Page 176 - Compact Numerical Methods For Computers

One-dimensional problems 165
TABLE 13.1. Values found in example 13.2.

b f(b)
0 -1·00001
0·1 -1·00001
0·2 -1·00001
0·3 -1·00001
0·4 -1·00001
0·41 -1·00001
0·42 -0·999987
0·43 -0·999844
0·44 -0·998783
0·45 -0·990939
0·46 -0·932944
0·47 -0·505471
0·48 2·5972
0·49 22·8404
0·5 98·9994
0·6 199
0·7 199
0·8 199
0·9 199
1 · 0 199

Example 13.3. Actuarial calculations
The computation of the premium for a given insurance benefit or the level of
benefit for a given premium are also root-finding problems. To avoid over-
simplifying a real-world situation and thereby presenting a possibly misleading
image of what is a very difficult problem, consider the situation faced by some
enterprising burglars who wish to protect their income in case of arrest. In their
foresight, the criminals establish a cooperative fund into which each pays a
premium p every period that the scheme operates. If a burglar is arrested he is
paid a fixed benefit b. For simplicity, let the number of members of the scheme be
fixed at m. This can of course be altered to reflect the arrests and/or admission of
new members. The fund, in addition to moneys mp received as premiums in each
period, may borrow money at a rate r to meet its obligations and may also earn
b
money at rate r . The fund is started at some level f . The scheme, to operate
e
0
effectively, should attain some target f after T periods of operation. However, in
T
order to do this, it must set the premium p to offset the benefits n b in each period
i
i, where n is the number of arrests in this period. If historical data are available,
i
then these could be used to simulate the behaviour of the scheme. However,
historical data may reflect particular events whose timing may profoundly in-
fluence the profitability of the scheme. That is to say, the equation
F (p ,n) -f = 0 (13.32)
T
may require a very much higher premium to be satisfied if all the arrests occur