Basic Financial Calculations

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1 I Corporate Finance Models

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3 Basic Financial Calculations. Overview This chapter aims to give you some finance basics and their Excel implementation. If you have had a good introductory course in finance, this chapter is likely to be at best a refresher. This chapter covers Net present value (NPV) Internal rate of return (IRR) Payment schedules and loan tables Future value Pension and accumulation problems Continuously compounded interest Almost all financial problems center on finding the value today of a series of cash receipts over time. The cash receipts (or cash flows, as we will call them) may be certain or uncertain. The present value of a cash CFt flow CF t anticipated to be received at time t is. The numerator t ( + r) of this expression is usually understood to be the expected time-t cash flow, and the discount rate r in the denominator is adjusted for the riskiness of this expected cash flow the higher the risk, the higher the discount rate. The basic concept in present-value calculations is the concept of opportunity cost. Opportunity cost is the return that would be required of an investment to make it a viable alternative to other, similar, investments. In the financial literature there are many synonyms for opportunity cost, among them discount rate, cost of capital, and interest rate. When the opportunity cost is applied to risky cash flows, we will sometimes call it the risk-adjusted discount rate (RADR) or the weighted average cost of capital (WACC). It goes without saying that this discount rate should be risk adjusted, and much of the standard finance literature discusses how to make this adjustment. As illustrated in this chapter, when we calculate the net present value, we use the investment s opportunity cost as a discount rate. When we calculate the internal rate of. In my book Principles of Finance with Excel (Oxford University Press, 00) I have discussed many basic Excel/finance topics at greater length.

4 Chapter return, we compare the calculated return to the investment s opportunity cost to judge its value.. Present Value and Net Present Value Both concepts, present value and net present value, are related to the value today of a set of future anticipated cash flows. As an example, suppose we are valuing an investment that promises $00 per year at the end of this and the next four years. We suppose that there is no doubt that this series of five payments of $00 each will actually be paid. If a bank pays an annual interest rate of 0 percent on a five-year deposit, then this 0 percent is the investment s opportunity cost, the alternative benchmark return to which we want to compare the investment. We may calculate the value of the investment by discounting its cash flows using this opportunity cost as a discount rate: 0 A B C D COMPUTING THE PRESENT VALUE Discount rate 0% Year Cash flow Present value <-- =B/(+$B$)^A 00. <-- =B/(+$B$)^A 00. <-- =B/(+$B$)^A 00.0 <-- =B/(+$B$)^A 00.0 <-- =B/(+$B$)^A Net present value Summing cells C:C Using Excel's NPV function Using Excel's PV function.0 <-- =SUM(C:C).0 <-- =NPV(B,B:B).0 <-- =PV(B,A,-00) The present value,.0, is the value today of the investment. In a competitive market, the present value should correspond to the market price of the cash flows. The spreadsheet illustrates three ways of obtaining this value: Summing the individual present values in cells C : C. To simplify the copying, note the use of to represent the power and the use of both

5 Basic Financial Calculations the relative and absolute references; for example: = B/( + $B$) A in cell C. Using the Excel NPV function. As we will soon show, Excel s NPV function is unfortunately misnamed it actually computes the present value and not the net present value (discussed in section..). Using the Excel PV function. This function computes the present value of a series of constant payments. PV(B,,-00) is the present value of five payments of 00 each at the discount rate in cell B. The PV function returns a negative value for positive cash flows; to prevent this unfortunate occurrence, we have made the cash flows negative... The Difference Between Excel s PV and NPV Functions The preceding spreadsheet may leave the impression that PV and NPV perform exactly the same computation. But this is not true whereas NPV can handle any series of cash flows, PV can handle only constant cash flows: 0 A B C D COMPUTING THE PRESENT VALUE In this example the cash flows are not equal Either discount each cash flow separately or use Excel's NPV function Excel's PV doesn't work for this case Discount rate 0% Year Cash flow Present value Present value of each cash flow <-- =B/(+$B$)^A 00. <-- =B/(+$B$)^A 00. <-- =B/(+$B$)^A 00.0 <-- =B/(+$B$)^A <-- =B/(+$B$)^A Net present value Summing cells C:C Using Excel's NPV function 0. <-- =SUM(C:C) 0. <-- =NPV(B,B:B). This somewhat strange property returning negative values for positive cash flows is shared by a number of otherwise impeccable Excel functions such as PMT and PV. The somewhat convoluted logic which led Microsoft to write these functions this way is not worth explaining.

