Finance Lecture Notes for the Spring semester V.71 of. Bite-size Lectures. the Time Value of Money (TVM) and

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1 Finance 2400 Lecture Notes for the Spring semester 2018 V.71 of Bite-size Lectures on the Time Value of Money (TVM) and the discounting of future cash flows. Sven Thommesen 2018 Last updated: Generated:

2 Sven Thommesen 2018 Lectures on the time value of money (TVM) and the discounting of future cash flows We live in a modern economy based on the use of money, in which specialization and the division of labor play a large role. Also playing a large role are the financial markets, in which people borrow and lend money in a large variety of financial transactions. One common feature of all financial transactions is that he who borrows money (whatever form the loan takes) expects to have to pay interest to the lender. Correspondingly, the lender expects to be able to earn interest on sums lent. Sometimes, to the lender, the transaction looks not so much like a loan as like an investment, where he purchases some kind of financial asset, holds it for a while, and then sells it. He will still expect to be able to earn some rate of return or yield on his investment. The following lectures give an introduction to simple financial theory (the time value of money and discounting) and the math used. They are called bite-size (a) because each lecture is small (most are limited to one page), and (b) because you are supposed to chew each lesson thoroughly before going on to the next one, in order to digest it properly ;) These lectures should be read in conjunction with lectures explaining the use of your specific financial calculator to solve practical problems. There is also a set of problems with worked-out solutions which you can use to test your understanding of the material. (The chapter references in blue refer to Ross, Westerfield and Jordan, Fundamentals of Corporate Finance, 9/e.) 2

3 V.71 TVM LECTURES: TABLE OF CONTENTS I: SIMPLE LOANS AND SIMPLE INTEREST Lecture 101: Simple Loans, Simple Interest Lecture 102: Simple Interest: calculating FV Lecture 103: Simple Interest: calculating PV Lecture 104: Simple Interest: solving for INT and N [m] [m] II: (3610) SIMPLE LOANS AND COMPOUND INTEREST Lecture 111: Simple Loans, Compound Interest 5.1 Lecture 112: Compound Interest: calculating FV 5.2 Lecture 113: Compound Interest: calculating PV [m] 5.3 Lecture 114: Compound Interest: calculating INT and N [m] 5.3 Lecture 115: Compound Interest: The Rule of 72 [m] III: PROBLEM SOLVING: SIMPLE LOANS Lecture 121: Different ways to solve simple loan problems Lecture 122: Solving simple loan problems using tables Lecture 123: Solving simple loan problems using a financial calculator VI: (3610) PROJECTS WITH MULTIPLE EVEN CASH FLOWS: ANNUITIES 6.1 Lecture 201: Cash Flows and Projects 6.2 Lecture 202: FV and PV for multiple cash flows: manual calculations [m] 6.1 Lecture 203: Even cash flows: Annuities VII: PROBLEM SOLVING: ANNUITIES Lecture 211: Different ways to solve annuity problems 6.2 Lecture 212: Solving annuity problems using formulas 6.2 Lecture 213: Solving annuity problems using tables 6.2 Lecture 214: Solving annuity problems using a financial calculator 6.2 Lecture 215: Normal Annuities vs. Annuities Due Lecture 216: Amortization 3

4 XI: (3610) INFLATION AND TAXES 7.6 Lecture 331: The Effects of Inflation on Interest Rates: the Fisher Equation 7.4 Lecture 332: The Effects of Taxes on After-Tax Bond Yields 7.4 Lecture 333: The Effects of Taxes on Pre-Tax Bond Yields 7.6 Lecture 334: The Combined Effects of Taxes and Inflation on Net Yields XV: (3700) MORE SPECIAL TOPICS Lecture 531: The Effects of Inflation on Purchasing Power: using the CPI Lecture 532: The Effects of Inflation on Purchasing Power: using average Π XVII: FORMULAS Lecture 991: A brief discussion of some calculators with financial functionality Lecture 999: Mathematical formulas [m] means: Accompanied by a mathematical explanation 4

5 PART I: SIMPLE LOANS AND SIMPLE INTEREST 5

6 LECTURE 101 SIMPLE LOANS, SIMPLE INTEREST We will start our discussion with the simplest possible type of financial transaction: A simple loan is a financial transaction where only two payments ever take place: one at the beginning, and another (in the opposite direction) at the end. A common example would be: you deposit a sum of money to your savings account today, leave it there for a year, then withdraw the balance, equal to the original deposit plus the interest you have earned. The interesting question is then: how much can you withdraw at the end? The answer depends on how your interest is calculated. There are two possibilities: simple interest and compound interest. Simple interest is based on the principle that interest gets paid on the original amount that was loaned or deposited (the principal). However, if the loan or deposit extends over several interest periods, you do not earn interest on the accumulating interest. In real life, simple interest is only used in cases where the interest you earn is in fact paid to the lender when it is earned, so that she may do something with it then (spend it, or invest it further). One example of such an arrangement is an interest-only mortgage. Another example is a coupon bond, where the bond issuer (borrower) makes regular interest payments to the bond holders (lenders, investors) in the form of coupon payments. Say the Treasury sells a $1000 face value 10-year 6% Treasury Note. You purchase the bond for $1,000, the Treasury sends you a check for $30 every six months for 10 years, then you get your $1,000 back. You get your interest sent to you as you earn it, and can do with it what you want as you receive each payment. 6

