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COMPOUND INTEREST AND ANNUITY TABLES
8 Percent
Present Future Increasing Decreasing 1 1.08 0.926 1.080 0.926 1.000 0.926 0. 2 1.17 0.857 0.561 1.783 2.080 2.641 2. 3 1.26 0.794 0.388 2.577 3.246 5.022 5. 4 1.36 0.735 0.302 3.312 4.506 7.962 8. 5 1.47 0.681 0.250 3.993 5.867 11.365 12. 6 1.59 0.630 0.216 4.623 7.336 15.146 17. 7 1.71 0.583 0.192 5.206 8.923 19.231 22. 8 1.85 0.540 0.174 5.747 10.637 23.553 28. 9 2.00 0.500 0.160 6.247 12.488 28.055 34. 10 2.16 0.463 0.149 6.710 14.487 32.687 41. 11 2.33 0.429 0.140 7.139 16.645 37.405 48. 12 2.52 0.397 0.133 7.536 18.977 42.170 55. 13 2.72 0.368 0.127 7.904 21.495 46.950 63. 14 2.94 0.340 0.121 8.244 24.215 51.717 71. 15 3.17 0.315 0.117 8.559 27.152 56.445 80. 20 4.66 0.215 0.102 9.818 45.762 78.908 127. 50 46.90 0.021 0.082 12.233 573.770 151.826 472. 100 2,199.76 0.000 0.080 12.494 27,484.516 168.105 1,093.
VALUE OF AN ANNUITY - ONE PER YEAR
PRESENT VALUE OF AN ANNUITY
COMPOUND INTEREST AND ANNUITY TABLES
COM- POUND
NO. OF YEARS HENCE
PRESENT VALUE OF $1.
AMORTIZ ATION
Interest and annuity tables provide a reference to enable the user to properly account for the effects of interest and time in making an economic analysis. The basic principles of the time value of money, and the use of interest factors in making comparisons between values that occur at different points in time are presented.
Interest and annuity problems have four elements in common: (a) an amount, (b) an interest rate, (c) a term, and (d) a payment. If any three of these elements are known, then the fourth can be derived from the tables.
Procedures for discounting future benefits and costs or otherwise converting benefits and costs to a common time basis are also presented.
BASIC DEFINITIONS
Value In economics, value represents any quantity expressing the worth of
something. In resource development projects, value is used to express benefits arising from effects of project measures, or it could be the cost for providing such measures.
Number of Years Hence This is the number of periods (years, months, or days) in which calculations
are considered. There may be many conditions which influence this determination: (1) a benefit may last a year or indefinitely (perpetuity), (2) the measures may have a short or long useful life, (3) the period of evaluation may be set by policy, (4) an individual may want to recover his costs in a certain time period, or (5) costs or returns may occur over varying time periods or at varying rates for the same period.
Annuity Annuity is a series of equal payments made at equal intervals of time. The
most common type of annuity is our paychecks, at least those that meet the equal payment requirement. Annuity may be a benefit (to those receiving equal sums of money) or a cost (to those making the payment).
Interest Interest is economic rent of money. When money is borrowed, the amount
borrowed must be repaid along with a use charge called interest. Or, said another way, interest is money paid for the use of money. The appropriate rate of interest will depend upon the situation or the reason for the analysis. Demand, time, and risk (includes inflation) determine the rate of interest charged or paid in commercial lending establishments. If personal money is used in lieu of borrowed or lent funds, an opportunity cost^1 should be taken into consideration. Interest rate is expressed as a percent of the principal amount and is understood to be an interest rate per year. There are two kinds of interest-- simple and compound
Simple Interest Simple interest is rent on the principle amount only. If $100 is loaned or
borrowed and a year later $108 is repaid, the $8 is interest and, since it was for one year, the interest rate is 8 percent. The amount of interest is computed by the following formula: i= (p)(r)(n), where i = interest, p = principle ($100), r = periodic interest rate (8%), and n = number of periods (1 year). i = ($100)(.08)(1) = $8.
Compound Interest Compound interest is interest that is earned for one period and immediately
added to the principle, yielding a larger principle on which interest is computed for the following period. This means the accrued unpaid interest is actually converted to additional principle. Problem: What will $500 grow to in 5 years at 8 percent interest? Solution: ($500)(1.46933^2 ) = $734. The following represents the compound interest factor Formula: (1 + i) n , where n is the number of periods, i is the periodic rate of interest, and 1 represents one dollar since the formula results in a factor that is multiplied by the principle dollar amount. NOTE: Compound interest factors are not shown by column heading in the tables, but the same answer can be obtained by dividing the appropriate "present value of 1" factor since the present value of 1 factor is the reciprocal of the compound interest factor. Using the preceding problem:
(^1) 0pportunity cost is the return forgone from the most likely alternative use of money. (^2) Compound interest, 5 years hence, and 8 percent interest (not shown in tables).
