PRESENT VALUE OF A LUMP-SUM WITH A COMPOUND INTEREST –EXPLAINED WITH 4 EXAMPLES

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Compound interest is the interest earned on principal plus accumulated interest.

Given an interest rate, the present value of compound interest is the amount that a future value of money will be worth now.

Alternatively, the present value can also refer to the amount that we need to invest now to earn a sum of money in the future (future value)

It is the present worth of a sum of money.

Discounting is the process of calculating the present value of a sum of money based on its future value.

It is the present of converting the future value of a lump sum of money to its present value 

The present value of an annual compounding/discounting sum of money is calculated as follows:

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

  Where FV is the future value

               i is the interest rate expressed as a percentage 

              n is the number of years

Example 1

What is the present value of N12,820.77 that will be received 6 years from now at a 4.8% annual interest rate?

Solution:

Remember that the present value is calculated in this way:

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

$PV=\frac{12,820.77}{(1+0.048)^6}$

$PV=\frac{12,820.77}{(1.048)^6}=N9677.13$

Example 2

What is the present value of N7500 to be received 9 years from today at an annual interest rate of 5.5%?

Solution:

Here, FV=7500, n=9, I=5.5% which is equivalent to 0.055

$PV=\frac{7500}{(1+0.055)^9}$

$PV=\frac{7500}{(1.055)^9}=N4632.22$

So far, we have assumed that interest discount annually.  However, the present value may be discounted semi-annually, monthly, weekly, quarterly, or even daily in some circumstances.

How do you calculate the value of a present value that discounts/compounds intra-yearly (semi-annually, monthly, quarterly etc?

In general, the present value is calculated as follows:

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

Where FV is the future value

               i is the interest rate expressed as a percentage 

               n is the number of years

              m is the number of times compounding occurs in a year 

Thus,

1.  The formula for annual compounding/discounting interest is:

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

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

This is what gave birth to the formula we used for the annual compounding interest earlier

2. The formula that applies to interest that compounds/discounts semi-annually is:

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

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

Note: m is 2 because a year is equivalent to 2 6-month.

3. The formula for quarterly compounding/discounting interest is::

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

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

Note: M is 4 because they are 4 quarters in a year

 4. For an interest that compounds/discounts monthly, the formula is

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

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

Note: m is 12 because they are 12 months in a year

5. The formula is as follows for a weekly compounding/discounting interest:

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

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

Note: m is 52 because they are 52 weeks in a year

6. For an interest that compounds/discounts daily, the formula that applies is

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

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

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

Let's look at a few examples to see how this formula works

Example 3

What is the present value of an N7500 to be received 9 years from today if it is compounded quarterly at a rate of 5.5%

Solution:

Keep in mind the following formula for quarterly compounding interest:

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

$PV=\frac{7500}{\left(1+\frac{0.055}{4}\right)^{9 \times 4}}$

$PV=\frac{7500}{(1.01375)^{36}}=N4587.23$

Examples 4

In eight years, James expects to receive N2000. Given that interest compounds semi-annually at a rate of 7%, how much should he invest to obtain such money?

Solution:

Remember that 

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

$PV=\frac{2000}{\left(1+\frac{0.07}{2}\right)^{2(8)}}$

$PV=\frac{2000}{\left(1+0.035\right)^{16}}$

$PV=\frac{2000}{\left(1.035\right)^{16}}=1153.41$

We know that calculating the present value of lump sum is cumbersome, that is why we have a created a calculator to help you.

That concludes this post. You'll learn how to calculate the future value of an ordinary annuity in the next post.

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