to understand before estimating unpaid claims. For an insurer operating in a stable environment, the average case outstanding should increase down each column of the triangle at the relevant annual inflation rate; averages rising faster are something to take back to the claims department. Development techniques assume that past development patterns will account for inflationary forces; frequency-severity techniques reflect inflation explicitly instead, at the price of being highly sensitive to the inflation assumption.
Social inflation is a significant rise in the propensity for lawsuits and the size of jury awards. Like general inflation, it has a disproportionate impact on increased-limits losses.
On Exam FM the word belongs to interest theory instead: the FM syllabus lists inflation and real rate of interest together among the Time Value of Money terms a candidate must define and recognize (see Real Rate of Interest).

Leveraged Effect of Inflation on a Deductible Ground-up losses are $X \sim \text{Exponential}(\theta = 1000)$ with an ordinary deductible $d = 500$ . Losses then inflate $10\%$ .

Compare the insurer's expected payment before and after.
Answer For an exponential, $E[(X-d)_+] = \theta e^{-d/\theta}$ . After $10\%$ inflation $X' \sim \text{Exponential}(\theta' = 1100)$ .
 $\begin{align*} E[(X - 500)_+] &= 1000\,e^{-0.5} \\ &= 606.5 \\[6pt] E[(X' - 500)_+] &= 1100\,e^{-500/1100} \\ &= 1100\,e^{-0.4545} \\ &= 698.2 \end{align*}$ 
$\frac{698.2}{606.5} - 1 = +15.1\%$

Ground-up losses rose $10\%$ ; the insurer's expected payment rose $15.1\%$ . The deductible has effectively fallen to $500/1.1 = \$454.55$ in real terms, and the insurer picks up both the extra severity and the claims that newly exceed the threshold.

Inflation as a Calendar Year Effect in Reserving A liability insurer's reported age-to-age factors, with the latest diagonal (year ending $12/31/2024$ ) in bold:

AY 12–24 24–36 36–48 48–60

$2020$ $1.42$ $1.18$ $1.09$ $1.11$

$2021$ $1.44$ $1.17$ $1.16$

$2022$ $1.41$ $1.28$

$2023$ $1.58$

Historical column averages excluding the latest diagonal: $1.42$ , $1.18$ , $1.09$ , $1.04$ . Average paid severity on closed claims rose $14\%$ during $2024$ against a $6\%$ historical trend.

Diagnose and respond.
Answer Every factor on the latest diagonal exceeds its column average: $1.58$ against $1.42$ , $1.28$ against $1.18$ , $1.16$ against $1.09$ and $1.11$ against $1.04$ . Measured against the development each column expects, the excess grows with maturity:
 $\begin{align*} \text{12--24:} \quad \frac{0.58 - 0.42}{0.42} &= 38\% \\ \text{24--36:} \quad \frac{0.28 - 0.18}{0.18} &= 56\% \\ \text{36--48:} \quad \frac{0.16 - 0.09}{0.09} &= 78\% \\ \text{48--60:} \quad \frac{0.11 - 0.04}{0.04} &= 175\% \end{align*}$ 
Each diagonal of the triangle is one valuation date, so a rise along the latest diagonal, rather than down one accident-year row, points to something that happened during $2024$ itself. Paid severity on closed claims rose too ( $14\%$ against a $6\%$ trend), so the payments themselves went up, not only the case reserves, which is consistent with an inflation shift.

Development factors averaged from the history assume that past development patterns will account for inflationary forces, and this diagonal says they no longer do. A frequency-severity method reflects inflation explicitly instead, but its estimate is highly sensitive to the inflation assumption, so test the rate selected. If the higher inflation is expected to persist, the severity trend used in pricing needs re-selecting as well.

AY	12–24	24–36	36–48	48–60
$2020$	$1.42$	$1.18$	$1.09$	$1.11$
$2021$	$1.44$	$1.17$	$1.16$
$2022$	$1.41$	$1.28$
$2023$	$1.58$