SURVEY OF FINANCE
SURVEY OF FINANCE FIN 3010
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Chapter 2 Time value of Money 40 Educational Objectives 0 Learn the importance of Time Value Concepts 0 Learn how time value of money concepts relate to businesses 0 Expose students to several kinds of problems that involve new ideas and concepts Businesses invest money today a cash out ow with the prospect of receiving cash from the investment in the future Comparing cash today to cash that is expected 0 be received in the future is like comparing apples and oranges Time value of money is the approach business analysts take to compare future cash ows to present cash ows To compare future expected cash ows to a present cash ow the business analyst moves all of the future expected cash ow stream back to the same point in time as the present cash ow Discounting is the process of moving cash ows from the future to the present Discounting is the process of taking interest out of a series of cash ows Whenever you move something you use up something in the process A trucking company moving freight uses energy As humans we use energy moving from one point to another Moving cash ows requires no energy but there is a cost and the cost is the return required each period by investors on the cash investment Business use of time value of money concepts The discount rate is one name given to the return required by investors The higher the discount the greater the return removed from the expected cash ows Higher discount rates mean lower values today of the future expected cash ows Present value is the value today of future cash ows It is the discounted value of future cash ows Future cash ows usually in ow into a company because of an investment When a business analyst calculates a present value of future cash ows it might be compared to an amount today The amount today might be the cost of an investment and the cost of an investment is a cash out ow The business analyst compares the present value of in ows to the cost of an investment out ow today to determine if an investment project provides a rate of return needed to compensate the providers of capital to fund the investment If the present value of the in ows exceeds the cost of the investment investors expect to earn a rate of return on their investment greater than the return they requ1re Personal use oftime value of money concepts In a capitalistic world everyone must deal with time value of money concepts Retirement planning is essential for everyone in our society and retirement planning involves among other things depositing money early in a career and compounding returns and interest over time A deposit of 500 each month for 30 years will accumulate to 50225752 immediately after the last deposit assuming interest compounds monthly at an annual rate of 6 The sooner the 500 monthly deposits begin the sooner you can accumulate 50225752 and the sooner this amount can grow to meet retirement needs The amount 50225752 is a future value and is an amount accumulated at the end of a series of payments 4l Even if someone ends up on skid row there remains a place for time value of money concepts when deciding to take a lump sum one payment or an annuity a series of payments at the same time interval with respect to lottery winnings To make this analysis compare the present value of the annuity payments future cash ows to the lump sum offered today The highest number should be the cash ow the winner should accept Almost everyone borrows money at some point in life and has the loan amortized so equal payments usually monthly that include principal and interest contribute to retirement of the loan at a specific date in the future If you know of someone making a monthly mortgage payment or a monthly car payment this is quite likely an amortized loan Knowledge of time value of money concepts will the individual to compare present values to future values calculate car and home payments and determine the amount owed on a mortgage at any point in time Time value of money concepts are essential to anyone with money in our society What has teaching time value of money taught the teacher and what can students learn from this While teaching time value of money concepts for over twenty years I have seen one big problem that students have Students do not pay attention to the detail associated with this topic Several details inputs are necessary for all time value of money problems I decided to include these details at the beginning of this chapter so as to emphasize their importance Very Important One of these details I have to tell students day after day is that t calculator key N is the number of annuity payments when working with an annuity and t is the number of compounding periods when working with a lump sum I will present a list of details to which you must pay attention Paying attention to detail will make the problems easier Time value of money is a technical topic that requires attention to several basic issues at the same time when solving problems To be good at this subject students do not have to be mathematically oriented students just need to pay attention to detail I believe that an understanding and integration of the basic ideas is so important that I am beginning the chapter with the most basic of these ideas and concepts and clearly pointing out that students must integrate these ideas and concepts in order to solve and understand the problems Below are the basic details and I am sure you have seen some of these in a math class You will have to use these together to solve problems so commit these to memory or write them on a reference sheet so that you can keep them handy while working problems These basic ideas are a checklist to use while working problems Problem practice is the only way to learn how to apply these ideas There is quite a bit to learn involving this material Don t even think about waiting until a few days before the exam to begin studying A word on test taking and studying for time value of money material Students taking an introductory finance course come from many different backgrounds and many different forms of study and testing Time value of money involves many basic ideas that are required to work the problems While you need to know these basic ideas I do not recall any teacher asking questions directly related to basic ideas The presentation of 42 