6 Chapter.. Excel s NPV Function Is Misnamed! In standard finance terminology, the present value of a series of cash flows is the value today of the cash flows starting in year : Present value = CF t + r t ( ) N t= The net present value is the present value and the cost of acquiring the asset (the cash flow at time zero): Net present value = CF t = t ( + r) N 0 t= 0 In many cases CF 0 < 0, meaning that it represents the price paid for the asset. N CF + CFt t t= ( + r) This is the present value, given by Excel s NPV function Excel s language about discounted cash flows differs somewhat from the standard finance nomenclature. To calculate the finance net present value of a series of cash flows using Excel, we have to calculate the present value of the future cash flows (using the Excel NPV function), taking into account the time-zero cash flow (this is often the cost of the asset in question)... The Net Present Value, NPV Suppose that the investment of section. is sold for $00. Clearly it would not be worth its purchase price, since given the alternative return (discount rate) of 0 percent the investment is worth only $.0. The net present value (NPV) is the applicable concept here. Denoting by r the discount rate applicable to the investment, the NPV is calculated as follows: NPV = CF + 0 N CFt t ( + r) t= where CF t is the investment s cash flow at time t and CF 0 is today s cash flow. Suppose, for example that the series of five cash flows of $00 is sold for $0. Then, as shown in the following spreadsheet, the NPV =.0.

7 Basic Financial Calculations 0 A B C D COMPUTING THE NET PRESENT VALUE Discount rate 0% Year Cash flow Present value <-- =B/(+$B$)^A <-- =B/(+$B$)^A 00. <-- =B/(+$B$)^A 00. <-- =B/(+$B$)^A 00.0 <-- =B/(+$B$)^A 00.0 <-- =B0/(+$B$)^A0 Net present value Summing cells C:C0 Using Excel's NPV function.0 <-- =SUM(C:C0).0 <-- =B+NPV(B,B:B0) The NPV represents the wealth increment that accrues to the purchaser of the cash flows. If you buy the series of five cash flows of 00 for 0, then you have gained.0 in wealth today. In a competitive market the NPV of a series of cash flows ought to be zero: Since the present value should correspond to the market price of the cash flows, the NPV should be zero. In other words, the market price of our five cash flows of 00 in a competitive market, assuming that 0 percent is the correct risk-adjusted discount rate ought to be.0... The Present Value of an Annuity Some Useful Formulas An annuity is a security that pays a constant sum in each period in the future. Annuities may have a finite or infinite series of payments. If the annuity is finite and the appropriate discount rate is r, then the value today of the annuity is its present value: C C C PV of finite annuity = n r ( + r) ( + r) n ( + r) = C r This formula can also be computed using Excel s PV function. The following illustration also shows the use of Excel s NPV function in valuing a finite annuity:. All the formulas in this subsection depend on some well-known but oft-forgotten high school algebra. See box on the Euler formula in Chapter (page ).

8 Chapter 0 A COMPUTING THE VALUE OF A FINITE ANNUITY Periodic payment, C,000 Number of future periods paid, n Discount rate, r % Present value of annuity Using formula,0. <-- =B*(-/(+B)^B)/B Using Excel's PV function,0. <-- =PV(B,B,-B) Annuity Period payment, <-- =B,000.00,000.00,000.00, Present value using Excel's NPV function B C,0. <-- =NPV(B,B0:B) If the annuity promises an infinite series of constant future payments, then this formula reduces to C C PV of infinite annuity = + + r ( + r) +... = C r A COMPUTING THE VALUE OF AN INFINITE ANNUITY Periodic payment, C,000 Discount rate, r % Present value of annuity,. <-- =B/B B C A growing annuity pays out a sum C that grows at a periodic growth rate g. If the annuity is finite, its value today is given by C C( + g) C( + g) PV of finite growing annuity = r ( + r) ( + r) C( + g) + ( + r) n n + g C + r = r g n +...

