Time Value of Money

The time value of money is a core concept in engineering economics that recognizes that the value of money changes over time.
The idea is that money has earning power and can generate more money over time through interest or investment.
Therefore, a sum of money is generally worth more today than the same sum will be worth in the future.
This principle applies to both money that is invested and money that is borrowed .
The change in the amount of money over a period of time is the time value of money.

Interest

Interest is the manifestation of the time value of money,
can be considered as the “rent” paid for the use of money.
Computationally, interest is the difference between an ending amount of money and the beginning amount.
There are always two perspectives to interest: interest paid and interest earned. Interest is paid when a person or organization borrows money, and interest is earned when a person or organization saves, invests, or lends money.

There are two basic ways of calculating interest, simple and compound. Simple interest is calculated only on the principal amount. Compound interest, on the other hand, is calculated on the principal amount and also on the accumulated interest from previous periods. The difference between simple and compound interest grows significantly over time. Compound interest is more commonly used in financial transactions .

$LaTeX: \text{Simple Interest}=\frac{P\times T\times R}{100}$

$LaTeX: \text{Compound Interest} = P [(1+i)^{n}-1]$

Nominal and Effective Interest Rate

When compounding occurs more than once a year, it is important to differentiate between nominal and effective interest rates. A nominal interest rate is a stated annual interest rate without considering the effect of compounding, while an effective interest rate is the actual interest rate earned or paid considering the effect of compounding.

Nominal Interest Rate:

The stated annual interest rate without considering compounding effects.
Also known as Annual Percentage Rate (APR).
Can be calculated for periods longer than a year.

Effective Interest Rate:

Reflects the actual interest earned or paid after accounting for compounding.
Also called Annual Percentage Yield (APY).
Always ≥ nominal rate.
Greater the frequency of compounding higher the effective interest rate.

The effective interest rate is the actual rate of interest earned or paid on an investment or loan, accounting for the effects of compounding over a given period. It reflects the true cost of borrowing or the true yield of an investment. The effective rate is typically higher than the nominal interest rate when compounding occurs more than once a year

Compounding Frequency:

The effective interest rate takes into account how often interest is compounded within a year (or other time period), such as monthly, quarterly, or daily. More frequent compounding results in a higher effective interest rate.

Calculation of effective interest rate:

$LaTeX: r = \left(1 + \frac{i}{n}\right)^n - 1$

where: – r is the effective interest rate,
– i is the nominal (stated) annual interest rate,
– n is the number of compounding periods per year

Continuous Compounding:

In continuous compounding, interest compounds infinitely over a period. The formula for continuous compounding simplifies to:

LaTeX: r = e^i - 1

Where i is the nominal annual interest rate as a decimal. This method maximizes potential returns by reinvesting continuously.

Example 1: Calculating Nominal Rate from Effective Rate –

Given: Effective rate = 10%, compounded monthly.
– Find: Nominal rate.

Using the formula:
$LaTeX: r = \left(1 + \frac{i}{12}\right)^{12} - 1$
where r = 0.10 ,

solve for i:
$LaTeX: 1.10 = (1 + \frac{i}{12})^{12}$
Solving gives: $LaTeX: i\approx 9.57\%$

Example 2: Calculating Effective Rate from Nominal Rate
– Given: Nominal rate = 21%, compounded monthly.
– Find: Effective rate.

Using the formula: $LaTeX: r = (1 + \frac{i}{n})^n - 1$
where i = 0.21, n = 12:
$LaTeX: r = (1 + \frac{0.21}{12})^{12} - 1$
Solving gives: $LaTeX: r \approx 23.14\%$
For daily compounding,

$LaTeX: r = (1 + \frac{0.21}{365})^{365} - 1$
Solving gives: $LaTeX: r \approx 23.360\%$

For the same example if continuous compounding is done:
LaTeX: r = e^i - 1

$LaTeX: r= e^{0.21} - 1 \approx 23.368\%$

Economic Equilibrium:

The concept of equivalence is also essential in engineering economics and is closely linked to the time value of money .
Economic equivalence means that different sums of money at different times can have the same economic value.
For example, if a person is indifferent between receiving Rs. 24.72 now and Rs. 100 in 10 years, then these two sums are equivalent if a common interest rate of 15% compounded annually is applied to both.
This concept allows us to compare different cash flow streams by bringing them to a common point in time using appropriate interest rates.

