Tag Archives: Mean Residual Life

Expected Values for Insurance

Before I begin, please note: I hated this chapter.  If there are any errors please let me know asap!

A deductible d is an amount that is subtracted from an insurance claim.  If you have a $500 deductible on your car insurance, your insurance company will only pay damages incurred beyond $500.  We are interested in the following random variables: (X - d)_+ and (X\wedge d).


  1. Payment per Loss: (X-d)_+ = \left\{ \begin{array}{ll} X-d &\mbox{ if } X>d \\ 0 &\mbox{ otherwise} \end{array} \right.
  2. Limited Payment per Loss:  (X\wedge d) = \left\{ \begin{array}{ll} d &\mbox{ if } X>d \\ X &\mbox{ if } 0<X<d \\ 0 &\mbox{ otherwise} \end{array} \right.
Expected Values:
  1. \begin{array}{rll} E[(X-d)_+] &=& \displaystyle \int_{d}^{\infty}{(x-d)f(x)dx} \\ \\ &=& \displaystyle \int_{d}^{\infty}{S(x)dx} \end{array}
  2. \begin{array}{rll} E[(X\wedge d)] &=& \displaystyle \int_{0}^{d}{xf(x)dx +dS(x)} \\ \\ &=& \displaystyle \int_{0}^{d}{S(x)dx} \end{array}
We may also be interested in the payment per loss, given payment is incurred (payment per payment) X-d|X>d.
By definition:
E[X-d|X>d] = \displaystyle \frac{E[(X-d)_+]}{P(X>d)}
Since actuaries like to make things more complicated than they really are, we have special names for this expected value.  It is denoted by e_X(d) and is called mean excess loss in P&C insurance and \displaystyle {\mathop{e}\limits^{\circ}}_d is called mean residual life in life insurance.  Weishaus simplifies the notation by using the P&C notation without the random variable subscript.  I’ll use the same.
  1. For an exponential distribution,
    e(d) = \theta
  2. For a Pareto distribution,
    e(d) = \displaystyle \frac{\theta +d}{\alpha - 1}
  3. For a single parameter Pareto distribution,
    e(d) = \displaystyle \frac{d}{\alpha - 1}
Useful Relationships:
  1. \begin{array}{rll} E[X] &=& E[X\wedge d] + E[(X-d)_+] \\ &=& E[X\wedge d] + e(d)[1-F(d)] \end{array}
Actuary Speak (important for problem comprehension):
  1. The random variable (X-d)_+ is said to be shifted by d and censored.
  2. e(d) is called mean excess loss or mean residual life.
  3. The random variable X\wedge d can be called limited expected value, payment per loss with claims limit, and amount not paid due to deductible.  d can be called a claims limit or deductible depending on how it is used in the problem.
  4. If data is given for X with observed values and number of observations or probabilities, the data is called the empirical distribution.  Sometimes empirical distributions may be given for a problem, but you are still asked to assume an parametric distribution for X.

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Filed under Coverage Modifications, Deductibles, Limits, Probability, Severity Models