7 LECTURE 102 SIMPLE INTEREST: CALCULATING FV Let us say that you loan $100 to your brother at 10% per week simple interest. (Maybe you don t like you brother much, given the rate you re charging him!) After 1 week, he will owe you $10 in interest. After two weeks, he owes $20. After 3 weeks, $30, and so on. In other words, interest is calculated on the original sum he borrowed, which in financial theory is called the principal. The general formula for simple interest is this: where: FV = PV * (1 + i*n) PV i n FV = present value = the sum lent (the principal); = the rate of interest charged or paid per time period (usually per year); = the number of time periods (years) involved; and = future value = the total sum owed or due at the end. Note that in some cases, interest is paid only at the end, while in other cases, the interest earned is paid as it accrues (weekly, in the case of your brother.) In the above example, how much would your brother owe you after a year? FV = PV * (1+i*n) = $100 * ( *52) = $620. (Ouch!) NOTE #1: The formula contains the 1+ part because the principal must also be paid back NOTE #2: In financial formulas we render interest rates as decimal fractions, not as percentages. Thus, i=12% becomes 0.12 and (1+i) becomes Another example: if you placed $1,000 in an investment yielding 12% simple interest per year, then after 5 years you would have: FV = PV * (1+i*n) = $1,000 * (1+0.12*5) = $1,600. 7

8 LECTURE 103 SIMPLE INTEREST: CALCULATING PV Staying with the topic of simple interest for a minute, we can of course use the formula given to calculate the other variables as well. First, let us ask about PV: if you locate a bank account belonging to a deceased uncle, which contains $81, and it has been paying 6% simple interest for 21 years, how much did your uncle originally deposit? From the basic formula, we can solve for PV: PV = FV / (1 + i*n) = $81, / ( *21) = $36,000. Another way to look at this example is to ask: if I could earn simple interest at a rate of 6% per year, and if I planned to leave my money in the bank for a period of 21 years, and if I needed to have $100,000 available to me at that point, how much would I have to deposit into that account today? The answer is: PV = FV / (1+i*n) = $100,000 / (1+0.06*21) = $44,248. 8

9 LECTURE 103 MATH Simple Interest: Solving for PV Our basic equation says that FV PV (1 i n) To solve for present value, all we have to do is divide by the interest factor on both sides: FV PV (1 i n) (1 i n) (1 i n) Which we can simplify to PV FV (1 i n) 9

10 LECTURE 104 SIMPLE INTEREST: SOLVING FOR i and n Next, we can solve for the interest rate (i): if the account contains $81,360 and it has been earning interest for 15 years, what would be the applicable interest rate? FV = PV * (1 + i*n) i FV PV PV n So we get: i = (81,360-36,000)/(15*36,000) = or 8.4% Finally, we can solve for the number of years (n): if the original deposit has earned interest at 9%, how many years has it been sitting in the bank? We have FV = PV * (1 + i*n) n FV PV PV i So we get: n = (81,360-36,000)/(.09*36,000) = 14 years. 10

11 LECTURE 104 MATH Simple interest: Solving for i Our basic equation says that FV PV (1 i n) FV 1 i n PV FV 1 i n PV FV 1 PV i n 1 FV PV ( ) i n PV PV i FV PV PV n Simple interest: Solving for n Our basic equation says that FV PV (1 i n) If you follow a similar process as the one above, you should get n FV PV PV i 11

12 II: SIMPLE LOANS AND COMPOUND INTEREST 12

13 LECTURE 111 SIMPLE LOANS, COMPOUND INTEREST Look back at the example used in Lecture 102, where you lend $100 to your brother at 10% simple interest per week. If this loan were like a coupon bond, he d have to send you a $10 interest check every week, which you could then spend or loan to someone else. If instead he pays you back the whole $620 owed after a full year, it s as if he has paid you an interest rate of 0% on the accumulating interest! For most financial transactions, therefore, (including the very simple one of a bank savings deposit), if the borrower does not send you regular interest payments as the interest is earned, we imagine that it is as if he has borrowed those interest payments as well when they come due, so that he needs to pay you interest-on-the-interest as we go along. This arrangement is referred to as compound interest. 13

14 LECTURE 112 COMPOUND INTEREST: CALCULATING FV [See also: CALC LECTURE #61] Let us use our previous example: say your brother borrows $100 from you at 10% per week, but this time using compound interest. Then: After one week, he owes you $10 in interest ($100 x 0.10), and his total debt is $110. The following week he owes you an additional $11 in interest ($110 x 0.10) for a total debt of $121. After yet another week, the debt is $ And so it goes. The general formula for compound interest is: where: FV = PV * (1+i) n PV i n FV = present value = the sum lent (the principal); = the rate of interest charged or paid per time period (usually per year); = the number of time periods (years) involved; and = future value = the total sum owed or due at the end. So how much does your brother owe you after one year? FV = PV * (1+i) n = $100 * ( ) 52 = $14, (double ouch!) Another example: you put $2,500 in a savings account that pays 4.6% per year, and leave it there for 12 years. How much is in your account now? Answer: FV = $2,500 * ( ) 12 = $4,

15 LECTURE 113 COMPOUND INTEREST: CALCULATING PV Now let us go through the same exercises as in lectures 103-4, but this time assuming compound interest. First, let us look backward and calculate PV: If FV = PV * (1+i) n then that gives us PV = FV / (1+i) n. So: if your uncle s bank account contains $81,360 and it has been earning 6% interest for 21 years, how much did your uncle originally deposit? Answer: PV = FV / (1+i) n = $81,360/(1.06) 21 = $23, (As you can see, compound interest accumulates money at a much faster rate than simple interest!) Another example: You are 22 years old. You want to have $1,000,000 in the bank the day you retire at 67. The bank will pay you 5% interest on your money over that time. How much would you have to deposit in the bank today to reach your goal? Answer: PV = $1,000,000 / (1.05) 45 = $111, If instead you could invest your money in the stock market and earn an average return of 12% per year, how much would you need to invest today to get there? Answer: PV = $1,000,000 / (1.12) 45 = $6, (Wow!) 15

16 LECTURE 113 MATH Recall that 1 x y can be expressed mathematically as x y. Thus, PV FV (1 i) n is the same as saying that PV FV (1 i) n. So $1,000,000 * (1.12) -45 = $6, [Do it on your calculator!] 16