basic ideas conceptual information relating to time value of money is for the purpose of problem solving The basic ideas are not individual pieces of information for memorization that might later appear on an exam as individual questions A nancial calculator facilitates calculation of time value of money problems and a presentation of calculator entries in this book will help with calculator usage There are many financial calculators on the market so the presentation of calculator entries will be generic and as best as possible apply to all calculators There are several ways to teach time value of money concepts These include using the formulas spreadsheets using table values that represent values of the formulas for a specific interest rate and number of periods and the financial calculator I believe that students need some association with the time value of money formulas regardless of the teaching method used This is especially true for lump sum formulas Annuity formulas are of little use and very difficult to use due to the layers of calculations necessary The important issue with annuity formulas is that you know when to apply a future value or present value of an annuity formula and how the annuity formulas are defined in terms of the number of compounding periods and payments This chapter offers exposure to basic ideas and equations As you encounter difficulty while working the problems go back in the chapter to reference the basic ideas Equations presented in this chapter relate a specific definition of how compounding periods and payments should fit a particular cash ow Formulas calculator key settings and cash flow linkages Very Important When we use the various methods formulas calculators table values etc to work time value of money problems what we are doing for the most part is moving cash ows around in time When we calculate the future value of an annuity it is turned into a lump sum at some point in the future so we are moving cash ows at different but regular points in time to one specific point in time in the future When we calculate the present value of an annuity or lump sum we move future cash ows to the present In business the present value of expected cash in ows is compared to the cost of an investment to determine the acceptability of an investment project As part of the process of moving cash ows around in time we must pay attention to the number of payments and the number of compounding periods Anyone solving time value of money problems must think about relating the cash ows in the problem to the facilities available to solve the problem The formulas table values calculator key settings and spreadsheet facilities functions offer specific definitions relating to compounding periods and payments which a problem solver must match to cash ows in a problem Future Value of a Lump Sum There are several concepts associated with calculating the future value of a lump sum A future value is an amount of money one will accumulate in the future One of the most important issues to students new to time value of money concepts is identifying a problem where a future value of a lump sum calculation is appropriate When a 43 problem involves one single cash ow amount and not a series of equal cash ows and you see wording in the problem relating to the dollar value an amount will accumulate or grow to this is a future value of a lump sum problem Basic Idea 1 relates to the calculation of the dollar amount of interest The first time I used this formula was in a 9 grade algebra class To nd the dollar amount of interest for a period of time multiply the annual rate of interest times the length of time the amount is on deposit and then multiply times the principal balance the amount in the bank Basic Idea 1 Dollar amount of interest Dollar amount of interest Balance outstanding Rate Time Suppose Dr Marsh deposits 10000 into an account for 6 months that ea1ns interest at an annual rate of 6 compounded semiannually Determine the amount Dr Marsh will have on deposit in 6 months Dollar amount of interest 10000 06 5 30000 Dr Marsh earned 300 interest for the first siX months The 300 amount represents interest on the principal amount only and interest on the principal amount Basic Idea 2 is that simple interest is interest on the principal amount only Basic Idea 2 Simple interest is interest on the principal amount only Basic Idea 3 The interest rate you are quoted is usually the nominal annual rate Basic Idea 4 A lump sum is one single amount as opposed to a series of cash flows Suppose Dr Marsh decides to leave 10000 in the account for the full year and let interest compound Dr Marsh decides to leave the 300 in interest in the bank How much will Dr Marsh accumulate Principal Deposited 1000000 For the first six months Dollar amount of interest 10000 06 5 30000 For the second six months Dollar amount of interest 10000 06 5 30000 For the second six months Dollar amount of interest 300 06 5 900 Total accumulation 10 60900 Time Value Concept 1 The higher the interest rate the greater the compounding or discounting effect A compounding effect means more interest is put into a cash ows A discounting effect means more interest is removed from the cash ows Dr Marsh will accumulate 1060900 in one year and you see that the interest of 30000 during the first months earns interest during the second siX months Basic Idea 5 is compound interest and this is interest on interest Basic Idea 5 Compound Interest is interest on interest 44 If you had to go through the steps above to calculate the amount of an accumulation for all of the problems it would take a very long time Formulas have been derived that will make the work easier These formulas derived for one unit of monetary value a dollar are appropriate for application to any dollar amount by multiplying the value of the formula evaluated at a certain interest rate and a certain number of compounding periods times the amount of the present value place on deposit The first equation in this chapter is the formula for future value of a lump sum The future value of a lump sum is the amount of the accumulation In the problem above it is the 1060900 that Dr Marsh accumulates in one year Variable De nitions very Important Quoted rate is the quoted annual rate of interest m is the frequency of compounding per year r is the interest rate per period r Quoted rate In so r m Quoted rate t