Time Value of Money

Abstract

The first steps toward understanding the relationship between the value of dollars today and that of dollars in the future is by looking at how funds invested will grow over time. This understanding will allow one to answer such questions as; how much should be invested today to produce a specified future sum of money?

Time Value of Money

In most cases, borrowing money is not free, unless it is a fiver for lunch from a friend. Interest is the cost of borrowing money. An interest rate is the cost stated as a percent of the amount borrowed per a period of time, usually one year. The current market rates are composed of three items.

The Real Rate of Interest is what compensates lenders for postponing their own spending during the term of the loan. An Inflation Premium is added to offset the possibility that inflation may eat into the value of the money during the term of the loan. In addition, various Risk Premiums are added to compensate the lender for risky loans such as unsecured loans made to borrowers with questionable credit ratings or loans that the lender may not be able to easily resell.

The first two components of the interest rate listed above, the real rate of interest and an inflation premium, together are referred to as the nominal risk-free rate. In the United States, the nominal risk-free rate is estimated by the rate of US Treasury bills.

Simple interest is calculated on the original principal only. Interest from prior periods is not used in calculations for the following periods. Simple interest is normally used for a single period loan of less than a year, such as 30 or 60 days loans. The formula is:

Simple Interest = p * i * n

where:

p = principal (original amount borrowed or loaned)

i = interest rate for one period

n = number of periods

Compound interest is calculated each period on the original principal and all interest accumulated during past periods. The interest rate normally is stated as a yearly rate. The compounding periods can also be yearly, semiannually, quarterly, or even continuously.

Think of compound interest as a series of back-to-back simple interest contracts. The interest earned in each period is added to the principal of the previous period to become the principal for the next period. For example, you borrow $10,000 for three years at 5% annual interest compounded annually. The following shows the compounding principle.

• Interest year 1 = p * i * n = 10,000 * .05 * 1 = 500

• Interest year 2 = (p2 = p1 + i1) * i * n = (10,000 + 500) * .05 * 1 = 525

• Interest year 3 = (p3 = p2 + i2) * i * n = (10,500 + 525) *.05 * 1 = 551.25

Total interest earned over the three years = 500 + 525 + 551.25 = 1,576.25. The power of compounding can have an amazing effect on the accumulation of wealth.

Money held denies its power to earn. Future Value is the amount of money that an investment made today (the present value) will grow to by some future date. Since money has time value, we obviously expect the future value to be greater than the present value. The difference between the two depends on the number of compounding periods involved and the going interest rate.

The relationship between the future value and present value can be expressed in the following formula:

FV = PV (1 + i)n

where:

FV = Future Value

PV = Present Value

i = Interest Rate per Period

n = Number of Compounding Periods

The time value of money concepts of present value and future value are like the concepts of compounding and discounting. Both are ways to describe how the value of an asset or cash at a specified