Exponential Trend

Study guide: Exam 5

An Exponential Trend models a ratemaking statistic — frequency, severity, pure premium, average premium, an exposure base — as changing by a constant percentage per unit of time. It is fitted by least-squares regression of the natural log of the statistic on time, and it is the standard model for loss, premium and exposure trend.

$\hat{y}_t = a\,e^{b t}$

$\ln \hat{y}_t = \ln a + b\,t$

$\hat{T} = e^{b} - 1$

$b$ is the least-squares slope of $\ln y_t$ on $t$ : $b = \sum (t - \bar{t})(\ln y_t - \overline{\ln y}) \big/ \sum (t - \bar{t})^2$ . With $t$ in years, $\hat{T}$ is the annual trend; with $t$ in quarters, the annual trend is $e^{4b} - 1$ .
The selected trend is applied as a factor $(1 + \hat{T})^{n}$ , where $n$ is the trend period from the average date of the experience period to the average date of the forecast period. Separately selected frequency and severity trends combine multiplicatively: $1 + T_{PP} = (1 + T_F)(1 + T_S)$ .
Why it is preferred: a constant rate of change compounds the way inflation does, and the fitted curve never goes negative. A linear trend on a decreasing series eventually projects negative frequencies or severities, which Werner & Modlin call its shortcoming.
How it is fitted in practice: for short-tailed lines, calendar-year paid data for the 12 months ending each quarter (the rolling year smooths seasonality); for long-tailed lines, accident-year losses developed to ultimate. Werner's example fits windows of 20, 16, 12, 8, 6 and 4 points, and $R^2$ is the usual diagnostic; the recent windows show whether the trend has shifted.
Catastrophes are excluded from trend data (with rolling years, one event contaminates four points) and large losses capped; benefit-level changes are removed or the fit will read them as trend. Selection is a judgment informed by external indices, and ASOP 13 requires it to be reasoned and disclosed.

Fitting and Applying an Annual Severity Trend Accident-year severities (developed, capped, excluding catastrophes): $2020$ : $\$9{,}800$ ; $2021$ : $\$10{,}150$ ; $2022$ : $\$10{,}700$ ; $2023$ : $\$11{,}150$ ; $2024$ : $\$11{,}800$ . New rates take effect for annual policies written in $2026$ .

Fit an exponential trend and trend the AY $2024$ severity to the forecast period.
Answer Centre time at $2022$ , so $t = -2, -1, 0, 1, 2$ and $\sum t^2 = 10$ . The logs are $9.19014$ , $9.22523$ , $9.27800$ , $9.31919$ , $9.37585$ .
 $\begin{align*} \sum t \ln y_t &= -2(9.19014) - 9.22523 \\ &\quad + 9.31919 + 2(9.37585) \\ &= 0.46538 \\ b &= 0.46538 / 10 = 0.046538 \\ \hat{T} &= e^{0.046538} - 1 = 4.76\% \end{align*}$ 
Select $4.8\%$ . The trend runs from the AY $2024$ average accident date, $7/1/2024$ , to the average accident date of policies written in $2026$ , $12/31/2026$ : $n = 2.5$ years.
 $\begin{align*} \text{Trend factor} &= 1.048^{2.5} = 1.1244 \\ \text{Trended severity} &= \$11{,}800 \times 1.1244 \\ &= \$13{,}268 \end{align*}$ 
The end-point shortcut $(11{,}800/9{,}800)^{1/4} - 1 = 4.75\%$ happens to agree here, but it uses only two of the five observations; one unusual year at either end would swing it, while the regression weighs them all.

Quarterly Frequency Fit Combined with Severity Frequencies for the 12 months ending six consecutive quarters: $0.0620$ , $0.0615$ , $0.0611$ , $0.0606$ , $0.0602$ , $0.0597$ . The selected severity trend is $+4.8\%$ a year.

Find the annual frequency trend, the pure premium trend, and the factor for a $4.5$ -year trend period.
Answer Number the quarters $t = 0, \dots, 5$ : $\bar{t} = 2.5$ and $\sum (t - \bar{t})^2 = 17.5$ . The logs are $-2.78062$ , $-2.78872$ , $-2.79524$ , $-2.80346$ , $-2.81008$ , $-2.81842$ , and $\sum (t - \bar{t}) \ln y_t = -0.13066$ .
 $\begin{align*} b &= -0.13066 / 17.5 = -0.007466 \text{ per quarter} \\ \hat{T}_F &= e^{4(-0.007466)} - 1 = -2.94\% \end{align*}$ 
Select $-2.9\%$ for frequency. Then
 $\begin{align*} 1 + T_{PP} &= (1 - 0.029)(1 + 0.048) \\ &= 1.0176 \\ \text{Trend factor} &= 1.0176^{4.5} = 1.0817 \end{align*}$ 
Pure premium trends at $+1.8\%$ a year, far less than severity alone suggests, because claims are becoming rarer as they become dearer. Trending pure premium directly would give a similar answer; fitting the components separately shows why, and flags when one of them shifts.