is the total number of compounding periods when working with a lump sum For a lump sum t m the number of years of compounding t is the total number of payments when working with an annuity Very Very important One of the rst concepts you want to know is the difference between t when dealing with a lumpsum and an annuity Equation 1 Formula for the future value of a lump sum of 1 I F V 1 X 1 r Basic Idea 6 Always use the interest rate per period r in formula calculations To find the future value of a lump sum multiply the value of the formula 1r plugging in r and t times the dollar amount placed on deposit the present value Equation 2 is a formula used to calculate the future value of a lump sum Exercise 01 gives an example of the use of Equation 2 Equation 2 1 r t Present value Future value of a lump sum Exercise 01 To use the formula to calculate the future value in one year of the 10000 on deposit today Dr Marsh will raise 103 to the 2quotd power On some financial calculators you use the yx key where y 103 and X 2 The value of the formula evaluated at an interest rate per period r of 3 and 2 t compounding periods is 10609 Future value of a lump sum formula evaluated at r 3 and t 2 is 10609 10609 10000 10609 45 Explanation The interest rate per period is 6 divided by 2 semiannual compounding periods per year and this equals 3 Dr Marsh will accumulate 10609 in one year two semiannual compounding periods Figure 1 illustrates a time line of the situation A time line shows cash ows in time perspective Time zero might be thought of as today and whole number time periods to the right of zero represent compounding periods or payments in the future For nancial mathematics purposes all cash ows occur today or in the future Time lines are very useful for diagramming a time value of money problem I have been teaching this topic for 22 years and I still use time lines because they help me to get the problem right You can begin a time line with the number one instead of zero but I like to begin with zero because beginning with zero aids in the calculation of compounding periods Time lines help you count compounding periods and payments and allow you to transfer a written problem to a time dimension Figure 1 Time Line 10000 10609 0 1 2 3 4 Present Values are on Future values are on the the left of a future value right of a present value Basic Idea 7 When a cash flow is compound or discounted it moves in time perspective into the future or bank to the present When you compound a present value you convert a present value into a future value When you discount a future value you convert a future value into a present value Also a present value is not necessarily determined as of today at time zero A present value is found when a future value is discounted Basic Idea 8 The effective interest rate is the nominal annual rate of interest adjusted for the frequency of compounding The effective annual rate will be greater than the annual rate if the frequency of compounding per year is greater than 1 Equation 3 The effective rate annual rate EAR 1r 1 Time Value Concept 2 The greater the frequency of compounding the more you will accumulate all other things remaining equal Dr Marsh is earning an effective annual rate of 1032 7 1 0609 or 609 The effective annual rate is greater than the quoted nominal annual rate because of compound interest The quoted annual rate is 6 the interest rate per period is 3 the semiannual rate and the effective rate is 609 If Dr Marsh has the 6 interest rate compounded quarterly instead of monthly the effective rate is 10154 7 1 61363 Basic Idea 9 indicates that the greater the frequency of compounding the higher the effective interest rate The difference between 46 semiannual and quarterly compounding appears small but over time the difference can be meaningful If you deposit 10000 in the bank that earns interest at an annual rate of 6 but compounded quarterly the accumulation in 40 years will be 10828461 If you deposit 10000 in the bank that earns interest at an annual rate of 6 but compounded semiannually the accumulation in 40 years will be 10640890 The 187570 difference is important Basic Idea 9 The greater the frequency of compounding the higher the effective interest rate Basic Idea 10 When dealing with a lump sum problem t is the number of compounding periods The nancial calculator You might ask at this point how to do the calculation on a financial calculator Most financial calculators have five keys that are associated with internal registers that hold values Calculator Example 1 Calculator Example 1 Financial calculator keys The PV PMT and FV keys with a heavy bold line in this example are the amount keys The N and IYR keys are the variable keys IN IWRI W IPMTI WI If you have a problem where the entry of two of the three amount keys is done in order to calculate the amount in the third key be very careful about the sign of the two amounts you enter and the expected sign of the amount key to be calculated The N key is the entry for the number of compounding periods if the PMT key entry is zero The N key is the entry for the number of payments if the PV or FV key entries are zero If you have a problem with an overlapping lump sum and annuity it cannot be worked unless N is equal for the number of payments and the number of compounding periods In terms of the notation in the textbook think of N on a financial calculator as t For introductory problems where one of the three initial amount key entries is zero the sign of the calculated amount key will be the opposite of the initial amount key entry The positive and negative signs indicate cash in ows and out ows The IYR key entry is always the periodic interest rate Financial calculators have the capability for you to enter the annual rate and change the number of payments per year so you can just enter the annual rate and let the calculator determine the periodic rate This is great for people working in an area like real estate where most of the payments are monthly But for students of nance working many different problems the payment per year PYR setting must me changed as the problem changes Both graduate and undergraduate students tell me that they would rather enter the periodic rate than change the PYR key for almost each problem 47 The PV PMT and FV keys are the amount keys For these keys you cannot enter the comma separating thousands but you can