# Time Value of Money Formula

### Calculating the Present and Future Values of a Single Sum The future or present value of money can be calculated given the variables of the number of periods (time), interest or discount rates, the amount invested, or the future value of money.

The future value (FV) calculation allows investors to predict, with a very high degree of accuracy, the amount of profit that can be generated by varying investments. The amount of growth earned by holding a given amount in cash will most likely be different than if that same amount were invested in stocks or other equities. The FV formula is used to compare multiple options and scenarios.

When applying the FV formula, the Present Value (PV) must first be calculated and then the rate and time period in which interest is earned, or compounded, is required to complete the calculation. As a result, the PV is multiplied times a constant of 1 plus the rate of return with an exponent of how many periods of interest is earned or compounded.

There is a formula for calculating future value.
$\text{FV}=\text{PV}\;(1+r)^n$
Where:
\begin{aligned}\text{FV}&=\text{Future Value}\\\text{PV}&=\text{Present Value}\\r&=\text{Interest Rate}\\n&=\text{Time Period}\end{aligned}
For example, assume Mary Nelson wants to calculate the future value of an investment and puts $1,000 into a bank for five years at 5 percent interest compounding annually, or $\1{,}000\;(1+0.05)^5=\1{,}276.28$. This means that Mary knows that making the investment at 5 percent will leave her with$1,276.28 at the end of five years. There is a formula for present value.
$\text{PV}=\frac{\text {FV}}{(1+r)^n}$
Where:
\begin{aligned}\text{PV}&=\text{Present Value}\\\text{FV}&=\text{Future Value}\\r&=\text{Interest Rate}\\n&= \text{Time Period}\end{aligned}
The present value calculation is useful in planning for a future expense, such as college tuition. For example, Jim and Susan Smith might want to know how much they need to invest in a certificate of deposit (CD) that pays 5 percent interest compounding annually to have 25,000 at the end of 10 years, which hypothetically is what will be needed to pay for their child's first year of college at that time. Present value can be used to determine the amount that’s needed to invest at the current time to have the necessary funds in 10 years. \begin{aligned}\text{PV}&=\frac{\25{,}000}{(1\;+\;0.05)^{10}}\\\\&=\15{,}347.83\end{aligned} Investors can calculate the future value or present value of money to better understand their investment options or to make a decision on whether or not to take out a loan. For example, consider the time value of money impact of saving for retirement if the retiree wants to have a certain lump sum saved at their retirement age of 65. An investor that saves2,400 annually from age 19 to 26 and then stops investing new money but lets the current investment continue to grow at a 12 percent annual growth rate will have a lump sum of $2,523,474 at age 65. The growth of the investment for ages 19 to 26 can be calculated using a spreadsheet by inputting numbers within the future value formula. Calculations can also be completed using a financial calculator or manually by using the future value formula. ### Calculating Future Value Using Spreadsheets Date Investment Future Value Formula Future Value Jan 1 − Age 19$2,400 $=\2{,}400\;(1+0.12)^1$ $2,688.00 Jan 1 − Age 20$2,400 $=\2{,}400\;(1+0.12)^1+\2{,}688\;(1+0.12)^1$ $5,698.56 Jan 1 − Age 21$2,400 $=\2{,}400\;(1+0.12)^1+\5{,}698.56\;(1+0.12)^1$ $9,070.39 Jan 1 − Age 22$2,400 $=\2{,}400\;(1+0.12)^1+\9{,}070.39\;(1+0.12)^1$ $12,846.84 Jan 1 − Age 23$2,400 $=\2{,}400\;(1+0.12)^1+\12{,}846.84\;(1+0.12)^1$ $17,076.46 Jan 1 − Age 24$2,400 $=\2{,}400\;(1+0.12)^1+\17{,}076.46\;(1+0.12)^1$ $21,813.64 Jan 1 − Age 25$2,400 $=\2{,}400\;(1+0.12)^1+\21{,}813.64\;(1+0.12)^1$ $27,119.28 A variety of formulas can be used in spreadsheets that simplify calculations such as determining future value. An investor who saves$2,400 annually from age 27 to age 65 and earns the same 12 percent annual growth rate will have a total of $1,838,619 by age 65. Even with years more of investing than the early investor, the late investor will have earned $\2{,}523{,}449-\1{,}838{,}619=\684{,}830$ less than if they had invested earlier. The growth of the investment for ages 27 to 65 can be calculated using a spreadsheet by inputting a numbers within the future value formula. Calculations can also be completed using a financial calculator or manually by using the future value formula. ### Using Spreadsheets to Calculate Cumbersome Future Value Calculations Date Investment Future Value Formula Future Value Jan 1 − Age 27$2,400 $=\2{,}400\;(1+0.12)^1$ $2,688.00 Jan 1 − Age 28$2,400 $=\2{,}400\;(1+0.12)^1+\2{,}688\;(1+0.12)^1$ $5,698.56 Jan 1 − Age 29$2,400 $=\2{,}400\;(1+0.12)^1+\5{,}698.56\;(1+0.12)^1$ $9,070.39 Jan 1 − Age 30$2,400 $=\2{,}400\;(1+0.12)^1+\9{,}070.39\;(1+0.12)^1$ $12,846.84 Jan 1 − Age 31$2,400 $=\2{,}400\;(1+0.12)^1+\12{,}846.84\;(1+0.12)^1$ $17,076.46 Jan 1 − Age 32$2,400 $=\2{,}400\;(1+0.12)^1+\17{,}076.46\;(1+0.12)^1$ $21,813.64 [As time passes, the formula is the same each year.] Jan 1 − Age 64$2,400 $=\2{,}400\;(1+0.12)^1+\1{,}461{,}193.28\;(1+0.12)^1$ $1,639,224.47 Jan 1 − Age 65$2,400 $=\2{,}400\;(1+0.12)^1+\1{,}639{,}224.47\;(1+0.12)^1$ 1,838,619.41 Spreadsheets are an often used method to calculate cumbersome future value calculations to help investors make investment decisions. ### Calculating the Present and Future Values of an Annuity The annual payout of an annuity can be calculated as long as the principal and interest rate are known. Calculating the future value of an annuity can also be helpful. An annuity is a fixed, regular payment stream paid in the same amount over a period of time. This is similar to how Social Security monthly payments work. When retirement planning, some people want to know how much they need to save to invest in an annuity in order to have a guaranteed monthly income stream at some point in the future. There is a formula for calculating the annual payout of an annuity. $\text{AP}=\frac{\text{Principal}\times r\times(1+r)^n}{\lbrack(1+r)^n-1\rbrack}$ Where: \begin{aligned}\text{AP}&=\text{Annual Payout}\\\text{Principal}&=\text{Amount Initially Invested}\\r&=\text{Rate}\\n&=\text{Time Period}\end{aligned} As an example, assume that a retiree would like to invest100,000 and receive it in equal annual payments for the next 10 years. Also, assume that the funds currently return a 5 percent rate of interest.
\begin{aligned}\text {Annual Payout}&=\frac{\100{,}000\times0.05\times(1+0.05)^{10}}{\lbrack(1+0.05)^{10}-1\rbrack}\\\\&=\12{,}950.46\end{aligned}
The retiree can invest $100,000 now and withdraw$12,950.46 annually for the next 10 years while in retirement.

