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ISE 211 Engineering Economy Interest and Equivalence (Chapter 3)
Introduction If economic consequences occur over time, we have to take into account the fact that the value of money changes with time. Also, not all costs or payments occur at one time. Therefore, we can use cash flows to visualize these issues. Example 1: The manager has decided to purchase a new $30k mixing machine. The machine may be paid for by one of two ways: Pay the full price now with a 3% discount Pay $5k now; $8k at the end of one year; $6k per year at the end of four subsequent years.
Example 2 A man borrowed $1000 from a bank at 8% interest. He agreed to repay the loan in two end-of-year payments. At the end of the first year, he will repay half of the $1000 principal amount plus the interest that is due. At the end of the second year, he will repay the remaining half of the principal amount plus the interest for the second year. Compute the borrower’s cash flow.
Time Value of Money Would you rather have: $100 today, or $100 a year from now? Basic assumption: Given a fixed amount of money, and A choice of having it now or in the future, Most people would prefer to have it sooner rather than later Reasons: ?
Time Value of Money (cont’d) A consequence: Suppose you are willing to exchange a certain amount now for some other amount later Then the later amount has to be ? What this means for us In this course, we will learn methods to: Compare different cash flows over time Using the interest rate or discount rate: How much more a dollar today is worth, compared to a dollar in one year For example, if the interest rate is 5%: Then $1 today is worth as much as $1.05 next year
Simple Interest Simple interest is interest that is computed on the original sum. Invest $100 in a savings account for 5 years, earning 7% simple interest per year.
Simple Interest (cont’d) Thus if you were to loan a present sum of money P to someone at a simple annual interest rate i for a period of n years, the amount of interest you would receive from the loan would be: Total Interest Earned = At the end of n years the amount of money due F would equal the amount of the loan P plus the total interest earned, or Amount of money due at the end of the loan, F =
Example You have agreed to loan a friend $5k for five years at a simple interest rate of 8% per year. How much interest will you receive from the loan? How much will your friend pay you at the end of five years? Note: We will deal extensively with compound interest calculations throughout the course.
Compound Interest Invest $100 in a savings account for 5 years, earning 7% compound interest per year. Compound interest is interest that is computed on the original principal (sum) as well as the interest added to it.
Four plans for repayment of $5000 in five years with interest at 8% 1)At end of each year, pay $1000 principal plus interest due. 2)Pay interest due at end of each year and principal at end of five years. 3)Pay in five equal end-of-year payments. 4)Pay principal and interest in one payment at end of five years.
Four plans for repayment of $5000 in five years with interest at 8%
What plan is best? Need to consider all costs moved to the same time period, because of the time value of money. We use the technique of EQUIVALENCE to do this. We will see that these plans are actually equivalent. For each plan, you pay a total amount of interest, which depends on what you still owe. Each plan has different amounts owed at different times, after the initial loan. Notice: the interest rate paid is the same for all plans.
Single Payment Compound Interest Formulas To facilitate equivalence computations, a series of interest formulas will be derived. Notation i = Interest rate per interest period n = Number of interest periods P = A present sum of money F = A future sum of money
Single Payment Compound Interest Formulas (cont’d) Derivation Amount at beginning of interest period Interest of PeriodAmount at end of interest period First Year Second Year Third Year. nth Year
The relationship between present sum and its equivalent future sum, F, is: F = P(1+i) n In functional notation, the single payment compound amount formula can be written as: F = P (F/P, i, n) This notation (i.e., F/P, i, n ) can be read as: Functional notation is used so that the compound interest factor may be written in an equation in algebraically correct form, which is dimensionally correct. F = P (F/P) Single Payment Compound Interest Formulas (cont’d)
Example 1 If $500 were deposited in a bank savings account, how much would be in the account three years hence if the bank paid 6% interest compounded annually? Solution:
Single Payment Present Worth Formula The equation is given as follows: In functional notation:
Example 2 If you wish to have $800 in a savings account at the end of four years, and 5% interest was paid annually, how much should you put into the savings account now? Solution:
Example 3 If $500 were deposited in a bank savings account, how much would be in the account three years hence if the bank paid 6% interest “compounded quarterly”? Solution:
Example 4 Consider the following situation: Year Cashflow 0+P 10 20 3-400 40 5-600 Solve for P assuming a 12% interest rate and using the Compound Interest Tables.