enter the decimal point Check your calculator manual for the location of the key because the key must be used to change the sign of the amount keys A common calculator error is a result of the calculation of the N and IYR keys where the amount key non zero initial entries are all of the same sign To correct this error think carefully about the sign of the cash ows entered into the amount keys and make sure you have one positive and one negative amount key If you initially enter amounts for all three of the amount keys you must think very carefully about the sign of each cash ow entry Calculator Example 2 Dr Marsh deposits 10000 into a CD and will accumulate 1060900 in one year with interest compounded semiannually at an annual rate of 6 How do I do this calculation on a nancial calculator The first thing to do with a nancial calculator is to set the PYR key to 1 This allows entry into the interest rate register IN R of the periodic rate Financial calculators provide a facility for the entry of the nominal annual rate into the IYR key but students will have to reset the PYR key each time a different problem is worked Please check your calculator manual to change the PYR setting On some calculators the key is labeled PY and on others the key is labeled PYR This facility might be fine for someone working in the real estate business where most all of the payments are monthly but in a finance class where students are working problems with several compounding frequencies per year there is a tendency for students to not change the PYR setting and miss problems In my opinion it is best for financial calculator users to always enter the periodic interest rate into the IYR key prior to working problems This forces students to think about the periodic rate for each problem and it is important for students to think Basic Idea 6 indicates to always use the interest rate per period in time value of money calculations This is really easy but critically important Calculator Hint 1 indicates that you should check your calculator manual to determine if there is a special key sequence associated with finding the answer Calculator Hint 1 Check your calculator manual to determine if there is a special key sequence associated with nding the answer On some calculators you just press the appropriate key and on others you must press a compute key before pressing the appropriate key l 2 l l 3 l 10000 I l 0 l E 1060900 I l N l l IYR l l PV l l PMT l l FV l 48 Calculator Hint 2 If you enter a positive number for PV or PMT and calculate the FV or the opposite the calculator will present the answer as the opposite sign Basic Idea 11 Any time the entry into the payment key is zero you know the problem is a lump sum Dr Marsh accumulates 1060900 in one year A good thing about a nancial calculator is that what if calculations are very easy to do For example to determine the amount of the accumulation in one year with interest compounded quarterly instead of monthly just change the IYR key to 15 and the N key to 4 You don t have to change the PV of PMT keys I do suggest that while learning the nancial calculator for each different problem that you make an entry for each key This is because the amount in the calculator for a certain key will remain unchanged until another key entry is made into that register Calculator Example 3 l 4 l l 15 l 10000 I l 0 l E 1061363 I Calculator Hint 3 You must enter a value for four of the five keys and calculate the value of the fifth key Calculator Hint 3 points out that four of the ve keys must have values entered in order to solve for the fth key value A special character may appear in the display of some calculators indicating the answer Calculator Hint 4 Enter the periodic interest rate into the IYR key as a number and the calculator will convert the rate into a percentage Any number in the IYR register is 5 percentage To convert to a decimal any number you see or enter into the IYR key register move the decimal point two places to the left and drop the percent sign Calculator Hint 4 points out that the calculator user should enter the interest rate into the IYR key as a number and not as a percentage In our example problem the number 3 was entered into the IYR key instead of 3 as a decimal which is 03 If the annual rate is 6 with interest compounded monthly the periodic rate is 6 12 5 Therefore 5 should be entered into the IYR key Notice that 6 12 06 12 005 as a decimal and 005 should not be entered into the IYR key Note however if you are funding the periodic interest rate to plug into a formula say Equation 1 005 will be used as the periodic interest rate 49 Basic Idea 12 Compounding intervals are usually daily monthly quarterly semiannually and annually The important idea is that the periodic rate is the rate for each compounding period If the annual rate is 12 The daily rate is 12365 00032877 032877 The monthly rate is 1212 01 1 The quarterly rate is 124 03 3 The semiannual rate is 122 06 6 The annual rate is 121 12 12 The daily monthly quarterly semiannual and annual rates all can be periodic rates The important idea is that the periodic rate is the rate for each compounding period Use the periodic rate always in formulas or in nancial calculators The calculations in the examples above show the accumulation of an amount on deposit the future value If you know four of the ve key values the fth can be calculated For example if you want to know the amount you must deposit today in order to accumulate 1060900 in one year with interest compounded semiannually at an annual rate of 6 just make the following key entries in Calculator Example 4 You must deposit 10000 today in order to accumulate 1060900 in one year two semiannual compounding periods The amount you must deposit today is the present value of the 1060900 expected in one year The calculation took 60900 of interest out of the future value so in order to calculate the present value we discounted the future value Calculator Example 4 I 2 Here s an example for you to try What monthly periodic interest rate will MsWashburn earn on a certi cate of deposit CD assuming she deposits 200000 today and the customer service representative at the bank indicates that the200000 deposited today will accumulate to 2254319552 in two years 3 I E 10000 0 I 1060900 Calculator Hint 5 When entering values into the PV PMT and FV keys enter at least one of the values as a