Interest and Time Value of Money - Theory & Concepts

Interest and Time Value of Money

Learning Objectives

Differentiate between simple and compound interest.
Apply the Rule of 72 for quick growth estimations.
Compute present and future worth using single payment compound interest formulas.
Understand the concept and application of continuous compounding.
Construct and interpret cash flow diagrams accurately.

Time Value of Money

The concept that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle holds that, provided money can earn interest, any amount of money is worth more the sooner it is received.

Principal

The initial amount of money borrowed or invested, before interest is added.

Interest Rate

The proportion of a loan that is charged as interest to the borrower, typically expressed as an annual percentage of the loan outstanding. It represents the "rent" paid for the use 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.

Interactive Simulation

Interact with the simulation below to explore the time value of money concept and see how a dollar today differs from a dollar in the future under varying interest rates.

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

Calculates the total interest earned and final amount for simple interest.

I = P \cdot n \cdot i

F = P + I = P(1 + n \cdot i)

Variables

Symbol	Description	Unit
$I$	Total interest earned	-
$F$	Future worth or total amount	-
$P$	Principal amount (present worth)	-
$n$	Number of interest periods	-
$i$	Interest rate per interest period	-

Ordinary vs. Exact Simple Interest

Ordinary Simple Interest

Uses a banker's year of 360 days. This assumes 12 months of 30 days each. ( $n = \frac{d}{360}$ )

Exact Simple Interest

Uses the exact number of days in a year, 365 or 366 for leap years. ( $n = \frac{d}{365}$ or $n = \frac{d}{366}$ )

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. Mathematically, it is derived from the compound interest formula $F = P(1+i)^n$ by setting $F=2P$ and solving for $n$ using natural logarithms. It is highly practical in engineering feasibility studies for quick mental math and rough estimations.

Rule of 72 Formula

Estimates the number of years required to double your money at a given annual fixed compound interest rate.

n \approx \frac{72}{i_{\%}}

Variables

Symbol	Description	Unit
$n$	Approximate number of years to double	-
$i_{\%}$	Interest rate expressed as a percentage (e.g., 5 for 5%)	-

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 (SPCAF)

Calculates the future worth given a present worth with compound interest.

F = P(1 + i)^n

Variables

Symbol	Description	Unit
$F$	Future worth	-
$P$	Present worth	-
$i$	Interest rate per compounding period	-
$n$	Number of compounding periods	-

This formula is the basis for all other compound interest factors. The term $(1+i)^n$ is known as the Single Payment Compound Amount Factor (SPCAF) and is commonly denoted in standard functional notation as $(F/P, i, n)$ . This is read as "Find $F$ , given $P$ , at interest rate $i$ for $n$ periods."

Conversely, to find the present worth given a future amount:

Single Payment Present Worth Factor (SPPWF)

Calculates the present worth given a future amount.

P = F(1 + i)^{-n} = F \left[ \frac{1}{(1 + i)^n} \right]

Variables

Symbol	Description	Unit
$P$	Present worth	-
$F$	Future worth	-
$i$	Interest rate per compounding period	-
$n$	Number of compounding periods	-

This factor is the Single Payment Present Worth Factor (SPPWF), denoted as $(P/F, i, n)$ .

Visualizing Compound Interest

The difference between simple and compound interest becomes significantly magnified over time.

Interactive Simulation

Use the simulator below to compare the linear growth of simple interest versus the exponential growth of compound interest over time.

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

Calculates the future or present worth assuming interest is compounded infinitesimally fast.

F = P \cdot e^{rt}

P = F \cdot e^{-rt}

Variables

Symbol	Description	Unit
$F$	Future worth	-
$P$	Present worth	-
$r$	Nominal annual interest rate	-
$t$	Time in years	-
$e$	Euler's number (approx. 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

Time Scale: Draw a horizontal line representing time, progressing from left to right. Mark periods (usually years, quarters, or months) $0, 1, 2, \dots, n$ . Period 0 represents the present time.
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).
End-of-Period Convention: Unless explicitly stated otherwise, all cash flows are assumed to occur at the end of the period.
Viewpoint: The diagram should consistently represent the point of view of a single person or organization. For example, a bank loan is a positive receipt (upward arrow) for the borrower, but a negative disbursement (downward arrow) for the lending bank. Mixing viewpoints will lead to incorrect signs in calculations.

Interactive Simulation

Use this interactive tool to visualize positive and negative cash flows drawn on a time scale.

Interactive Cash Flow Diagram

Visualize inflows, outflows, and calculate Present Worth.

Net Present Value (NPV)$0.00

Interest Rate (MARR) %

10%

Cash Flows

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 $e$ .
Single Payment Factors: The foundation formulas: $(F/P, i, n)$ to find future worth, and $(P/F, i, n)$ 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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