Understanding the Time Value of Money

The time value of money is a financial concept that states that a dollar is worth more today than it will be worth in the future. Money you have now can be invested for a financial return and the impact of inflation will reduce the future value of the same amount.

What Is the Time Value of Money?

Assume that you've won a cash prize and you have two options available to you. You can receive $10,000 now or you can have $10,000 in three years.

You'd choose to receive the $10,000 now if you're like most people. Why would any rational person defer payment into the future when they could have the same amount of money now? Taking the money in the present is just plain instinctive for most people. The time value of money demonstrates that it seems better to have money now rather than later.

A $100 bill has the same value as a $100 bill one year from now...or does it? The bill is the same but you can do much more with the money now. You can earn more interest on it over time. You're poised to increase the future value of your money by investing and gaining interest over three years if you receive $10,000 today.

A payment received in three years would be your future value. It would still be only $10,000.

Future Value Basics

The future value of your investment at the end of the first year would be $10,450 if you choose Option A and take the money now, investing the total amount at a simple annual rate of 4.5%. We arrive at this sum by multiplying the principal amount of $10,000 by the interest rate and then adding the interest gained to the principal amount:

$\begin{aligned} &\$10,000 \times 0.045 = \$450 \\ \end{aligned}$​$10,000×0.045=$450​

$\begin{aligned} &\$450 + \$10,000 = \$10,450 \\ \end{aligned}$​$450+$10,000=$10,450​

You can also calculate the total amount of a one-year investment with a simple manipulation of the above equation:

$\begin{aligned} &\text{OE} = ( \$10,000 \times 0.045 ) + \$10,000 = \$10,450 \\ &\textbf{where:} \\ &\text{OE} = \text{Original equation} \\ \end{aligned}$​OE=($10,000×0.045)+$10,000=$10,450where:OE=Original equation​

$\begin{aligned} &\text{Manipulation} = \$10,000 \times [ ( 1 \times 0.045 ) + 1 ] = \$10,450 \\ \end{aligned}$​Manipulation=$10,000×[(1×0.045)+1]=$10,450​

$\begin{aligned} &\text{Final Equation} = \$10,000 \times ( 0.045 + 1 ) = \$10,450 \\ \end{aligned}$​Final Equation=$10,000×(0.045+1)=$10,450​

The manipulated equation above is a removal of the like-variable $10,000 (the principal amount) by dividing the entire original equation by $10,000.

How much would you have if the $10,450 left in your investment account at the end of the first year remains untouched and you invested it at 4.5% for another year? Take the $10,450 and multiply it again by 1.045 (0.045 +1). You would have $10,920.25 at the end of two years.

Calculating Future Value

The above calculation is equivalent to the following equation:

$\begin{aligned} &\text{Future Value} = \$10,000 \times ( 1 + 0.045 ) \times ( 1 + 0.045 ) \\ \end{aligned}$​Future Value=$10,000×(1+0.045)×(1+0.045)​

The rule of exponents states that the multiplication of like terms is equivalent to adding their exponents. The two like terms in this equation are (1+ 0.045) and the exponent on each is equal to 1. The equation can therefore be represented like this:

$\begin{aligned} &\text{Future Value} = \$10,000 \times ( 1 + 0.045 )^2 \\ \end{aligned}$​Future Value=$10,000×(1+0.045)2​

The exponent is equal to the number of years for which the money is earning interest in an investment. The equation for calculating the three-year future value of the investment would therefore look like this:

$\begin{aligned} &\text{Future Value} = \$10,000 \times ( 1 + 0.045 )^3 \\ \end{aligned}$​Future Value=$10,000×(1+0.045)3​

We don't have to keep calculating the future value after the first year, however. You can figure it all at once. You can calculate the future value (FV) of that amount if you know the present amount of money you have in an investment, its rate of return, and how many years you would like to hold that investment:

$\begin{aligned} &\text{FV} = \text{PV} \times ( 1 + i )^ n \\ &\textbf{where:} \\ &\text{FV} = \text{Future value} \\ &\text{PV} = \text{Present value (original amount of money)} \\ &i = \text{Interest rate per period} \\ &n = \text{Number of periods} \\ \end{aligned}$​FV=PV×(1+i)nwhere:FV=Future valuePV=Present value (original amount of money)i=Interest rate per periodn=Number of periods​

Present Value Basics

Its present value would be $10,000 if you received $10,000 today because the present value is what your investment gives you now if you were to spend it today. The present value of the amount wouldn't be $10,000 if you were to receive $10,000 in one year because you don't have the money in your hand now in the present.