negative number if you are solving for the N key or the IYR key The positive and negative numbers represent cash in ows and outflows For calculations involving any two of the three amount keys PV PMT and FV it doesn t matter which key is positive or negative For calculations involving all three of the amount keys the calculator user must be very careful as to the appropriate keys to enter as positive or negative For calculations involving multiple cash ows for example a bond pays a periodic cash interest PMT and a cash maturity value FV the correct sign 50 of the cash flow is critical to a correct answer In the case of a bond pricing problem both the PMT and the FV keys will be positive when calculating the present value price of the bond Calculator Example 5 quot 39 39 39 39 39 39 quotI l 24 l 5 2000 l l 0 l l 2254319552 l I I The periodic interest rate found in Calculator Example 5 is 5 Just add the sign to the value you see in the IYR key As a decimal this is 005 To find the nominal annual rate multiply the periodic rate of 005 times 12 and the answer is 06 or 6 The effective rate Ms Washburn is earning on her CD can be found b raising 1005 to the 123911 power 1005lz and subtracting 1 This rate is 06167781 or 6167781 The effective rate is 00167781 higher then the 6 nominal annual rate and this is due to the compounding effect Here are a few questions before moving on to present value problems 1 Determine the amount John will accumulate in 4 years assuming he deposits 20000 today and earns interest at an annual rate of 4 compounded quarterly Answer 2345157 2 Determine the semiannual interest rate required for Jane to accumulate 4956144107 in 40 years assuming Jane earns compound interest with interest compounded semiannually and Jane places 10000 on deposit today Answer 5 3 Determine the nominal annual rate Jane is quoted by a nancial institution Answer 10 5 2 semiannual periods in a year 4 Determine the effective annual rate Jane is earning on her deposit Answer 1052 1 1025 Use 2 as the exponent because there are 2 semiannual periods in a year and in the formula 1in 1 n represents the number of compounding periods in one year the frequency of compounding per year Present Value of a Lump Sum A present value is the amount one is willing to pay today for a future cash ow If you have 10 in your pocket the present value is 10 You know you can go and buy 10 worth of goods today but suppose you feel the price of the same goods in ten years 51 will be 40 The present value today of the goods that will cost 40 in ten years is the amount of money needed today to pay 40 in ten years for the goods Present value analysis is very widely used in business because business involves an outlay of cash today for investment purposes with the expectation of receiving cash in the future because of the investment Present values can be found at any point on a time line and do not necessarily have to be determined as of today The only requirement for nding a present value is that it is to the left on a time line of the future value To find a present value a future expected cash ow is discounted at the return required by the investor and this required return is the cost of capital A higher cost of capital will result in a lower present value One of the most important fundamentals in uencing the cost of capital is the risk of the cash flows the investor expects This is the principle of risk and return and the higher the risk the higher the cost of capital The concept of future value is more easily understood than present value because almost everyone has thought about accumulating money Future value begins with an amount today and finds the value of that amount in the future assuming the investment today earns some rate of return compounded per period Present value analysis begins with an amount expected in the future and finds the amount someone will pay today for that amount Future value analysis involves moving compounding cash ows into the future and present value analysis moves discounts future cash ows to the future Equation 4 shows the Present value of a lumpsum factor of 1 used to discount future cash ows to today Equation 4 Present value of a lumpsum factor of 1 PV 1gtlt11rt Time Value Concept 3 The present value of a lump sum is the amount one is willing to pay today for a future cash ow The higher the discount rate the lower the amount one is willing to pay To find the present value of a lump sum multiply the value of the formula 11rt plugging in r and t times the dollar expected to be received in the future Equation 5 1lrt X Future value Present value of a lump sum Equation 5 is a formula used to calculate the present value of a lump sum Exercise 02 gives an example of the use of Equation 5 Exercise 02 Suppose Mike is on the reality show Wrestle Smack Up and wins 121550 but the show s producer tells Mike that the money will be available to him in four years Mike knows he can earn a return of five percent per year on his money As Mike s friend help him determine the amount he wins today the present value of the amount he will receive in four years 52 11rt with r equal to 05 and t equal to 4 8227 11054 Future Value Present Value 82270247 121550 1000 Explanation Mike actually won 1000 for being on the show and not 121550 Suppose the producer of Wrestle Smack Up offers Mike the option of 121550 in four years or 950 today with both amounts set aside in a trust account at a bank Present value analysis offers Mike a way to compare amounts at two points in time The present value ofthe 121550 today is 100000 and this amount is greater than the 950 being offered today by the producer so Mike should take the 121550 in four years Calculator Example 6 shows the nancial calculator key entries needed to work the problem The present value calculate is 1000 This amount shows up as a negative number because you would have to deposit cash into a bank today a cash out ow in order to receive a cash in ow of 121550 Calculator Example 6 l 4 l l 5 l I 1000 l l 0 l E 121550 l N l l IYR l l PV l l PMT l l FV l Figure 2 illustrates a time line that shows the location in time of the future value and the present value Think of a present value as on the left of the time line and a present value on the right of a time line Figure 2 Time Line 1000 121550 0 1 2 3 4 Present Values are on Future values are on the the left of a time line right of a time line Just