Page 257 - Plant design and economics for chemical engineers

INTEREST AND INVESTMENT COSTS 22’7

amount of capital, or receive a lump m of capital that is due in periodic
installments as in some life-insurance 3ls. Engineers often encounter annu-
pl
ities in depreciation calculations, where the decrease in value of equipment with
time is accounted for by an annuity plan.
The common type of annuity involves payments which occur at the end of
each interest period. This is known as an ordinary annuity. Interest is paid on all
accumulated amounts, and the interest is compounded each payment period.
An annuity term is the time from the beginning of the first payment period to
the end of the last payment period. The amount of an annuity is the sum of all
the payments plus interest if allowed to accumulate at a definite rate of interest
from the time of initial payment to the end of the annuity term.

Relation between Amount of Ordinary Annuity
and the Periodic Payments
Let R represent the uniform periodic payment made during n discrete periods
in an ordinary annuity. The interest rate based on the payment period is i, and
S is the amount of the annuity. The first payment of R is made at the end of the
first period and will bear interest for n - 1 periods. Thus, at the end of
the annuity term, this first payment will have accumulated to an amount of
R(l + V-i. The second payment of R is made at the end of the second period
and will bear interest for n - 2 periods giving an accumulated amount of
R(1 + i)n-*. Similarly, each periodic payment will give an additional accumu-
lated amount until the last payment of R is made at the end of the annuity
term.
By definition, the amount of the annuity is the sum of all the accumulated
amounts from each payment; therefore,
S = R(l + i)n-l + R(l + i)n-2 + R(l + i)n-3 + *.. +R(l + i) + R (19)
To simplify Eq. (19), multiply each side by (1 + i> and subtract Eq. (19)
from the result. This gives
Si = R(l + i)” - R (20)
o r
s = R (1 + v - 1
i (21)
The term [(l + i)” - l]/i is commonly designated as the discrete uniform-
series compound-amount factor or the series compound-amount factor.

Continuous Cash Flow and Interest Compounding
: The expression for the case of continuous cash flow and interest compounding,
equivalent to Eq. (21) for discrete cash flow and interest compounding, is
developed as follows:
As before, let r represent the nominal interest rate with m conversions or
interest periods per year so that i = r/m and the total number of interest