# Chapter 1 Summary

### Key Concepts Summary

• 1.1 Compound Interest Terminology
• How compounding works.
• How to calculate the periodic interest rate.
• 1.2 Calculating the Future Value
• The basics of taking a single payment and moving it to a future date.
• Moving single payments to the future when variables change.
• 1.3 Calculating the Present Value
• The basics of taking a single payment and moving it to an earlier date.
• Moving single payments to the past when variables change.
• 1.4 Equivalent Payments
• The concept of equivalent payments.
• The fundamental concept of time value of money.
• The fundamental concept of equivalency.
• Applying single payment concepts to loans and payments.
• 1.5 Calculating the Interest Rate
• Solving for the nominal interest rate.
• How to convert a variable interest rate into its equivalent fixed interest rate.
• 1.6 Calculating the Term
• Figuring out the term when n is an integer.
• Figuring out the term when n is a non-integer.
• 1.7 Equivalent and Effective Interest Rates
• The concept of effective rates.
• Taking any nominal interest rate and finding its equivalent nominal interest rate.

## Glossary of Terms

Compound Interest. A system for calculating interest that primarily applies to long-term financial transactions with a time frame of one year or more. Interest is periodically converted to principal throughout a transaction, with the result that the interest itself also accumulates interest.

Compounding Frequency. The number of compounding periods in a complete year.

Compounding Period. The amount of time that elapses between the dates of successive conversions of interest to principal.

Discount Rate. An interest rate used to remove interest from a future value.

Effective Interest Rate. The true annually compounded interest rate that is equivalent to an interest rate compounded at some other (non-annual) frequency.

Equivalent Payment Streams. Equating two or more alternative financial streams such that neither party receives financial gain or harm by choosing either stream.

Equivalent Interest Rates. Interest rates with different compounding that produce the same effective rate and therefore are equal to each other.

Focal Date. A point in time to which all monies involved in all payment streams will be moved using time value of money calculations.

Fundamental Concept of Equivalency. Two or more payment streams are equal to each other if they have the same economic value on the same focal date.

Fundamental Concept of Time Value of Money. All monies must be brought to the same focal date before any mathematical operations, decisions, or equivalencies can be determined.

Nominal Interest Rate. A nominal number for the annual interest rate, which is commonly followed by words that state the compounding frequency.

Periodic Interest Rate. The percentage of interest earned or charged at the end of each compounding period.

Present Value Principle for Loans. The present value of all payments on a loan is equal to the principal that was borrowed.

### Formula & Symbol Hub

#### Symbols Used

• $FV$ = Future value or maturity value
• $PV$ = Present value of principal
• $m$ or $C/Y$ = Number of compounds per year or compounding frequency
• $i$ = Periodic interest rate
• $j$ or $I/Y$ = Nominal interest rate per year
• $n$ or $N$ = Total number of compound periods for the term
• $f$ or $EFF$ = Effective interest rate

### Formulas Used

• ##### Formula 1.1 – Total Number of Compounds

$n=m \times \mbox{time in years}$

• ##### Formula 1.2 – Periodic Interest Rate

$i=\frac{j}{m}$

• ##### Formula 1.3 – Future Value

$FV=PV \times (1+i)^n$

• ##### Formula 1.4 – Present Value

$PV=FV \times (1+i)^{-n}$

• ##### Formula 1.5 – Effective Interest Rate

$f=(1+i)^m-1$

• ##### Formula 1.6 – Equivalent Interest Rate

$i_2=(1+i_1)^{\frac{m_1}{m_2}}-1$

## Calculator

• ### Time Value of Money Functions

• The time value of money buttons are the five buttons located on the third row of your calculator.
 Characteristic Data Entry Requirements Calculator Symbol N The number of compounding periods. An integer or decimal number; no negatives. I/Y The nominal interest rate per year. Percent format without the % sign (i.e., 7% is just 7). PV Present value or principal. An integer or decimal number. PMT Used for annuity payment amounts (covered in Chapter 11) and is not applicable to lump-sum amounts; it needs to be set to zero for lump-sum calculations. An integer or decimal number. FV Future value or maturity value. An integer or decimal number.

To enter any information into any one of these buttons or variables, called loading the calculator, key in the information first and then press the appropriate button.

• The frequency function is logically placed above the $I/Y$ button and is labeled $P/Y$. This function addresses compound interest frequencies, such as the compounding frequency. Access the function by pressing $2\text{nd} P/Y$ to find the following entry fields, through which you can scroll using your arrow buttons.
 Characteristic Data Entry Requirements Calculator Symbol P/Y Annuity payments per year (payment frequency is introduced in Chapter 11); when working with lump-sum payments and not annuities, the calculator requires this variable to be set to match the C/Y. A positive, nonzero number only. C/Y Compounds per year (compounding frequency). A positive, nonzero number only.

To enter any information into one of these fields, scroll to the field on your screen, key in the data, and press ENTER. When you enter a value into the $P/Y$ field, the calculator will automatically copy the value into the $C/Y$ field for you. If in fact the $C/Y$ is different, you can change the number manually. To exit the $P/Y$ window, press 2nd QUIT.

• To key in a question, you must load the calculator with six of the seven variables. To solve for the missing variable, press CPT followed by the variable.
• The cash flow sign convention is used for the $PV$, $PMT$, and $FV$ buttons. If money leaves you, you must enter it as a negative. If money comes at you, you must enter it as a positive.

Compound Interest (Present and Future Values) by Joshua Emmanuel [6:56] (transcript available).

• ### Interest Conversions

• To find the effective interest rate.
1. Press 2nd ICONV (the number $2$ key).
2. At the NOM screen, enter the nominal interest rate $I/Y$ and press ENTER. The interest rate is entered as a percent (without the percent sign). For example $10\%$ is entered as $10$.
3. Press the up arrow.
4. At the $C/Y$ screen, enter the compounding frequency for the nominal interest rate entered in the previous step and press ENTER.
5. Press the up arrow.
6. At the EFF screen, press CPT to calculate the effective interest rate.
• To find the equivalent rate.
1. Find the effective interest rate for the original interest rate following the steps above.
2. Press the down arrow.
3. At the $C/Y$ Screen, enter the new compounding frequency and press ENTER.
4. Press the down arrow.
5. At the NOM screen, press CPT to calculate the new interest rate.

Nominal and Effective Rate Conversions by Joshua Emmanuel [5:50] (transcript available).

Chapter 9 Summary” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.