# What is the time value of money?

The time value of money refers to the fact that money we receive in the future is worth less to us than money we receive today. If you loaned us \$100 today and we paid you back the \$100 two years from now, it would not be fair to you because we have had the use of your money for two years and paid nothing to use it. If we borrowed your car for two years, you would expect us to compensate you for its use.

The correct thing for us to have done would have been to have paid you some amount of money for the use of your money for two years. Professional lenders, banks and others, do this all the time. The amount that they charge for the use of their money is called interest, and it is calculated from what is called the interest rate.

Besides knowing how much banks and other lenders will charge us for the use of their money, we also need to know the value of money we will receive in the future, the future value, when it is compared to money we receive today, the present value. For convenience we make the adjustment using the same interest rate.

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It is very important for project managers to understand the time value of money. Projects of almost any size have cash flows that occur in the future. Generally, the timing of these cash flows is far enough in the future that an adjustment of these cash flows to their present values is important enough to be considered.

The concept in this adjustment forms the same basis for the payment of compound interest for money that is loaned. When we lend money to someone for a period of time, we expect him to pay for the use of that money. Compound interest is just the amount of money we have at the end of a given period of time if we do not use any of the money or the interest it earns. After the first amount of interest is paid, we will earn additional interest on the interest as well as the money we started with.

We should not only be paid for the use of our money, but we should also be paid for the risk that we are taking by lending the money. There is a possibility that the person to whom we are lending the money will never pay us back. The most conservative estimate of the interest rate, the risk-free investment, is usually equivalent to the U.S. Government Treasury bill rate of interest. The U.S. Government is considered a risk- free investment since it can almost always be expected to pay its bills and is not expected to go out of business any time soon.

When we put a sum of money into a bank account, there is an interest rate associated with the account. The interest is usually called the annual interest rate. This is the amount of interest that will be paid if the money is left in the account for one year. The interest is usually compounded, which means that the interest earned is left in the account so that the next time the interest is calculated, the interest that was paid last time earns interest as well.

For example, if we had \$100 that was put into a 10 percent annual interest account, it would yield \$110 the first year it was left there. The second year the interest would be calculated on the \$110, so at the end of two years we would have \$121 and so on. The formula in Figure 4-3 calculates the future value of money if a sum of money, the present value, earns compound interest at an interest rate r for n periods. It should be noted that the value of n and r must be compatible. If the time periods are measured in years, then r must be in terms of the annual interest rate. If the value of n is in months, days, weeks, or any other time period, the interest rate r must be in terms of interest per month, day, week, and so on. Most bank interest rates are given in annual interest rates, but the interest is paid more often, monthly, quarterly, weekly, or even daily.

Suppose we put \$10 in a bank account that pays 10 percent annual interest monthly. What amount of interest would we earn at the end of two years? Since the interest is paid each month, we must calculate it each month. If the interest is 10 percent per year, one month of interest is 10/12 percent. In the first month our \$10 would earn \$0.083. The second month we would calculate the interest on \$10.083 instead, and we would earn \$0.084 for a total of \$10.167, and so on for the next twenty- two periods. We would earn \$22.29 in interest. Figure 4-3: Compound interest: Calculating the future value of money

If we take the equation for the future value of money and solve it Figure 4-4: Discounting: Calculating the present value of money

for the present value of money, we will be able to calculate the present value of money that we will receive in the future (see Figure 4-4). This is a little more abstract. What it says is that if I receive money in the future, the value of that money to me today is going to be somewhat less. Suppose I am to receive a \$10,000 payment from a customer two years from now and the interest rate that is being used to borrow money is 10 percent. The future value is \$10,000, r is 10 percent and the value of n is 2 (see Figure 4-5).

Let us make one more example. Suppose we have a project that will require an investment of \$1,000,000. The project has estimated cash inflows of \$750,000 per year for the next three years. If we calculate the present value for these cash flows, we end up with the numbers in Figure 4-6.

In this example we have a net cash flow at the end of the third year of \$1,250,000. This is a pretty good return on our million-dollar investment. The net cash flow is the sum of the inflows and the outflows.

Suppose we adjust the cash flows to their present values, as shown in Figure 4-7.

The net cash flow adjusted for net present value is only \$865,100. This value is substantially less than the \$1,250,000 that we had without adjusting the future values for present values. It is important to note that this adjustment is very important for anticipating the cash flows that actually occur in projects. Figure 4-5: Discounting: Calculating the present value of money

These adjustments are very real. Since most companies depend on borrowed money or stockholder investments for their operating funds, it is correct to assume that they have to pay something for the use of these funds. When we make an investment in a project because we expect to receive some future returns on that investment, we should adjust the money being received in the future to something less because of the Figure 4-6: Example: Simple cashflow Figure 4-7: Example: Cashflow adjusted to present value

time that we do not have the money. Remember that any money received in the future will be worth less than money we receive today.