17 LECTURE 114 COMPOUND INTEREST: CALCULATING i AND n [See also: CALC LECTURE #62] Question: you want to double your money over 7 years. What yearly return on your money do you need to earn to reach this goal? We have: FV = PV * (1+i) n i FV n 1 PV We get: i $2 7 1 = or 10.4% [Notice: 10.4 * 7 = 72.8] $1 Question: You want to double your money. You invest in a really rotten project which yields only a meager 2% return on your investment. How many years does your money have to stay invested? FV = PV * (1+i) n n FV ln( ) PV ln( FV ) ln( PV ) ln(1 i) ln(1 i) We get: n = (ln(2.0) ln(1.0)) / ln(1.02) = years [Notice: 2.0 * 35 = 70] 17

18 LECTURE 114 MATH Compound interest: calculating i Our basic equation states that FV PV (1 i) n from which we get these steps: FV (1 i) n PV n n FV n (1 i) (1 i) PV i FV n PV 1 Compound interest: calculating n Our basic equation states that FV PV (1 i) n from which we get these steps: FV (1 i) n PV FV n ln( ) ln((1 i) ) [Rule: if a=b then ln(a)=ln(b)] PV FV a ln( ) n ln(1 i) [Rule: ln( x ) a ln( x) ] PV FV ln( ) PV ln( FV ) ln( PV ) n ln(1 i) ln(1 i) The natural logarithm ln(x) [Rule:ln( x y) ln( x) ln( y) ] x [Rule: ln( ) ln( x) ln( y) y ] The natural log function ln(x) is the inverse of the e x function this way: If y x e then x ln( y). [Note that 0 e 1 and ln( ) 1 e and ln(1) 0.] 18

19 LECTURE 115 COMPOUND INTEREST: THE RULE OF 72 (A SHORTCUT) Notice from the two examples in Lecture 114 that if you multiply the number of years by the yearly interest rate, you get approximately 72 in both cases. Financial practitioners have come to use this as a quick approximation: to double your money, (# periods required) x (period interest rate in %) = 72 OR: IF FV = 2*PV, then N * %INT ~ 72 So to figure out how many years it will take you to double your money, divide 72 by the expected yearly return. Or, to figure out what % monthly return you need, divide 72 by the number of months you have to do it. ADVANCED: What if you want to more than double your money? Let us say you can earn 8% interest. By the rule of 72, it will take you 9 years to double your money. After another 9 years, you will have doubled it again i.e. it takes 9x2=18 years to quadruple your money, 27 years to grow your investment to 8 times its original size, and so on. 19

20 LECTURE 115 MATH THE RULE OF 72: WHY IT WORKS If we are using continuous compounding (see Lecture 143), we have that: FV PV e in Since we want to double our money, we get: FV 2 e PV From which we get (using natural logs): in i n ln(2) And if we measure interest rates in percent, that gives us: i n 100*ln(2) 69.3 which we can solve for either i or n. (We use 72 instead of 69.3 in our shortcut formula, since 72 is easily divisible.) For yearly compounding, we get: FV PV (1 i) n From which we get: FV PV n ln(2) 2 (1 i) ln(2) n ln(1 i) n ln(1 i) Not easy to solve directly for the interest rate, but we can easily construct a little table showing combinations of i and n. Examples: i=8% -> n=9.01; i=12%->n=6.12. If you are dealing with compounding more often than once a year ( m times per year) and you want to be as accurate as possible, the above expression is modified as follows, where i/m is the periodic interest rate, and m*n is the total number of compounding periods: ln(2) m n i ln(1 ) m 20

21 III: PROBLEM SOLVING: SIMPLE LOANS 21

22 LECTURE 121 DIFFERENT WAYS TO SOLVE SIMPLE LOAN PROBLEMS There are several different ways to solve financial problems involving simple loans or similar types of projects. Some of them are: 1. You can solve them manually using the mathematical formulas directly. Advantage: you don t need a special financial calculator to do this, just a reasonably competent $15 scientific one. We discussed this way of doing things in Lectures above. [See Lecture 999 for relevant formulas.] 2. You can use pre-computed tables of compounded-interest factors [ (1+i) n and its inverse]. Advantage: as long as you have access to such tables, you only need a very simple calculator. Disadvantage: some tables give only a limited number of decimals, thus reducing the accuracy of your answers. (In finance, we sometimes need to be accurate to the nearest penny!) This solution method is discussed below in Lecture 122. [Your textbook, RWJ, has such tables in Appendix A.] 3. You can use a financial calculator. There are several such calculators on the market, by Hewlett-Packard, Texas Instruments, and others. (See Lecture 991 for a list of some available models.) The use of a financial calculator to solve simple loan problems is discussed below in Lecture 123. There is a separate set of Lectures available discussing the operation of specific calculator models (for now, only for the HP-10BII). The following methods are also available, though they are not discussed in these Lectures: 4. You can use a software emulator: a program that runs on a PC that looks like and acts like a specific financial calculator. For example, there is an excellent emulator for the HP-12C available for the Android platform. 5. You can use a spreadsheet such as Microsoft Excel or OpenOffice/LibreOffice Calc, which have built-in financial functions. 6. You can use one of the many specialized online calculator programs available on various web sites; these are usually tailored to solving some specific class of problems. It is to your advantage to become proficient in all these methods if you are going to need to solve a lot of financial problems in your career. 22

