The time value of money (TVM) is the basic mathematics of investing and is the basis of all financial calculations. Most importantly, it explains the concept of compound returns, which causes investments to grow exponentially over time. An understanding of the time value of money is crucial to understanding the expected and actual returns on investments and analyzing past returns on investments. It's something every investor should know well, and being armed with a good financial calculator will help. Financial calculators are indispensable to investors when it comes to doing anything that involves the time value of money.

Return can be the interest earned on a bond, the rate of return realized on a stock or mutual fund over its holding period, the rate of return a corporation earns on invested capital and more. In finance, the letter r is used to represent the rate of return. In some references you will find the letter k is used instead of r. Unfortunately, no one told the calculator manufacturers about this. You'll usually find that i is used instead of r on financial calculators. In the interest of continuity and because r is a more logical representation of return than k, I have used r.

The basic time value of money formula, which applies the concept of compound return, is:

_{n}= PV(1 + r)

^{n}

PV = Present Value

r = Periodic rate of return

n = Number of compounding periods

This formula computes the future value (FV) of a current amount (PV) earning a compound rate of return r for n periods. Compounding can also be continuous, but that requires a formula that is derived by solving an infinite series. You'll find that formula farther down this page.

Although we tend to think of returns as being annual returns, the compounding period can be any fixed duration, such as days, weeks, months or years. For instance, you've probably noticed that the APR of monthly credit card interest is greater than 12 times the monthly rate. That's explained by the fact that the compounding period is monthly. If the monthly rate is 1.5%, then the APR will be 19.56% which is greater than 12 x 1.5% = 18.00%.

(1 + 0.015)^{12}
-1 = 0.1956 = 19.56%

The 1 in the formula represents the beginning value, thus it must be subtracted to find the compound return, which in this case is the credit card company's return.

This formula can be used to compare investment opportunities.
Suppose you have the option of investing in one of two mutual funds
which are identical in every respect except the management fees they
charge. Both funds have a ten year historical rate of return of 12% and
they also have identical standard deviations (risk). However, fund B
charges an annual management fee of 1.25% but fund A only charges
0.75%. (Of course, the usual disclaimer applies: Past performance is
not necessarily indicative of future results.) As the two funds are
equal in every way except the management fee and management fees must
be deducted from performance to determine return, you will obviously
choose fund A. So lets see how much better you'll do over the years
with fund A assuming your initial investment is $10,000, that the
funds' future performance is equivalent to their past performance and
all distributions are reinvested.

There are a few things you should note about this example.

- Both curves are upward-sweeping, i.e., they are exponential curves.
- In both cases, the value of your investment increases at an increasing rate, which is expected given the basic time value of money formula for compound return.
- The 0.50% difference in management fees results in an 11.9% difference in the value of the investments after 25 years.
- The 11.25% compound rate of return causes fund A to grow to $143,714 in 25 years, which is a total return of $133,714 or 1337%, compared to 1184% for fund B.

Here's how the total return on fund A looks in the compound return formula, where ROI = return on investment:

Fund A has an
11.25% annual ROI for 25 years.

_{n}= PV(1 + r)

^{n}= $10,000(1 + 0.1125)

^{25}

= $10,000 x 14.37 = $143,714

$143,714 - $10,000 = $133,714 Total ROI

$133,714 ÷ $10,000 = 13.37

13.37 x 100% = 1337% Total ROI

That, ladies and gentlemen, is the power of compounding. With time, compounding will turn a modest investment into a tidy sum and a small difference in the rate of return can become a significant gap. I can't overemphasize the importance of these two points to you as investors.

The following chart shows how the value of the S&P 500 index has grown exponentially since 1950. (The value of the index is net of dividends.)

Click on the chart for a full-screen view. (pdf file opens in new window.)

The red line was plotted from the actual values of the S&P 500 index at the end of each month. The blue line is what the index would have been if the rate of return had been constant over the period January 1950 through September 2007 and reflects a compound annual rate of return of 8.13% computed from monthly returns. The average annual return for the period was 8.84%, also computed from monthly returns. (The difference in these two numbers is explained in the subsection on mutual fund returns.) In 57 years and nine months, the S&P 500 grew from a value of 16.71 to 1526.75, which is an excellent example of the power of compounding. The dividend yield would add approximately 2% to each of these returns, thus making the compound growth of an investment in the index even greater than growth of the index itself.

