12.3: The Time Value of Money

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Time Value of Money

In engineering economics, this foundational concept is known as the Time Value of Money (TVM): the principle that a dollar available today is worth more than a dollar received in the future due to its earning potential. Whether evaluating a personal vehicle purchase or comparing complex industrial investments, understanding cash flows, interest rates, and opportunity costs enables engineers to make sound decisions that optimize value over an asset's entire life cycle.

Interest Accrual

Interest can work in two ways:

Earned Interest: Money made by allowing someone else (such as a bank) to use our money.
Paid Interest: Money spent when borrowing someone else's money for our use.

Simple Interest Example

Suppose I lend my friend $1000 for a term of 5 years at 5% annual simple interest:

Annual Interest: 5% of $1000 = $50 per year.
Total Interest (5 Years): $50 \times 5 = $250.
Total Repayment: After 5 years, she pays back $1250 ($1000 principal + $250 interest).

Compound Interest Example

Now suppose I lend my friend $1000 for 5 years at 5% interest compounded annually:

Year 1: Interest owed is 5% of $1000 = $50. Total owed at end of Year 1 = $1050.
Year 2: Interest is calculated on the new balance ($1050), not the original principal ($1000). The interest owed for Year 2 is 5% of $1050 = $52.50.

Annual Compounding Schedule

Table showing 5 years of compounding interest
Year	Owed at Beginning of Year	Annual Interest (5%)	Owed at End of Year
1	$1000.00	$50.00	$1050.00
2	$1050.00	$52.50	$1102.50
3	$1102.50	$55.13	$1157.63
4	$1157.63	$57.88	$1215.51
5	$1215.51	$60.78	$1276.29

With annual compounding, my friend owes $1276.29 after 5 years versus $1250.00 with simple interest—a difference of $26.29 strictly due to compounding.

Mathematical Model for Compound Interest

Interest may be compounded yearly, monthly, daily, or continuously. The smaller the compounding period, the faster interest accumulates.

Compound interest is modeled using the formula:

$$F = P(1 + i)^N$$

Where:

F= Future worth

P=Present worth

i= interest in decimal form for the compounding time period

N=Number of periods

Varying Compounding Periods

1. Monthly Compounding

Let’s look at the scenario where I lend my friend $1000 for a period of 5 years one more time. What happens if we have the same loan of $1000 and the same annual interest rate of 5% but we now compound it monthly?

P=$1000

i=.05/12 since the compound period is monthly

N= 12(5) =60 months

\[ F = P(1+i)^N \]

\[ F = \$1000\left(1 + \frac{0.05}{12}\right)^{60} = \$1283.36 \]

2. Daily Compounding

What if we now compound it daily?

$ P = \$1000 $

$ i = \frac{0.05}{365} $ since the compound period is daily

$ N = 365(5) = 1825 $ days

\[ F = P(1+i)^N \]

\[ F = \$1000\left(1 + \frac{0.05}{365}\right)^{1825} = \$1284 \]

Summary Comparison

Scenario: $1000 loan at 5% annual interest for 5 years.

Table showing different compounding frequencies
Loan Principal	Annual Rate	Term	Compounding Frequency	Total Owed After 5 Years	Total Interest Paid
$1000	5%	5 Years	None (Simple)	$1250.00	$250.00
$1000	5%	5 Years	Yearly	$1276.29	$276.29
$1000	5%	5 Years	Monthly	$1283.36	$283.36
$1000	5%	5 Years	Daily	$1284.00	$284.00

Good to Know

Credit card companies typically compound interest on a daily basis, which accelerates balance growth if unpaid.

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