To establish equivalence between cash flows, it is necessary to choose a reference point in time and use interest formulas to find the value of each cash flow at that point. The choice of the reference point will not affect the result of the equivalence calculations, but choosing the most suitable point in time can simplify the calculations.

Different Factors:

To facilitate time value of money calculations, several interest factors and formulas are used. These factors and formulas allow us to convert cash flows to equivalent values at different points in time. Some of the common factors are:

1. Single-Payment Compound Amount Factor (F/P, i, n):

LaTeX: F = P (1 + i)^n

This formula calculates the future value F of a single present amount P after n periods at an interest rate i .

2. Single-Payment Present Worth Factor (P/F, i, n):

$LaTeX: P = F \frac{1}{(1 + i)^n}$

This formula calculates the present value P of a single future amount F .

3. Uniform-Series Present Worth Factor (P/A, i, n):

$LaTeX: P = A \left[ \frac{(1 + i)^n - 1}{i (1 + i)^n} \right]$

This formula calculates the present value P of a series of equal payments A over n periods.

4. Capital Recovery Factor (A/P, i, n):

$LaTeX: A = P \left[ \frac{i (1 + i)^n}{(1 + i)^n - 1} \right]$

This formula calculates the equal annual payment A required to repay a present sum P over n periods.

5. Uniform-Series Compound Amount Factor (F/A, i, n):

$LaTeX: F = A \left[ \frac{(1 + i)^n - 1}{i} \right]$

This formula calculates the future value F of a series of equal payments A over n periods.

6. Sinking Fund Factor (A/F, i, n):

$LaTeX: A = F \left[ \frac{i}{(1 + i)^n - 1} \right]$

This formula calculates the equal periodic payment A that will accumulate to a future sum F after n periods.

7. Gradient Present Worth Factor (P/G, i, n):

$LaTeX: P = G \left[ \frac{(1 + i)^n - i n - 1}{i^2 (1 + i)^n} \right]$

This formula calculates the present value of a cash flow that changes by a constant amount G each period.

8. Geometric Gradient Series Factor (P/A, g, i, n):

$LaTeX: P = A \left[ \frac{1 - \left( \frac{1 + g}{1 + i} \right)^n}{i - g} \right]$

This formula calculates the present worth of a series of cash flows that increase or decrease by a constant percentage g per period.

These factors and formulas are essential tools for making economic evaluations of engineering projects, helping to convert all cash flows to a common time basis.

Present Worth

also known as present value (PV) or net present value (NPV)
method used to convert future cash flows to their equivalent value in the present time
future cash flows are discounted back to the present using an appropriate interest rate, often referred to as the discount rate.
commonly used to evaluate and compare different investment alternatives .
A future amount of money converted to its equivalent present worth will have a lower value than the actual future cash flow.

The present worth method can be used to analyze alternatives with equal or unequal lives, as well as independent projects. The present worth of a cash flow is the sum of the present worths of each individual cash flow.

Annual Worth

Converts all cash flows to an equivalent uniform annual series over the life of the project.
It is the equivalent uniform annual cash flow that, if received or paid over the life of the project, would have the same value as all of the actual project cash flows.
The annual worth method is especially useful for comparing mutually exclusive alternatives with different lives, and for determining the unit cost of an activity.

Some Important Points

End-of-period convention: For simplicity, it is assumed that all cash flows occur at the end of the time period in which they occur.
Cash flow diagrams: A visual representation of cash flows over time to understand the timing and magnitude of payments and receipts.
Spreadsheets: Spreadsheet software such as Excel or Google Sheets can be used to quickly solve time value of money problems.
Inflation: The rate at which the purchasing power of money decreases over time. Inflation can be incorporated into the analysis using inflation-adjusted interest rates.

Curriculum

Time Value of Money

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