# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

A derived distribution useful in actuarial science

Contributed by:
Seth J. Chandler

ResourceFunction["TimeShiftedDistribution"][ represents a distribution | |

ResourceFunction["TimeShiftedDistribution"][{ represents a multivariate truncation of the distribution |

The translation works by subtracting *x*_{min} from the outputs so that the lowest outcome from the distribution is now 0.

A time-shifted distribution can be used to model the longevity of a person who had a certain longevity distribution at birth but who has survived to some attained age.

A BinomialDistribution, time-shifted so that its value must lie above 3 and be translated to the left by 3:

In[1]:= |

Out[1]= |

The probability density (mass) function of that distribution:

In[2]:= |

Out[3]= |

The distribution of the remaining life of a person aged 61, where longevity at birth is distributed according to a GompertzMakehamDistribution:

In[4]:= |

Out[4]= |

The survival function of that time-shifted distribution:

In[5]:= |

Out[5]= |

The mean of a time-shifted distribution of a ProductDistribution of two symbolic BetaDistributions:

In[6]:= |

Out[6]= |

A random variable drawn from a time-shifted MultinormalDistribution:

In[7]:= |

Out[7]= |

The numeric probability of a draw from a time-shifted CopulaDistribution falling into a particular set of values:

In[8]:= |

Out[8]= |

Show survival functions of a BetaDistribution for different time-shift values:

In[9]:= |

Out[9]= |

Show survival functions of a GompertzMakehamDistribution for different time-shift values:

In[10]:= |

Out[10]= |

Some distributions, such as the ExponentialDistribution, remain unchanged after time shifting :

In[11]:= |

Out[11]= |

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:

In[12]:= |

In[13]:= |

Out[13]= |

In[14]:= |

Out[14]= |

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:

In[15]:= |

Out[15]= |

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:

In[16]:= |

Out[16]= |

- 1.0.0 – 07 August 2019

This work is licensed under a Creative Commons Attribution 4.0 International License