We present a geometric approach to studying greatest accuracy credibility theory. Our main tool i... more We present a geometric approach to studying greatest accuracy credibility theory. Our main tool is the concept of orthogonal projections. We show, for example, that to determine the Biihlmann credibility premium is to find the coefficients of the minimum-norm vector in an affine space spanned by certain orthogonal random variables. Our approach is illustrated by deriving various common credibility formulas. Several equivalent forms of the credibility factor Z are derived by means of similar triangles.
Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate ... more Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment
The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of th... more The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the...
For a general fully continuous life insurance model, the variance of the loss-at-issue random var... more For a general fully continuous life insurance model, the variance of the loss-at-issue random variable is the expectation of the square of the discounted value of the net amount at risk at the moment of death. In 1964 Jim Hickman gave an elementary and elegant derivation of this result by the method of integration by parts. One might expect that the method of summation by parts could be used to treat the fully discrete case. However, there are two difficulties. The summation-by-parts formula involves shifting an index, making it somewhat unwieldy. In the fully discrete case, the variance of the loss-at-issue random variable is more complicated; it is the expectation of the square of the discounted value of the net amount at risk at the end of the year of death times a survival probability factor. The purpose of this note is to show that one can indeed use the method of summation by parts to find the variance of the loss-at-issue random variable for a fully discrete life insurance po...
... by a task force of the American Academy of Actuaries (AAA 1997), Lee (2002), Mitchell and Sla... more ... by a task force of the American Academy of Actuaries (AAA 1997), Lee (2002), Mitchell and Slater (1996), Streiff and DiBiase (1999), and Tiong (2000a, b ... It remains to check that its derivative, with respect to u, is Gu(t). Note that the variable u appears five times in (8). Thus, the ...
... in the securities market in order to optimize, in a certain sense, his wealth at a fixed fu-t... more ... in the securities market in order to optimize, in a certain sense, his wealth at a fixed fu-ture time T(T 0). Because T represents the stra-tegic planning horizon of ... INVESTING FOR RETIREMENT: OPTIMAL CAPITAL GROWTH AND DYNAMIC ASSET ALLOCATION 45 ...
We present a geometric approach to studying greatest accuracy credibility theory. Our main tool i... more We present a geometric approach to studying greatest accuracy credibility theory. Our main tool is the concept of orthogonal projections. We show, for example, that to determine the Biihlmann credibility premium is to find the coefficients of the minimum-norm vector in an affine space spanned by certain orthogonal random variables. Our approach is illustrated by deriving various common credibility formulas. Several equivalent forms of the credibility factor Z are derived by means of similar triangles.
Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate ... more Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment
The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of th... more The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the...
For a general fully continuous life insurance model, the variance of the loss-at-issue random var... more For a general fully continuous life insurance model, the variance of the loss-at-issue random variable is the expectation of the square of the discounted value of the net amount at risk at the moment of death. In 1964 Jim Hickman gave an elementary and elegant derivation of this result by the method of integration by parts. One might expect that the method of summation by parts could be used to treat the fully discrete case. However, there are two difficulties. The summation-by-parts formula involves shifting an index, making it somewhat unwieldy. In the fully discrete case, the variance of the loss-at-issue random variable is more complicated; it is the expectation of the square of the discounted value of the net amount at risk at the end of the year of death times a survival probability factor. The purpose of this note is to show that one can indeed use the method of summation by parts to find the variance of the loss-at-issue random variable for a fully discrete life insurance po...
... by a task force of the American Academy of Actuaries (AAA 1997), Lee (2002), Mitchell and Sla... more ... by a task force of the American Academy of Actuaries (AAA 1997), Lee (2002), Mitchell and Slater (1996), Streiff and DiBiase (1999), and Tiong (2000a, b ... It remains to check that its derivative, with respect to u, is Gu(t). Note that the variable u appears five times in (8). Thus, the ...
... in the securities market in order to optimize, in a certain sense, his wealth at a fixed fu-t... more ... in the securities market in order to optimize, in a certain sense, his wealth at a fixed fu-ture time T(T 0). Because T represents the stra-tegic planning horizon of ... INVESTING FOR RETIREMENT: OPTIMAL CAPITAL GROWTH AND DYNAMIC ASSET ALLOCATION 45 ...
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