Chapter 1 Motivating Example

Mr. (55) is an actuary – a grumpy old actuary: He dislikes all his heirs, so when he dies he “will consider” his remaining savings wasted. He just celebrated his 55th birthday and decides to invest $S_0= \$100,000$ to secure his future financial situation; he picks his 70th birthday as the pay-out year. Since he has taken all actuarial exams, he knows how to calculate his expected payoff.

His first thought is to put all his money in a savings account, where he earns an annual interest rate of $i=2.5\%$. Thus, after 15 years he would have \[ S_{15} = S_0 \,\left(1+i\right)^{15} = \$100,000 \cdot 1.025^{15} = \$144,829.80. \] The probability to survive until the age of 70 when being 55 now is approximatively $0.75$, so the expected amount at his disposal is \[\begin{eqnarray*} p_{\text{alive}} \,S_{15} + p_{\text{dead}} \, 0 &=& p_{\text{alive}} \, S_{15} + (1- p_{\text{alive}}) \, 0\\ &=& 0.75 \cdot \$144,829.80 + 0.25 \cdot 0\\ &=& \$108,622.40. \end{eqnarray*}\]

On his way to the bank he meets his colleagues $(55)^*$ and $(55)^{**}$ who are in the same situation as him. While sitting together for a coffee, they come up with the following idea: ``How about putting all our savings of $3S_0 = \$300,000$ in a shared account and those who survive share the accumulated capital $3S_{15} = \$434,489.40$ equally." (55) excuses himself and runs to his car, where he has a calculator, to decide whether this scheme would be beneficial for him. The expected amount at his disposal is now \[\begin{eqnarray*} && p_{\text{(55) is alive}} \,\text{pay-out} + p_{\text{(55) is dead}} \, 0 \\ &=& p_{\text{(55) is alive}} \left(p_{\text{$(55)^*$ and $(55)^{**}$ died}} \, \frac{3S_{15}}{1} + p_{\text{$(55)^*$ died but $(55)^{**}$ is alive}} \, \frac{3S_{15}}{2} \right.\\ &\;& \left. + p_{\text{$(55)^{**}$ died but $(55)^*$ is alive}} \,\frac{3S_{15}}{2} + p_{\text{both, $(55)^*$ and $(55)^{**}$, are alive}}\, \frac{3S_{15}}{3} \right) \end{eqnarray*}\] \[\begin{eqnarray*} &=& 0.75 \left( (1-0.75)(1-0.75) \frac{3S_{15}}{1} + (1-0.75) \,0.75 \, \frac{3S_{15}}{2} \right. \\ &\;& \left. + 0.75 \, (1-0.75) \, \frac{3S_{15}}{2} + 0.75\, 0.75\, \frac{3S_{15}}{3} \right)\\ &=& 0.984375 \, S_{15} = \$142,566.90. \end{eqnarray*}\]

But instead of returning to his colleagues, he wonders now how good he could do if he finds 4, 5 or even more persons to join the scheme. Say he finds $l_{55}$ people who are in the same situation as him and of which $l_{70}$ until the age of 70. The expected amount at his disposal will then be \[ p_{\text{(55) is alive}} \, \frac{l_{55} S_{15}}{l_{70}} + p_{\text{(55) is dead}}\, 0 = S_{15} \, 0.75 \, \frac{l_{55}}{l_{70}} \] and by the strong law of large numbers we have \[\begin{eqnarray*} \frac{l_{70}}{l_{55}} = \frac{1}{l_{55}} \sum_{k=1}^{l_{55}} \underbrace{1_{\left\{\text{Person k is alive}\right\}}}_{\text{iid Bernoulli random variables}} &\rightarrow& E\left[1_{\left\{\text{Person k is alive}\right\}}\right] \\ && = Pr(\text{Person 1 is alive}) = 0.75. \end{eqnarray*}\] Therefore, \[ S_{15} \, 0.75\, \frac{l_{55}}{l_{70}} \rightarrow S_{15} \, 0.75 \, \frac{1}{0.75} = S_{15}, \] and hence the expected discounted amount at his disposal is \[ \frac{S_{15}}{(1+i)^{15}} = S_0 = \$100,000. \] That is insurance, and clearly (55) should have known that.

This type of contract is called , and a combination of several endowment contracts is called . In contrast, contracts that pay upon death if the insured has heirs he wants to secure are called contacts.

In the example, $S_0$ is * the amount the insured pays initially, i.e. the (single upfront) ; * the expected discounted amount at the insured’s disposal. We will see that this is not a coincidence. However, note that this in general not the present value of the pay-out. In fact, in our example (55) either gets \[ \frac{S_{15}}{0.75} = \$ 193,106.4 > S_{15} \] with a present value of \[ \frac{\$ 258,037.65}{(1+i)^{15}} = \$ 258,037.65 \, v^{15} = \$ 133,333.33 > S_0 \] or nothing if he dies, i.e. the situation is not the same as in interest rate theory as we have non-deterministic payments (random variables).

This basic example communicates a good bit of the intuition behind this class: (Life) insurance is all about sharing risk. Doing the actuarial calculations requires information/models for future lifetimes, here the probability for (55) to survive. Chapter 2 will cover this in-depth. Afterwards, Chapters 3, 4, and 5 will use this information to analyze a variety of life-contingent payoffs.