You must pretend that the $10,000 is the total future value of an amount you invested today to find the present value of the $10,000 you'll receive in the future. We must determine how much we would have to invest today to receive that $10,000 in one year if we want to find the present value of the future $10,000.

Subtract the hypothetical accumulated interest from the $10,000. We can discount the future payment amount of $10,000 by the interest rate for the period. All you're doing is rearranging the future value equation above so you may solve for the present value (PV). The above future value equation can be rewritten as follows:

$\begin{aligned} &\text{PV} = \frac{ \text{FV} }{ ( 1 + i )^ n } \\ \end{aligned}$​PV=(1+i)nFV​​

An alternate equation would be:

$\begin{aligned} &\text{PV} = \text{FV} \times ( 1 + i )^{-n} \\ &\textbf{where:} \\ &\text{PV} = \text{Present value (original amount of money)} \\ &\text{FV} = \text{Future value} \\ &i = \text{Interest rate per period} \\ &n = \text{Number of periods} \\ \end{aligned}$​PV=FV×(1+i)−nwhere:PV=Present value (original amount of money)FV=Future valuei=Interest rate per periodn=Number of periods​

Calculating Present Value

Remember that the $10,000 to be received in three years is the same as the future value of an investment. We would discount the payment back one year if we had a year to go before receiving the money. The present value of the $10,000 to be received in one year would be $10,000 x (1 + .045)^-1 = $9569.38 using our present value formula at the current two-year mark.

The above $9,569.38 would be considered the future value of our investment one year from now if we were at the one-year mark today.

We would be expecting at the end of the first year to receive the payment of $10,000 in two years. The calculation for the present value of a $10,000 payment expected in two years would be $10,000 x (1 + .045)^-2 = $9,157.30 at an interest rate of 4.5%

We don't have to calculate the future value of the investment every year counting back from the $10,000 investment in the third year because of the rule of exponents. We could use the $10,000 as FV:

$\begin{aligned} &\$8,762.97 = \$10,000 \times ( 1 + .045 )^{-3} \\ \end{aligned}$​$8,762.97=$10,000×(1+.045)−3​

The present value of a future payment of $10,000 is worth $8,762.97 today if interest rates are 4.5% per year. Choosing Option B is like taking $8,762.97 now and investing it for three years. These equations illustrate that Option A is better not only because it offers you money right now but also because it offers you $1,237.03 ($10,000 - $8,762.97) more in cash.

Your choice gives you a future value that's $1,411.66 ($11,411.66 - $10,000) greater than the future value of Option B if you invest the $10,000 that you receive from Option A.

Present Value of a Future Payment

What if the future payment is more than the amount you'd receive right away? You could receive either $15,000 today or $18,000 in four years. The decision is now more difficult. You may end up with an amount of cash in four years that's less than $18,000 if you choose to receive $15,000 today and invest the entire amount.

Let's find the present value of $18,000 in this situation. We'll assume that interest rates are currently 4%. Remember that this is the equation for the present value:

$\begin{aligned} &\text{PV} = \text{FV} \times ( 1 + i )^{-n} \\ \end{aligned}$​PV=FV×(1+i)−n​

All we're doing in this equation is discounting the future value of an investment. The present value of an $18,000 payment in four years would be calculated as $18,000 x (1 + 0.04)^-4 = $15,386.48.

We now know that our choice today is between opting for $15,000 or $15,386.48.

The Bottom Line

These calculations demonstrate that time is literally money. The value of the money you have now isn't the same as it will be in the future. It's important to know how to calculate the time value of money so you can distinguish between money-related options offered to you now and in the future. These options could be investment opportunities, loan transactions, mortgage payment options, or even donations to charities.