Lecture 3: Interest and Equivalence
Introduction: The Value of Time
Most engineering costs, revenues, and benefits occur over time. We cannot simply add up sums of money at different times due to the time value of money.
I. The Time Value of Money
Money is valuable because it can be used, invested, or consumed now. The charge for using money is called interest—like rent for money.
If offered $1000 today or $1000 ten years from now, you’d choose today’s money. This preference is measured by the interest rate.
A. Simple Interest
- Computed only on the original principal
- Interest does not compound
- \[\text{Total Interest} = P \times i \times n\]
- \[F = P + Pin = P(1 + in)\]
- Rarely used in practice
B. Compound Interest
- Interest is calculated on the unpaid balance, including prior interest
- \[F = P(1 + i)^n\]
- Used in all engineering economic analysis unless stated otherwise
II. Equivalence
Equivalence means two different cash flows are equal at a specified interest rate if you are indifferent between them. - Example: At \(8\%\), \(100 today \equiv 108\) in one year - Key Principle: Equivalence depends on the stated interest rate
III. Single Payment Compound Interest Formulas
Standardized formulas relate a single present sum (\(P\)) and a single future sum (\(F\)).
A. Notation
| Symbol | Definition |
|---|---|
| \(i\) | Interest rate per period |
| \(n\) | Number of periods |
| \(P\) | Present sum (Time 0) |
| \(F\) | Future sum (end of \(n\) periods) |
B. Future Worth Factor (F/P)
- \[F = P(1 + i)^n\]
- \((F/P, i, n)\): Find \(F\) given \(P\), \(i\), \(n\)
C. Present Worth Factor (P/F)
- \[P = F(1 + i)^{-n}\]
- \((P/F, i, n)\): Find \(P\) given \(F\), \(i\), \(n\)
IV. Nominal and Effective Interest Rates
A. Nominal Interest Rate (\(r\))
- Stated annual rate, not accounting for compounding
- Also called Annual Percentage Rate (APR)
B. Effective Interest Rate (\(i_a\))
- Actual annual rate after compounding
- Also called Annual Percentage Yield (APY)
- For \(m\) compounding periods/year: \[i_a = \left( 1 + \frac{r}{m} \right)^m - 1\]
C. Continuous Compounding
- \[i_a = e^r - 1\]
- Future Worth: \(F = P(e^{rn})\)
- Present Worth: \(P = F(e^{-rn})\)
V. Solved Examples
Example 1: Simple Interest Calculation
- Loan: \(5000\) for 5 years at \(8\%\) simple interest
- Total Interest: \(5000 \times 0.08 \times 5 = 2000\)
- Amount Due: \(5000 + 2000 = 7000\)
Example 2: Compound Interest Calculation
- Deposit: \(500\) at \(6\%\) compounded annually for 3 years
- \[F = 500(1 + 0.06)^3 = 500(1.191) = 595.50\]
- Amount after 3 years: \(595.50\)