331. Time of Compounding Interest. Interest is said to be com pounded when it is added to the principal. In the problem on page 239 the interest is compounded at the end of each year or yearly. Interest may be compounded yearly, semiannually, or quarterly. The actual interest is greater the more frequently the interest is compounded. Thus the interest on $100 at 4% for one year when compounded annually is $4, and when compounded quarterly it is $4.06. 4% yearly interest when compounded quarterly is computed like 1% interest compounded yearly. Similarly 4% yearly interest compounded semiannually is computed the same as 2% interest compounded annually. PART OF COMPOUND INTEREST TABLE 332. Amount of $1 at compound interest compounded annually: On page 240 is a part of a compound interest table such as is used by bankers. Example. Find the amount of $500 at 5% compound interest for 6 years compounded annually. Solution. From the table the amount of $1.00 for this time and rate is 1.3401. Hence the amount of $500 is 500 X $1.34010 $670.050. = If the rate is 5% compounded semiannually for 6 years, we take 2% for 12 years. If the rate is 4% compounded quarterly for 5 years, we take 1% compounded annually for 20 years. WRITTEN EXERCISES Solve the first four exercises without using the table. For the 333. Sinking Fund Table. The table below shows the accumulation at compound interest of an annual investment of one dollar. Such tables are used by all who have much computing of this kind to do and especially by bankers. 6 7 8 9 6.434283 6.662462 10 6.898294 7.019152 7.142008 7.393838 8.214226 9.582795 9.802114 10.026564 10.491316 11.006107 11.288209 11.577893 12.180795 11.168715 11.807796 12.486351 12.841179 13.206787 13.971643 8.380014 8.549109 8.897468 11 12.412090 13.192030 14.025805 14.464032 14.917127 15.869941 12 13.680332 14.617790 15.626838 16.159913 16.712983 17.882138 13 14.973938 16.086324 17.291911 17.932109 18.598632 20.015066 14 16.293417 17.598914 19.023588 19.784054 20.578564 22.275970 15 17.639285 19.156881 20.824531 21.719337 22.657492 24.672528 16 19.012071 20.761588 22.697512 23.741707 24.840366 27.212880 17 20.412312 22.414435 24.645413 25.855084 27.132385 29.905653 18 21.840559 24.116868 26.671229 28.063562 29.539004 32.759992 19 23.297370 25.870374 28.778079 30.371423 32.065954 35.785591 20 24.783317 27.676486 30.969202 32.783137 34.719252 38.992727 21 26.298984 29.536789 33.247970 35.303378 37.505214 42.392298 22 27.844963 31.452884 35.617884 37.937030 40.430475 45.996822 23 29.421862 33.426470 38.082604 40.689196 43.501999 49.815570 24 31.030300 35.459264 40.645909 43.565210 46.727099 53.864517 25 32.670906 37.553042 43.311745 46.570645 50.113454 58.156383 The payment at a future date of an indebtedness such as municipal bonds is frequently provided for by what is called a sinking fund. Thus, to pay $50,000 at the end of twenty years, a certain sum may be invested at compound interest each year. The above table shows that one dollar invested each year at 4% compound interest will amount to $30.969202 in twenty years. Example 1. What amount must be invested yearly at 4% interest compounded yearly to amount to $100,000 in 16 years? Solution. By the table $1.00 invested each year at 4% amounts to $22.697512 in 16 years. Hence the required yearly investment is 100,000 22.697512 Example 2. What amount must be invested semiannually at 4% interest compounded semiannually to amount to $75,000 in 12 years? Suggestion. The required amount is the same as the annual investment at 2% compounded yearly for 24 years. WRITTEN EXERCISES In the following, find the investment required on the dates on which the interest is compounded: |