Exercise-1: Computation of simple and compound interest

Learning objective:
This exercise illustrates the computation of simple interest and compound interest when interest is compounded annually, semi-annually, quarterly and monthly.

A company is considering to start a new product line which requires the installation of new machines and equipment. For this purpose, company wants to borrow money by issuing bonds of $10,000 for 12-year period. The interest on these bonds is to be paid at a rate of 8% per year.

Required: Compute the amount of interest to be paid to bondholders over 12-year period:

1. if the simple interest is charged.
2. if the compounded interest is charged and the interest is compounded:
   (i). annually,
   (ii). semi-annually,
   (iii). quarterly
   (iv). monthly

Solution:

(1) If simple interest is charged:

I = Pin
= $10,000 × 8% × 12
= $10,000 × 0.08 × 12
= $9,600

(2) If compound interest is charged:

To compute compound interest for 12-years, first we would compute the compound amount using the compound amount formula and then deduct the initial principal amount from the compound amount to obtain the total amount of interest over the life of the bonds.

Compound amount formula is:

A = P(1 + i)ⁿ

• A = Compound amount
• P = Principal amount
• Interest rate of the loan
• Number of periods involved

Let’s do the computations using all four compounding frequencies given in the question.

(i). If interest is compounded annually:

i = 8%
n = 12

= $10,000 × (1 + 8%)¹²
=$10,000 × 2.518*
= $25,180

Total interest to be paid over 12-year period if the interest is compounded annually:

$25,180 – $10,000 = $15,180

*(1.08)^12 = 2.518; alternatively, we can get this value from future value of $1 table: 8% interest rate, 12 periods.

(ii). If interest is compounded semi-annually:

Interest rate (i) = 8%/2 = 4%
Number of periods (n) = 12 × 2 = 24

= $10,000 × (1 + 4%)²⁴
= $10,000 × 2.563*
= $25,630

Total interest to be paid over 12-year period if the interest is compounded semi-annually:

$25,630 – $10,000 = $15,630

*(1.04)^24 = 2.563; alternatively, we can get this value from future value of $1 table: 4% interest rate, 24 periods.

(iii). If interest is compounded quarterly:

Interest rate (i) = 8%/4 = 2%
Number of periods (n) = 12 × 4 = 48

= $10,000 × (1 + 2%)⁴⁸
= $10,000 × 2.587*
= $25,870

Total interest to be paid over 12-year period if the interest is compounded quarterly:

$25,870 – $10,000 = $15,870

*(1.02)^48 = 2.587; alternatively, we can get this value from future value of $1 table: 2% interest rate, 48 periods.

(iv). If interest is compounded monthly:

Interest rate (i) = 8%/12 = 0.6667%
Number of periods (n) = 12 × 12 = 144

= $10,000 × (1 + 0.6667%)¹⁴⁴
= $10,000 × 2.604*
= $26,040

Total interest to be paid over 12-year period if the interest is compounded monthly:

$26,040 – $10,000 = $16,040

*(1.006667)^144 = 2.604