23 LECTURE 122 SOLVING SIMPLE LOAN PROBLEMS USING TABLES CALCULATING FV: You deposit $2,350 in the bank at 7% interest and leave it there for 15 years. How much is in your account now? The mathematical formula is FV = PV * (1+i) N = $2,350 * (1.07) 15 To use the table, we can look up the value of the interest factor (1.07) 15 = (Find the number at the intersection of the 7% column and the 15-year row.) Thus, the answer to our problem is $2,350 * = $6, CALCULATING PV: How much do you have to invest today if you want to have $100,000 available 25 years from now, and you can earn 12% interest? The math says that FV = PV * (1+I) N so PV = FV / (1+I) N or PV = FV * (1/(1+I) N ) So we can use the same table as above: at N=25, i=12% we find the factor Then: PV = FV / int-factor = $100,000 / = $5, Or, we can use the table for the inverse factor: at N=25, i=12% we find Then: PV = FV * (inverse-int-factor) = $100,000 * = $5, We see that this second method is a bit less accurate, since the tables give us fewer effective decimals for the inverse interest factor. CALCULATING 'i' OR 'n': We do not have tables to calculate either N or i. 23

24 LECTURE 123 SOLVING SIMPLE LOAN PROBLEMS USING A FINANCIAL CALCULATOR Lecture 991 provides a list of some financial calculator models currently available, with their approximate prices. Accompanying this set of Lectures on the theory of TVM and cash flow problems is a separate set of Lectures describing the operations of specific financial calculators. (Currently limited to the HP-10BII.) Lecture 214 below describes how to use a financial calculator to solve annuity problems. Problems involving simple loans and compound interest are solved on the financial calculator just like annuity problems, with the exception that PMT is always set equal to zero. Thus, go read Lecture 214. Or, continue through these lectures until you have completed Lecture 216, then come back here. Solving simple loan problems should now seem trivial to you. 24

25 LECTURE 146 TERMINOLOGY: DIFFERENT CONCEPTS OF YIELD and RETURN APR: the annualized yearly interest rate on a loan, found by taking the periodic interest rate and multiplying by the number of periods in a year. Example: a credit card account charging 1.5% interest per month would report an APR of 1.5% * 12 = 18%. EFFECTIVE ANNUAL RETURN (EAR): the true compounded yearly return to some project or investment which compounds interest more often than once a year. As opposed to the APR, the EAR = (1+i/m) m 1, or (1.015) 12 1 = 19.56% for the same credit card. REQUIRED RETURN: the rate of return we use as the discount rate in computing the NPV of some financial instrument or project. It is intended to represent the opportunity cost of the funds used. Thus, in most cases we use the market return for comparable investments as our required return. However, in special cases when we have access to above-market returns in an alternate investment, we may use that project s return as our required return. YIELD TO MATURITY: the total yield you would earn from purchasing a bond or other debt instrument at today s price, and holding it until it matures (when you receive the face value as your last payment.) YIELD TO CALL: the total yield you would earn from purchasing a bond or other debt instrument at today s price, and holding it until the earliest date the issuer can call (pay off) the bond. The bond indenture specifies what price you would receive when/if the bond is called (which may not be the same as face value.) MARKET RATE (MARKET RETURN): the yield or interest rate currently earned by market participants on the type of financial instrument you are contemplating. This depends, among other factors, on the current risk-free rate, the risk and liquidity premiums for your specific instrument, and the time to maturity. The market rate for a given type of financial instrument is a function of supply and demand, and fluctuates daily. COUPON RATE: the interest rate used by a bond issuer to determine the yearly coupon payment, which is fixed from then on. Yearly coupon payments = face value * coupon rate. TOTAL RETURN: the return you have earned or will earn from a given investment, including all cash flows from coupons or dividends, as well as the purchase and selling prices. It is found by calculating the project s IRR. Sometimes approximated by Current Yield + capital gain. APY: the yearly yield on a savings account or investment. Equals the EAR (see above.) QUOTED RATE: the yearly rate quoted on a loan or other debt. Equals the APR (see above.) CURRENT YIELD: a simplified yield calculation for coupon bonds. See Lecture 314. YIELD ON A DISCOUNT BASIS: a simplified yield for discount bonds. See Lecture 312. BOND-EQUIVALENT YIELD: a simplified yield for discount bonds. See Lecture

26 PART VI: PROJECTS WITH MULTIPLE EVEN CASH FLOWS 26

27 LECTURE 201 CASH FLOWS AND PROJECTS So far we have been discussing simple loans, that is, financial transactions involving only two payments. From here on, we discuss the more general concept of projects consisting of an arbitrary number of cash flows. CASH FLOWS A cash flow is a single payment that: (a) has a size (in dollars), (b) has a direction (you either receive the money, or pay it out), and (c) takes place at a specific point in time (usually at the beginning or end of a given time period, which can be a month, a year, or something else). In our putting-money-in-the-bank example, my deposit today is one cash flow, and my withdrawal a year later is a second cash flow. PROJECTS In finance we will often need to deal with projects which involve multiple cash flows. A set of cash flows may consist of a number of different size payments (flows), or it may consist of a sequence of identically sized payments. The cash flows may be spaced out at equal intervals, or may happen at varying intervals. A construction project might be an example of a project with cash flows of varying sizes and intervals, while a mortgage or a coupon bond would be examples of equal-size cash flows spaced out equally over time. CASH FLOW DIAGRAMS The cash flows associated with a given financial transaction or project may be illustrated with a cash flow diagram. See Lecture 304 for some examples. THE CASH FLOW SIGN CONVENTION When we do computations with cash flows, we need to observe the cash flow sign convention: Amounts which you have received or will receive are represented as positive numbers; Amounts which you have paid or will pay out are represented as negative numbers. We see that the signs depend on whether we are looking at the project from the point of view of the borrower or the lender/investor. On the other hand, the sizes and timings of the cash flows are the same in either case. 27

