7.g 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.

Hive Chat

Hi, I'm Hive Chat, an AI assistant created by CollegeHive.
How can I help you today?

🎶