form a geometric progression (i.e., the ratio of a given payment to the preceding payment
is constant). In this instance, the deposits form a uniform-rate series because each deposit
is 1.25 times the preceding deposit. Apply Eq. 8: URSCA = [r" - (1 + /)"]/(> - / - 1),
where r = ratio of given payment to preceding payment, n = number of payments, and i =
interest rate for the payment period. With r = 1.25, n = 8, and i = 7 percent, URSCA =
[(1.25)
8
- (1.07)
8
]/(1.25 - 0.07 - 1) = 23.568.
- Compute the future value of the set of deposits
Use the relation S = ^ 1 (URSCA), where ^ 1 = first payment. Then S = $1000(23.568) =
$23,568.
DETERMINATION OF PAYMENTS UNDER
UNIFORM-RATE SERIES
At the beginning of year 1, the sum of $30,000 was borrowed with interest at 9 percent
per annum. The loan will be discharged by payments made at the end of years 1 to 6, in-
clusive, and each payment will be 95 percent of the preceding payment. Find the amount
of the first and sixth payments.
Calculation Procedure:- Compute the URSPW value of the uniform-rate series
Apply Eq. 9: URSPW = {[r/(l + i)]n - 1 }/(r - i - 1), where the symbols are as defined in
the previous calculation procedure. With r = 0.95, n = 6, and i = 9 percent, URSPW =
[(0.95/1.09)^6 - 1]/(0.95 - 0.09 - 1) - 4.012. - Find the amount of the first payment
Use the relation P = ,K 1 (URSPW), where ^ 1 = first payment. Then $30,000 = /^(4.012),
or #! = $7477.60. - Find the amount of the sixth payment
Use the relation Rm = Rf "^^1 , where Rm = mth payment. Then ^ 6 = $7477.60(0.95)^5 =
$5786.00.
CONTINUOUS COMPOUNDING
If $1000 is invested at 6 percent per annum compounded continuously, what will it
amount to in 5 years?Calculation Procedure:Apply the continuous compounding equation
Use the relation SPCA - e>
n
, where e = base of the natural logarithm system = 2.71828
..., j = nominal interest rate, n = number of years. Substituting gives SPCA = (2.718)°
30
- 1.350. ThCnS = P(SPCA) = $1000(1.350) = $1350.