28 LECTURE 202 FV AND PV FOR MULTIPLE CASH FLOWS: MANUAL CALCULATIONS We have defined a financial project as a set of current and future cash flows. We have seen above, in Lecture 113, how to compute the present value PV of a single future cash flow. The Net Present Value (NPV) for a financial project as a whole is equal to the sum of the present values (PV) of each participating cash flow, discounted at an appropriate discount rate (interest rate) i. Example: a project promises to pay us $1,000 per year for 3 years, at the end of each year. The total present value of these three future payments is (assuming a discount rate of 8%): PV of CF1: $1,000 / (1.08) 1 = $ PV of CF2: $1,000 / (1.08) 2 = $ PV of CF3: $1,000 / (1.08) 3 = $ TOTAL $2, If this project required an initial investment (cash flow CF0, at time t=0) of $1,200, the net present value (NPV) of the whole project is: NPV = $2, $1, = $1, Similarly, the Net Future Value (NFV) of this project is equal to the sum of the future values (FV) of all the participating cash flows, at a given point in time, and given the applicable rate of interest. For the above project, if the applicable interest rate i=8% then the NFV at time t=3 (at the same time as we receive the last payment) is: FV of CF0: -$1,200 * (1.08) 3 = $1, FV of CF1: $1,000 * (1.08) 2 = $1, FV of CF2: $1,000 * (1.08) 1 = $1, FV of CF3: $1,000 * (1.08) 0 = $1, TOTAL $1, (Negative cash flows retain their signs in the process.) We note that for a financial project as a whole, NFV = NPV * (1+i) n Verify for yourself that 1, * (1.08) 3 = 1, [See also Lecture 541 on Net Future Value.] 28

29 LECTURE 202 MATH FV AND PV FOR MULTIPLE CASH FLOWS The net present value (NPV) for a financial project can be expressed in two different ways. First, if we think of it as the sum of the present values of all the participating cash flows, we have: NPV CFt 0 (1 i) n where i is the appropriate discount rate. t t Second, we can think of it as the present value of all future cash flows, less the initial investment (CF0): NPV CF 1 (1 i) n t CF t t 0 (notice that t goes from 1 to n here). We use the form that best suits the problem at hand. (See Lectures 431, 435 for an example of when the second form is used, in calculating the Profitability Index.) The net future value (NFV) for a project is equal to the sum of the future values of all the participating cash flows, at a given future point in time, given an applicable interest rate i : n NFV CF (1 i ) t 0 t n t As noted above, for a given project this means that NFV NPV (1 i) n 29

30 LECTURE 203 EVEN CASH FLOWS: ANNUITIES The manual calculations shown in the preceding Lecture works for any arbitrary combination of cash flows in a project. However, if we impose certain restrictions on the cash flow profile of the project, we have available to us special formulas and calculator procedures we can use. The restrictions are: (a) the cash flows of the project must all be of the same SIZE and DIRECTION, and they must be SPACED OUT EVENLY over time; (b) there may be a single cash flow of a different size and/or direction that happens at the beginning of time (t = 0); (c) there may be a single cash flow of a different size and/or direction that happens at the end of time (t = n), along with the last regular payment. Examples of financial projects that fit such a profile would be: car loans, mortgages with balloon payments, and coupon bonds. Financial projects that fit this cash flow profile are referred to as annuities. A cautionary note: insurance companies sell a financial product that they call annuities, which involve regular payments by the insurance company to some beneficiary. These products do fit the profile of an annuity in the present sense, but so do various other financial instruments and relationships. 30

31 VII: PROBLEM SOLVING: ANNUITIES 31

32 LECTURE 211 DIFFERENT WAYS TO SOLVE ANNUITY PROBLEMS There are several different ways to solve financial problems involving projects that take the form of annuities. Some of them are: 1. You can solve for NPV or NFV manually, cash flow by cash flow, since the NPV for an annuity is equal to the sum of the PV s for each cash flow taken by itself. And we have already seen how to calculate the PV for a single future payment (lump sum). The same goes for the NFV of an annuity. Lecture 202 above showed how to do this. 2. You can solve them using the mathematical formulas directly. Advantage: you don t need a special financial calculator to do this. This procedure is described below in Lecture 212. [See Lecture 999 for the relevant formulas for FVA, PVA, PMT, SFP.] 3. You can use pre-computed tables of annuity factors [for FVA and PVA]. Advantage: as long as you have access to such tables, you only need a very simple calculator. Disadvantage: some tables give only a limited number of decimals, thus reducing the accuracy of your answers. (In finance, we sometimes need to be accurate to the nearest penny!) This solution method is described below in Lecture 213. [Your textbook, RWJ, has such tables in Appendix A.] 4. You can use a financial calculator. There are several such calculators on the market, by Hewlett-Packard, Texas Instruments, and others. (See Lecture 991 for a list of some available models.) The use of a financial calculator to solve annuity problems is discussed below in Lecture 214. There is a separate set of Lectures available discussing the operation of specific calculator models (for now, limited to the HP-10BII). The following methods are also available, though they are not discussed in these Lectures: 5. You can use a software emulator: a program that runs on a PC that looks like and acts like a specific financial calculator. There is an excellent emulator for the HP-12C financial calculator available for the Android platform, for example. 6. You can use a spreadsheet such as Microsoft Excel or OpenOffice/LibreOffice Calc. 7. You can use one of the many specialized online calculator programs available on various web sites; these are usually tailored to solving some specific class of problems. It is to your advantage to become proficient in all these methods if you are going to solve a lot of financial problems in the future. 32

