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A derived distribution useful in actuarial science
ResourceFunction["TimeShiftedDistribution"][xmin,dist] represents a distribution dist that has been truncated to lie above xmin and then translated so that the lowest outcome is 0. | |
ResourceFunction["TimeShiftedDistribution"][{xmin,ymin,…},dist] represents a multivariate truncation of the distribution dist, translated so the lowest outcome from the distribution is a vector of 0s. |
A BinomialDistribution, time-shifted so that its value must lie above 3 and be translated to the left by 3:
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The probability density (mass) function of that distribution:
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The distribution of the remaining life of a person aged 61, where longevity at birth is distributed according to a GompertzMakehamDistribution:
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The survival function of that time-shifted distribution:
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The mean of a time-shifted distribution of a ProductDistribution of two symbolic BetaDistributions:
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A random variable drawn from a time-shifted MultinormalDistribution:
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The numeric probability of a draw from a time-shifted CopulaDistribution falling into a particular set of values:
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Show survival functions of a BetaDistribution for different time-shift values:
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Show survival functions of a GompertzMakehamDistribution for different time-shift values:
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Some distributions, such as the ExponentialDistribution, remain unchanged after time shifting :
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Consider a life insurance product in which the time a person dies and the time a person lets their policy lapse are given by the ProductDistribution of a GompertzMakehamDistribution (for longevity) and an ExponentialDistribution (for lapse). The insured has neither died nor lapsed before age 61. Compute the probability that an insurer will have to pay a death benefit between k and k+1 years thereafter and that the insurance policy will not have lapsed by that time:
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Compute the actuarial present value of a death benefit of $1 for such a person, assuming the policy pays nothing if the policy lapses before death and assuming a 5% effective rate of interest:
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Compute the maximum amount the insurer would be willing to pay a 75-year-old who holds such a policy to let it lapse, again assuming a 5% effective rate of interest:
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