# Insurance and Investment

Planning a Scheme

The actual operation of a modern life insurance company is extremely complex, involving all the aids of modern technology and especially, of course, the computer. But the essential principle is simple. Let us assume that we were planning a scheme whereby 1,000 men all aged 45 would agree to pay into a common fund each year of their lives enough to pay out £111,000 to the dependants of any of them who died during any year. From Table 1 on p. 3, it is possible to determine fairly precisely just how many of the 1,000 will die in any year. In Year 1, on the basis of the mortality experience underlying that table, we could expect four to die, in Year 2 five and so on. So in Year 1, to provide the £14,000 necessary to meet the claims of the dependants of the unlucky four, each of the original 1,000 would have to pay £14. In Year 2 the remaining 996 would have to pay in £15.02 each to meet the £15,000 needed to meet the death claims for that year. But if we continue in this way, deriving the "premium" from the year's actual mortality, then by the time 15 years have passed there will be only 861 of the original 1,000 left. And each of them will have to pay in over £121 to meet the outgoings of the sixteenth year. Since their income would quite probably be falling just as the contributions rose, such a scheme would be extremely unattractive.

What the founders of life insurance discovered, however, was that with a lot of mathematical calculation and a little guesswork, they could work out a premium rate which each of the 1,000 would agree to pay throughout their lives in return for the guarantee that all claims would be met when they fell due. The graph in Fig. 1 shows how it works. The annual premium is far higher than the payments required in the early years of the "pay-as-you-go" scheme, because a reserve is being accumulated in the early years which will cover the excess of claims over premiums in the later ones. The fund is investing the surplus of the early years at interest to build up reserves which will meet future claims.

To devise such a system there are two principal calculations: the mortality rate and the interest rate. The first determines how much is likely to be required each year to meet claims. The second is needed to work out how much the premium can be reduced to allow for the effect of accumulation of reserves at interest. In the early years of life insurance, neither calculation could be made as precisely as they can be today. In fact, some early companies made substantial errors - the funds accumulated significant surpluses after meeting all the relevant claims, and various means had to be devised for returning this surplus to the policyholders. It might appear difficult to work out what proportion of the surplus was attributable to any class of policyholder if the company was continually taking on new policyholders of different ages, but in fact the actuaries rapidly developed ways of relating the value of the company's assets to its "liabilities", the latter being the amounts due on death at dates which could be predicted from mortality tables.