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When Tx is a discrete random variable, the definition of the first moment is N E[Tx ] = x i Pr[Tx = x i ]. 3 provides a sense of how the expected remaining lifetime at birth E[T 0 ] varies throughout the world. 3 years. Despite this variation, life expectancy has been steadily improving throughout the world. 4 illustrates the trend that has been observed since 1950. Note that there is a difference between E[T 0 ] and E[T 1 ], for example. The former is life expectancy at birth, while the latter is life expectancy at the age of x = 1.

The median remaining lifetime (MRL) will be less than the expected remaining lifetime (ERL) in all cases. 8 Exponential Law of Mortality 45 average of a collection of positive numbers is always greater than the median of the same numbers. ” This could be either a median value M [T 0 ] or a mean value E[T 0 ], and the former is less than the latter. 8 Exponential Law of Mortality I’ve shown that modeling Tx using the normal distribution does not create a realistic approximation of the remaining lifetime.

3 Present Value of Consumption Now let’s examine the retirement period in terms of income, spending, and consumption. Imagine you have reached retirement with the nest egg FV(S, R, N ) and now intend to spend or consume C dollars per year from your accumulated savings. At the end of each year, you withdraw C dollars from your nest egg or investment account to finance your retirement needs. I will now compute the present value (where the word “present” refers to the exact date of your retirement) of your consumption and spending needs.

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