In 2014, the Society of Actuaries released the
RP-2014 mortality tables and the MP-2014 mortality improvement scale. 3 Companies that adopt
these updated assumptions may see their pension
liabilities increase substantially if the prior mortality assumption was outdated before adopting this
new mortality table. The communication challenge in this situation usually falls into the “skewed
curve” category—the adverse consequences of understating mortality improvement.
In the following sections, we move beyond the
definitions of longevity risk to examine some additional relevant factors in the understanding and
communication of longevity risk.
Mathematical Relationships Underlying
Different Notions of Longevity Risk
The prevailing view in understanding risk among
experts and practitioners is the financial economics view. Risk is represented as variability. There is
a simple mathematical expression that unifies the
individual and systematic longevity risks.
Assume the remaining lifetime of an individual
is a random variable that follows the probabilities
of death in a mortality table, and the mortality table itself is a collection of age-dependent random
variables that depend on estimated parameters.
Let L denote the remaining lifetime of an individual and
P denote the set of estimated parameters. An important
identity in the theory of mathematical probability is
Var [L] = E [Var [L | P]] + Var [ E [ L | P]],
where Var is the variance, E is the expected value and [ . | . ]
is the conditional expectation over a suitable sample space. 4
In words, this identity is commonly stated as
The total variability is equal to the mean of the
variability plus the variability of the mean.
Intuitively, the conditional expectation is the mathematical way of saying “holding certain variables constant.”
Unpacking this identity, we have
The mean of the variability (E [Var [L | P]]) is the
expected variability given mortality rates—i.e., the
individual longevity risk. With a large group, the ratio
of the variance to the mean inside the expected value is
reduced due to the law of large numbers. Thus, this is a
component of variability that can be managed by pooling.
The variability of the mean (Var [ E [ L | P]]) is
the variance of the life expectancy over uncertain
mortality rates—i.e., the systematic longevity risk. 5
This component cannot be managed by pooling.
Thus the total longevity risk is the sum of individual
longevity risk and systematic longevity risk. This formu-
lation gives the two types of longevity risk a more precise
mathematical definition and provides a mathematical
framework for calculating them. The calculation of variabil-
ity is fundamental to the pricing and hedging of longevity
risk with longevity swaps and other capital market solutions.
The above identity treats risk as variability (i.e., the “flat
curve” and the “skinny curve” pictures). However, the expected level of mortality is also a factor when we speak
of longevity risks (i.e., the “skewed curve” picture). This
happens when the mortality rates have been estimated incorrectly or when inadequate mortality assumptions are
used. In practice, the composition of a group under consideration will be different from the population from which
a mortality table or a mortality index is derived. This is
known as the basis risk.
Finally we have the possibility that the model specifying the dynamics of the mortality rates is itself inadequate.
This is known as the model risk.
Combination of Different Categories
of Longevity Risk
In practice, the longevity risk under consideration usually
does not belong to a single category, but is rather a combination of several categories discussed above. The four
categories of longevity risks provide a way to think about
Risk: Variability in an
individual’s life span
Pooled Longevity Risk:
Uncertainty in mortality
Additional Cost to a
Society or a Pension
System When Mortality
Adverse Consequences of
Living a Long Time