Present Value of an Present value of an annuity of 1 per year also referred to as constant
Annuity of 1 Per Year annuity, present worth of an annuity, or capitalization factors.
This factor represents the present value or worth of a series of equal deposits over a period of time. It tells us what an annual deposit of $1.00 is worth today. If a fixed sum is to be deposited or earned annually for "n" years, this factor will determine the present worth of those deposits or earnings. Problem: If $600 will be placed in your savings account each year for 10 years, what is the present worth of the total amount or if you want to give $600 per year to someone for 10 years what sum would be required at present? The interest rate is 8 percent. Solution: $600 x 6. 71008 2 = $4, This is the present value of receiving $600 per year for 10 years or it is the amount that would need to be deposited today to make annual withdrawals of $600 for 10 years. The present value of an annuity of 1 per year factor is expressed as follows: (1 + i) n
1 (1 + i) n
NOTES: a. The factor is the reciprocal of the "amortization" factor. Therefore, the same answer can be obtained by dividing by the amortization factor: Solution: $600 /. 1 = $4,
b. "Amount of an annuity of 1 per year" multiplied by "present value of 1" = "present value of an annuity of 1 per year."
Amount of an Amount of an Annuity of 1 per year is also called "accumulation of
Annuity of 1 Per Year an annuity.” As stated earlier an annuity is a sequence of equal payments
made at uniform intervals with each payment earning compound interest during it’s respective earning term. The amount of an annuity of 1 per year factor shows how much an annuity, invested each year, will grow over a period of years. It can also show how much it is worth to provide protection against losing the opportunity of investing an annuity each year. For example, preventing erosion that causes a loss of net income of $25 per year for 50 years has an accumulated value of $14,344 at the end of 50 years at 8 percent interest (25 x 573.77016^1 ). The factor for an amount of an annuity of 1 per year is expressed as follows:
(1 + 1) n
1 Another example is a person establishing a retirement reserve. If $500 is placed in a savings account, each year earning 8 percent, what is the amount of the reserve at the end of 20 years? Solution: ($500)(45. 76196^2 ) = $22,
(^1) Amount of an annuity of 1 per year, SO years hence, and 8 percent interest. (^2) Amount of an annuity of 1 per year, 20 years, 8 percent interest
Sinking Fund This factor is used to determine what size annual deposit will be required to
accumulate to a certain amount (given) in a certain number (given) of years at compound interest. Problem: If $25,000 is needed to meet a term note due in 10 years, what amount will need to be deposited each year at 8 percent compound interest to reach the goal?
Solution: ($25,000)(.069029^1 ) = $1, The sinking fund factor is expressed as: Sinking Fund = 1 (1 + i) n -l NOTE: The sinking fund factor is not shown by column heading in the tables but the same answer can be obtained by dividing by the appropriate "amount of an annuity of 1 per year" factor since the amount of an annuity of 1 per year factor is the reciprocal of the sinking fund factor. Solution: $25,000 / 14.48656^2 = $1,
Present Value of an This factor shows how much something is presently worth that will provide
Increasing Annuity increasing sums of money over a period of years.
It should be noted that to meet the definition of an annuity (equal payments at equal intervals) the increases must be uniform. For example, seeding a field to grass may eventually be worth $18 per acre in increased net income, but it will take 6 years before this value is realized. The annuity is increasing $3 per year (18 / 6) so the definition of an annuity is satisfied. The total annuity we will receive is not level or the same each year since
We will receive $3 the first year, $6 the second year, and $9 the third year until at the end of the sixth year the $18 per year will be realized. At 8 percent interest the present value or present worth of something that will provide this amount of flow or build up of funds is $45.44 ($3 x 15.14615^3 ).
The factor for "present value of an increasing annuity" is expressed in the following formula:
(1 + i) n + 1
- (1+ I) – n(i) (1 + i) n (i) 2
Present Value of a This factor tells us how much something is presently worth that will provide
Decreasing Annuity an annuity that gets less each year.
The decrease must be uniform to meet the definition of an annuity. For example, a sediment pool will return $1,000 recreation benefits the first year but at the end of 40 years, because of sediment accumulation, it will have no value. The flow of funds is decreasing uniformly at $25 per year ($1,000 / 40 years). Therefore, the basic definition of annuity is met. Benefits would be $1,000 the first year, $975 the second year, $950 the third year, etc., until at the end of 40 years there would be no returns.
(^1) Sinking fund factor 10 years hence, 8 percent interest (not shown in tables). (^2) Amount of an annuity of 1 per year, 8 percent interest, 10 years hence. (^3) Present value of an increasing annuity, 6 years hence, 8 percent interest.
