TABLE. The present worth of an annuity of $1 for any number of years from 1 to 25. YR. 2% 3% 4% 5% 6% 1$.980392 $ .970874 $ .961538 $.952381 $ .943396 2 1.941561 1.913470 1.886095 1.859410 1.833393 3 2.883883 2.828611 2.775091 2.723248 2.673012 4 3.807729 3.717098 3.629895 3.545951 3.465106 5 4.713460 4.579707 4.451822 4.329477 4.212364 6 5.601431 5.417191 5.242137 5.075692 4.917324 76.471991 6.230283 6.002055 5.786373 5.582381 8 7.325481 7.019692 6.732745 6.463213 6.209794 9 8.162237 7.786109 7.435332 7.107822 6.801692 10 8.982585 8.530203 8.110896 7.721735 7.360087 11 9.786848 9.252624 8.760477 8.306414 7.886875 12 10.575341 9.954004 9.385074 8.863252 8.383844 13 11.348374 10.634955 9.985648 9.393573 8.852683 14 12.106249 11.296073 10.563123 9.898641 9.294984 15 12.849264 11.937935 11.118387 10.379658 9.712249 16 13.577709 12.561102 11.652296 10.837770 10.105895 17 14.291872 13.166118 12.165669 11.274066 10.477260 18 14.992031 13.753513 12.659297 11.689587 10.827603 19 15.678462 14.323799 13.133939 12.085321 11.158116 20 16.351433 14.877475 13.590326 12.462210 11.469921 21 17.012269 15.415024 14.029160 12.821153 11.764077 22 17.658048 15.936917 14.451115 13.163003 12.041582 23 18.292204 16.443608 14.856842 13.488574 12.303379 24 18.913926 16.935542 15.246963 13.798642 12.550358 25 19.523456 17.413148 15.622080 14.093945 12.783356 To find the amount of an annuity. 1. What is the amount of an annuity of $500, at 5% compound interest, which has remained unpaid for 5 years? $1.055-1 = × 500 $2762.815, the amount required. .05 ANALYSIS.—The amounts of the 5 payments form a series in geometrical progression, of which $500 is the first term, the amount of $1 for one year ($1.05) is the ratio, and 5, the number of years, is the number of terms. The amount of the annuity, therefore, is the sum of the series. By using the tables showing the amount of an annuity of $1, the work is very much diminished: thus, the amount of $1 at 5% for 5 years=$5.525631, which being multiplied by 500 gives $2762.815 as the value required. RULE. Multiply the amount of $1 for the given time, as found in the table, by the number denoting the annuity, and the product will be the required amount. 2. What is the value of an annuity of $1200 which has remained unpaid for 10 years, at 5%? 3. $120 a year invested in a loan association which earns 6% compound interest will amount to how much in 9 years? 4. What is the amount of an annuity of $750, at 3%, which has remained unpaid for 9 years? 5. If a man deposits $50 a year in a bank that pays 5% compound interest, how much will he save in 20 years? 6. If an annuity of $125 remains unpaid for 10 years, what sum will discharge the debt with compound interest at 6% ? To find the present worth of an annuity. 1. What is the present worth of an annuity of $500, which is to continue for 5 years, interest compounded at 5%? = $4.329477 x 500 $2164.7385, the present worth required. ANALYSIS.-We find in the table, the present worth of an annuity of $1 for five years, and multiply by the number denoting the value of the given annuity (500), thus obtaining $2164.7385 as the required present worth. RULE. Multiply the present worth of an annuity of $1 for the given time, as found in the table, by the number denoting the annuity, and the product will be the required present worth. 2. What is the present worth of an annuity of $150, for 10 years, at 6% compound interest? 3. How much should be paid for an annuity of $50 a year, for 20 years, at 5% compound interest? 4. I wish to secure the payment of $100 a year for 15 years; what sum must I pay, if interest is compounded at 4%? To find the value of a perpetuity. 1. What is the value of a ground-rent of $60, interest at 6%? $60÷.06 = $1000, the value required. ANALYSIS. To produce $.06 requires $1, and to produce $60 will require as many dollars as .06 is contained times in 60, or $1000. RULE. Divide the given annuity by the number denoting the rate of interest, expressed decimally. 2. What is the value of a perpetual income of $1000 a year, interest at 9%? 3. Find the value of a ground-rent of $300, one-half of which is to be paid every six months, interest at 8%. 4. What sum will extinguish a ground-rent of $20 a year, interest at 10%? To find the annuity, the present worth, or the amount, the time, and the rate being given. 1. The present worth of an annuity, to continue for 5 years, at 5% compound interest, is $2164.7385; what is the annuity? = $2164.73854.329477 $500, the required annuity. ANALYSIS.-If the present worth of an annuity of $1 at the given rate is $4.329477, the required annuity will be as many dollars as 4.329477 is contained times in $2164.7385, or $500. RULE. Divide the given present worth by the present worth of $1, or the given amount by the amount of $1, for the given time and rate, and the quotient will be the required annuity. 2. The amount of a ground-rent, which has been forborne for 5 years, is $2762.815; what is the groundrent, interest being at 5%? 3. The present value of an annuity to be continued for 15 years, at 6% compound interest, is $9712.249; what is the annuity? To find the value of an annuity in reversion. 1. What is the present worth of an annuity of $60, to commence in 4 years, and to continue 4 years, compound interest, at 4%? ANALYSIS.—The present worth of an annuity of $1, payable for 8 years, at 4%, is $6.732745 3.629895 $3.102850 $3.102850 × 60=$186.171, the value of the reversion. RULE. Find the present worth of an annuity of $1 for the full time, also for the time during which payment is deferred; the difference of these present worths multiplied by the number which denotes the annuity will be the value of the reversion. 2. I have given the reversion of the rent of a house, worth $200 a year, to my son, to commence after my wife has had the use of it for 10 years, and to continue for 10 years. What is the present value of my son's reversion? 3. What is the value of a ground-rent of $500 per annum, to commence after 20 years? BUILDING ASSOCIATIONS. Building Associations are organized for the purpose of accumulating a fund to be loaned to the members, on approved security. The stockholders are thus enabled to purchase real estate or other property, and to invest their savings safely and speedily. The shares are estimated at a fixed sum, usually $200 each, and are paid for in monthly installments of $1 each. |