DISCOUNT. 408. Discount is defined to be "the abatement made for the payment of money before it becomes due", and although in ordinary business the quantity of such abatement is generally according to private contract, there is besides a true mathematical discount which affords exact justice both to the payer and to the receiver. It is clear that if A receives from B a sum of money n years before it is due, n being whole or fractional, A is benefited by the interest of that money for the time; and therefore, in justice, B ought to receive an abatement such, that the sum thus diminished, paid to A, would, if put out to interest until the proper time of payment arrives, amount to the sum due. This sum is called the Present Value of the debt. Hence if D be the discount, and V the present value, of a debt of £P due at the expiration of n years, V=P-D, and I would amount at the end of n years to P, i.e. V(1+nr)=P, reckoning simple interest. COR. 1. If n be sufficiently great that compound interest may be reckoned, then COR. 2. If I be the interest on £P for the given time, we shall have P+I=P(1+nr), or P(1+r)", according as simple or compound interest is reckoned; From this result we see that the discount on any sum is always less than the interest. COR. 3. Since D=P-V, if these be put out to interest for the time in question, we shall have amount of D=amount of P-amount of V =amount of P-P =interest on P. EQUATION OF PAYMENTS. 409. When various sums of money due at different times are to be paid, it may be required to know the time at which they may all be paid together, without injury to either debtor or creditor. To determine this time, which is called the Equated Time, it is clear that we must suppose the interest of the sums paid after they are due to be together equal to the discount of the sums paid before they are due, the debtor being entitled to discount for that which is paid before, and the creditor to interest for that which is paid after, it becomes due. 410. To find the equated time of payment of two sums due at different times, reckoning simple interest. Let P, p, be two sums due at the end of times T, t, respectively; r the rate of interest, and a the equated time; then supposing T>t, the interest of p for the time x-t must be equal to the discount of P for the time T−x, or 411. Since the above method does not furnish any simple Rule, and is more complicated as the number of payments is increased, another method is generally used, although incorrect, which is founded on the supposition that the interest of the sums paid after they are due should be equal to the interest (not the discount) of those which are paid before they are due. Thus, if P and p be the sums due at the end of times T and t, and the equated time required, p(x−t)r= P(T−x)r; Or, more generally, let P1, P2, P3, &c. be the sums due after the equated time, at the end of times T1, T2, T3, &c. and p1, P2, P3, &c. the sums due before, at the end of times t1, ta, ts, &c. then we have P1(x−t,) r+p2(x−t2)r+p ̧(x−t ̧)r+&c. =P1(T,−x)r+P2(T2−x)r+P ̧(T ̧−x)r+&c. 2 2 P,T,+P2T2+ P3T,+&c.+P11+ Pata + Pats+&c. and .. x= P+ P+P+&c.+p1+P2+P2+&c. which furnishes a simple rule easy of application. By this rule a small advantage is given to the payer, because he reckons on his side the interest, instead of the discount, of those sums which he pays before they are due, whilst the opposite side of the account is confined to strict accuracy; and it has been shewn in Art. 408, Cor. 2. that the interest of any sum is greater than the discount. 412. Another method of finding the Equated time is to find the Present Value of each payment, and make the sum of them equal to the Present Value of the sum of the several payments supposed due at the Equated time. Thus, if P, p, be due at the end of times T, t, respectively, r the rate of interest, and a be the equated time, from which simple equation with respect to a we get COR. If the quantities multiplied by r be neglected, since r is generally a very small fraction, we have 413. ANNUITIES. To find the Amount of an annuity or pension left unpaid any number of years, allowing simple interest upon each sum or pension from the time it becomes due. Let A be the annuity; then at the end of the first year A becomes due, and at the end of the second year the interest of the first annuity is rA (Art. 401); at the end of this year the principal becomes 24, therefore the interest due at the end of the third year is 2rA; in the same manner, the interest due at the end of the fourth year is 3r▲; &c. Hence the whole interest at the end of n years is rA+2rA+3rA......+n-1.rA = n. n-1 rA (Art. 282); and the sum of the annuities is nA, therefore the whole amount 414. Required the Present Value of an annuity which is to continue a certain number of years, allowing simple interest for the money. Let P be the present value; then if P, and the annuity, at the same rate of interest, amount to the same sum, they are upon the whole of equal value. The amount of P in n years is P+ Pnr (Art. 401); and the amount of the annuity in the same From this equation, any three of the four quantities P, A, n, r being given, the other may be found. 415. COR. Let n be infinite, then P= nA an infinite quan tity; therefore for a finite annuity to continue for ever, an infinite sum ought, according to this calculation, to be paid; a conclusion which shews the necessity of estimating the Present Value of an annuity upon different principles. 416. To find the Amount of an annuity in any number of years, at compound interest. Let A be the annuity, or sum due at the end of the first year; then 1 : R :: A : RA, its amount at the end of the second year; therefore A+RA is the sum due at the end of the second year; in the same manner, 1 : R :: (1+ R)×A : (R+R2)×A, the amount of the two payments at the end of the third year; and (1+R+R)-A is the whole sum due at the end of the third in the same manner, year; (1+R+ R2... + R* −1)×A is the sum due at the end of n years, that is, the amount required R"-1 M= XA. 417. COR. 1. From this equation, any three of the quantities being given, the fourth may be found. COR. 2. If interest be paid q times per annum, and be each pay 9 ment per 1£, the amount of the annuity in n years, reckoning compound interest, will be COR. 3. If the annuity (4) be payable m times per annum, each of the payments being A " m amount in n years will be and ρ be the annual rate of interest, the and if the interest also be payable q times a year, each payment of interest for every 1£ being, A m this amount becomes (1+2)-1 418. To find the Present Value of an annuity to be paid for n years, allowing compound interest. Let P be the Present Value, A the annuity; then since PR" is the amount of P in n years, and -xA the amount of A in R2-1 PR"= -×A, .. P= --A, or 419. COR. 1. Any three of the quantities P, A, R, n being given, the fourth may be found. |