(600)(9.81815^1 ) = $5,
Step 2 - Discount the benefits that are delayed 30 years. Multiply the $5, by the discount factor "present value of 1" corresponding to the length of delay (30 years). ($5.891)(.09938^2 ) = $ Step 3 - Determine the annual equivalent value of the capital sum. Multiply the $585 by the "amortization" factor corresponding to the length of the evaluation period (50 years). ($585)(. 1 ) = $ Conclusion: The average annual value is $48.
Clue 3 Situation: A return will begin to build from $0 in year 0 to $600 in 20 years
then cease. What is the average annual benefit at 8 percent interest for a 50-year evaluation period?
Step 1 - To meet the definition of an annuity where values are uniformly increasing, divide the ending value reached by the number of years to reach the value. $ 20 = $30 per year
Step 2 - Determine present worth of the income stream. Multiply the $30 per year by the "present value of an increasing annuity" corresponding to the number of years the return is received (20 years). ($30) (78.90794^3 ) = $2,
Step 3 -Determine annual equivalent value of the income stream. Multiply the $2,367 by the “amortization" factor corresponding to the length of the evaluation period (5O years).
($2.367)(.08174^4 ) = $
Conclusion: The average annual benefit is $193.
Clue 4 Situation: A return will begin to build from $0 in year 30 to $600 in year 50.
What is the average annual benefit at 8 percent interest for a 50-year evaluation period?
(^1) Present. value of an annuity of 1 per year, 20 years, and 8 percent interest. (^2) Present value of 1, 30 years, 8 percent interest. (^3) Present value of an increasing annuity of 1 per year, 20 years, -8 percent interest (^4) Amortization. 50 years, 8 percent interest.
0 20 Years 50
$
0 30 Years 50
$
Step 1 - To meet the definition of an annuity where values are uniformly increasing, divide the ending value reached by the number of years to reach the value. $ 20 = $30 per year Step 2 - Determine present worth of the income stream beginning in year
- Multiply the $30 per year by the "present value of an increasing annuity" corresponding to the number of years the return is received (20 years). ($30)(78.90794^1 ) = $2, Step 3 -Discount the income stream that is delayed 30 years. Multiply the $2,367 by the discount factor (present value of 1) corresponding to the length of delay (30 years). ($2,367)(.09938^2 ) = $ Step 4 - Determine the annual equivalent value of the income stream. Multiply the $235 by the "amortization” factor corresponding to the length of the evaluation period (50 years).
($235)(. 1 ) = $ Conclusion: The average annual benefit is $
Clue 5 Situation: A present return of $600 will uniformly decrease to $0 at year 20.
What is the average annual value of 8 percent interest rate for a 50-year evaluation period?
Step 1 -To meet the definition of an annuity where values are uniformly decreasing, divide the beginning value by the number of years to reach $0. $ 20 = $30 per year
Step 2 - Determine the present worth of the income stream. Multiply the $ per year by the "present value of a decreasing annuity" corresponding to the number of years the return is to be received (20 years).
($30)(127.27316^3 ) = $3,
Step 3 - Determine the annual equivalent value of the income stream. Multiply the $3,818 by the "amortization" factor corresponding to the length of the evaluation period (50 years).
($3.818)(.08174^4 ) = $
Conclusion: The average annual benefit is $312.
(^1) Present value of an increasing annuity, 20 years, 8 percent interest. (^2) Present value of 1, 30 years, 8 percent interest (^3) Present value of a decreasing annuity, 20 years, 8 percent interest. (^4) Amortization, 50 years, 8 percent interest.
0 20 Years 50
$
Example Problems
The following problems illustrate the use of the compound interest and annuity tables. The purpose of the problems is to illustrate the principles used in applying each of the annuity factors described in the previous section
Problem 1 A net return will increase in uniform amounts to $3,000 for a 15-year period.
Thereafter, through the remaining 35 years of the evaluation period, the benefits will remain constant at 8 percent interest. Determine the annual equivalent value over the 50-year evaluation period. First diagram the problem.
Solution: a. $3,000 / 15 = $200 per year increase ($200)(56.44514^1 ) = $11,
b. ($3,000)(.31524^2 )(11.65457^3 ) = $11,
Annual Equivalent Value: $11,289 + $11,022(/ $22, ($22,311)(.0817^4 ) = $1, Shortcut -Table A (page 14) provides straight-line lag discount factors that can be used directly for selected percents of interest. The illustration of the shortcut method in solving the above problem is as follows: ($3,000)(0.608^5 ) = $1,
(^1) Present value of an increasing annuity, 15 years hence, 8 percent interest (^2) Present value of 1I, 15 years hence, 8 percent interest (^3) Present value of an annuity of 1 per year, 35 years hence, 8 percent interest. (^4) Amortization, 50 years, 8 percent interest. (^5) Table I, page 14,15 year lag, 50-year evaluation period, and 8 percent interest.
