Interest Rate and Rate of Return

Interest  is the manifestation of the time value of money. Computationally, interest is the difference between an ending amount of money and the beginning amount. If the difference is zero or negative, there is no interest. There are always two perspectives to an amount of interest—interest paid and interest earned. These are illustrated in  Figure 1–2 . Interest is   paid  when a person or organization borrowed money (obtained a loan) and repays a larger amount over time. Interest is  earned  when a person or organization saved, invested, or lent money and obtains a return of a larger amount over time. The numerical values and formulas used are the same for both perspectives, but  the interpretations are different.  

Interest paid  on borrowed funds (a loan) is determined using the original amount, also called the principal,

  Interest = amount owed now - principal                            [1.1]

When interest paid over a   specific time unit  is expressed as a percentage of the principal, the result is called the interest rate.

The time unit of the rate is called the   interest period.  By far the most common interest period used to state an interest rate is 1 year. Shorter time periods can be used, such as 1% per month. Thus, the interest period of the interest rate should always be included. If only the rate is stated, for example, 8.5%, a 1-year interest period is assumed.

  Figure 1–2   (a) Interest paid over time to lender. (b) Interest earned over time by investor.

From the perspective of a saver, a lender, or an investor,  interest earned  ( Figure  1–2  b ) is the  final amount minus the initial amount, or principal.

  Interest  earned total amount now principal           [1.3]

Interest earned over a specific period of time is expressed as a percentage of the original amount and is called rate of return (ROR).

 The time unit for rate of return is called the interest period,  just as for the borrower’s perspec-
tive. Again, the most common period is 1 year.

The term  return on investment (ROI)  is used equivalently with ROR in different industries and  settings, especially where large capital funds are committed to engineering-oriented  programs.  

The numerical values in Equations [1.2] and [1.4] are the same, but the term   interest rate paid
is more appropriate for the borrower’s perspective, while the   rate of return earned  is better for
the investor’s perspective.

In Examples 1.3 to 1.5 the interest period was 1 year, and the interest amount was calculated  at the end of one period. When more than one interest period is involved, e.g., the amount of interest after 3 years, it is necessary to state whether the interest is accrued on a simple or compound  basis from one period to the next. This topic is covered later in this chapter.

Since   inflation   can  significantly increase an interest rate, some comments about the funda- mentals of inflation are warranted at this early stage. By definition, inflation represents a decrease  in the value of a given currency. That is, $10 now will not purchase the same amount of gasoline  for your car (or most other things) as $10 did 10 years ago. The changing value of the currency affects market interest rates.

In simple terms, interest rates reflect two things: a so-called real rate of return   plus  the  expected  inflation rate. The real rate of return allows the investor to purchase more than he or she could  have purchased before the investment, while inflation raises the real rate to the market rate that  we use on a daily basis.

The safest investments (such as government bonds) typically have a 3% to 4% real rate of  return built into their overall interest rates. Thus, a market interest rate of, say, 8% per year on a  bond means that investors expect the inflation rate to be in the range of 4% to 5% per year.  Clearly, inflation causes interest rates to rise.

From the borrower’s perspective, the rate of inflation is another interest rate   tacked on to the  real interest rate . And from the vantage point of the saver or investor in a fixed-interest account, inflation   reduces the real rate of return  on the investment. Inflation means that cost and revenue  cash flow estimates increase over time. This increase is due to the changing value of money that  is forced upon a country’s currency by inflation, thus making a unit of currency (such as the dollar) worth less relative to its value at a previous time. We see the effect of inflation in that money  purchases less now than it did at a previous time. Inflation contributes to

   •     A reduction in purchasing power of the currency
   •     An increase in the CPI (consumer price index)
   •     An increase in the cost of equipment and its maintenance
   •     An increase in the cost of salaried professionals and hourly employees
   •     A reduction in the real rate of return on personal savings and certain corporate investments

In other words, inflation can materially contribute to changes in corporate and personal economic  analysis.
  
Commonly, engineering economy studies assume that inflation affects all estimated values  equally. Accordingly, an interest rate or rate of return, such as 8% per year, is applied throughout  the analysis without accounting for an additional inflation rate. However, if inflation were explicitly taken into account, and it was reducing the value of money at, say, an average of 4% per year,  then it would be necessary to perform the economic analysis using an inflated interest rate.

EXAMPLE 1.3

An employee at LaserKinetics.com borrows $10,000 on May 1 and must repay a total of  $10,700 exactly 1 year later. Determine the interest amount and the interest rate paid.

Solution

The perspective here is that of the borrower since $10,700 repays a loan. Apply Equation [1.1]  to determine the interest paid.

EXAMPLE 1.4

Stereophonics, Inc., plans to borrow $20,000 from a bank for 1 year at 9% interest for new recording equipment. (  a ) Compute the interest and the total amount due after 1 year. (  b )  Construct a column graph that shows the original loan amount and total amount due after 1 year used to compute the loan interest rate of 9% per year.

Solution

  Figure 1–3  Values used to compute an interest rate of 9% per year. Example 1.4.

Comment

Note that in part (  a ), the total amount due may also be computed as

     Total  due = principal(1 + interest rate) = $20,000(1.09) = $21,800

Later we will use this method to determine future amounts for times longer than one interest period.

EXAMPLE 1.5

(a)    Calculate the amount deposited 1 year ago to have $1000 now at an interest rate of 5%  per year.  (b)    Calculate the amount of interest earned during this time period.  

Solution