This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1878 Excerpt: ...= 79.20 Present value required = difference = 910.80 The same result would have been obtained if we had first discounted $1000 certain at eight per cent., and then multiplied the avails by 0.99, which expresses the probability of repayment. In the case of annuities, or sums to be received periodically, if the risk is assumed to be constant, the contingent value may be found by computing the value as certain at the given rate, and then multiplying the result by the fraction which expresses the probability of receiving the sums stipulated. Example 3.--What is the value, subject to contingencies, of an annuity of $100 to be received at the end of each year during twenty years, so as to realize a net rate of six per cent, per annum upon the investment, the probability of loss each year being 1852 in 100,000? The present value of an annuity of $100 certain for twenty years, reckoning interest at 6 per cent, per annum, is found from Table XXV. to be 81146.99. The fraction which expresses the risk is, in the case supposed, 0.01852, and hence that which expresses the probability of receiving the several sums to mature is 0.98148. Hence we have: Present value required = $1146.99X0.98148 =$1125.75. The correctness of this rule is evident when it is remembered that the risk is not supposed to be dependent upon the elapsed time, but that it is constant. Hence the present value of each annual sum to be received will be multiplied by this constant factor, and then the total sum taken; or, what is equivalent, the total sum certain may be taken, and then multiplied by the given factor. If the risk were supposed to be proportional to the number of years elapsed, and the initial risk that assumed in the example just given, then the nominal rate of interest would first be obt...
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