Present Value and Single Amount

This section discusses how to calculate the present value of a future single-period payment, the return on a multi-period investment over time, and what real-world costs to the investor comprise an investment’s interest rate. It also addresses what a period is in terms of present value calculations and distinguishes between the formula for present value with simple interest and compound interest.

Multi-Period Investment

Multi-period investments are investments with more than one period, so $n$ (or $t$ ) is greater than one.

LEARNING OBJECTIVE

Calculate the present value of a multi-period investment

KEY TAKEAWAYS

Key Points

Finding the PV for a multi-period investment is the same as for a single-period investment: plug $FV$ , the interest rate, and the number of periods into the correct formula.
$PV$ varies jointly with $FV$ , and inversely with $i$ and $n$ .
When $n>1$ , simple and compound interest cease to provide the same answer (unless the interest rate is 0).

Key Terms

compound interest: Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest.
multi-period: More than one unit of time.

Multi-Period Investments

Things get marginally more complicated when dealing with a multi-period investment. That is, an investment where $n$ is greater than 1.

Suppose the interest rate is 3% per year. That means that the value of $100 will be 3% more after one year, or $103. After the second year, the investment will be 3% more, or 3% more than $103. That means the original investment of $100 is now worth $106.09. The investment is not worth 6% more after two years. In the second year, you earn 3% interest on your original $100, but you also earn 3% interest on the $3 you earned in the first year. This is called compounding interest: interest accrues on previously earned interest.

As such, $PV$ and $FV$ are related exponentially, which is reflected in. Using the formula in is relatively simple. Just as with a single-period investment, you simply plug in the $FV$ , $i$ and $n$ in order to find the $PV$ . $PV$ varies jointly with $FV$ and inversely with $i$ and $n$ , which makes sense based on what we know about the time value of money.

The formula may seem simple, but there is one major tripping point: units. Sometimes, the interest rate will be something like 5% annually, and you are asked to find the PV after 24 months. The number of periods, however, is not 24--it is 2. If the interest rate is written as "percent per year" your periods must also be measured in years. If your periods are defined as "days", your interest rate must be written as "percent per day".

Consider the $100 investment mentioned above. How much would the investment be worth in 5 years with yearly compounding interest? An expanded explanation of the calculations is as follows:

$\mathrm{FV}=\mathrm{PV} \cdot(1+\mathrm{i}) \mathrm{n}$

$\mathrm{FV}=100 \cdot(1+0.03) 5$

$\mathrm{FV}=100 \cdot 1.159$

$\mathrm{FV}=\$ 115.93$

Using a Financial Calculator for a Multi-Period Investment

In order to perform the calculation for the above investment, follow these steps:

We know that $PV$ is $100, $i$ is 3% and $n$ is 5. When using a financial calculator, we enter our known values followed by their corresponding function key.
Enter 5 and then press the " $n$ " key (or whatever function key corresponds with the number of periods).
Enter 0.03 and press the " $i$ " key (or whatever function key corresponds with the investment rate).
Enter 100 and press the " $PV$ " key.
Press the "compute" (or "solve") key and then the " $FV$ " key to solve for future value.