ارزش گذاری منصفانه قراردادهای مشارکت بیمه عمر وابسته به مسیر

Fair valuation of insurance contracts, and of options embedded in them, is an important, incompletely understood issue. With the coming IAS insurance contract standard, the valuation of liabilities in life insurance is due to a drastic change. We present a computationally tractable model for fair valuation of participating life insurance contracts with given, almost arbitrary bonus policies. Unlike traditional valuation methods, our model captures several essential features of participating life insurance contracts, such as fair values of interest rate guarantees and of various bonus policies. In the model, fair value of life insurance contracts is understood as the arbitrage free price in the presence of liquid markets for liabilities. In addition to numerical results, the model gives solutions in closed form.

nsurance accounting is experiencing a radical shift from traditional valuation of insurance liabilities towards fair valuation of liabilities. This has occurred partly because one side on the balance sheet is valued in market values—the assets—while the other—the liabilities—is not. Another important reason for the change of emphasis on accounting is the need for a better understanding of financial risks, such as interest rate guarantees, and, in general, of valuation of various elements in insurance contracts. This is especially timely now, since low interest rates have made interest rate guarantees an important issue for life insurance contracts. Several insurance companies, such as Nissan Mutual Life, have run into trouble partially as a result of an underestimation of embedded options in their written insurance contracts. To address these issues, the forthcoming IAS standard for insurance contracts has adopted a radical way of valuation: insurance liabilities are to be valued as if they were traded in huge numbers among well-informed, independent investors in a liquid marketplace. This is called fair valuation, which is formally defined as “the amount for which an asset could be exchanged or a liability settled between knowledgeable, willing parties in an arm’s length transaction”, see IASB (2001). According to the draft IAS standard, liabilities including embedded options, should be valued with stochastically estimated future cash flows and discounted with riskless interest, which is often interpreted as interest of 30-year government bonds. Although International Accounting Standards Board (IASB) seems to be moving away from demanding use of strict fair valuation, importance of understanding fair valuation is not reduced, since the original motivation for the valuation method—reliable valuation of embedded options and indirect obligations—still remains. To better understand how fair valuation is feasible, we construct a model for valuation of participating life insurance contracts, by extending the work of Grosen and Jørgensen (2000), and by deriving an analytical solution for the fair value of the contract. For certain simple bonus mechanisms our model produces nice analytic results, but for most bonus policies we get iterated integral representations that yield results only after relying on numerics. Furthermore, the formulation of the model allows easy incorporation of various kinds of bonus mechanisms, thus leading to a more practical and comprehensive approach to the understanding of the value of participating contracts. We shall also briefly consider how to include a known term structure of riskless interest to the model. It is well known that option valuation models capture well some aspects of market’s valuation process, hence we use them as a starting point. We take it as given that a fair value is the arbitrage free price1 for assets and liabilities, as in option valuation models. We assume that liabilities are valued as if a liquid market for the underlying contracts existed.

The traditional method of accounting of insurance liabilities is very different from the method presented here. Traditionally, option-like elements embedded into insurance contracts have been taken into account implicitly using prudently estimated discount factors, without explicit estimation of the value of each embedded option. To gain better understanding of the financial risks associated with insurance contracts, one has to estimate the value of embedded options explicitly. We developed a model for the estimation of the fair value of path-dependent, participating life insurance contracts and option elements embedded in them, assuming that liabilities are valued as if they were traded on a liquid market. The model produces analytic expressions for contracts of very short duration, and gives an iterative method for evaluation of contracts of longer duration. As the bonus functions can be chosen fairly freely, our model adapts well to several different situations. For instance, we could use it to analyze how solvency requirements affect the fair valuation of life insurance policies. Using numerics, our model yields explicit values for various kinds of contracts and their option elements. The model behaves numerically relatively well, although we do not believe that the numerical procedure we used in the examples would be optimal. One possible improvement could be to employ a logarithmic scale instead of the linear one for the (a,p)-lattice. To the best of our knowledge, the model presented here has not really been used before, despite its remarkable simplicity. The model sheds light on the problem of fair valuation of path-dependent liabilities, and it offers a basis for the modeling of liabilities of several types of path-dependent, participating insurance contracts.