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Ira R o b b i n , P h D , S e n i o r P r i c i n g Actuary, P a r t n e r R e


This paper presents three related measures of the return on a Property-Casualty insurance policy. These measures are based on a hypothetical Single Policy Company model. Accounting rules are applied to project the Income and Equity of the company and the flows of money between thecompany and its equity investors. These are called Equity Flows. The three measures are: i) the Internal Rate of Return (]RR) on Equity Flows, h) the Return on Equity (ROE), and iii) the Present Value of Income over the Present Value of Equity (PVI/PVE). The IRR is the yield achieved by an equity investor in the Single "Policy Company. The ROE is the Growth Model Calendar Year ROE computed on abook of steadily growing Single PoLicy business. The PVI/PVE is computed by taking present values of the projected Income and Equity of the Single Policy Company. The paper includes new results relating the PVI/PVE and ROE to the IRR. Beyond developing the foundation and theory of these

return measures, the other main goal of the paper is to demonstrate how to use the measures to obtainrisksensitive prices. To do this, Surplus during each calendar period is set to a theoretically required amount based on the risk of the venture. The main source of risk arises from uncertainty about the amount and timing of subsequent loss payments. With the IRR and PVI/PVE, the indicated prices are those needed to achieve a fixed target return. The indicated price using the Growth Model is that neededto hit the target return at a specified growth rate. With the Growth Model, one can also compute the premium-to-surplus leverage ratio for the Book of Business when it achieves equilibrium. The ability to relate indicated pricing to a leverage ratio, growth rate, and return is an advantage of Growth Model and could lead to greater acceptance of its results. The paper includes sensitivity analysison the returns and on the indicated profit provisions. In the presentation, the analysis of return is initially done for a single loss scenario. Later, there is discussion on how to model the return when losses are a random variable instead of a single point estimate. this paper versus that of the Discounted Cash Flow model. Keywords: ROE, IRR, PVI/PVE Finally, there is a comparison of the approachin


INTRODUCTION In this p a p e r , w e will p r e s e n t three related w a y s to m e a s u r e the r e t u r n o n an i n s u r a n c e

policy. T h e three m e a s u r e s are: • T h e I n t e r n a l Rate o f R e t u r n o n E q u i t y F l o w s (IRR)



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The Growth Model Calendar Year Return on Equity (ROE) The Present Value of Income OverPresent Value of Equity (PVI/PVE)-

Then we will demonstrate how to use these measures to price Proper~:Casualty insurance products. We will do this from the perspective of a pricing actuary conducting analysis for a stock insurance company. Whether any of these methods is appropriate in another context is a subject outside the scope of our discussion.

There is nothing novel about usingmeasures of return to priceproducts. The idea is simple enough: any venture with return above a given target hurdle rate is piesumably profitable enough to be undertaken. The indicated price for a product can then be'defined as the one at
• • , , .

which its expected return hits the target. Within the context of internal Corporate pricing analysis, corporate management usually sets the targetreturn and a common target is generally used for all insurance ventures.

A significant problem in Property and Casualty insurance pricing applications is that there is no one universally accepted measure of return. The sale of an insurance policy leads to cash flows, underwriting income, investment income, income taxes, and equity commitments that may span several years. How do we distill all this...
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