as with future value problems if the present value and a future value is given the interest rate and number of periods can be found Mike knows he can earn an interest rate of 5 at the bank if he invests money today If Mike takes the 950 today what rate of return must Mike earn for four years in order to grow 950 to 121550 Calculator Example 7 shows the calculator entries necessary to calculate the interest rate Be sure to enter the amount in the PV 0r FV key but not both as a negative number 53 Calculator Example 7 l 4 l E 63549 l 950 l l 0 l l 121550 l N l l IYR l l PV l l PMT l l FV l Mike needs to earn 63549 notice that I added the sign to the value in the IYR key on the 950 investment today in order to grow the 950 to 121550 Now Mike has an interest rate to compare to the bank interest rate of 5 Since Mike can t earn 63549 at the bank he is better offtaking the 121550 in four years than the 950 today Revisiting time line analysis Think about time line analysis as leftright analysis Present values are on the left and future values are on the right As part of the method of solving time value of money problems construction of a time line is critical particularly when a problem has more than one part How do you know if a problem is present value or future value or best worked as a present value or future value problem The key to determining if a problem is best viewed or worked as present value or future value is to examine the cash ows in the problem Financial calculators have five basic entry keys and the used must have four of the five variables in order to solve a problem In a lump sum problem the payment variable is always zero If you are given the future value of the cash ow s the problem is best worked as a future value and the amount of the future value is entered into the FV key on the calculator If you are given the amount today or if you need to determine the amount today the problem is best worked as present value Problems where you need to know the amount today relate to finding the amount you owe on a mortgage and the amount one is willing to pay for an investment today Both of these situations involve discounting a future cash ows Determine if a problem is present value or future value To determine if a problem is present value or future value draw a time line and place the cash ows appropriately Begin the time line at time zero and enter the cash ows associated with the problem Remember all cash ows will occur today or in the future The problem will be present value or future value depending on the cash ows given If the problem involves a series of payments and the amount to which these will accumulate the problem is future value The problem might give the future value amount and ask for the calculation of the interest rate amount of the payment or the number of payments No matter what variable is missing if the problem gives payments and 54 indicates in some way the amount to which these payments must accumulate the problem is best viewed and worked as a future value problem Determine the amount or the yearly payment required for Hank to accumulate 10000 in 4 years The 10000 is the future value of the annuity so the problem is best viewed or worked as a future value of an annuity If a problem involves a series of payments and in some way asks for the amount someone might be willing to pay today for these payments the problem is best viewed or worked as a present value problem Pay attention to detail For example suppose Flextron is considering purchase of a piece of equipment that will pay a 10000 cash ow in 10 years Determine the amount Flextron might pay today for the cash ow given that the riskiness of the project is high and investors require a return of 20 per year The amount Flextron is willing to pay today is the present value and the amount Flextron expects to receive is the future value Another example Determine the amount John owes today on a mortgage with 240 monthly payments and compounding periods remaining until maturity Assume the monthly payment John makes is 500 and the annual interest rate compounded monthly is 6 5 is the monthly or periodic rate The amount John owes today is the present value of the cash ows Time Value Concept 4 Future value problems give or ask for a future amount Present value problems give or ask for an amount today Annuities An annuity is a series of equal cash ows arriving at regular time intervals Some examples of annuities are monthly auto and home payments the yearly payment into an Individual Retirement Account if made at the same time each year and in the same amount and a payment stream associated with lottery winnings An annuity can be thought of as a series of lump sums in the same amount The future value of an annuity is the amount accumulated given the deposit of equal cash ows at the same point in time prior to the accumulation The present value of an annuity is the value today of a series of equal future cash arriving at the same point in time Annuities usually have a finite life but if an annuity continues forever it is called a perpetuity Time Value Concept 5 A perpetuity is an equal cash flow that never stops Since businesses are formed with the assumption that they will last forever perpetuities are very useful for valuing businesses What is the difference between a present value of an annuity and a future value of an annuity problem What are some key words that help differentiate between the two First remember that all cash ows occur today or in the future With this in mind a present value of an annuity problem will ask for or give the amount today the present value Present value of an annuity problems won t come right out and say that the problem is present value but the wording of the problem will help with identification For example the present value of a series of payments on a home loan is the amount an individual borrows today to buy a house Also a company might wish to find the amount 55 it is willing to pay today for an investment that is expected to return a series of equal cash ows in the future This amount is the present value of the annuity Future value of an annuity problems are associated with a situation where a series of cash ows accumulate to some amount in the future The amount accumulated in the future is the future value of the annuity There are a couple of strict definitions of which you need to be aware relating to