Interest and Time Value of Money

The most fundamental concept in engineering economy is that money has a time value. A dollar today is worth more than a dollar one year from now because of the interest it could earn if invested.

Time Value of Money (TVM) Explorer

Present Value (PV)1,000 $
Interest Rate (r)5 %
Years (t)10 yrs

Future Value (FV)

$1,628.89

$FV = PV \times (1 + r)^t$

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Simple Interest

Simple Interest

Interest that is calculated only on the principal amount or on that portion of the principal amount that remains unpaid. It does not earn interest on previously accumulated interest.

Simple Interest Formula

The total interest earned (II) is:
I=P⋅n⋅i I = P \cdot n \cdot i
The total amount (FF) at the end of nn periods is:
F=P+I=P(1+n⋅i) F = P + I = P(1 + n \cdot i)
Where:
  • PP = Principal amount (present worth)
  • nn = Number of interest periods
  • ii = Interest rate per interest period

Ordinary vs. Exact Simple Interest

Checklist

The Rule of 72

The Rule of 72 is a quick, useful heuristic to estimate the number of years required to double your money at a given annual fixed compound interest rate. By dividing 72 by the annual rate of return, investors obtain a rough estimate of how many years it will take for the initial investment to duplicate itself.

Rule of 72 Formula

Years to double≈72Interest Rate (% ) \text{Years to double} \approx \frac{72}{\text{Interest Rate (\% )}}

Compound Interest

Compound Interest

Interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. This leads to exponential growth of the investment, often called "interest on interest."

Compound Interest Formula

The total amount (FF) at the end of nn periods is:
F=P(1+i)n F = P(1 + i)^n
Where:
  • PP = Present worth
  • FF = Future worth
  • ii = Interest rate per compounding period
  • nn = Number of compounding periods
This formula is the basis for all other compound interest factors. The term (1+i)n(1+i)^n is known as the Single Payment Compound Amount Factor (SPCAF) and is commonly denoted in standard functional notation as (F/P,i,n)(F/P, i, n). This is read as "Find FF, given PP, at interest rate ii for nn periods."
Conversely, to find the present worth given a future amount:
P=F(1+i)−n=F[1(1+i)n] P = F(1 + i)^{-n} = F \left[ \frac{1}{(1 + i)^n} \right]
This factor is the Single Payment Present Worth Factor (SPPWF), denoted as (P/F,i,n)(P/F, i, n).

Visualizing Compound Interest

The difference between simple and compound interest becomes significantly magnified over time. Use the simulator below to compare the linear growth of simple interest versus the exponential growth of compound interest.

Compound Interest Visualizer

Principal Amount ($P$)1,000 $
Interest Rate ($i$)5 %
Time Period ($n$)20 years

After 20 Years

Simple Interest Total:$2,000
Compound Interest Total:$2,653
Difference:+$653
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Continuous Compounding

While discrete compounding occurs at distinct intervals (monthly, annually, etc.), continuous compounding assumes that interest is compounded infinitesimally fast. As the number of compounding periods per year approaches infinity, the formula for future value changes.

Continuous Compounding Formulas

For continuous compounding with a nominal annual interest rate rr and time tt in years:
Future Worth (FF):
F=P⋅ert F = P \cdot e^{rt}
Present Worth (PP):
P=F⋅e−rt P = F \cdot e^{-rt}
Where ee is Euler's number (approximately 2.71828).

Cash Flow Diagrams

A cash flow diagram is a graphical representation of cash flows drawn on a time scale. It is an essential, highly recommended tool for solving engineering economy problems.

Drawing Cash Flow Diagrams

  1. Time Scale: Draw a horizontal line representing time, progressing from left to right. Mark periods (usually years, quarters, or months) 0,1,2,…,n0, 1, 2, \dots, n. Period 0 represents the present time.
  2. Cash Flows: Use vertical arrows to represent cash flows.
    • Upward Arrows (↑\uparrow): Represent positive cash flows (receipts, income, savings, benefits).
    • Downward Arrows (↓\downarrow): Represent negative cash flows (disbursements, costs, expenses, investments).
  3. End-of-Period Convention: Unless explicitly stated otherwise, all cash flows are assumed to occur at the end of the period.
  4. Viewpoint: The diagram should consistently represent the point of view of a single person or organization. (A loan is a positive receipt for the borrower but a negative disbursement for the bank).

Interactive Cash Flow Diagram

Visualize inflows, outflows, and calculate Present Worth.

Net Present Value (NPV)$0.00
0$10001$3002$3003$3004$3005$500
10%
0
1
2
3
4
5
Key Takeaways
  • Simple vs. Compound: Simple interest grows linearly; compound interest grows exponentially. Compound interest is the standard assumption in engineering evaluations.
  • Continuous Compounding: Represents the theoretical limit of infinite compounding periods, using base ee.
  • Single Payment Factors: The foundation formulas: (F/P,i,n)(F/P, i, n) to find future worth, and (P/F,i,n)(P/F, i, n) to discount back to present worth.
  • Cash Flow Diagrams: The universal visual language for modeling complex economic problems. Always draw one before calculating.
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