Calculating life expectancies

Life Insurance Products Valuation in R

Roel Verbelen, Ph.D.

Statistician, Finity Consulting

The curtate future lifetime

  • $K_x = \lfloor T_x \rfloor$, the number of whole years lived
    $\quad \,$ by $(x)$ in the future.

$$ \large \begin{aligned} Pr(K_x = k) &= Pr(k \leq T_x < k+1) \\ &= {}_kp_x \cdot q_{x+k} \\ &= {}_kp_x - {}_{k+1}p_x, \end{aligned} $$

  • Compute $Pr(K_{65} = 5) = {}_5p_{65} \cdot q_{70} = {}_{5}p_{65} - {}_{6}p_{65}$.
# 5-year deferred mortality probability of (65) 
prod(px[(65 + 1):(69 + 1)]) * qx[70 + 1]

0.02086664

# Alternatively
prod(px[(65 + 1):(69 + 1)]) - 
  prod(px[(65 + 1):(70 + 1)])

 0.02086664
Life Insurance Products Valuation in R

The curtate life expectancy

  • The expected value of $K_x$ is called the curtate life expectancy:

$$ \begin{aligned} E[K_x] &= \sum_{k=0}^{\infty} k \cdot Pr(K_x=k) \\ &= \sum_{k=0}^{\infty} k \cdot ({}_kp_x - {}_{k+1}p_x) \\ &= \ldots \\ &= \sum_{k=1}^{\infty} {}_kp_x. \end{aligned} $$

Life Insurance Products Valuation in R

The life expectancy of a superhero

 

Mr. Incredible is 35 years old and lives in Belgium.

As an independent superhero he needs to take care of his financial planning.

What is a good estimate of his curtate future lifetime?

Can you help?

Life Insurance Products Valuation in R

The life expectancy of a superhero in R

Compute $E[K_{35}]= \sum_{k=1}^{\infty} {}_kp_{35}$.

# one-year survival probabilities
head(px[(35 + 1):length(px)])

0.99883 0.99896 0.99902 0.99879 0.99887 
0.99824

# k-year survival probabilities of (35)
kp35 <- cumprod(px[(35 + 1):length(px)])
head(kp35)

0.9988300 0.9977912 0.9968134 0.9956072
0.9944822 0.9927319

# curtate expected future lifetime of (35)
sum(kp35)

43.53192
Life Insurance Products Valuation in R

The expected future lifetime

  • How to step from $E[K_x]$ to $E[T_x]$?

  • Thus: from the curtate to the complete life expectancy?

  • Use:

    $$ E[T_x] \approx E[K_x] + \frac{1}{2}, $$

Life Insurance Products Valuation in R

The life expectancy of a superhero in R

Compare $E[K_{35}]$ to $E[T_{35}]$.

# Curtate life expectancy of (35)
sum(kp35)

43.53192

# Complete life expectancy of (35)
ex <- life_table$ex
ex[35 + 1]

44.03
Life Insurance Products Valuation in R

Let's practice!

Life Insurance Products Valuation in R

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