Present Value

Introduction

The present value (PV) is the current worth of a sum of money to be received or paid in the future, discounted at a specific interest rate. It helps determine how much a future amount is worth in today's terms. The concept of present value is fundamental in finance and investment, as it accounts for the time value of money – the idea that money today is worth more than the same amount in the future due to its potential earning capacity.

### 1. Present Value with Simple Interest

The formula to calculate present value when simple interest is used is:

[ PV = \frac{A}{1 + (R \times T)} ]

Where:

( PV ) = Present value

( A ) = Accumulated future amount

( R ) = Annual interest rate (decimal form)

( T ) = Time period (in years)

2. Present Value with Compound Interest

The formula to calculate present value when compound interest is used is:

[ PV = \frac{A}{\left(1 + \frac{R}{n}\right)^{n \times T}} ]

Where:

( PV ) = Present value

( A ) = Accumulated future amount

( R ) = Annual interest rate (decimal form)

( n ) = Number of compounding periods per year

( T ) = Time period (in years)

Example 1: Calculating Present Value with Simple Interest

Problem

You want to know the present value of $5,000 that you will receive in 3 years if the interest rate is 6% per annum with simple interest.

Solution

Given:

( A = 5000 ) (Future amount)

( R = 6% = 0.06 ) (Annual interest rate)

( T = 3 ) years (Time period)

Using the present value formula for simple interest:

[ PV = \frac{5000}{1 + (0.06 \times 3)} ]

[ PV = \frac{5000}{1 + 0.18} ]

[ PV = \frac{5000}{1.18} \approx 4237.29 ]

Explanation

The present value of $5,000 to be received in 3 years, at a 6% simple interest rate, is approximately $4,237.29. This means that $4,237.29 today is equivalent to $5,000 in 3 years if the annual interest rate is 6%.

Example 2: Calculating Present Value with Compound Interest

Problem

You want to determine the present value of $10,000 that you will receive in 5 years if the interest rate is 8% per annum, compounded quarterly.

Solution

Given:

( A = 10000 ) (Future amount)

( R = 8% = 0.08 ) (Annual interest rate)

( n = 4 ) (Compounded quarterly)

( T = 5 ) years (Time period)

Using the present value formula for compound interest:

[ PV = \frac{10000}{\left(1 + \frac{0.08}{4}\right)^{4 \times 5}} ]

[ PV = \frac{10000}{\left(1 + 0.02\right)^{20}} ]

[ PV = \frac{10000}{1.485947} \approx 6730.68 ]

Explanation

The present value of $10,000 to be received in 5 years, at an 8% annual interest rate compounded quarterly, is approximately $6,730.68. This implies that $6,730.68 today is equivalent to $10,000 in 5 years under the given interest rate and compounding frequency.