## Concepts

Time Value of Money (TVM) is an important concept in financial management. It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities.

TVM is based on the concept that a dollar that you have today is worth more than the promise or expectation that you will receive a dollar in the future. Money that you hold today is worth more because you can invest it and earn interest. After all, you should receive some compensation for foregoing spending. For instance, you can invest your dollar for one year at a 6% annual interest rate and accumulate \$1.06 at the end of the year.  You can say that the future value of the dollar is \$1.06 given a 6% interest rate and a one-year period. It follows that the present value of the \$1.06 you expect to receive in one year is only \$1.

A key concept of TVM is that a single sum of money or a series of equal, evenly-spaced payments or receipts promised in the future can be converted to an equivalent value today.  Conversely, you can determine the value to which a single sum or a series of future payments will grow to at some future date.

You can calculate any one of the following five TVM values when the other four are known: Interest Rate, Number of Periods, Payments, Present Value, and Future Value.  The first three are defined below, PV and FV will be further illustrated by examples.

### Interest

Interest is a charge for borrowing money, usually stated as a percentage of the amount borrowed over a specific period of time.

Simple interest is computed only on the original amount borrowed. It is the return on that principal for one time period.

In contrast, compound interest is calculated each period on the original amount borrowed plus all unpaid interest accumulated to date.  Compound interest is always assumed in TVM problems.

### Number of Periods

Periods are evenly-spaced intervals of time. They are intentionally not stated in years since each interval must correspond to a compounding period for a single amount or a payment period for an annuity.

### Payments

Payments are a series of equal, evenly-spaced cash flows.  In TVM applications, payments must represent all outflows (negative amount) or all inflows (positive amount).

Interest

Simple interest may be thought of as rent paid on borrowed money. Simple interest is calculated only on the beginning principal. For instance, if someone were to receive 5% interest on a beginning value of \$100, the first year they would get:

 .05 × \$100   – or –   \$5 in interest

If they continued to receive 5% interest on the original \$100 amount, over five years the growth in their investment would look like this:

 Year 1:  5% of \$100 = \$5 + \$100 = \$105 Year 2:  5% of \$100 = \$5 + \$105 = \$110 Year 3:  5% of \$100 = \$5 + \$110 = \$115 Year 4:  5% of \$100 = \$5 + \$115 = \$120 Year 5:  5% of \$100 = \$5 + \$120 = \$125

Compound interest is another matter. It’s good to receive compound interest, but not so good to pay compound interest. With compound interest, interest is calculated not only on the beginning interest, but on any interest accumulated in the meantime.

For instance, if someone were to receive 5% compound interest on a beginning value of \$100, the first year they would get the same thing as if they were receiving simple interest on the \$100, or \$5. The second year, though, their interest would be calculated on the beginning amount in year 2, which would be \$105. So their interest would be:

 .05 × \$105   – or –   \$5.25 in interest

This provides a balance at the end of year two of \$110.25

If this were to continue for 5 years, the growth in the investment would look like this:

 Year 1:  5% of \$100.00 = \$5.00 + \$100.00 = \$105.00 Year 2:  5% of \$105.00 = \$5.25 + \$105.00 = \$110.25 Year 3:  5% of \$110.25 = \$5.51 + \$110.25 = \$115.76 Year 4:  5% of \$115.76 = \$5.79 + \$115.76 = \$121.55 Year 5:  5% of \$121.55 = \$6.08 + \$121.55 = \$127.63

Note that in comparing growth graphs of simple and compound interest, investments with simple interest grow in a linear fashion and compound interest results in geometric growth. So with compound interest, the further in time an investment is held the more dramatic the growth becomes.

 Simple Interest Compound Interest Present and Future Values

Source: http://teachmefinance.com/timevalueofmoney.html
Accessed: 04/24/2017.

You accumulate \$100 and find an investment that will pay 10% interest per year for 2 years:

• The Present Value = \$ 100
 Year 1:  10% of \$100.00 = \$10.00 + \$100.00 = \$110.00 Year 2:  10% of \$110.00 = \$11.00 + \$110.00 = \$121.00
• Future Value = \$121.
 FV= PV (1 + i )N
• FV = Future Value
• PV = Present Value
• i = the interest rate per period
• n= the number of compounding periods

What is the future value of \$34 in 5 years if the interest rate is 5%?

• FV= PV ( 1 + i ) N
• FV= \$ 34 ( 1+ .05 ) 5
• FV= \$ 34 (1.2762815)
• FV= \$43.39.

Let’s now look at the opposite situation, where you know some future value and want to know what you need to invest today (PV) to get there.

If your goal is to have \$1,000 (FV) available in 5 years, how much money should you invest today in an investment that is paying 6%?

• FV= PV ( 1 + i ) N
• \$1000 = PV ( 1 + .06) 5
• \$1000 = PV (1.338)
• \$1000 / 1.338 = PV
• \$ 747.38 = PV