values, the interest rate built into annuity contracts, and the length of time it takes to reach a financial goal using an annuity. Growing annuities are often used in the area of financial planning. Their analysis is more complex and often easier solved using a financial spreadsheet, so we will limit our discussion here to the first two types of annuities.

The future value of an ordinary annuity, FVA ${?}_N,$ is the total amount one would have at the end of the annuity period if each payment $($PMT$)$ were invested at a given interest rate and held to the end of the annuity period. The equation is:

$FV{A}_N=PMT[\frac{(1+I{)}^N-1}{\text{I}}]$

Each payment of an annuity due is compounded for one

$3$ period, so the future value of an annuity due is equal to the future value of an ordinary annuity compounded for one

a period. The equation is:

$FV{A}_{\text{d}\text{u}\text{e}}=FV{A}_{\text{o}\text{r}\text{d}\text{i}\text{n}\text{a}\text{r}\text{y}}(1+I)$

The present value of an ordinary annuity, PVAN, is the value today that would be equivalent to the annuity payments $($PMT$)$ received at fixed intervals over the annuity period. The equation is:

$PV{A}_N=PMT[\frac{1-\frac{1}{(1+{)}^N}}{\text{I}}]$

Each payment of an annuity due is discounted for one I period, so the present value of an annuity due is equal to the present value of an ordinary annuity multiplied by $(1+I).$ The equation is:

$PV{A}_{\text{d}\text{u}\text{e}}=PV{A}_{\text{o}\text{r}\text{d}\text{i}\text{n}\text{a}\text{r}\text{y}}(1+I)$

One can solve for payments $($PMT$),$ periods $($N$),$ and interest rates $($I$)$ for annuities. The easiest way to solve for these variables is with a financial calculator or a spreadsheet.

Quantitative Problem $1$: You plan to deposit $$1,800$ per year for $4$ years into a money market account with an annual return of $3\%.$ You plan to make your first deposit one year from today.

a$.$ What amount will be in your account at the end of $4$ years? Do not round intermediate calculations. Round your answer to the nearest cent.

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b$.$ Assume that your deposits will begin today. What amount will be in your account after $4$ years? Do not round intermediate calculations. Round your answer to the nearest cent.

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