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F V = $9,000<br />

F V = $10,000<br />

F V = $15,000<br />

F V = $20,000<br />

I = 8%<br />

N = 3 years<br />

To solve for the various present values, follow the instructions in the previous PV exercise and<br />

review the comments regarding the inputs into each cell as seen in Excel example 5 (Exhibit 4.8).<br />

Notice in Excel example 5 that the PVs are negative; this reflects that a cash outflow is required.<br />

Exhibit 4.8 Solving for various PVs with a fixed time and interest rate: Excel example 5.<br />

Annuities<br />

While discussing present values and future values, we considered single payments at the beginning or<br />

at the end of a fixed time interval. Annuities differ, as they require or provide a cash flow (CF) or<br />

series of payments over a specific time period. The time period may be annually, quarterly, monthly,<br />

or even weekly.<br />

Annuities meet three criteria: They (1) pay an equal amount, (2) at a specific time interval, (3) for a<br />

specific time period.<br />

If we are placing money into an annuity, we need to consider in which type of annuity to invest.<br />

Two types of annuities are the ordinary annuity and the annuity due.

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