You should note that the compound return formula works in reverse.

^{1/n}-1 = r

(14.3714)

^{1/25}=1.1125

($143,714 ÷ $10,000)

^{1/25}=(14.3714)

^{1/25}=1.1125

1 represents the original investment, so it must be

subtracted to determine the rate of return:

0.1125 x 100% = 11.25%

You'll need either a financial or scientific calculator, or a spreadsheet, to perform calculations involving the time value of money. All of them should have the functions used in the formulas above.

Continuous compounding falls under the heading of the time
value of money. The formula for continuous compounding is:

_{n}= PV(e

^{r n})

PV = Present Value

r = Periodic rate of return

n = Number of compounding periods

e is the base of natural logarithms,2.71828

Continuous compounding was big in the early 1980s when the
T-Bill rate was as high as 17%. At that time, inflation was so high
that it was extremely important to keep your cash someplace where it
would earn a reasonable rate of interest. Many banks offered continuous
compounding on their interest-bearing checking accounts in an attempt
to lure new customers, and it worked because people wanted to get the
very highest rate of return possible in an attempt to keep up with
inflation.

The formula for the present value of a future amount is derived by simply rearranging the terms of the future value formula:

PV = FV_{n}
÷ (1 + r)^{n}

The other time value of money formulas also are based on the
principle of compound return. You need to know how to use the preceding
formulas because they are used for much more than just computing
present and future value. However, you shouldn't ever have to use these
additional formulas, as there are calculator and spreadsheet functions
and tables that can be used for all calculations involving the time
value of money. Some of them are used in the valuation of bonds and
also in some stock valuation models. Those used in bond valuation are
included in the bond functions in financial calculators, but it's good
to know how those functions work. The other formulas are as follows:

Future Value of an Ordinary Annuity

(Payments due end of period.)

_{n}= PMT[((1 + r)

^{n}- 1) ÷ r]

Future Value of an Annuity Due

(Payments due at beginning of period.)

_{n}= PMT[((1 + r)

^{n}- 1) ÷ r](1 + r)

This formula computes the FVA at the time the nth payment is made.

Present Value of an Ordinary Annuity

(Payments due at end of period.)

_{n}= (1 - (1 + r)

^{-n}) ÷ r

Present Value of an Annuity Due

(Payments due at beginning of period.)

_{n}= [(1 - (1 + r)

^{-n}) ÷ r](1 + r)

The value of a bond is the sum of the present value of an ordinary annuity (the coupon payments) and the present value of a future amount (the return of your principal). Like many things in finance, bond calculations involve computing a missing piece of the puzzle with the rest of the pieces. If you know the time to maturity, coupon payment, face value and current market interest rate, you can compute the current value of the bond. It's all based on the time value of money.

Perpetuities are an interesting but archaic instrument involving the time value of money. They make a constant periodic payment forever. Say you sell 1000 acres of your farm to the state fish and game commission and they pay you with a perpetuity, which will provide income to your heirs forever. The value of the perpetuity is the payment you and your heirs will receive divided by the opportunity rate of return, i.e., the rate of return you could make on investments if the state paid you a lump sum rather than a perpetuity. The payment might look good now, but in 50 years it will be peanuts. Why? Plug a future value, r = the current rate for intermediate bonds and n = 50 into the formula for the present value of a future amount and see what happens to that payment 50 years hence. Which brings us to the final point.

Corporations commonly use net present value for evaluating investments or estimating the value of their assets. But present value analysis is only valid in the short term, as you should know if you tried what I suggested in the prior paragraph. If a company owns assets like forest land that is harvested on a 30 year rotation, harvests 30 years in the future don't appear to be worth much and harvests 60 years in the future have almost no value by present value analysis, which is counterintuitive when one considers the basic concept of the time value of money. Many analysts picked up on this in the 1980s and found some real gems, like railroads that had tens of thousands of acres that they carried on their books at their 19th century acquisition costs. And there are undoubtedly more to be found. Also in the 1980s, a lot of timber land was sold to developers by forest products companies that used present value to assess the value of their holdings. So beware of companies making such suboptimal decisions and keep an eye out for those undiscovered gems. This is an example of the inappropriate use of the time value of money.