A Continuous Version of Sherman’s Inverse Power Curve Model with Simple Cumulative Development Factor Formulas

1. Introduction

The inverse power curve model for loss development factors was introduced by Sherman (1984). Several papers have commented on the lack of a closed form representation and/or tail convergence information (Boor 2006; CAS Tail Factor Working Party 2013; Lowe and Mohrman 1985). The conditions for tail factor convergence and estimates of the rate of tail factor convergence have been determined (Evans 2014). However, simple or closed-form formulas for finite and infinite products of the discrete incremental development factors still appear elusive.

This paper will demonstrate that a continuous version of the inverse power curve model captures the relevant properties of the discrete model. At the same time, this continuous model leads to simple cumulative development factor, including tail factor, formulas. These simple formulas facilitate various practical calculations, such as fitting or calculating development factors using intermediate time values or fitting the model to a preselected tail factor. Basic real analysis is used throughout this paper, as described in standard textbooks such as Rudin (1976).

Section 2.1 defines the continuous version of the model. Convergence conditions are proved in Section 2.2. Several basic analytical properties are proven in Section 2.3. Empirical fitting, and a comparison with the discrete model, is discussed in Section 2.4. Further numerical comparisons are shown in Sections 2.5. Section 3 contains concluding remarks. Appendix A contains the proof of two lemmas and Theorem 2 from Section 2.3.

2. Analysis and results

In the remainder of this paper t is used for age or time, whether discrete or continuous.

2.1. Continuous inverse curve model definition

The discrete version of the inverse power curve model can be defined in terms of a cumulative development factor F_t(a, b, c) from time 1 to t.

Definition:

For real numbers a, b, and c, where a > 0 and c ≥ 0,

i. If t ≥ 2 is an integer, then

$$F_{t}(a, b, c)=\prod_{k=1}^{t-1}\left(1+a(k+c)^{b}\right) .$$

ii. If t = 1, then F₁(a, b, c) = 1.

F_t(a, b, c) obeys the finite difference equation

$$F_{t+1}(a, b, c)-F_{t}(a, b, c)=a(t+c)^{b} F_{t}(a, b, c) .$$

If the boundary value of F₁(a, b, c) = 1 is included, then this equation, along with the previous parameter restrictions, is an equivalent definition of F_t(a, b, c).

A corresponding continuous version of the inverse power curve model can similarly be defined in terms of a cumulative development factor F_t*(a, b, c) from time 1 to t, but with much simpler closed-form expressions.

Definition:

For a > 0, b, c ≥ 0, and t ≥ 0

i. $F_{t}^{*}(a, b, c)=\exp \left(\frac{a(c+t)^{1+b}-a(c+1)^{1+b}}{1+b}\right) \text { if } b \neq-1$

if b ≠ −1

ii. $F_{t}^{*}(a, b, c)=\left(\frac{c+t}{c+1}\right)^{a} \text { if } b=-1$

Using an analogous boundary value F₁*(a, b, c) = 1, F_t*(a, b, c) is the solution to a differential equation in continuous time t,

$$\frac{d F_{t}^{*}(a, b, c)}{d t}=a(t+c)^{b} F_{t}^{*}(a, b, c),$$

that is analogous to the finite difference equation satisfied by the discrete model.

2.2. Tail convergence

The tail of F_t*(a, b, c) converges when b < −1 and diverges when b ≥ −1, which are exactly the same as the conditions for convergence of F_t(a, b, c) as shown in Evans (2014).

Theorem 1

i. $\text { If } b \geq-1, \lim _{t \rightarrow+\infty} F_{t}^{*}(a, b, c)=+\infty \text {. }$

ii. $\text { If } b<-1, \lim _{t \rightarrow+\infty} F_{t}^{*}(a, b, c)=\exp \left(\frac{-a(c+1)^{1+b}}{1+b}\right) .$

Proof:

i. If b = −1, clearly $\lim _{t \rightarrow+\infty} \frac{c+t}{c+1}=+\infty$ and since a > 0 by Lemma A.1 $\lim _{t \rightarrow+\infty}\left(\frac{c+t}{c+1}\right)^{a}=+\infty .$ If b > −1, then b + 1 > 0, so again by Lemma A.1 $\lim _{t \rightarrow+\infty} \frac{a(c+t)^{1+b}}{1+b}=+\infty$ and since exp(x) is an increasing function of x, $\lim _{t \rightarrow+\infty} \exp \left(\frac{a(c+t)^{1+b}-a(c+1)^{1+b}}{1+b}\right)=+\infty .$

ii. If b < −1, then b + 1 < 0, so by Lemma A.1$\lim _{t \rightarrow+\infty} \frac{a(c+t)^{1+b}}{1+b}=0$ and consequently since exp(x) is continuous $\lim _{t \rightarrow+\infty} \exp \left(\frac{a(c+t)^{1+b}-a(c+1)^{1+b}}{1+b}\right)$ $=\exp \left(\frac{-a(c+1)^{1+b}}{1+b}\right) .$

2.3. Some basic analytical properties

For convenience we will first set up notational definitions of the one-period development factors, f_t(a, b, c) for the discrete model and f_t*(a, b, c) for the continuous model.

