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# Time value of money

Money has a time value. This means a dollar received today has more value to us than a dollar received far in the future. Other than a desire for instant gratification, there is a very rational reason for this phenomenon. If we have a dollar today, we can put it to work by making an investment and have more than a dollar at some future date. This concept of putting the money to work has important implications later in this section when discount rates are discussed.

## Concept of equivalence

Another important concept is the concept of equivalence between a current lump sum of money and a lump sum to be received in the future. Offering someone a choice between receiving \$100 today and receiving \$101 one year from today can demonstrate this. Most people will opt for the \$100 today. If we increase the amount of future money to \$115 or \$125 or perhaps \$200 and guarantee payment, there will be a point at which the future sum of money will become more attractive than the current \$100. The amount of future money necessary to sway the person to choose the future sum is dependent upon many things, which include, among others:

• The inflation rate
• Current opportunities to invest the \$100
• Perceived risk

No matter what the amount of money necessary to tip the scales, the concept that money has a time value is established.

In the case just discussed, if the person is indifferent to receiving \$125 one year from now or \$100 now, we say that the two sums are "equivalent." This concept of equivalence is fundamental to the evaluation of all engineering projects. We are often faced with the choice of having a certain sum of money now or receiving various sums of money in the future. By determining the equivalence between money received today and money received in the future, we can make an informed decision.

In the previous example, making the choice is relatively simple. There are only two sums to compare, and the time period is one year. This is usually not the case in oil/gas property evaluations, so we need a mechanism to handle complex choices. The mechanism that works best is interest.

We can define interest as the amount of money that must be added to our current sum to make an equivalent future sum. The amount of interest necessary to create equivalence is dependent upon the period under consideration. We may be indifferent to receiving:

• \$100 now
• \$112.50 six months from now
• \$125 one year from now

In that case, the \$12.50 or \$25.00 is the amount of interest. To easily compare all three alternatives, interest is compared as a rate. It may be expressed as 12.5% per six months or 25% per year. The interest rate is calculated by dividing the amount of interest paid per period by the principal amount at the beginning of the period. Often, interest rate is expressed on an annual basis such as 25% per annum.

For loans or bank deposits that exceed one year, the interest is usually compounded. That is, the interest earned during the first period is added to the original principal to form the principal for the second period. The compound interest concept will be used when calculating the equivalence between a sum of money today and future sums of money.

### Future worth of a lump sum

If we have a present sum of money, P, and we put it to work at a compound interest rate, i, we will have a future lump sum of money, F, at the end of n periods. The relationship between these parameters is expressed in equation form as ....................(1)

The term (1 + i)n is called the single payment compound amount factor in many texts and is often tabulated.

#### Example 1: Future value

\$1,000 is placed in a bank paying 12% per compounding period. How much money will be in the account after five periods?

F = P(1 + i)n.

F = \$ 1,000(1 + 0.12)5.

F = \$ 1,762.

### Present worth of lump sum

Present worth of lump sum is by far the most important equation in discussing the time value of money. This one equation allows the creation of an equivalence between future projected net cash flows and current sums of money, which can then be compared to the amount to be invested to obtain those net cash flows.

If an amount, F, is going to be received n periods in the future, then its present value, P, can be calculated for a given interest rate, i, by rearranging Eq.1. ....................(2)

This is the inverse of the single payment compound amount formula. The justification for this formula lies in the equivalence concept. The sums P and F are equivalent to each other because P could be invested at i for n periods to become F. The term (1 + i)n is often referred to as the:

• "Single payment present worth factor"
• "Discount factor"

The value of n in the previous equations does not have to be an integer. Although there are some theoretical difficulties, it is quite practical to use a value of 2.5 for n to create an equivalence between a future lump sum received 2.5 periods in the future and a present lump sum. This technique is used quite often when calculating the present value of annual cash flow streams.

#### Example 2: Present value

\$1,762 will be received five periods from now. What is the present value of this amount at an interest rate of 12% per period? ....................(3)

When cash flows are calculated for several periods, as done in Fig.1, Eq.2 can be used repeatedly to find the equivalent present value of each of the future cash flows.

### Annuities and loans

There are a number of specialized equations that can be used when particular types of repetitive cash flows are projected. In particular, when the same cash flow, A, is received at the end of every period, the present value of the cash flow stream can be calculated from the equation, ....................(4)

Eq.3 is often called the annuity equation because it can be rearranged to calculate the value of A, which is the amount of money one would receive at the end of every n period if one invested P at an interest rate of i. It is also used to calculate loan payments where P is the principal amount.

#### Example 3: Loan

What are the monthly payments on a \$100,000 loan with a term of 360 months (30 years) at an interest rate of 1% per month compounded monthly? ....................(5) ....................(6)

There are a number of other specialized equations, but they are of limited use in today’s era of fast computers. See Thompson and Wright, Chap. 2, for examples.

### Annual vs. monthly interest rates

Interest rates are normally expressed on an annual basis or per annum. As the previous equations show, when working with monthly cash flows, it is necessary to convert the annual interest rate to a monthly interest rate. There are two ways to do this:

• Divide the annual interest rate by 12
• Calculate the equivalent effective monthly interest rate

These two methods will result in different answers.

Consumer lending groups use the divide-by-12 method to comply with the Truth-in-Lending Act. The Truth-in-Lending Act required disclosures from lenders, which include, among other things:

• The note interest rate
• Any points or origination costs
• Most lender imposed fees, such as underwriting and processing fees

These fees are all rolled into the calculation of an annual percentage rate (APR) for the loan. As the name implies, this is an annual rate. Because most consumer loans are paid on a monthly basis, the monthly rate, used in Eq.3 to determine the monthly payments, is obtained by dividing the annual rate by 12. Fig.2 from the Federal Reserve website shows some sample monthly payments. The next example illustrates the calculation.

#### Example 4: Monthly payment calculation

Loan principal = \$6,000. APR = 15%. Monthly interest rate = 15%/12 = 1.25%/month. Loan term = 4 years (48 months). ....................(7)

The effective-monthly-rate method, although more complicated to calculate, has some advantages as discussed later in Sec. 16.5. In this method, the annual interest rate is converted to a monthly rate, which, when compounded 12 times, results in the annual interest rate. The derivation of the effective monthly interest rate begins with the relationship (1+i) = (1+im)12.

Rearranging this equation gives ....................(8)

The divide-by-12 method leads to a higher monthly payment than the effective-monthly-rate method, as shown in the next example.

#### Example 5: Monthly payments with effective monthly interest rates

Using the data from Example Four, the effective monthly interest rate is calculated as ....................(9)

The loan payment is then calculated. ....................(10)

## Nomenclature

 A = constant periodic payment F = future lump sum of money i = the periodic interest rate n = the number of periods for interest calculations or the hyperbolic exponent for decline curve equations P = present lump sum of money