A loan is amortized if the repayment of the loan is made through a fixed repayment schedule in regular payment over some time.

Each payment includes both the interest on the outstanding loan and principal and amortisation of the loan is done by applying the concept of the present value of an annuity.

To calculate the periodic payment of an amortized loan, we used the formula:

$$C=\frac{PV}{\frac{1-\left(1+\frac{i}{m}\right)^{-nm}}{\frac{i}{m}}}$$

Where PV is the present value or the total amount of the loan

i is the annual interest rate

m is the **manner through which the periodic payment will be made**. For example, if repayment is on a monthly basis, then m will be 12

n is the number of years

A loan is amortized if both the capital and interest rate are paid by a sequence of equal periodic payments.

__Example 1__

Kenny took a loan of N80,000 payable in 10 semi-annual instalments with 8% compounded semi-annually. Find the amount of each instalment.

**Solution:**

Recalled that periodic cash flow is calculated as:

$$C=\frac{PV}{\frac{1-\left(1+\frac{i}{m}\right)^{-nm}}{\frac{i}{m}}}$$

Here, I=0.08, m is 2 because the loan is to be paid **semi-annually**. n=5 because 10 semi-annuals period is equivalent to 5 years

$C=\frac{80,000}{\frac{1-\left(1+\frac{0.08}{2}\right)^{-5(2)}}{\frac{0.08}{2}}}$

$C=\frac{80,000}{\frac{1-\left(1+0.04 \right)^{-5(2)}}{0.04}}$

$C=\frac{80,000}{\frac{1-\left(1.04 \right)^{-10}}{0.04}}$

$C=\frac{80,000}{\frac{1-0.6755641688}{0.04}}$

$C=\frac{80,000}{\frac{0.3244358312}{0.04}}$

$C=\frac{80,000}{8.110895779}=N9863.275546$

__Example 2__

Kenny requires N200,000 to purchase a house after 5 years. He has an opportunity to invest the funds in an account that earn 6% per annum compounded quarterly. Find how much should be deposited at the end of each quarter to have the required amount at the end of 5 years.

**Solution:**

$$C=\frac{PV}{\frac{1-\left(1+\frac{i}{m}\right)^{-nm}}{\frac{i}{m}}}$$

$C=\frac{200,000}{\frac{1-\left(1+\frac{0.06}{4}\right)^{-5(4)}}{\frac{0.06}{4}}}$

Note: m is four because the money is to be paid quarterly.

$C=\frac{200,000}{\frac{1-\left(1+0.015\right)^{-20}}{0.015}}$

$C=\frac{200,000}{\frac{1-0.7424704182}{0.015}}$

$C=\frac{200,000}{\frac{0.2575295818}{0.015}}$

$C=\frac{200,000}{17.16863879}$

$C=N11649.14717$

__Example 3__

Daniel borrowed N1,000,000 from a bank to purchase a house and decided to amortize the loan through monthly equal instalments in 10 years. The bank charges interest 9% compounded monthly. Calculate his monthly payment.

**Solution:**

$$C=\frac{PV}{\frac{1-\left(1+\frac{i}{m}\right)^{-nm}}{\frac{i}{m}}}$$

$C=\frac{1,000,000}{\frac{1-\left(1+\frac{0.09}{12}\right)^{-10(12)}}{\frac{0.09}{12}}}$

$C=\frac{1,000,000}{\frac{1-\left(1+0.0075\right)^{-120}}{0.0075}}$

$C=\frac{1,000,000}{\frac{1-\left(1.0075\right)^{-120}}{0.0075}}$

$C=\frac{1,000,000}{\frac{1-0.407937305}{0.0075}}$

$C=\frac{1,000,000}{\frac{0.592062695}{0.0075}}$

$C=\frac{1,000,000}{78.94169267}=N12668$

__Example 4__

A loan of N10,000 is amortized by making equal payments at the end of every six months for three years and interest is 6% compounded semi-annually. Calculate the amount of money that will be paid in each period.

**Solution:**

$$C=\frac{PV}{\frac{1-\left(1+\frac{i}{m}\right)^{-nm}}{\frac{i}{m}}}$$

$C=\frac{10,000}{\frac{1-\left(1+\frac{0.06}{2}\right)^{-3(2)}}{\frac{0.06}{2}}}$

$C=\frac{10,000}{\frac{1-\left(1+0.03\right)^{-6}}{0.03}}$

$C=\frac{10,000}{\frac{1-(0.8374842567)}{0.03}}$

$C=\frac{10,000}{\frac{0.1625157433}{0.03}}$

$C=\frac{10,000}{5.417191443}=N1845.975005$

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