33 LECTURE 212 SOLVING ANNUITY PROBLEMS USING MATHEMATICAL FORMULAS Lecture 999 contains mathematical formulas which you can use to calculate certain annuity variables, using only a simple scientific calculator. The variables we can calculate are: PVA: the present value of an annuity, i.e. the present value of a sequence of future equal size payments. You only need to supply the size of the recurring payment (PMT), the periodic interest rate (i) and the number of periods (n). FVA: the future value of an annuity, i.e. the value of the annuity at the same time as the very last payment is made or received. Again, you supply the size of the payment (PMT), the periodic interest rate (i), and the number of periods (n). SFP: the required size of a sinking fund payment, i.e. the amount of money you have to contribute each period for n periods, earning a periodic interest rate of i, in order to have the amount FVA available at the end. You supply FVA, n, and i. PMT: the required size of an annuity loan payment, i.e. the amount of money you have to contribute each period for n periods, paying a periodic interest rate of i, in order to completely amortize (pay off) an original loan principal of PVA. You supply PVA, n, and i. WARNING: you have to know the operating characteristics of your calculator to calculate these formulas correctly! In particular, you may need to enter parentheses at appropriate places to let the calculator understand which operations go with which number. Some calculators are smarter than others about understanding what you mean when you skimp on the parentheses. Suggestion: in the beginning, obtain an answer by one of the other methods discussed here as a check, until you are comfortable with the proper operation of your calculator. 33

34 LECTURE 213 SOLVING ANNUITY PROBLEMS USING TABLES OF ANNUITY VALUES There are many sources available [such as Appendix A of your RWJ textbook] where you can find tables of annuity factors, such as: PVAF: the present value of a $1 annuity, given the number of periods and the interest rate; and FVAF: the future value of a $1 annuity, given the number of periods and the interest rate. Using such tables, we can calculate the following variables: PVA: the present value of an annuity, given the size of the payment, the number of periods, and the periodic interest rate. Example: a particular financial investment promises to pay you $50,000 per year for 12 years. Your required rate of return is 9%. What is the present value of this stream of payments? Answer: look in the table for (n=12, I=9%). You find the factor PVAF = Multiply this factor by the size of the payment, and we get: PVA = $50,000 * = $358, FVA: the future value of an annuity, given the size of the payment, the number of periods, and the periodic interest rate. In the above example (n=12 and I/YR=9%) we find in the table an FVAF of , which gives us FVA = PMT * FVAF = 50,000 * = $1,007, (If you refer to the formulas in Lecture 999, you see that the textbook tables supply the values for the expression in brackets, which you multiply by PMT.) 34

35 LECTURE 214 SOLVING ANNUITY PROBLEMS USING A FINANCIAL CALCULATOR Every financial calculator will have a set of TVM functions which can be used to calculate one of the following variables related to an annuity-type financial project: PV: the present value of the annuity FV: the future value of the annuity PMT: the recurring payment N: the number of payments or compounding periods I/YR: the yearly interest rate or yield You can give the calculator the values for 4 of these variables, then ask it to calculate the value for the missing 5 th variable. In order for this to work properly, you need to first set these variables, which will be different from problem to problem: BEG/END: P/YR: normally set to END; set to BEGIN if your project is an annuity due (see Lecture 215 below); number of payments or compounding periods per year. Example: For the simplest type of annuity problem, calculating the required monthly payment on a mortgage loan, you would enter: CLEAR ALL : clear the calculator of data from the previous problem (how to do this will vary from calculator to calculator); BEG/END P/YR N I/YR PV FV = END (since a mortgage is a regular annuity); = 12 (since we will be making monthly payments); = 360 payments (30 years x 12 payments/year) = 5.65 (or whatever yearly interest rate you will be charged); = 300, (the amount of the loan); = 0 (since we plan to pay off the loan completely); Then tell the calculator to calculate: PMT => -1, (negative, since it s money you ll be paying out). To calculate a savings ( sinking fund ) problem, PV would be zero and FV would be the target amount you re saving towards. NOTE that when entering dollar values for FV, PV, or PMT, you need to obey the cash flow sign convention: positive for amounts you receive, negative for amounts you pay out. 35

36 LECTURE 215 NORMAL ANNUITIES VS ANNUITIES DUE [See also CALC LECTURE 12] An annuity in this context is any stream of future cash flows of even size, spaced out equally across time. (Not necessarily the same thing as the products of the same name sold by insurance companies.) The majority of cash flow projects are such that cash flows take place at the end of the corresponding time period. We call such projects normal annuities. For example, your basic home mortgage or car loan is such that each monthly payment is due at the end of the month. Coupon bonds pay coupon payments at the end of the 6-month period during which the interest was earned. And so on. In some cases, however, cash flows happen (are due) at the beginning of the corresponding time period. We refer to such projects as annuities due. A typical example of an annuity due would be a lease, where payment for a given month is due at the beginning of that month. (Remember your apartment lease: rent for the first month was due in advance. Car leases work the same way.) Another example of an annuity due is the scenario where you have won some huge sum in a state lottery, and you are offered either a lump sum or a number of yearly payments. The yearly payments would be in the form of an annuity due, since the first payment will be received right away (at the beginning of year 1.) USING FORMULAS The formulas given in Lecture 999 for FV and PV for an annuity assumes normal annuities. USING TABLES The annuity factors given in the textbook tables assume normal annuities. A NOTE ABOUT CALCULATOR USAGE Your calculator needs to know whether a particular problem refers to a normal annuity or an annuity due, since the results will be different. Every financial calculator will have a function key that toggles between END mode (normal annuities) and BEGIN mode (annuities due). Make sure this toggle is set correctly for the problem you are working on. HINT: Since the majority of financial calculator problems deal with normal annuities, it is customary to leave the calculator set for END mode unless the problem specifically gives you reason to think it is an annuity due. 36

