Future Worth (or Value) of One Per Period: Financial Growth Concept

The Future Worth (or Value) of One Per Period refers to the accumulation of a series of equal cash flows over time, compounded at a specific interest rate.

The Future Worth (or Value) of One Per Period, also known as the Compound Amount of One Per Period, is a fundamental concept in financial mathematics. It refers to the future sum accumulated when regular, equal payments (or deposits) are made at the end of each period, compounded at a specified interest rate.

Key Formula

The formula to calculate the Future Worth (or Value) of One Per Period (F) is given by:

$$ F = P \left( \frac{(1 + r)^n - 1}{r} \right) $$

where:

• \( F \) = Future worth of the series of payments
• \( P \) = Payment amount per period
• \( r \) = Interest rate per period
• \( n \) = Number of periods

Types of Future Worth (or Value) Calculations

Ordinary Annuity

This refers to payments made at the end of each period. The formula above applies directly to ordinary annuities.

Annuity Due

For annuities where payments are made at the beginning of each period, the formula is modified as follows:

$$ F_{\text{due}} = P \left( \frac{(1 + r)^n - 1}{r} \right) (1 + r) $$

Special Considerations

Compounding Frequency

The frequency of compounding can significantly affect the future value. Common compounding frequencies include annual, semi-annual, quarterly, and monthly.

Interest Rate Variability

Interest rates may change over time, necessitating adjustments in calculations.

Payment Timings

Changing the timing of payments from end-of-period to beginning-of-period (annuity due) affects the future value.

Examples

Example 1: Calculating Future Worth of Ordinary Annuity

Suppose you deposit $1,000 at the end of each year for 5 years into an account paying 5% annual interest. Using the formula:

$$ F = 1000 \left( \frac{(1 + 0.05)^5 - 1}{0.05} \right) $$

Example 2: Calculating Future Worth of Annuity Due

If the $1,000 deposits are made at the beginning of each year, the future value is:

$$ F_{\text{due}} = 1000 \left( \frac{(1 + 0.05)^5 - 1}{0.05} \right) (1 + 0.05) $$

Historical Context

The concept of compounding dates back to antiquity, with significant developments in the field by mathematicians such as Fibonacci and later as part of modern financial theory in the 20th century.

Applicability

The concept is crucial in various financial disciplines, including investments, retirement planning, mortgage calculations, and any scenarios involving regular periodic payments.

Comparisons

Future Value of a Single Sum

Unlike the future worth of one per period, which involves multiple cash flows, the future value of a single sum deals with one-time investments.

Present Value of Annuity

The present value focuses on the current worth of a series of future payments, essentially reversing the future worth calculation.

Related Terms

• Annuity: A series of equal payments made at regular intervals.
• Present Value: The current value of a future sum of money or stream of cash flows.
• Compound Interest: Interest calculated on the initial principal, which also includes all the accumulated interest.

FAQs

Q1: How does compounding frequency affect future value?

Higher compounding frequencies result in larger future values for the same nominal interest rate and number of periods.

Q2: Can the interest rate change over the investment period?

Yes, if the interest rate is variable, periodic recalculations are necessary to determine the accurate future value.

References

1. Ross, S. A., Westerfield, R., & Jaffe, J. (2008). Corporate Finance. McGraw-Hill Education.
2. Fabozzi, F. J., & Peterson Drake, P. (2009). Finance: Capital Markets, Financial Management, and Investment Management. Wiley.

Summary

The Future Worth (or Value) of One Per Period is a powerful concept allowing individuals and businesses to understand the growth of regular investments over time with compound interest. Whether for retirement planning, loan amortizations, or investment strategies, mastering this concept is essential for effective financial management.