Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have some underlying financial implications, either direct or indirect. Also, financial concepts arise in the everyday management of your personal resources. It s important, therefore, to understand the basics of finance. For example, the time value of money and the analysis of financial statements are basic components of finance that will be used throughout the remainder of this course and in future finance classes. It s essential that you take the time to master these concepts. As you work your way through this course, you ll learn the importance of finance to the success of every entity, both personal and professional. In Lesson 1, you ll learn some important fundamentals of finance. Some of this material is analytical in nature, requiring you to understand some mathematical calculations. Example problems in both your textbook and study guide will help you to master these calculations. Some of the problems can be completed manually or with the help of tables; however, you ll find that some calculations are much easier to perform with the aid of a financial calculator. A financial calculator is a special type of calculator that s designed to perform specific financial functions. A financial calculator is a useful professional tool that can be used throughout this course and in future finance classes. If you prefer, use an electronic spreadsheet such as Microsoft Excel to perform financial calculations. Professionals in finance generally use electronic spreadsheets more than calculators, although they require more time to learn. Your textbook provides instructions for both financial calculators and spreadsheets. You ll find financial calculator instructions starting on pages 111; Appendix E, starting on page 587, provides instructions for using Excel. Either a financial calculator or an electronic spreadsheet is required to complete this course. Lesson 1 7
OBJECTIVES When you complete this lesson, you ll be able to Explain the importance of financial decision making to the business community Describe the importance of financial statement analysis Explain the concepts of compounding and future value Discuss the concepts of discounting and present value Calculate the present value and future value of an annuity Read and understand the principal components of a balance sheet Perform ratio calculations to determine liquidity, activity, and profitability ASSIGNMENT 1 Read the following assignment. Then read pages 3 8 and 106 129 in your textbook. Be sure to complete the self-check to gauge your progress. The Time Value of Money One of the most important concepts in the study of finance is the time value of money. As this phrase implies, this concept covers how time impacts the value of money. One dollar today isn t equal in value to one dollar 10 years from now. The difference in the value of these two dollars can be explained by the time value of money. The Future Value of a Dollar The future value of one dollar is the amount that one dollar will grow to at some point in the future. The process of compounding takes into account the earning of interest on interest, and is the process of finding the future value of some initial amount. 8 Financial Management
Let s look at an example problem. Example: Suppose you begin with $100 today and deposit it in an account that pays 10 percent annually. How much will you have in the account after 1 year? Solution: The calculation is relatively simple. In this example, you ve been given three variables. The present value (PV) is the amount you begin with, which is $100. The number of time periods (N) is 1 year. The interest rate (I) is 10 percent annually. The missing variable that you need to calculate is the future value (FV), which is the value of the investment at the end of 1 year. You would use the following formula to calculate the future value of the investment. FV = PV (1 + I.) N Substitute the known values of PV, I, and N into the formula and solve. FV = $100 (1 + 10 percent) 1 FV = $100 (1 + 0.10) 1 FV = $100 (1.10) 1 FV = $100 1.1 FV = $110 Thus, the value of the $100 investment after 1 year will be $110. Now, let s consider the same problem over a 5-year period. Example: Today, suppose that you deposit $100 into an account that pays 10 percent annually. How much will you have in the account after 5 years? Solution: In this problem, you re given the following variables: PV = $100 N = 5 years I = 10 percent annually You would again use the following formula to calculate the future value of the investment (FV). FV = PV (1 + I.) N Lesson 1 9
