Compound interest is the interest earned on the initial investment plus all the interest that has accumulated over time.

It can also be defined as interest earned on the sum of principal and the accumulated interest.

The idea behind compound interest is that you will earn interest on the interest you earned the previous year in the second year.

To put it another way, the interest you earn in the first year is combined with the principal, and you earn interest on the combined sum in the second year.

To calculate compound interest, you must understand two concepts: future value and present value

**The future value of a lump sum** is the value of the original sum of money at a future point in time at a given rate of interest.

In other words, the future value of a lump sum/single-period investment is the cash value of an investment at some time in the future.

The future value of a lump sum investment is the amount an investment will grow to over some period given a particular interest rate.

When calculating a future value (FV), you are calculating how much a given amount of money today will be worth some time in the future and this is **compounding**.

**The present value of compound interest** is the present worth of a sum of money. It is the initial investment or deposit.

Present value is sometimes known as the **principal**.

When calculating the present value, you are calculating how much a given amount of money in the future will be worth now and this is called **discounting**.

For our topic today, we will be focusing on the future value of compound interest. The present value of compound interest will be discussed in a subsequent post.

Forthwith, let's get started

It's easy to calculate the future value of a sum of money that compounds annually via:

$$FV=PV(1+i)^n$$

Where FV is the **future value**

PV is the **present value or principal**

n denotes **the number of years**.

i is the **interest rate** expressed as a percentage.

__Example 1__

__Example 1__

How much money do you have after investing N7000 for 5 years in a savings account that earns 9% compound interest per year?

**Solution:**

This question is about compound interest's future value. Hence, we employ the formula:

$$FV=PV(1+i)^n$$

Here PV=7000, i=9% which the is same as 0.09, and n=5

$FV=7000(1+0.09)^5$

$FV=7000(1.09)^5$

$FV=7000(1.538624)$

$FV=N10,770.368$

Hence, the future value of the N7000 at 9% per annum is N10,770.368

__Example 2__

__Example 2__

If we place N5000 in a savings account that yields 4.5% compounded annually, what would be the future value of the investment after 7 years

**Solution:**

Here, PV=N5000, i=4.5% or 0.045 and n=7

$FV=5000(1+0.045)^7$

$FV=5000(1.045)^7$

$FV=5000(1.36086)$

$FV=N6804.309$

Until now, we have assumed that interest compounds annually. However, in real-life situations, interests do compound monthly, quarterly, weekly and even daily.

This takes us to the next heading

**How To Calculate The Future Value Of A Compound Interest Payable Intra-yearly**

Generally, the formula for the future value of compound interest is:

$$FV=PV\left(1+\frac{i}{m}\right)^{n ×m}$$

Where PV is the present value

i is the interest rate expressed as a percentage

M is the number of times compounding occurs in a year

n is the number of years.

1. Therefore, for an interest that **compounds annually**, the formula is computed as:

$$FV=PV\left(1+\frac{i}{1}\right)^{n × 1}$$

$$FV=PV(1+i)^n$$

This is where the formula we used earlier to solve yearly compound interest came from.

2. For an interest that **compounds quarterly**, the formula is

$$FV=PV\left(1+\frac{i}{4}\right)^{n×4}$$

$$FV=PV\left(1+\frac{i}{4}\right)^{4n}$$

** Note:** m is four because there are four quarters in a year.

3. For an interest that **compounds monthly**, we apply the formula

$$FV=PV\left(1+\frac{i}{12}\right)^{n ×12}$$

$$FV=PV\left(1+\frac{i}{12}\right)^{12n}$$

** Note:** m is 12 because there are 12 months in a year.

4. For an interest that **compounds weekly**, we apply the formula

$$FV=PV\left(1+\frac{i} {52}\right)^{n × 52}$$

$$FV=PV\left(1+\frac{i}{52}\right)^{52n}$$

** Note:** m is 52 because there are 52 weeks in a year and it's only natural that compounding occurs 52 times in a year.

5. For an interest that **compounds daily,** we apply the formula

$$FV=PV\left(1+\frac{i}{365}\right)^{n × 365}$$

$$FV=PV\left(1+\frac{i}{365}\right)^{365n}$$

** Note:** m is 365 because a year is equivalent to 365 days.

6. For an interest that compounds **semi-annually**, it is obtained via

$$FV=PV\left(1+\frac{i}{2}\right)^{n × 2}$$

$$FV=PV\left(1+\frac{i}{2}\right)^{2n}$$

** Note:** Semi-annually means interest compound every six months.

To better appreciate these formulas, let's take some examples

__Example 3__

__Example 3__

A principal of N8000 is invested for three years at a rate of 12%. If the interest is compounded monthly, calculate the future value.

**Solution:**

Recalled that the future value of monthly compound interest is:

$$FV=PV\left(1+\frac{i}{12}\right)^{12n}$$

Here, PV=N8000, I=0.12 and n=3

$FV=8000(1+\frac{0.12}{12})^{12(3)}$

$FV=8000(1+0.12)^{36}$

$FV=8000(1.12)^{36}$

$FV= N11.446.15$

__Example 4__

__Example 4__

A principal of N8000 is invested at 12% interest for 3 years. Determine the future value if the interest is compounded semi-annually

**Solution:**

Recalled that

$FV=PV\left(1+\frac{i}{2}\right)^{2n}$

$FV=8000(1+\frac{0.12}{2})^{2(3)}$

$FV=8000(1.06)^6$

$FV=N11.348.15$

__Example 5__

__Example 5__

A principal of N8000 is invested at 12% interest for 3 years. Determine the future value if the interest is compounded daily.

**Solution:**

$$FV=PV(1+\frac{i}{365})^{365n}$$

$FV=8000(1+\frac{0.12}{365})^{365(3)}$

$FV=8000(1+0.000328767)^{1095}$

$FV=8000(1.000328767)^{1095}$

$FV=8000(1.4332444)$

$FV=11465.9552$

So far, we've looked at the future value of compound interest with definite periods such as yearly, annually, quarterly, and monthly.

However, sometimes, interest can compound for an infinite period. This is called** continuous compound interest**.

More appropriately, continuous compound interest is one where the number of years (n) is infinite.

The formula for calculating the future value of continuous compound interest is as follows:

$$FV=PV(e^{in})$$

Where FV is the future value

PV is the present value

e is Euler's number which is approximately 2.71827

n is the number of years

i is the interest rate expressed as a percentage

__Example 6__

__Example 6__

Supposed that N10,000 is deposited at 4% compounded continuously, find the compound amount after 7 years.

**Solution**

Recalled that $$FV=PV(e^{in})$$

$FV=10,000(e^{0.04 ×7})$

$FV=10,000(2.71827^{0.28})$

$FV=13,231.3$

Got difficult questions on the future value of lump sums, you can use our calculator to solve it

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