annuities These are the ordinary annuity and the annuity due and the these definitions have confounded students in introductory finance classes for years An ordinary annuity occurs at the end of the period and an annuity due occurs at the beginning of the period Since the end of one period is the beginning of the other period there certainly is room for confusion Payments for an auto or home loan set up on an amortization schedule are usually an ordinary annuity A lease where the initial payment is the same as the remainder of the payments is often an annuity due For all of the trouble the two basic definitions of annuities have caused most annuities can be worked as an ordinary annuity with just a bit of adjustment I approach the definition of annuities not in terms of beginning and ending of period payments because problems can be worded that might mislead students I approach the definition of annuities in terms of the number of compounding periods and payments associated with the series of cash ows Time Value Concept 6 When you find the present value or future value of an annuity you essentially turn an annuity into a lump sum Time Value Concept 7 A present value is to the left on a time line relative to a future value A future value is to the right on a time line relative to a present value Present Value of an Annuity The present value of an annuity is the value today of a series of equal cash ows To identify a present value of an annuity problem look for wording that identifies a series of cash ows that are associated with an amount today On both present value and future value of annuity problems you might see the words payments withdrawals or deposits On present value annuity problems you should see some wording relating to the present value that will aid in identifying the problem as a present value of an annuity Here are some examples of present value of an annuity problems Biggest Enterprises expects cash ows of 4000 each year from a risky investment project and the company requires a return on invested capital of 14 for this risk class project The company expects the first 4000 cash ow to occur one year from today What is the most Biggest Enterprises will pay today for the project The amount the company will pay today is the present value of the cash ows To pay off a mortgage loan in 15 years Cathy will make 180 equal monthly payments of 600 beginning one month after she receives a check from the bank to purchase a house Determine the amount the bank is willing to loan Cathy to purchase the house The amount the bank is willing to loan Cathy to purchase the house is the present value of the cash ows The present value is to the left on the time line and the 56 cash ows are to the right on the time line A home loan is usually amortized where a series of equal payments at the same time each month are made for a speci ed number of months to pay off the loan balance to zero after the last payment Each payment has an interest and principal component unless the first of the series of equal payments is made the day a loan is taken out As with the calculation of the future value of an annuity where the number of payments and the number of compounding periods can be different present value of an annuity problems can have differences in the number of payments and the number of compounding periods A present value of an annuity problem is defined by the number of compounding periods and annuity payments After recognizing a problem as a present value of an annuity determine whether or not interest is associated with the initial payment If there is no interest charged on the principal amount associated with the initial payment there is one less interest compounding period than payment If the first payment is made after a period of time passes between the present value and the first payment the cash ow series will have the same number of compounding periods as payments Equation 6 gives the formula for the present value of an annuity where we account for the same number of quot periods as navments Equation 6 Present value of an annuity where we account for the same number of compounding periods as payments This formula is often referenced when calculating the future value of an ordinary annuity Calculator Setting If the LED screen on your calculator shows no reference to a beginning setting you can assume the ending setting applies 1 MW 7quot Annuity Pr esent Value C x C the cash ow the equal amount of the annuity payment Figure 3 Illustration of compounding periods and payments associated with Equation 6 The present value is found at the arrow 1 1000 1000 1000 1000 0 1 4 Compounding Compounding interva 1 39 Compounding interval 2 39 interval 3 rval 4 Compounding 39nte To find the present value of an annuity multiply the value of the formula evaluated at some interest rate r and t number of annuity payments times the amount of the annuity Equation 7 illustrates this concept and 57 Exercise 03 gives an example of its use To count compounding periods when working a present value problem start at the right of the time line and count compounding intervals to the left To determine the number of annuity payments just count them Equation 7 1 11 r t r C Present Value of an Annuity Exercise 03 Determine the amount Jane should pay today for a fourmonth annuity of 1000 and Jane will earn interest at an annual rate of 4 compounded monthly The initial 1000 payment begins one month from today Below is a time line representation of this problem By counting compounding intervals you should see that there are four compounding intervals periods to time zero Just count the payments and you see that there are four payments on the time line 1 11 r t r C Present value of an Annuity with one less compounding period than payment 1 1100333 4r 1000 396688 39688 1000 396688 Another application of a present value of an annuity formula is to nd the amount of the equal payment To nd the amount of the equal payment of an annuity divide the value of the formula into the amount of the present value of the annuity evaluated at some interest rate r and t number of payments This is illustrated in Equation 8 Exercise 04 applies Equation 8 Equation 8 Algebraic rearrangement of Equation 7 Present Value of an Annuity 1 11 r t r Amount ofthe equal payment Exercise 04 Determine the equal annuity payment required to accumulate a future amount Jane has 396688 in the