Interest rate can be classified into simple interest rate and compound interest rate.

**Interest Formulas **

**Interest Formulas **

Interest rate can be classified into simple interest rate and
compound** **interest rate.

ü In simple interest, the interest is calculated, based on the
initial deposit for every interest period. In this case, calculation of
interest on interest is not applicable.

ü In compound interest, the interest for the current period is
computed based on the amount (principal plus interest up to the end of the
previous period) at the beginning of the current period.

The notations which are used in
various interest formulae are as follows: *P *= principal amount

*n *=
No. of interest periods

*i *= interest rate
(It may be compounded monthly, quarterly,* *semiannually or annually)

*F *=
future amount at the end of year* n*

*A *=
equal amount deposited at the end
of every interest period

*G *= uniform
amount which will be added/subtracted period after period* *to/from the
amount of deposit A1 at the end of period 1

**Time Value
of Money - **. It represents the growth of
capital per unit period. The period** **may be a month, a quarter,
semiannual or a year.

An interest rate
15% compounded annually means that for every hundred rupees invested now, an
amount of Rs. 15 will be added to the account at the end of the first year. So,
the total amount at the end of the first year will be Rs. 115.

At the end of the
second year, again 15% of Rs. 115, i.e. Rs. 17.25 will be added to the account.

Hence the total
amount at the end of the second year will be Rs. 132.25. The process will
continue thus till the specified number of years.

If an investor
invests a sum of Rs. 100 in a fixed deposit for five years with an interest
rate of 15% compounded annually, the accumulated amount at the end of every
year will be as shown in Table

**Compound Amounts**

** (amount
of deposit = Rs. 100.00)**

** **

** Year end Interest Compound
amount**

** (Rs.) (Rs.)**

0 100.00

1 15.00 115.00

2 17.25 132.25

3 19.84 152.09

4 22.81 174.90

5 26.24 201.14

The formula to find the future worth in the third column is

*F *=* P *(1 +* i*)^{n}

where

*P *= principal
amount invested at time 0,* F *= future amount,

*i *= interest
rate compounded annually,* n *= period of deposit.

The maturity value at the end of the
fifth year is Rs. 201.14. This means that the amount Rs. 201.14 at the end of
the fifth year is equivalent to Rs. 100.00 at time 0 (i.e. at present). This is
diagrammatically shown in Fig. 3.1. This explanation assumes that the inflation
is at zero percentage.

**Single-Payment
Compound Amount - **Here, the objective is to find the
single future sum (*F*) of** **the initial payment (*P*) made at
time 0 after *n* periods at an interest rate *i* compounded every

Cash flow diagram of single-payment compound amount.

The formula to obtain the single-payment compound amount is

*F *=* P*(1 +* i*)* ^{n}
*=

where

(*F*/*P*, *i*, *n*) is called as
single-payment compound amount factor.

**Single-Payment
Present Worth Amount**** - **Here, the objective is to find the present worth** **amount (*P*)
of a single future sum (*F*) which will be received after *n* periods
at an interest rate of *i *compounded at the end of every interest period.

Cash flow diagram of single-payment present worth amount.

The formula to
obtain the present worth is

Where

(*P*/*F*, *i*, *n*) is termed as *single-payment
present worth factor.*

**Equal-Payment Series Sinking Fund**

In this type of investment mode, the
objective is to find the equivalent amount (*A*) that should be deposited
at the end of every interest period for *n* interest periods to realize a
future sum (*F*) at the end of the *n*th interest period at an
interest rate of *i*.

*A *= equal
amount to be deposited at the end of each interest period* n *= No. of
interest periods

*i *= rate of interest

*F *= single future amount at the end of
the* n*th period

The formula to get *F* is

Where

(*A*/*F*, *i*, *n*) is called as *equal-payment
series sinking fund factor.*

**Equal-Payment
Series Present Worth Amount**** **The objective of this mode of investment is to** **find the
present worth of an equal payment made at the end of every interest period for
n interest periods at an interest rate of i compounded at the end of every
interest period.

The corresponding cash flow diagram is shown in Fig. 3.8. Here,

*P *= present worth

*A *= annual
equivalent payment* i *= interest rate

*n *= No. of interest periods

The formula to
compute *P* is

Where

*(P/A, i, n) is
called equal-payment series present worth factor*

** Equal-Payment Series
capital recovery** The objective of this
mode of investment
is to find the
annual equivalent amount (A) which is to be recovered at the end of
every interest period for n interest periods for a loan (P) which is sanctioned
now at an interest

Cash flow diagram of equal-payment series capital recovery
amount.

*P *= present worth (loan amount)

*A *= annual
equivalent payment (recovery amount)* i *= interest rate

*n *= No. of interest periods

The formula to
compute *P* is as follows:

Where,

(*A*/*P*, *i*, *n*) is called *equal-payment
series capital recovery factor*.

**Uniform Gradient series annual equivalent**** **The objective of this mode of investment is to** **find the annual equivalent amount of a series with an amount *A*1
at the end of the first year and with an equal increment (*G*) at the end
of each of the following *n* – 1 years with an interest rate *i* compounded annually.

The corresponding
cash flow diagram is shown in Fig

Cash flow diagram of uniform gradient series annual equivalent
amount.

The formula to
compute *A* under this situation is

here Where

(*A*/*G*,
*i*, *n*) is called *uniform gradient series factor*

**Effective Interest rate**** **Let** ***i*** **be the nominal interest rate compounded annually. But, in** **practice, the compounding may occur less than a year. For
example, compounding may be monthly, quarterly, or semi-annually. Compounding
monthly means that the interest is computed at the end of every month. There
are 12 interest periods in a year if the interest is compounded monthly. Under
such situations, the formula to compute the

effective interest rate, which is compounded annually, is Effective interest rate,

*R* = 1 + *i*/*C ^{C} *−

where,

*i *= the nominal interest rate

*C *= the number of interest periods in a year.

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Mechanical : Engineering Economics & Cost Analysis : Value Engineering : Interest rate Formulas |

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