Definition:

$f_{t}(a, b, c)=\frac{F_{t+1}(a, b, c)}{F_{t}(a, b, c)}=1+a(c+t)^{b}$

Definition:

i. For $b \neq-1, f_{t}^{*}(a, b, c)=\frac{F_{t+1}^{*}(a, b, c)}{F_{t}^{*}(a, b, c)}$ $=\exp \left(\frac{a(c+t+1)^{1+b}-a(c+t)^{1+b}}{1+b}\right)$

ii. For $b=-1, f_{t}^{*}(a, b, c)=\frac{F_{t+1}^{*}(a, b, c)}{F_{t}^{*}(a, b, c)}$ $=\left(\frac{c+t+1}{c+t}\right)^{a}$

Lowe and Mohrman (1985) list several analytical properties for a curve of one-period loss development factors to be “well-behaved.”

Definition:

A curve of one-period loss development factors, f(t) for t ≥ 0, is said to be well-behaved if it has all of the following properties:

i. f(t) ≥ 1

ii. $\lim _{t \rightarrow+\infty} f(t)=1$

iii. f′(t) < 0

iv. $\lim _{t \rightarrow+\infty} f^{\prime}(t)=0$

v. f″(t) > 0

vi. $\lim _{t \rightarrow+\infty} f^{\prime \prime}(t)=0$

If b < 0 then f_t(a, b, c) and f_t*(a, b, c) are both well-behaved.

See Appendix A for details of the proof.

2.4. Fitting to empirical data

Table 1 includes an example from Sherman’s original paper (1984) of the discrete model fit to empirical data. Also shown are one-period development factors from the continuous model, first using the same parameter values as the discrete model and then using another set of parameter values refit for the continuous model itself. The continuous model development factors using the discrete model parameter values are fairly close to the discrete model development factors. When the parameters values are refit, the resulting development factors for the continuous model are very close to the discrete model development factors.

Definition:

f^a(t) will denote an empirical observation of a one-period development factor from time t to t + 1.

The fits and goodness-of-fit (R²) numbers in Table 1 are determined using the squared error function

$$\begin{aligned} \sum_{t=1}^{n} & \left(\log \left(f^{a}(t)-1\right)-\log \left(f_{t}(a, b, c)-1\right)\right)^{2} \\ & =\sum_{t=1}^{n}\left(\log \left(f^{a}(t)-1\right)-\log (a)-b \log (c+t)\right)^{2} \end{aligned}$$

for the discrete model.

Correspondingly, for the continuous model,

$$\begin{array}{l} \sum_{t=1}^{n}\left(\log \left(f^{a}(t)-1\right)-\log \left(f_{t}^{*}(a, b, c)-1\right)\right)^{2} \\ \quad=\sum_{t=1}^{n}\left(\log \left(f^{a}(t)-1\right)\right. \\ \left.\quad-\log \left(\exp \left(\frac{a(c+t+1)^{1+b}-a(c+t)^{1+b}}{1+b}\right)-1\right)\right)^{2} \end{array}$$

A simpler squared error function for the continuous model would be

$$\begin{array}{l} \sum_{t=1}^{n}\left(\log \left(f^{a}(t)\right)-\log \left(f_{t}^{*}(a, b, c)\right)\right)^{2} \\ \quad=\sum_{t=1}^{n}\left(\log \left(f^{a}(t)\right)-\left(\frac{a(c+t+1)^{1+b}-a(c+t)^{1+b}}{1+b}\right)\right)^{2} . \end{array}$$

This is still a fairly complicated function, likely not having a simple formulaic solution to minimize the parameters a, b, and c. One of the few apparent advantages of the discrete model is a somewhat simpler error function. It may be advantageous to use a numerical optimization program (like Solver in Excel) to first optimize a, b, and c for the discrete model. Those values can then be used as a starting point, or initial values, for the optimizer to search for values optimal for the continuous model.

2.5. More numerical comparisons with the discrete model

Table 2 follows the basic layout of Table 1, except that the fitting targets a one-period development factor of 1.01 from time 1 to 2 and a development factor of 1.30 from time 1 to 101, rather than fitting to a set of actual empirical development factors. The b parameter runs through the set of values {−2.0, −1.5, −1.1, −1.0, −0.9, −0.6} in these examples. Similar to what happened in Table 1, in Table 2 the continuous model is fairly close to the discrete model using the same parameter values and very close—identical up to three digits past the decimal in some examples—when the parameter values continuous model are refit. Note, in the table the time values extend upward to very large numbers, irrelevant to any realistic actuarial application, simply to illustrate asymptotic properties of the models.

3. Conclusions

The continuous inverse power curve model presented in this paper has the same tail convergence conditions and “well-behaved” analytical properties as the discrete model. Unlike the discrete model, it is known to have very simple closed formulas for cumulative development factors, including tail factors. It tends to produce numerical values extremely close to the discrete value when fit to the same data. Squared error functions for fitting the parameters of the continuous model tend to be more complex, but fits borrowed from the discrete model can be used as initial values to facilitate fitting the continuous model.