Substitute the known values of PV, I, and N into the formula and solve. FV = $100 (1 + 10 percent) 5 FV = $100 (1 + 0.10) 5 FV = $100 (1.10) 5 FV = $100 (1.6105) FV = $161.05 Thus, the value of the $100 investment after 5 years will be $161.05. In this example problem, note that the compounding process (the process of earning interest on interest) has produced total interest of $61.05, which is greater than the total simple interest of $50. The Present Value of a Dollar Finding the present value of a dollar is the opposite of calculating its future value. The present value of a dollar is the amount that a future dollar is worth today. You would calculate the present value when you need to determine how much money to invest today to obtain some future goal. Discounting is the process of finding the present value of some future amount. Let s look at another example problem. Example: Suppose you want to know how much money to invest today to reach a future goal of $100. You want to invest the money for 1 year in an account that pays 10 percent interest annually. Solution: This calculation is relatively simple. You ve been given the following three variables: future value (FV) = $100 number of time periods (N) = 1 year interest rate (I) = 10 percent annually 10 Financial Management
The missing variable that you need to calculate is the present value (PV), which is the amount of money you ll need to invest today to reach your future goal. You would use the following formula to calculate the present value of the investment. PV = FV [(1 + I.) N.] Next, substitute the known values of FV, I, and N into the formula and solve. PV = $100 [(1 + 10 percent) 1 ] PV = $100 [(1 + 0.10) 1 ] PV = $100 [1.101] PV = $100 1.10 PV = $90.91 Thus, you ll need to invest $90.91 today to have $100 after one year. Now, consider the same problem over a five-year period. Example: Suppose you want to know how much money to invest today in order to reach a future goal of $100. You want to invest the money for 5 years in an account that pays 10 percent interest annually. Solution: In this problem, you re given the following three variables. FV = $100 N = 5 years I = 10 percent annually You would again use the following formula to calculate the present value of the investment (PV). PV = FV [(1 + I.) N ] Lesson 1 11
Substitute the known values of PV, I, and N into the formula and solve. PV = FV [(1 + 10 percent) 5 ] PV = $100 [(1 + 0.10) 5 ] PV = $100 [(1.10) 5 ] PV = $100 [1.6105] PV = $62.09 Thus, the process of discounting tells you that you ll need to invest $62.09 today. After 5 years of earning 10 percent interest annually, your investment will have a value of $100. The Future Value of an Annuity An annuity is a series of equal payments made at equal time intervals (for example, annually). An annuity that s paid annually is called an ordinary annuity. Let s look at some example problems that demonstrate how to calculate the value of an annuity. Note: The equations we provide in the study guide for calculating the time value of annuities take a different form than the equations in the textbook. We think you ll find that the study guide equations are simpler. Example: What will be the future value of an ordinary annuity after 3 years, if $100 is deposited annually and the account earns an interest rate of 10 percent annually? Solution: In this problem, you re given the following three variables. value of each payment (PMT) = $100 number of annuity payments (N) = 3 annual payments interest rate (I) = 10 percent annually The missing variable that you need to calculate is the future value of the annuity (FV), which is the value of the investment after 3 years. 12 Financial Management
You would use the following formula to calculate the future value of the investment (FV). FV = PMT [(1 + I.) N 1] I Substitute the known values of PMT, I, and N into the formula and solve. FV = $100 [(1 + 10 percent) 3 1] 10 percent FV = $100 [(1 + 0.10) 3 1] 0.10 FV = $100 [(1.10) 3 1] 0.10 FV = $100 [1.331 1] 0.10 FV = $100 0.331 0.10 FV = 33.1 0.10 FV = $331.00 Thus, the value of the annuity after 3 years will be $331.00. The Present Value of an Annuity Now let s examine how to calculate the present value of an annuity, which is the amount of money you ll need to invest today to reach a future goal. Example: What is the present value of an annuity that will pay $100 a year, at the end of each of the next 3 years, at an interest rate of 10 percent annually? Solution: In this problem, you re given the following three variables. PMT = $100 N = 3 yearly payments I = 10 percent annually The missing variable that you need to calculate is the present value of the annuity (PV). You would use the following formula to calculate PV. PV = PMT {1 [1 (1 + I ) N ]} I Lesson 1 13