$
0 15 Years 50
(a) (b)
Problem 2 L and use will become more intensive in a flood prevention-drainage project
as the operators accept and utilize the protection made possible by the project. Presently net benefits per acre are $50. They are expected to uniformly increase to $95 per acre in 20 years. Thence, to $149 by the end of 50 years and then remain uniform for the remainder of the project life. Determine the average value per acre if the interest rate is 8 percent and the evaluation period is 100 years.
Diagram the problem:
Solution: a. ($50)(12. 1 ) = $ b. $95 -$50 = $ 45 / 20^2 = $2.25 per year ($2. 25)(78.90794^3 ) = $ c. ($45)(12.47351^4 )(.21455^5 ) = $
d. $149 -$95 = $ 54 / 03^6 = $1.80 per year ($1.80)(114.71358^7 )(. 5 ) = $ e. ($54)(12.23348^8 )(.02132^9 ) = $
The present value of receiving the above benefits is $ (625 + 178 + 120 + 44 + 14). The average annual value per acre is $78.52 (981 x .08004^10 )
(^1) Present value of an annuity of 1 per year, 100 years, and 8 percent interest. (^2) Necessary to meet the definition of an annuity. (^3) Present value of an increasing annuity, 20 years, 8 percent interest. 4 Present value of an annuity of 1 per year, 80 years, and 8 percent interest. (^5) Present value of 1, 20 years, 8 percent interest. (^6) Necessary to meet the definition of an annuity, (50 -20) = 30 years. (^7) Present value of an increasing annuity, 30 years, 8 percent interest. (^8) Present value of an annuity of 1 per year, 50 years, and 8 percent interest. (^9) Present value of 1, 50 years, 8 percent interest. (^10) Amortization, 100 years, 8 percent interest.
0 50 Years 100
(a)
(b) (c)
(d) (e) $
Problem 5 This problem combines all of the previous principles into one example.
Assume this problem involved benefits that would accrue in the following manner: begin with $300; grow to $800 in 5 years; grow to $1500 by year 15; remain constant until year 25, then decrease to $800 by year 35; decrease to $300 by year 50 and remain constant to year 100.
To solve this complicated problem, first break the problem into segments that can be easily computed.
a. Is a constant annuity^1 for the duration of the evaluation period. b. Is a buildup period or increasing annuity for 5 years with no lag or delay. c. Is a constant annuity 1 for the duration of the evaluation period. d. Is a decreasing annuity for 15 years delayed 35 years. e. Is an increasing annuity for 10 years delayed 5 years. f. Is a constant annuity 1 for 10 years delayed 15 years. g. Is a decreasing annuity for 10 years delayed 25 years.
Dollar values listed in the problem and the factors from the tables are multiplied together. The values for (a) through (g) are then added together and amortized over the evaluation period to determine the average annual values. Remember where increasing and decreasing annuities are involved, you must divide the dollar value by the number of years, to get an annual rate of increase or decrease.
(^1) Constant annuity is referred to in the tables as "present value of an annuity of 1 per year."
0 5 15 25 35 Years 50 100
$
(a)
(b) (c)^ (d)
(e) (f)^ (g
Solution: The average annual value at 8 percent interest is as follows:
a. ($300)(12.49432^1 ) = $3,
b. $800 - $300 = $
$500 / 5^2 = $
($100)(11.36514^3 ) = $1,
c. ($500)(11.25778^4 )(68058^5 ) = $3,
d. $500 / 15 2 = 33. (33.33)(80.50652^6 )(.06763^7 ) = $
e. $1.500 - $800 = $
$700 / 10 2 = $ ($70)(32.68691^8 )(. 5 ) = $1,
f. ($700)(6.71008^9 )(.31524^10 ) = $1,
g. $700 / 10 = $
($70)(41.1239^11 )(.14602^12 ) = $
($12,355)(.08004^13 ) = $989 Average Annual Value
(^1) Present value of an annuity of 1 per year, 100 years hence, 8 percent
interest. (^2) Necessary to meet the definition of an annuity (^3) Present value of an increasing annuity, 5 years hence, 8 percent interest. (^4) Present value of an annuity of 1 per year, 30 years hence, 8 percent
interest. (^5) Present value of 1, 5 years, 8 percent interest. (^6) Present value of a decreasing annuity, 15 years hence, 8 percent interest. (^7) Present value of 1, 35 years hence, 8 percent interest. (^8) Present value of an increasing annuity, 10 years hence, 8 percent interest. (^9) Present value of an annuity of 1 per year, 10 years hence, 8 percent
interest. (^10) Present value of 1, 15 years, 8 percent interest. (^11) Present value of a decreasing annuity, 10 years hence, 8 percent interest. (^12) Present value of 1, 25 years, 8 percent interest. (^13) Amortization, 100 years, 8 percent interest.