37 LECTURE 216 AMORTIZATION [See also CALC lectures 26 and 29.] The key characteristic of an annuity such as a mortgage loan is that the recurring payments are all of the same size. (Also, under normal circumstances we plan to fully amortize the loan over the given number of payments, so that FV = 0.) What this means is that part of each payment will go to pay interest on the previous period s ending balance, while the rest goes towards reducing the outstanding balance (the principal). Since we are continually reducing the principal, the interest part of the payments will be shrinking as we proceed through paying off the loan. If we want to know, for each payment of the loan, how much goes to pay interest and how much to reduce the principal, we need to create an amortization schedule. This is done most efficiently using a spreadsheet such as Microsoft Excel, since doing it manually for a 360-payment loan would be very tedious. We can also use the financial calculator to calculate, for a given range of payments, INT PRIN BAL total amount of interest paid with those payments; total amount by which the principal has been reduced; and the remaining principal balance after the last payment. One use for this would be to calculate how much interest you can deduct from your tax return for each tax year. See the separate set of Lectures for your specific calculator to see how to use the amortization functions. 37

38 XI: INFLATION AND TAXES 38

39 LECTURE 331 THE EFFECTS OF INFLATION ON INTEREST RATES: THE FISHER EQUATION REAL VS. NOMINAL INTEREST RATES Inflation and the Fisher effect We distinguish between nominal and real interest rates. The nominal rate is the observed market rate of interest. The real rate is the nominal rate, adjusted for inflation. According to (American economist Irving) Fisher, the relationship is this: People s demand for, and supply of, loanable funds depends on the real (inflation-adjusted) interest rate r (interest measured in terms of purchasing power). If the economy experiences inflation, this reduces the purchasing power of the money the borrower pays back to the lender later, and if the inflation is expected when a loan is granted the lender is going to want to be compensated for this. The market rate of interest therefore incorporates compensation for expected inflation. Ex ante [before the loan is made] we have [using the Greek letter π to stand for inflation]: (1+i) = (1+r)*(1+E(π) ) -> Nominal rate i = (1+r)(1+E(π)) - 1 Ex post [after the loan has been paid back] we have (1+r) = (1+i) / (1+ π) -> Real rate r = (1+i)/(1+ π) - 1 In daily use, we often simplify the above equations this way (for small values of i, r, π): i = r + E(π) and r = i π [The nominal or market interest rate = the real interest rate + the expected rate of inflation; The real interest rate or return = the nominal or market rate the actual inflation rate ] Question: Can we know what the market expects future inflation to be? A: Not normally; the interest rates we actually observe in financial markets are nominal interest rates, which incorporate the different inflation expectations of all borrowers and lenders. However, a few years back (around 1997) the Treasury started selling so-called TIPS bonds, which pay a fixed interest rate plus after-the-fact compensation for inflation. These bonds will therefore carry a nominal market rate equal to the real interest rate r, since buyers know they ll get compensated for inflation if we have any. Thus, if we subtract the interest rate on TIPS bonds (r) from the market rate on comparable standard bonds (i) we get an estimate of what the market currently expects the inflation rate to be over the given time horizon. (I suspect this is exactly why the Treasury started offering TIPS in the first place to make economists happy!) 39

40 LECTURE 332 THE EFFECTS OF TAXES ON AFTER-TAX BOND YIELDS When you invest in financial instruments, you are subjected to (at least) two forms of taxation: a) Any interest, dividends or coupon payments you receive are subject to income taxes at the rates applicable to the individual investor. (As you know, we have a progressive graduated income tax system in the United States.) b) If you buy a financial instrument today and sell it at a higher price later, you are subject to the capital gains tax. (However, if you hold the investment for less than a year, you pay regular income taxes on the gain instead.) Here we discuss the income tax only. Fact 1A: federal income taxes can amount to up to 35% in the top income bracket (2008). Fact 1B: state income taxes vary greatly, from zero in some states (Florida, Nevada) up to as much as 9.9% in high-tax states (CA, IA, NJ, OR, RI, VT, DC). Fact 2A: income from federal (Treasury and agency) bonds is not taxed by the states. Fact 2B: income from municipal bonds is not taxed by the federal government (which is why they are sometimes referred to as tax-free bonds.) Fact 2C: income from corporate bonds is taxed by everybody! Result: for the same nominal yield (bond coupon rate), the after-tax yield is highest for municipal bonds, lower for T-Bills, and lowest for corporate bonds. If the effective tax rate for a given investment is teff, then the after-tax yield you earn is: iat = inom * (1 teff) So for federal bonds, we have: iat = inom * (1 tfed) [ using the taxpayer s relevant marginal rate ] For municipal bonds, we have: iat = inom * (1 tstate) [ using the applicable state tax rate, if any ] For corporate bonds, we have: iat = inom * (1 tfed) * (1 tstate) [ using both ] 40

41 LECTURE 333 THE EFFECTS OF TAXES ON PRE-TAX BOND YIELDS Rational investors care about the net after-tax yield they will earn from their investments, rather than the nominal pre-tax yield. This means that arbitrage activity in financial markets will see to it that the after-tax yield for bonds of the same quality is equalized. By same quality we mean similar maturity and degrees of liquidity and risk. We can use the above equations to solve for inom, given the common iat and the applicable tax rates. What this means is, for example, that municipal bonds can offer a lower nominal yield than T- Bills and still be competitive -- even though municipal bonds are both riskier and less liquid Example If you want an after-tax yield of 6%, and your marginal federal income tax rate is 33%, and your applicable state income tax rate is 10%, what nominal yields do you have to be offered on federal bonds, municipal bonds, and corporate bonds, respectively? Municipal bonds: 6% / (1 0.1) = 6.67% Federal bonds: 6% / (1 0.33) = 9.00% Corporate bonds: 6% / (1 0.1)*(1-0.33) = 10.00% 41