### Using Spreadsheets to Perform Financial Calculations Investors can use financial calculators to perform various financial calculations. These calculations can also be done easily using spreadsheets.
Using spreadsheets allows investors to try different savings amounts and interest rates to see how certain choices can affect their ultimate return. To use spreadsheets for these calculations, investors need to know which spreadsheet functions are equivalent to the financial calculator keys.

Formula Being Calculated Spreadsheet Function Calculator Key
Determine future value FV (rate, nper, pmt, pv, type) FV
Determine present value PV (rate, nper, pmt, fv, type) PV
Determine annuity payment PMT (rate, nper, pv, fv, type) PMT

Spreadsheets can be used as financial calculators and can make it easier for investors to evaluate various investments.

For example, if Mary has $500 to invest for five years at an interest rate of 8 percent per year, how much will she accumulate by the end of the period? A spreadsheet can be used to easily calculate the result. ### Spreadsheet File A B C 1 Present Value$500
2 # of Years 5
3 APR 0.08%
4 Future Value \$735 = FV (B3, B2, B1)

Spreadsheets can be used to easily calculate future value using present value, number of years, and APR. In some situations “type” or when payments are due can be included in future value calculations, but it is not always relevant if payment due dates are not being considered.

Mary is able to make this calculation using the formula FV (rate, nper, pmt, pv, type). Typing in the formula and then selecting the appropriate cells and hitting enter returns the result. By using a spreadsheet, Mary can easily see what the impact would be of a different annual rate, a different present value, or saving for more years. This can aid in the decision-making process by allowing Mary to substitute in different numbers and easily to try out different scenarios when making an investment decision.