# Calculating the Compound Rate of Interest

The nominal rate of interest is the interest rate per year. The rate of compound interest is commonly expressed as a nominal rate of interest. For example in 10% compounded quarterly, 10% refers to the nominal rate of interest.

To calculate the nominal rate of interest, simply multiply the rate of interest per period by the number of periods per year.

##### Nominal Rate = Rate per Period x Periods per Year

In order to compute the nominal rate of interest, the rate of interest per period should be determined.

Here are the steps in order to solve the nominal rate of interest:

• 1) Future amount, principal, periods per year and number of years should be given.
• 2) Divide the future amount by the principal amount.
• 3) Get the nth root of the quotient where n is the number of periods.
• 4) Subtract the root by 1.
• 5) Multiply the difference by the periods per year.

The formula to find the nominal rate of interest can be derived using the formula,

As we know that,

So,

##### text(Rate per Period) = text(Nominal Rate)/text(Periods per Year)

Replacing "Rate per period" in Equation (I):

##### Future Amount=Principal x (1+text(Nominal Rate)/text(Periods per Year))^text(Number of Periods)

text(Future Amount)/text(Principal) = (1+text(Nominal Rate)/text(Periods per Year) )^text(Number of Periods) --- (II)

Let n be the number of periods, then,

text(Future Amount)/text(Principal)= (1+text(Nominal Rate)/text(Periods per Year))^n

(text(Future Amount)/text(Principal))^(1/n)=1+text(Nominal Rate)/text(Periods per Year)

n sqrt(text(Future Amount)/text(Principal))-1=text(Nominal Rate)/text(Periods per Year)

(n sqrt(text(Future Amount)/text(Principal))-1) text( x Periods per Year) = text(Nominal Rate)

Nominal Rate of Interest = Periods per Year x (n sqrt(text(Future Amount)/text(Principal))-1)

Where n is the total number of periods.

## Find the nominal rate of interest if 1.5% interest is added to the amount every month.

##### Explanation:

Interest rate per period is 1.5%.

Number of periods per year is 12 (since monthly).

Determining the nominal rate of interest;

Nominal Rate of Interest = Rate per Period x Periods per Year

Nominal Rate of Interest = 1.5% x 12

Nominal Rate of Interest = 18%

Therefore, the interest rate is 18% compounded monthly.

## Elizabeth loaned $1,000 from a bank and compound interest is done quarterly. If she paid a total of$1,194.83 after 2 years, what was the interest rate charged by the bank?

##### Explanation:

Principal amount is $1,000. Future amount is$1,194.83.

Number of periods per year is 4 (since quarterly).

Number of years is 2.

Determining the total number of periods;

Total number of Periods = n = Periods per Year x Number of Years

Total number of Periods = n = 4 x 2 = 8

Determining the nominal rate of interest;

Nominal Rate of Interest = Periods per Year x (n sqrt(text(Future Amount)/text(Principal)) - 1)

Nominal Rate of Interest = 4 x (8 sqrt(text($1,194.83)/text($1,000)) - 1)

Nominal Rate of Interest = 4 x (8 sqrt((1.194)) - 1)

Nominal Rate of Interest = 4 x (1.0225 - 1)

Nominal Rate of Interest = 4 x 0.0225

Nominal Rate of Interest = 0.09 or 9%

Therefore, the bank charges 9% interest compounded quarterly.

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