Intro to Finance: Time Value of Money - Annuities Ordinary vs. Annuity, Due PV and FV Formulas

What This Is

An annuity is a series of equal cash flows received or paid at regular intervals. In finance, annuities are crucial for calculating present and future values of cash flows, which is essential for investment and financing decisions. For example, consider a $1,000 investment in a 5-year certificate of deposit (CD) with a 5% annual interest rate, compounded annually. The future value of this investment can be calculated using annuity formulas.

Key Formulas & Symbols

• PV = PMT × [(1 - (1 + r)^(-n)) / r] where PV = present value, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• FV = PMT × [(1 + r)^n - 1] / r where FV = future value, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• S = PMT × [(1 + r)^n - 1] / r where S = sinking fund, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• PV of an annuity due = PMT × [(1 + r)^n - 1] / r × (1 + r) where PV = present value, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• FV of an annuity due = PMT × [(1 + r)^n - 1] / r × (1 + r) where FV = future value, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• r = (FV/PV)^(1/n) - 1 where r = periodic interest rate, FV = future value, PV = present value, n = number of periods.
• n = ln(FV/PV) / ln(1 + r) where n = number of periods, FV = future value, PV = present value, r = periodic interest rate.

Step-by-Step Calculation

1. Determine the periodic payment (PMT) and the periodic interest rate (r).
2. Choose the number of periods (n) for the annuity.
3. Select the type of annuity (ordinary or annuity due).
4. Apply the relevant formula to calculate the present value (PV) or future value (FV) of the annuity.
5. Use the calculated PV or FV to make investment or financing decisions.

Common Mistakes

• Mistake: Confusing the PV and FV formulas for annuities.
• Correction: Remember that the PV formula calculates the present value of a future cash flow, while the FV formula calculates the future value of a present cash flow.
• Mistake: Forgetting to adjust the interest rate for compounding frequency.
• Correction: Ensure that the interest rate (r) is adjusted for the compounding frequency (e.g., monthly, quarterly, annually).
• Mistake: Using the wrong type of annuity (ordinary or annuity due).
• Correction: Choose the correct type of annuity based on the specific problem or scenario.

Exam / CFA Tips

• Tip: Be careful with the wording of the question, as it may specify an ordinary or annuity due.
• Tip: Use the correct formula for the type of annuity specified in the question.
• Tip: Pay attention to the compounding frequency and adjust the interest rate accordingly.

Quick Practice Problem

A 5-year certificate of deposit (CD) has a face value of $1,000 and a 5% annual interest rate, compounded annually. What is the future value of this CD after 5 years?

Answer: $1,276.78 Explanation: Using the FV formula for an ordinary annuity, FV = $1,000 × [(1 + 0.05)^5 - 1] / 0.05 = $1,276.78.

Last-Minute Cram Sheet

• The PV formula for an annuity assumes equal periodic payments.
• The FV formula for an annuity assumes equal periodic payments.
• The periodic interest rate (r) must be adjusted for compounding frequency.
• The annuity due formula assumes the first payment is made immediately.
• The number of periods (n) must be a positive integer.
• The PV formula for an annuity due assumes the first payment is made immediately.
• The FV formula for an annuity due assumes the first payment is made immediately.
• The sinking fund formula assumes equal periodic payments.
• The periodic interest rate (r) must be adjusted for compounding frequency.
• The annuity formulas assume equal periodic payments and a fixed interest rate.