Substitute the known values of PMT, I, and N into the formula and solve. PV = $100 {[1 [1 (1 + 10 percent) 3 ]} 10 percent PV = $100 {[1 [1 (1 + 0.10) 3 ]} 0.10 PV = $100 {[1 [1 (1.10) 3 ]} 0.10 PV = $100 {[1 [1 1.331]} 0.10 PV = $100 {1 0.7513} 0.10 PV = $100 0.2487 0.10 PV = 24.87 0.10 PV = $248.70 Thus, you ll need to invest $248.70 today to receive payments of $100 per year for 3 years. Practice Problems Now, in this section, we ll examine some more practice problems. Work through each of the practice problems to make sure you understand the calculations that are represented. Example: At 5 percent interest compounded annually, how many years will be needed for an investment of $200 to grow to $255? Solution: In this problem, you re given the following three variables. FV = $255 PV = $200 I = 5 percent annually The missing variable that you need to calculate is the number of time periods (N). You would use the following formula to calculate N. FV = PV (1 + I ) N 14 Financial Management
Substitute the known values into the formula and solve for N. $255 = $200 (1 + 5 percent) N $255 = $200 (1 + 0.05) N $255 = $200 (1.05) N $255 $200 = (1.05) N 1.275 = (1.05) N log 1.275 = N (log 1.05) 0.1055 = N (0.021189) 0.1055 0.021189 = N 4.97 = N N = 5 years (rounded) Example: A widow currently has a $75,000 investment that yields 7 percent annually. Can she withdraw $15,000 a year for the next 10 years? Solution: This problem requires you to find the amount of money that can be withdrawn from her account annually. In other words, you re looking for the annuity payment for this investment. In this problem, you re given the following three variables. PV = $75,000 N = 10 yearly payments I = 7 percent annually The missing variable that you need to calculate is the value of each payment (PMT). You would use the following formula to calculate PMT. PV = PMT {1 [1 (1 + I ) N ]} I Lesson 1 15
Substitute the known values into the formula and solve for PMT. $75,000 = PMT {1 [1 (1 + 7 percent) 10 ]} 7 percent $75,000 = PMT {1 [1 (1 + 0.07) 10 ]} 0.07 $75,000 = PMT {1 [1 (1.07) 10 ]} 0.07 $75,000 = PMT {1 [1 1.9672]} 0.07 $75,000 = PMT {1 0.5083} 0.07 $75,000 = PMT 0.4917 0.07 $75,000 = PMT 7.024 $75,000 7.024 = PMT $10,677.67 = PMT No, she can withdraw only $10,677.67 per year for the next 10 years. Example: Imagine that you re 30 years old and inherit $75,000 from your grandfather. You want to invest your inheritance and increase the total amount to $100,000 after 4 years. What compound annual interest rate of return must you earn to achieve your goal? Solution: This problem requires you to find the interest rate that will produce a certain future value. You re given the following three variables: FV = $100,000 PV = $75,000 N = 4 You would use the following formula to calculate the interest rate (I). FV = PV (1 + I ) N 16 Financial Management
Substitute the known values into the formula and solve for I. $100,000 = 75,000 (1 + I.) 4 $100,000 75,000 = (1 + I.) 4 1.3333 = (1 + I.) 4 log 1.3333 = 4 log (1 + I.) 0.1248 = 4 log (1 + I.) 0.0312 = log (1 + I.) 1.075 = 1 + I 0.075 = I I = 7.5 percent (rounded) An interest rate of 6 percent will increase the amount of your inheritance to $100,000 after 4 years. Example: Suppose that you want an investment of $1,000 to double within a period of 3 years. At what annual rate of growth must your investment increase to achieve your goal? Solution: You need to find the interest rate that will cause your investment of $1,000 to double to $2,000 within 3 years. You re given the following three variables: FV = $2,000 PV = $1,000 N = 3 You would use the following formula to calculate the interest rate (I). FV = PV (1 + I ) N Lesson 1 17
Substitute the known values into the formula and solve for I. $2,000 = $1,000 (1 + I.) 3 $2,000 $1,000 = (1 + I.) 3 2 = (1 + I.) 3 log 2 = 3 log (1 + I.) 0.3010 = 3 log (1 + I.) 0.3010 3 = [3 log (1 + I.)] 3 0.1003 = log (1 + I.) 1.26 = 1 + I 0.26 = I I = 26 percent An interest rate of 26 percent will cause an investment of $1,000 to double within 3 years. Note that this answer will be the same no matter what value you choose for your present value. You can prove this to yourself if you wish by resolving the problem with a different present value. For example, try PV = $10,000 and FV = $20,000. 18 Financial Management
Self-Check 1 At the end of each section of Financial Management, you ll be asked to pause and check your understanding of what you ve just read by completing a Self-Check. Writing the answers to these questions will help you review what you ve studied so far. Please complete Self-Check 1 now. Indicate whether each of the following statements is True or False. 1. Compounding is the process of determining the amount of simple interest on an investment. 2. At an annual interest rate of 7 percent, it will take 16.2 years for an investment of $100 to triple to $300. 3. The future value of one dollar increases with lower interest rates. 4. The future value of one dollar increases with longer periods of time. 5. An ordinary annuity is a series of equal payments made at the beginning of each time period. Complete Problems 1, 2, 4, 5, 6, 7, 9, 12, 13, 14, 15, 18, and 19 on pages 131 132 in the textbook. Check your answers with those on page 121. Lesson 1 19