EXAMPLE X.-What is the present value of a perpetual annuity which falls due in annual instalments of £20 eachthe rate of interest being 3 per cent.? If the instalments were £1 each, the present value would be The required present value, therefore, is £- X 20 I *035 I *035 EXAMPLE XI. What is the present value of a perpetual annuity which falls due in half-yearly instalments of £35 10s. each-the rate of interest being 4 per cent.? £ If the instalments were £1 each, the present value would be The required present value, therefore, is £ I *02125 I *02125 EXAMPLE XII.-What is the present value of a perpetuity which falls due in quarterly instalments of £144 12s. 6d. each -the rate of interest being 34 per cent.? If the instalments were £1 each, the present value would be I £ *009375 I The required present value, therefore, is £.00937 X144·625=144625=£15,426 138. 4d. *009375 316. To determine the present value of a "deferred" annuity: Find what the present value would be, if the annuity were immediate, (a) for the time which is to elapse before the annuity comes into possession, and (b) for the time which is to elapse before the annuity terminates; the difference between these two present values will be the present value required. Thus, in determining the present value of a deferred annuity which to come into possession 5 years hence, and then to continue for 25 years, we regard the annuity as immediate, and find its present values for 5 years and (5-+25=) 30 years, respectively; the difference between these two present values is evidently the present value of the deferred annuity. In practice, the answers to exercises like the preceding would be taken from an annuity table, which, at the present day, is always employed in the calculation of Contingent or Life annuities. The principles involved in the construction of Life-annuity tables cannot be fully entered into here. It may be observed, however, that the probable duration of the life of any particular person is determined, by what is called the "Doctrine of Probabilities," from such materials as are supplied by the records of births and deaths in different localities; those materials being considered in connexion with the person's age, health, habits, occupation, &c. For instance, it has been ascertained that, of 10,000 infants born alive in Great Britain and Ireland, only— on an average Taking, therefore, the case of an ordinarily healthy person 30 years old, we find that his chances of living to the age of So that ABSOLUTE CERTAINTY being represented by unitythe probability of his reaching the age of In order to understand the use which is made of such fractions as these, let us suppose that the person whose case we are considering purchases a life annuity-to be paid in annual instalments of £50 each. Now, if there were an absolute certainty of his being alive at the end of 10 years, the present value of the 10th instalment-the rate of interest being, say, 50 3 per cent.—would be £10310; but as the payment of this instalment is contingent upon his living to claim it, and as the probability of his completing his 40th year is represented by 5075, it is evident that the present value of the 10th instalment 5642 but) 5642 not £5310, but) ; of the 50 4397. 103205642 present value of the 20th instalment is £- X 30th instalment, £53643; &c. The sum of the present 103305642 values, found in this way, of all the instalments which the annuitant might possibly live to claim, constitutes the present value of the life annuity. The present value of an annuity falling due in annual instalments of 1 each, and continuing during the annuitant's life, is technically spoken of as the VALUE of that life. DIFFERENT SYSTEMS OF NOTATION. It is by no means improbable that, instead of the DECIMAL system of notation, we should have a In order to realize any of these systems, we have merely to return to the illustration employed at page 3, and suppose that each of the boys there referred to has exactly the number of fingers indicated by the "base" of the system. Thus, if the boys had only two fingers each, a finger held up by the 1st boy would represent one nut four &c.. So that, in a BINARY system, the values of the digit I would be— If the boys had only three fingers each, a finger held up by the So that, in a TERNARY system, the values of I would be If the boys had only four fingers each, a finger held up by the So that, in a QUATERNARY system, the values of I would be =B3 B2 It is unnecessary to proceed further with these illustrations. We see that, in any system similar in principle to the decimal, I would have the following values-if the base were represented by B: And by doubling, or trebling, or quadrupling, &c. these values, we should, of course, obtain the corresponding values of 2, of 3, or of 4, &c.-as the case may be. The number of different digits in any system would be less by I than the base of the system; the base indicating the number of different figures, and one of those figures being, in every instance, a cipher.* Thus, as the figures employed in the decimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9—so, the figures employed in a In a system having a higher base than 10, one or more new digits would be required. For instance: in an undenary. system [base eleven] there should be a character for ten; in a duodenary system [base=twelve] two new characters would be necessary-one for ten, and one for eleven; &c. It is obvious that, in any system, the removal of the units' point (we can no longer say "decimal" point) n places, to the right or left, would multiply or divide-as the case may bea combination of figures by B"; the base of the system being represented by B. * In the illustration at page 3, each boy is supposed, on finding all his fingers up, to "begin again"-being relieved (so to speak) by his neighbour, who puts up one finger. If, therefore, the boys were furnished with B fingers each, no boy would, at the close of the reckoning, have more than B-1 fingers up; so that, in a system of notation having B for base, the value of the highest digit would, in the units' place, be B-1♪ |