42 LECTURE 334 COMBINED EFFECTS OF TAXES AND INFLATION ON NET YIELDS If you need to know the after-tax real return to an investment, the rule is: first subtract the taxes (since you pay taxes on the nominal yield), and then adjust the remainder for inflation. Using simplified math, we get: r i (1 t) AT NOM Example: If your fabulous investment scheme yielded a nominal return of 25%, and your marginal income tax rate is 35%, your nominal after-tax yield is: i 25% (1 0.35) 16.25% AT And if inflation that year ran 12% (must have been under President Carter, this project ;) your real net return was: r i 16.25% 12% 4.25% AT AT which is not quite as impressive as 25%! NOTE: Referring back to Lecture 331, the more correct way of dealing with inflation is multiplicative rather than additive: (1 r )*(1 ) (1 i (1 t)) AT NOM In this formulation, the correct answer to the above example becomes: r [( (1 0.35))/(1 0.12)] % AT 42

43 LECTURE 531 THE EFFECTS OF INFLATION ON PURCHASING POWER: USING THE CPI We know from Macroeconomic theory that over the long run, the average price level in the economy will rise roughly at the same rate as the increase in the money supply less the rate of real growth. [MV=PQ so dp = dm dq when dv=0.] The federal government, in the guise of the Bureau of Labor Statistics (BLS) tries to measure this average price level P through the calculation of various price indexes such as the Consumer Price Index (CPI), the Producer Price Index (PPI), and the GDP Deflator. They also calculate different versions of the CPI, such as [CPI-U, CPI-W, C-CPI-U, core CPI]. Each such index sets the price level in a specific year, called the base year, equal to 100, and the index for subsequent years is calculated from this basis. If for example prices increase by 5% from the base year till the next year, next year s index will be 105. If inflation in 3 subsequent years is 5%, 6%, and 8.5%, the index will increase by: (1.05)*(1.06)*(1.085) = So if we started at an index of 134.7, after those 3 years the index would be * = To compare prices in two different years, we have the following rule: P CPI P CPI Year1 Year 2 Year1 Year 2 From this relationship, if we know 3 values we can calculate the fourth. Example 1: A given basket of goods cost $1, in The CPI in 1977 was and the CPI in 1985 was (1967=100). What should this basket of goods cost in 1985? Answer: $1, * (322.3 / 181.8) = $1,244.5 * = $2,206.25! Example 2: A given basket of goods cost $2, in 1980 and $3, in The CPI in 1980 was What must the CPI have been in 1990? Answer: * ($3, / $2,500.00) = * =

44 LECTURE 532 THE EFFECTS OF INFLATION ON PURCHASING POWER: USING AN AVERAGE INFLATION RATE Example 3: Using the data in the lecture above, what was the average rate of inflation from 1977 till 1985? From 1980 till 1990? Answer: We see that from 1977 until 1985, prices increased by a factor of , or by 77.28%. This is over a period of 8 years. The yearly compounded rate, then, is: or 7.42% From 1980 until 1990, the average inflation rate was: or 4.62% Now, we can use these average inflation rates to calculate price changes: Example 4 (forward): If you spent $375 per month on groceries in 1978, what could you expect to spend on average for the same groceries in 1979? Answer: $ * (1 + Π) = * = $ Example 5 (backward): If you purchased a car for $15,000 in 1990, how much would that car have cost back in 1985 [assuming car prices follow the average inflation rate]? Answer: $ 15,000 / (1 + Π) 5 = $15,000 / (1.0462) 5 = $11,968. We can also use them to calculate real rates of return from nominal rates [see Lecture 331]: Example 6: Real investment yields. If you invested in some shares of stock in 1985 and sold them again in 1990, and you earned a nominal annual return of 12.7% on those shares for that time period, what was your real return? Answer: Using the Fisher equation, (1+r) = (1+i)/(1+p) = 1.12 / = i.e. r = 7.72% The simplified shortcut solution is: r = i Π = 12.7% % = 8.08%. Note: The above discussion uses geometric averaging for the yearly inflation rate. See lecture 323 for a discussion of geometric vs. arithmetic averages. 44

45 XVII: FORMULAS 45

46 LECTURE 991 A BRIEF DISCUSSION OF CALCULATORS WITH FINANCIAL FUNCTIONS The following is a list of calculator models available as of May 2007 that are advertised as financial calculators, with an indication of their approximate price. To be classified as a financial calculator, a model must have both TVM functions and CFLO functions. In addition, some of these models also have specialized bond functions or other functions. Any of these calculators is usable for FINC-2400, FINC-3610 and FINC-3700, except the ones from CASIO. We base our lectures on the HP-10BII. Hewlett-Packard HP-10BII $30 (entry level model) Hewlett-Packard HP-12C $70 (popular with financial professionals) Hewlett-Packard HP-12C Platinum $80 (faster than the 12C and more functions) Hewlett-Packard HP-17BII Plus $100 (menu driven) Hewlett-Packard HP-19BII (discontinued; about $130 on ebay) Hewlett-Packard HP-30B $50 (latest model) Texas Instruments TI-BAII Plus $35 (entry level model) Texas Instruments TI-BAII Plus Pro $45 (more functions) Sharp EL-733A $25 (entry level) CASIO FC-200V $32 (entry level) Other scientific or graphing calculators sometimes have financial functions (most often TVM functions, sometimes also CFLO functions). Some of these are: CASIO CFX-9850GC Plus $117 TVM, CFLO Sharp EL-9900C $99 TVM Hewlett-Packard HP-39GS $80 TVM (CFLO by download) Hewlett-Packard HP-48GII $110 TVM ( ) Hewlett-Packard HP-49G+ TVM ( ) Hewlett-Packard HP-50G $150 TVM ( ) Texas Instruments TI-83,84 series TVM Texas Instruments TI-89, 92, Voyage200 TVM, CFLO (by download) Additionally: - Some general calculators will take plug-in program cards and/or downloadable programs, which may contain financial functions. - Some general calculators (esp. Hewlett-Packard) have a generic SOLVER function where you can enter an equation or set of equations, and have the calculator solve for unknown variables; this functionality can be used for financial functions. 46