420. COR. 2. If the number of years be infinite, R" is infinite, and R- vanishes; therefore P= A or R-1' A Ex. If the annual rent of a freehold estate be 1£, what is its value, allowing 5 per cent. per ann. compound interest? P= In this case, A=1, R-1=0.05; therefore the Present Value A COR. 3. The Present Value of an Annuity of A£ payable m times per annum for n years, each of the payments being rate of interest, will be and p the annual m and if the interest also be payable q times a year, each payment of interest for every 1£ being, the Present Value will be COR. 4. If the annuity is to continue for ever, this Present Value becomes 421. The Present Value of an annuity, to commence at the expiration of p years, and to continue q years, is the difference between its present value for p+q years, and its present value for p years, COR. If the annuity commences after p years, and continues for ever, the Present Value will be AR-P Ex. What is the Present Value of an annuity of 1£, for 14 years, to commence at the expiration of 7 years, allowing 5 per cent. per ann. compound interest? hence the value of the annuity for fourteen years after the expiration of 7 is 7.03£, or 7 years' purchase, nearly. The preceding Article contains the whole Theory of the RENEWAL OF LEASES. 422. To determine the fine which ought to be paid for renewing any number of years lapsed in a lease. Let p+q be the number of years for which the lease was originally granted; p the number lapsed; and 4 the clear annual value of the estate, after deducting reserved rent (if any), taxes, and all other fixed annual charges. Then it is clear that the lessee has to purchase an annuity of A£ to commence at the expiration of q years, and to continue p years, the Present Value of which is the Present Value for p+q years - Present Value for q years, = A .{R~4_R−(P+9)}. Ex. In a lease of 21 years 7 years lapsed are to be renewed, the reserved rent is 10£, and the estate is really worth 150£ a year, what fine ought to be paid for the renewal, reckoning interest at 5 per cent.? In this case the lessee has to pay for an annuity of 140£ to commence at the end of 14 years and to continue 7 years; therefore the fine required is .. the required fine=2800×{0.50505-0·35895}, =2800×0.1461, =409£, very nearly. [Exercises Zs.] SCHOLIUM. 423. The method of determining the present value of an annuity at simple interest, given in Art. 414, has been decried by several eminent Arithmeticians, and in its stead a solution of the question has been proposed upon the following principle; "If the present value of each payment be determined separately, the sum of these values must be the value of the whole annuity." Let a be the value or price paid down for the annuity, a the yearly payment, n the number of years for which it is to be paid, r the interest of 1£ for one year. The present value of the first payment is a (Art. 402); the present value of the second pay ment, or of a£ to be paid at the end of two years, is These different conclusions arise from a circumstance which the opponents seem not to have attended to. According to the former solution, no part of the interest of the price paid down is employed in paying the annuity, till the principal is exhausted. Let the annuity be always paid out of the principal a as long as it lasts, and afterwards out of the interest which has accrued; then x, x-a, x-2a, x-3a, &c. are the sums in hand, during the first, second, third, fourth, &c. years, the interest arising from which ra, ræ-ra, rx -2ra, rx-3ra, &c. that is the whole in n-1 2 terest, is nræ−{1+2+3...(n−1)} ×ra, or, nrx-n. ra, which, together with the principal a, is equal to the sum of all the annuities; therefore According to the other calculation, part of the interest, as it arises, is employed in paying the annuity, but not the whole. Thus, the first payment is made by a part of the principal, and the interest of that part, which together amount to the annuity; and the other payments are made in the same manner; this is, in effect, allowing interest upon that part of the whole interest which is incorporated with the principal. According to either calculation, the seller has the advantage, since the whole or a part of the interest will remain at his disposal till the last annuity is paid off. If the whole interest, as it arises, be incorporated with the principal, and employed in paying the annuity, compound interest is, in effect, allowed upon the whole. Let a be the price paid for the annuity, n the number of years for which it is granted, and R=1£ together with its interest for one year. Then a in one year amounts to Ra, out of which the annuity being paid, Rx-a is the sum in hand at the end of the first year; Rx - Ra is the amount of this sum at the end of the second year, therefore Rx-Ra-a is the sum in hand at the end of the second year; in the same manner, Rx - R-1a - R-2a... - a is the sum left, after paying the last annuity, which ought to be nothing; hence 424. Chance, or Probability, has two meanings; the one a popular meaning, without any very distinct signification; the other a mathematical meaning, pointing out a real value existing in the circumstances. DEF. Most questions of probabilities will fall under one of two classes, called direct and inverse probabilities. A question of probability is termed direct, when, certain causes being given as existent, from which a certain event may proceed, the probability of that event happening is required. A question of probability is termed inverse, when, an event being given as existent, and proceeding from one of several causes, the probability of one proposed cause being the true one is required. Some more complex questions may partake of the nature of both kinds of probability. I. DIRECT PROBABILITIES. 425. If an event may take place in n different ways, and each of these be equally likely to happen, the probability that it 1 will take place in a specified way is properly represented by n certainty being represented by 1. Or, which is the same thing, if the value of certainty be 1, the value of the expectation that the event will happen in a specified way is 1 n For the sum of all the probabilities is certainty, or 1, because the event must take place in some one of the ways; and the probabilities are equal: therefore each of them is n 426. COR. If the value of certainty be a, the value of the expectation is a n But in the following Articles we suppose the value of certainty to be 1. 427. If an event may happen in a ways, and fail in b ways, any of these being equally probable, the chance of its happening The chance of its happening must, from the nature of the supposition, be to the chance of its failing, as a b; therefore the chance of its happening: chance of its happening together with the chance of its failing: a: a+b. And the event must either happen or fail; consequently the chance of its happening together with the chance of its failing is certainty. Hence the chance of its happening certainty (1) a: a+b; or the chance of its hap Also, since the chance of its happening together with the chance α of its failing is certainty, which is represented by 1, 1 that a+b' |