APPENDIX THE SIXTH.
ON A QUESTION CONNECTED WITH THE VALUATION OF
THE ASSETS OF AN INSURANCE OFFICE.

IF an insurance office were about to close its doors, and to buy up all the policies of its members, the proof valuation would only require the assets to be expressed by the amount of money which they would actually produce at the time of valuation. In this case, the profit or surplus is properly expressed by A+ P— C, or the amount of assets increased by the present value of all the premiums, and diminished by that of all the claims.

But valuation is not usually made with reference to an immediate settlement ; but for the purpose of ascerwhat sum can be set apart as profit, and deto belong to existing policies, without anticipatory injustice to future members. The preceding formula, with allowance for expences of management, still repre_ sents the sum which may be called profit, provided that the stock belonging to the office can really be improved at the rate of interest assumed in the valuation. For the sufficiency of this stock to answer all demands depends upon its increasing at that rate of interest upon which the values of P and C were found.

Now, it generally happens, that the property of an insurance office consists of funds invested at different rates of interest, the consequence of which is, that there is no absolutely rigorous method of determining the profit, except by prospective calculation of the state of the office for every year of the tabular duration of the life of its youngest member. Supposing the insured to die precisely in the manner indicated in the table, and assigning the order in which the different principals are to be touched, when necessary, it is then possible to calculate the amount which will remain when all claims are paid. The present value of this amount (the speof stock in which it is to be left being known) is all that can be called profit at the time of the valuation. This process, however, is exceedingly laborious ; and, in all probability, where yearly valuations are made, the expence of making the calculation would be greater than the loss prevented by taking the more simple, but less accurate, method.
If money made only simple interest, and computations were performed accordingly, no difficulty would arise : for IS improving at r per pound, and IS' at r' per pound, is at all times equivalent to 1(S t S^{'), imat (Sr+S' r') - (S+S') per pound : so that all the different stocks might be considered as lying at one average rate of interest. Such, however, is not the case with compound interest.
To introduce the question in a simple form, let us sup-pose that all the stock of the office makes r per pound, the rate assumed in the valuation, except only one sum, II, which makes r' (less than r) per pound. If, then, this sum were set down as H in the item A, the profit would be overrated ; nor can we answer the question, how much should it be estimated at, without some reto the time at which H, with its accumulations, is to become necessary. If this will not be wanted for}

ON OFFICE VALUATIONS.    XXXIII
The best method of treating this case is to suppose H to stand for such a sum, that there will be no loss arising from a lower rate of interest before the next valuation. Accordingly, in the preceding formula, it must be the number of years intervening between two valuations. If such a process should give too little profit at one valuation, the same item will be larger in the next, and vice versil: so that there will be a contendency to correctness. If, for instance, the valuations be made yearly (for which this very circumstance is one reason among many), then H (1 + r^{') =(1 -F r) should be taken for H, and the existing polimay have the benefit when r' is greater than r.
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