GENERAL FORMULA. 165 2o. Any power (integral or fractional) of the amount of £1 for any rate and time is the amount of £1 at the given rate for the same multiple or fraction of the time. Thus the square of amount of £1 for 2 years is the amount for 4 years; and the cube root of amount of £1 for 6 years is the amount for 2 years. Hence the general rule is easily established: The compound amount of any principal for n years at a given rate = the principal x (amount of £1 for 1 year at given rate)". The convenience of this rule in connection with tables like those above is obvious. Also its use with the method of approximation reduces the labour of finding the compound amount for many years to a minimum. The principle involved is the squaring and resquaring, etc. of successive powers of the compound amount for £1. An example is given below. 14. To find the compound amount for a large number of years. First find the compound amount of £1 for the no. of years at given rate correct to the number of places required by the size of the given principal to ensure the final answer correct to 3 places. If principal is below 100-6 places are sufficient. If principal is between 100 and 1000—7 places are sufficient, and so on. The compound amount of £1 for n years = (amount of £1 for 1 year)". Thus the practical problem is to raise the amount of £1 for 1 year to a given power correct to 6, 7, or 8 decimal places, by means of the method of approximation for multiplication. Example. Find compound amount of £640 for 20 years at 4 p.c. Compound amount of £1 for 1 year = £1.04. .. we have to find (1.04)20 correct to 7 (or at any rate 6 places). C. Amt. of £1 for 20 years 2-1912397 20th power 046 13147438 876496 £1402-3934 (78. 10d.) Note. It has been discovered by observation that "70÷by given rate" gives the no. of years during which any sum will double itself very nearly. ANNUITY TABLES. 15. Annuities. 167 Annuities are calculated from tables which are made on these principles. 1o. The amount of £1 at compound interest is calculated for 1, 2, 3, etc. years at a given rate. 2o. The present worths of these amounts are placed in the second column. 3°. The amount of annuity, if allowed to accumulate at compound interest, is placed in the third column. 4°. The present worths of these accumulated annuities are placed in the 4th column. Model of Annuity Table, Annuity £1, Rate 5 p.c. The first column is formed by multiplying the previous row by 5, placing result two places to the right and adding in previous row as you proceed. When 8 places are reached commence to multiply at the 6th, making allowance for the 7th and adding result to 8th place of the previous row for new 8th place, and so on. The 2nd column is formed by dividing the previous present-worth by 105, putting result two places to the left and at the end using division-approximation. The 3rd column is formed by adding previous row to collinear amount (1st col.) of previous row. The 4th column is formed by adding previous row to collinear present-worth (2nd col.) of previous row. The chief questions are these: Q. 1. What annuity will a given sum buy at given rate for a given no. of years? Find present-worth of annuity of £1 for given time at given rate. Divide this into the given sum. Q. 2. What sum will buy a given annuity for a given no. of years at a given rate? Find present worth of annuity of £1 for given time at given rate. Multiply this by the given sum. Q. 3. What will a given annuity amount to, forborne for a given no. of years at a given rate p.c.? Find amount of £1 annuity for given no. of years at given rate p.c. Multiply by the given annuity. Q. 4. What is the present worth of an annuity forborne for a given no. of years at a given rate p.c.? Find present worth of £1 annuity for given no. of years at given rate p.c. Multiply by the given annuity. Example 1. What annuity will £1000 buy for 4 years at 5 p.c.? 3-5,4,5,9,5 | 1000-000 £282-012 1000.000 282.012 290.810 50.000 14.101 Example 2. What sum will buy an annuity of £100 for 5 years at 5 p.c.? Present worth of £1 ann. for 5 yrs. at 5 p.c.=£4·32947664. Example 3. What sum will buy an annuity of £282-012 for 4 yrs. at 5 p.c.? Present worth of £1 ann. for 4 yrs. at 5 p.c. = £3.545950. 282.012 595-453 846-036 141.006 11.280 1.410 •254 14 £1000.000 Example 4. What will annuity of £100 amount to forborne for 5 yrs. at 5 p.c.? Amount of annuity of £1= £5.52563125. .. amount of annuity of £100-£552·563 (11s. 31d.). Example 5. What will be present worth of annuity in Ex. 4? .. present worth of annuity of £100 = £432·948 (188. 11 d.). EXAMPLES. 1. Construct annuity tables for £1 at 3, 4, 3, 4 p.c. for 10 yrs. 2. Extend annuity table for £1 at 5 p.c. to 20 years. 3. What annuity can be bought for 20 yrs. at 5 p.c. with £1250? 4. What sum will buy £100 a year for 10 yrs. at 31 p.c.? 5. What is present worth of an annuity surrendered now of £356 which has still 8 yrs. to run at 4 p.c. ? 6. Find amount of annuity for £1 at 3 7. Find amount of annuity for £240 at p.c. for 10 yrs. 5 p.c. for 18 yrs. 8. Find amount of annuity for £1 at 2 9. Find present worth of an annuity for £100 at 57 p.c. for 4 yrs. 10. Find present worth of an annuity for £250 at 33 p.c. for 15 yrs. |