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How to Value Your Rich Elderly Aunt

Philip Cooper, 26 November 2002, 10 am – 3.30 pm

SECTION 2

Probabilities of living and dying

Just to remind you of our original problem, we want to find out how much you might expect to get for selling your interest in Auntie’s Trust, so what we basically need to know is the probability that you outlive her. We are going to see how information from Life Tables can be used to work this out.

Now suppose you and Auntie both live from now until some future time, t years later, then Auntie dies very soon afterwards, before time t + dt years later (as shown on the time line above).

Then

P(you outlive Auntie, and she dies in interval between time, 60 + t,
and time, 60 + t + dt) =

P(you both live t more years) x P(Auntie dies in time 60 + t +dt)

assuming these are independent events, and

P(you both live t more years)
= P(you live to time 20 + t) x P(Auntie lives to time 60 + t)

again assuming these are independent events.

Using $\ell_x$ to denote the number of people alive at age x, we can build up a table like this, where we are looking at the number of people still alive from 10,000 births:

Age, x	$\ell_x$
20	98 946
30	97 645
40	96 500
...	...

So the number of deaths expected for people between 20 and 30 years of age is
98 946 - 97 645 = 1 301

Probability = Relative frequency of events

$\begin{displaymath} P(alive at age 20, survive to age 30) = \frac{\ell_{30}}{\ell_{20}} = \frac{97645}{98946} = 0.98685 \end{displaymath}$

Now generalising this, we have:

$\begin{displaymath} P(you both live t more years) \end{displaymath}$

$\begin{displaymath}= P(you live to time 20 + t) x P(Auntie lives to time 60 + t) \end{displaymath}$

$\begin{displaymath}= \frac{\ell_{20+t}}{\ell_{20}} x \frac{\ell_{60+t}}{\ell_{60}} \end{displaymath}$

$\begin{displaymath} P(you outlive Auntie, and she dies in time between time 60 + t, and time 60 + t + dt) \end{displaymath}$

$\begin{displaymath}= P(you both live t more years) x P(Auntie dies in interval between 60+t and 60 + t + dt) \end{displaymath}$

$\begin{displaymath}= \frac{\ell_{20+t}}{\ell_{20}} x \frac{\ell_{60+t}}{\ell_{60}} x P(Auntie dies in interval between 60+t and 60 + t + dt) \end{displaymath}$

Now $\mu_x$ , which you were looking at in the Life Tables just now, gives the rate of mortality measured very close to your xth birthday. To change the rate to a probability, we just need to multiply by the time interval.

$\begin{displaymath}P(Auntie dies in interval between 60+t and 60 + t + dt) = \mu_{60+t} dt \end{displaymath}$

and

$\begin{displaymath} P(you outlive Auntie, and she dies in interval between time 60 + t, and time 60 + t + dt) \end{displaymath}$

$\begin{displaymath}= P(you both live t more years) x P(Auntie dies in interval between time 60 + t, and 60 + t + dt) \end{displaymath}$

$\begin{displaymath}= \frac{\ell_{20+t}}{\ell_{20}} x \frac{\ell_{60+t}}{\ell_{60}} x \mu_{60+t} dt \end{displaymath}$

We can find values for all of these except dt from Life Tables for various values of t, and we could therefore plot $\frac{\ell_{20+t}}{\ell_{20}} x \frac{\ell_{60+t}}{\ell_{60}} x \mu_{60+t}$ against t.

Here is a table of the values we get by doing this, and a graph of those values.

t	males $\ell_{20+t}$	females $\ell_{60+t}$	females $\mu_{60+t}$	$\frac{\ell_{20+t}}{\ell_{20}} x \frac{\ell_{60+t}}{\ell_{60}} x \mu_{60+t}$
0	98 496	91 732	0.007 86	0.007 86
7	97 900	94 567	0.016 09	0.014 80
14	97 273	72 048	0.032 03	0.024 84
21	96 334	52 293	0.064 64	0.036 04
28	94 668	27 017	0.131 74	0.037 29
35	92 217	7 206	0.257 32	0.018 72
42	84 173	663	0.437 69	0.002 70
49	70 598	12	0.731 91	0.000 07

The total probability that Auntie dies before you is therefore the area under this curve.

[Have a go at Task 2 now]

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