bank today and four more months of law school remaining Jane would like to withdraw the 396688 in four equal payments with the rst payment beginning one month from now Determine the monthly payment Jane can withdraw beginning in one month assuming she earns interest at an annual rate of 4 compounded monthly 396688 1 11003334 1000 396688 396688 1000 Jane can withdraw 1000 each month for the next four months Let s use the nancial calculator to solve some present value of an annuity problems 58 Exercise 05 Suppose Thrifty Investments is considering the purchase of an investment project that is expected to pay an annuity cash ow of 900 each year for the next ve years Thrifty requires a return on this risk class investment of 6 per year Determine the amount Thrifty should pay for the investment today assuming the rst 900 cash ow is expected to arrive one year from now 5 lNlllYRPVPMTFV Thrifty Investments will be willing to pay 379112 today for the investment that is expected to provide four end of year cash ows of 900 Thrifty Investments will expect to earn a return of 6 in invested capital Notice that Thrifty Investments gets back 5 900 4500 The difference between the 4500 expected to be received over four years and the 379112 cash paid out today represents the total return to Thrifty Investments in dollars The price Thrifty Investments will be willing to pay today for the investment 379112 is the price paid that will result in a compound rate of return per year of 6 6 I 53791125 I 900 I I 0 I I I Exercise 06 Mary and Feliciajust bought a house for 250000 They paid 50000 down and borrowed 200000 on a 15 year amortized loan The rst payment will be one month after the house closes All of the nancial and legal agreements begin at closing They were able to obtain nancing for the 200000 at a 55 annual rate compounded monthly Determine the amount of the equal monthly payments Mary and Felicia will pay I 180 I I45833 I E 200000 E I1634166 I I 0 I I I N I I IYR I I PV I I PMT I I FV I People who owe money on a mortgage frequently ask for the loan payoff or the balance as of a certain date If you know time value of money concepts this calculation is relatively easy The amount owed on a mortgage at any one time is the discounted value of the future expected cash ows mortgage payments To nd the amount owed at any time discount the remaining cash ows Also to nd the amount you owe on a mortgage immediately after any payment you can examine an amortization schedule Let s rst show how to calculate the amount you owe and then let s explore the amortization schedule 59 Time Value Concept 8 The amount you owe on a mortgage at any point in time is the discounted value of the future expected cash flows Exercise 07 Suppose Mary and Felicia just made their 49Lh payment Determine the amount Mary and Felicia owe on the mort age Mary and Felicia have made 180 7 49 131 payments just after making the 49 payment Immediately after the 49Lh payment is made Mary and Felicia wi11 owe 16068220 The amount 16068220 is the discounted value of 131 payments of 1634166 discounted at a rate of 5512 per month l 131 l I 45833 I E 16068220 E l 1634166 l l 0 l I l N l l IYR l l PV l l PMT l l FV l The Amortization Schedule When a loan is structured such that making a speci c number of equal payments will bring the loan balance to zero this is an amortize loan Each of the equal payments will have a principal and interest component The interest component of the payment will be greatest at the beginning of the loan and the principal component will be the greatest near the end of the loan An amortized loan presents the opportunity to build an amortization schedule The amortization schedule shows the time period amount of the equal payment the interest component the principal component and the loan balance Time Value Concept 9 An amortized loan When a loan is structured such that making a specific number of equal payments will bring the loan balance to zero this is an amortize loan Mary and Feliciajust bought a house for 250000 They paid 50000 down and borrowed 200000 on a 15 year amortized loan The rst payment was one month after the house closed They were able to obtain nancing for the 200000 at a 55 annual rate compounded monthly Prepare an amortization schedule for Mary and Felicia for the rst three months The amortization schedule is Figure 4 is the result Note that all of the numbers in the amortization 39 J 39 were 39 39 J on a J J 39 The numbers you see are formatted to two decimal places so some of your calculator calculations might be just a few cents or in some cases a dollar or so off 60 Figure 4 Amortization Schedule 15year loan rst three months Time Payment Interest Principal Balance 0 20000000 1 163417 91667 71750 19928250 2 163417 91338 72079 19856171 3 163417 91007 72409 19783762 From the amortization schedule in Figure 4 you see the division of the equal payments into interest and principal components and that the interest component is greater than the principal component at the beginning of the loan If a loan is really long say for 30 years the interest component will be considerably greater than the principal component An amortization schedule for the rst three months of a 30 year loan is shown in Figure 5 Figure 5 Amortization Schedule 30year loan rst three months Time Payment Interest Principal Balance 0 20000000 1 113558 91667 21891 19978109 2 113558 91566 21991 19956117 3 113558 91466 22092 19934025 Someone might take out a thirty year loan in order to lower the monthly payment Also notice that the interest component is the same in the 15 year and the 30 year mortgage for the rst month Why This is because the monthly interest rate and the outstanding balance are the same for the rst month To point out the differences in the interest and principal components at the beginning and end of the loan Figure 6 shows the amortization schedule for a 30 year loan for the last three months Notice in Figure 6 that the interest component is very small in the last year of a loan Only the interest component is taX deductible and this small interest component will provide very little income taX bene t to an individual Figure 6 Amortization Schedule 30year loan last three months Time Payment Interest Principal Balance 358 113558 1547 112011 225564 359 113558 1034 112524 113040 360 113558 518 113040 000 61
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