## Additional Detail on Present and Future Values

This section gives more detail on computing present and future values. It shows you how to compute more complex problems involving future and present values when there are multiple compounding periods and when the time duration of those problems are longer or are less than one year.

### The Relationship Between Present and Future Value

Present value ($PV$) and future value ($FV$) measure how much the value of money has changed over time.

#### LEARNING OBJECTIVE

• Discuss the relationship between present value and future value

#### KEY TAKEAWAYS

##### Key Points
• The future value ($FV$) measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return. The $FV$ is calculated by multiplying the present value by the accumulation function.
• $PV$ and $FV$ vary jointly: when one increases, the other increases, assuming that the interest rate and number of periods remain constant.
• As the interest rate (discount rate) and number of periods increase, $FV$ increases or $PV$ decreases.

##### Key Terms
• discounting: The process of finding the present value using the discount rate.
• present value: a future amount of money that has been discounted to reflect its current value, as if it existed today
• capitalization: The process of finding the future value of a sum by evaluating the present value.

The future value ($FV$) measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return. The $FV$ is calculated by multiplying the present value by the accumulation function. The value does not include corrections for inflation or other factors that affect the true value of money in the future. The process of finding the $FV$ is often called capitalization.

On the other hand, the present value ($PV$) is the value on a given date of a payment or series of payments made at other times. The process of finding the $PV$ from the $FV$ is called discounting.

$PV$ and $FV$ are related , which reflects compounding interest (simple interest has $n$ multiplied by $i$, instead of as the exponent). Since it's really rare to use simple interest, this formula is the important one.

$F V=P V(1+i)^{n}$
FV of a single payment: The PV and FV are directly related.

$PV$ and $FV$ vary directly: when one increases, the other increases, assuming that the interest rate and number of periods remain constant.

The interest rate (or discount rate) and the number of periods are the two other variables that affect the $FV$ and $PV$. The higher the interest rate, the lower the $PV$ and the higher the $FV$. The same relationships apply for the number of periods. The more time that passes, or the more interest accrued per period, the higher the $FV$ will be if the $PV$ is constant, and vice versa.

The formula implicitly assumes that there is only a single payment. If there are multiple payments, the $PV$ is the sum of the present values of each payment and the $FV$ is the sum of the future values of each payment.