Introduction to Annuities

Section 3.1 Introduction to Annuities

In the discussion above, investments considered a single "deposit" at the beginning of the time period and a single "payout" at the end of the time period. In many real-life situations, one may desire an option where there is perhaps a single deposit at the beginning of time and then a series of (perhaps equal) payouts over the time period. Such a sequence of payments is called an annuity.

Theorem 3.1.1. Annuity Future Value.

An annuity of R per period over t years, paying out m times per year with yearly rate i yields \(\displaystyle A = R\frac{(1+r)^n-1}{r}\)

Proof.

We will presume this sequence of regular and equal payments of amount R starts in 1 period from the beginning of time and then ends on the last day of the allotted time period. To determine the future value of this accumulated promise, take each payment and find its future value at the end of the time period. Note that this means the first payment must be "moved forward" n-1 time periods, the second n-2 time periods, etc. until the last payment which is actually at the final time and so is not moved at all. This gives

\begin{align*} A & = R(1+r)^{n-1} + R(1+r)^{n-2} + R(1+r)^{n-3} + ... R(1+r)^{1} + R\\ & = R \left [ (1+r)^{n-1} + (1+r)^{n-2} + (1+r)^{n-3} + ... (1+r)^{1} + 1 \right ]\\ & = R \displaystyle \frac{1-(1+r)^n}{1-(1+r)} \\ & = R \displaystyle \frac{(1+r)^n-1}{r} \end{align*}

Corollary 3.1.2. Annuity Present Value.

An annuity of R per period over t years, paying out m times per year with yearly rate i yields \(\displaystyle P = R\frac{1-(1+r)^{-n}}{r}\)

Proof.

Apply the present value formula to the future value formula above.