Skip to main content
Mathematics LibreTexts

11.S: Summary

  1. Last updated
  2. Save as PDF
  • Page ID
    34111

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Key Concepts

    11.1: Fundamentals of Annuities

    • Understanding what an annuity is
    • The four different types of annuities
    • The difference between annuities and single payments
    • The annuity timeline format

    11.2: Future Value of Annuities

    • The future value of ordinary annuities
    • Variable changes in future value annuity calculations
    • The future value of annuities due

    11.3: Present Value of Annuities

    • The present value of both ordinary annuities and annuities due
    • Variable changes in present value annuity calculations
    • Applying both future value and present value calculations to loans
    • Determining loan balances
    • Selling loan contracts between companies

    11.4: Annuity Payment Amounts

    • Calculating the annuity payment amount for both ordinary annuities and annuities due

    11.5: Number of Annuity Payments

    • Calculating the number of annuity payments (term) for both ordinary annuities and annuities due
    • What to do when N has decimals

    11.6: Annuity Interest Rates

    • Calculating the interest rate for both ordinary annuities and annuities due

    The Language of Business Mathematics

    annuity

    A continuous stream of equal periodic payments from one party to another for a specified period of time to fulfill a financial obligation.

    annuity payment

    The dollar amount of the equal periodic payment in an annuity environment.

    due

    Annuity payments that are each made at the beginning of a payment interval.

    future value of any annuity

    The sum of all the future values for all of the annuity payments when they are moved to the end of the last payment interval.

    general annuity

    An annuity in which the payment frequency and compounding frequency are unequal.

    general annuity due

    An annuity where payments are made at the beginning of the payment intervals and the payment and compounding frequencies are unequal. The first payment occurs on the same date as the beginning of the annuity, while the end of the annuity is one payment interval after the last payment.

    ordinary

    Annuity payments that are each made at the end of a payment interval; this is the most common form of an annuity payment.

    ordinary general annuity

    An annuity where payments are made at the end of the payment intervals and the payment and compounding frequencies are unequal. The first payment occurs one interval after the beginning of the annuity, while the last payment is on the same date as the end of the annuity.

    ordinary simple annuity

    An annuity where payments are made at the end of the payment intervals and the payment and compounding frequencies are equal. The first payment occurs one interval after the beginning of the annuity while the last payment is on the same date as the end of the annuity.

    payment interval

    The amount of time between each continuous and equal annuity payment.

    payment frequency

    The number of annuity payments in a complete year.

    present value of any annuity

    The sum of all the present values for all of the annuity payments when they are moved to the beginning of the first payment interval.

    simple annuity

    An annuity in which the payment frequency and compounding frequency are equal.

    simple annuity due

    An annuity where payments are made at the beginning of the payment intervals and the payment and compounding frequencies are equal. The first payment occurs on the same date as the beginning of the annuity, while the end of the annuity is one payment interval after the last payment.

    The Formulas You Need to Know

    Symbols Used

    \(CY\) = compounding per year or compounding frequency

    \(FV_{DUE}\) = future value of annuity due

    \(FV_{ORD}\) = future value of an ordinary annuity

    \(IY\) = nominal interest rate

    \(i\) = periodic interest rate

    \(N\) = number of annuity payments

    \(PY\) = payments per year or payment frequency

    \(PMT\) = annuity payment amount

    \(PV_{DUE}\) = present value of annuity due

    \(PV_{ORD}\) = present value of ordinary annuity

    Years = the term of the annuity

    Formulas Introduced

    Formula 11.1 Number of Annuity Payments:

    \[N = PY × \text{ Years} \nonumber \]

    Formula 11.2 Ordinary Annuity Future Value:

    \[FV_{ORD}=PMT\left[\dfrac{\left[(1+i)^{\frac{CY}{PY}}\right]^{N}-1}{(1+i)^{\frac{CY}{PY}}-1}\right] \nonumber \]

    Formula 11.3 Annuity Due Future Value:

    \[FV_{DUE}=PMT\left[\dfrac{\left[(1+i)^{\frac{CY}{PY}}\right]^{N}-1}{(1+i)^{\frac{CY}{PY}}-1}\right] \times(1+i)^{\frac{CY}{PY}} \nonumber \]

    Formula 11.4 Ordinary Annuity Present Value:

    \[PV_{ORD}=PMT\left[\dfrac{1-\left[\frac{1}{(1+i)^{\frac{C Y}{PY}}}\right]^{N}}{(1+i)^{\frac{CY}{PY}-1}}\right] \nonumber \]

    Formula 11.5 Annuity Due Present Value:

    \[PV_{DUE}=PMT\left[\dfrac{1-\left[\frac{1}{(1+i)^{\frac{CY}{PY}}}\right]^{N}}{(1+i)^{\frac{CY}{PY}}-1}\right] \times(1+i)^{\frac{CY}{PY}} \nonumber \]

    Technology

    Calculator

    clipboard_efb0226c0a654bf94c2fc988d661c59f2.png

    Annuity Type Settings

    • The calculator default is for END mode, which is the ordinary annuity.
    • The annuity type (payment timing) setting can be found on the second shelf above the PMT key. This function works as a toggle.
    • To toggle the setting, complete the following sequence:
      • 2nd BGN (the current payment timing of END or BGN is displayed)
      • 2nd SET (it toggles to the other setting)
      • 2nd Quit (to get out of the window)
    • When the calculator is in annuity due mode, a tiny BGN is displayed in the upper right of your calculator.

    Contributors and Attributions

    This page titled 11.S: Summary is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jean-Paul Olivier via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.