Understanding the Time Value of Money (TVM)

Investors prefer to receive money today rather than the same amount of money in the future because a sum of money, once invested, grows over time. For example, money deposited into a savings account earns interest. Over time, the interest is added to the principal, earning more interest. That's the power of compounding interest. 

If it is not invested, the value of the money erodes over time. If you hide $1,000 in a mattress for three years, you will lose the additional money it could have earned over that time if invested. It will have even less buying power when you retrieve it because inflation has reduced its value.

As another example, say you have the option of receiving $10,000 now or $10,000 two years from now. Despite the equal face value, $10,000 today has more value and utility than it will two years from now due to the opportunity costs associated with the delay. 

In other words, a payment delayed is an opportunity missed.

Formula for Time Value of Money

Depending on the exact situation, the formula for the time value of money may change slightly. For example, in the case of annuity or perpetuity payments, the generalized formula has additional or fewer factors. But in general, the most fundamental TVM formula takes into account the following variables:

• FV = Future value of money
• PV = Present value of money
• i = interest rate
• n = number of compounding periods per year
• t = number of years

Based on these variables, the formula for TVM is:

FV = PV x [ 1 + (i / n) ] ^{(n x t)}

Time Value of Money Examples

Assume a sum of $10,000 is invested for one year at 10% interest compounded annually. The future value of that money is:

FV = $10,000 x [1 + (10% / 1)] ^ (1 x 1) = $11,000

The formula can also be rearranged to find the value of the future sum in present day dollars. For example, the present day dollar amount compounded annually at 7% interest that would be worth $5,000 one year from today is:

PV = $5,000 / [1 + (7% / 1)] ^ (1 x 1) = $4,673

Effect of Compounding Periods on Future Value

The number of compounding periods has a dramatic effect on the TVM calculations. Taking the $10,000 example above, if the number of compounding periods is increased to quarterly, monthly, or daily, the ending future value calculations are:

• Quarterly Compounding: FV = $10,000 x [1 + (10% / 4)] ^ (4 x 1) = $11,038
• Monthly Compounding: FV = $10,000 x [1 + (10% / 12)] ^ (12 x 1) = $11,047
• Daily Compounding: FV = $10,000 x [1 + (10% / 365)] ^ (365 x 1) = $11,052

This shows TVM depends not only on the interest rate and time horizon but also on how many times the compounding calculations are computed each year.