9 Basic Financial Calculations 0 This formula can easily be implemented in Excel, and as shown here can also be computed using the Excel NPV function: A First payment, C,000 Growth rate of payments, g % Number of future periods paid, n Discount rate, r % Present value of annuity Using formula,00. <-- =B*(-((+B)/(+B))^B)/(B-B) B COMPUTING THE VALUE OF A GROWING FINITE ANNUITY Annuity Period payment, <-- =B,00.00 <-- =$B$*(+$B$)^(A-),.0 <-- =$B$*(+$B$)^(A-),.0 <-- =$B$*(+$B$)^(A-),. <-- =$B$*(+$B$)^(A-) Present value using Excel's NPV function C,00. <-- =NPV(B,B0:B) Taking the previous formula and letting n, we can compute the value of an infinite growing annuity: C C( + g) C( + g) PV of infinite growing annuity = r ( + r) ( + r) C + g =, provided < r g + r Here is an illustration in Excel: A Periodic payment, C,000 <-- Starting at date Growth rate of payments, g % Discount rate, r % Present value of annuity,. <-- =B/(B-B) B C +... COMPUTING THE VALUE OF A GROWING INFINITE ANNUITY. The Internal Rate of Return (IRR) and Loan Tables The internal rate of return (IRR) is defined as the compound rate of return r that makes the NPV equal to zero: CF 0 N CFt + ( t + r ) t= = 0

10 0 Chapter To illustrate, consider the following example given in rows 0. A project costing 00 in year zero returns a variable series of cash flows at the end of years. The IRR of the project (cell B0) is. percent. 0 Year A B C INTERNAL RATE OF RETURN Cash flow Internal rate of return.% <-- =IRR(B:B) Note that the Excel IRR function includes as arguments all the cash flows of the investment, including the first in this case negative cash flow of Determining the IRR by Trial and Error There is no simple formula to compute the IRR. Excel s IRR function uses trial and error, which can be simulated as shown in the following spreadsheet: 0 A B C Discount rate % Year INTERNAL RATE OF RETURN Cash flow Net present value (NPV) 0. <-- =B+NPV(B,B:B0) By playing with the discount rate or by using Excel s Goal Seek, we can determine that at. percent the NPV in cell B is zero:

11 Basic Financial Calculations 0 A B C Discount rate.% Year INTERNAL RATE OF RETURN Cash flow Net present value (NPV) 0.00 <-- =B+NPV(B,B:B0) Here s the way the Goal Seek screen looked before we got the correct answer: FPO.. Loan Tables and the Internal Rate of Return The IRR is the compound rate of return paid by the investment. To understand this point fully, it helps to make a loan table, which shows the division of the investment s cash flows between investment income and the return of the investment principal:

12 Chapter The loan table divides each of the cash flows of the asset into an income component and a return-of-principal component. The income component at the end of each year is IRR times the principal balance at the beginning of that year. Notice that the principal at the beginning of the last year ($. in the example) exactly equals the return of principal at the end of that year. We can actually use the loan table to find the internal rate of return. Consider an investment costing $,000 today that pays off the cash flows indicated at the end of years,,. At a rate of percent (cell B), the principal at the beginning of year is negative, indicating that too little has been paid out in income. Thus the IRR must be larger than percent: 0 0 A B C D E F INTERNAL RATE OF RETURN Year Cash flow Internal rate of return.% <-- =IRR(B:B) USING THE IRR IN A LOAN TABLE Division of cash flow between investment =-B =$B$0*B income and return of principal Year Investment at beginning of year Cash flow at end of year Income Return of principal <-- =C-D =B-E The remaining investment principal in the year after the last cash flow is zero, indicating that all the principal has been repaid.

13 Basic Financial Calculations 0 A B C D E F IRR?.00% Year USING A LOAN TABLE TO FIND THE IRR Principal at beginning of year Division of cash flow between investment income and return of principal Cash flow at end of year Income Principal, <-- =C-D =$B$*B =B-E If the interest rate in cell B is indeed the IRR, then cell B should be 0. We can use Excel s Goal Seek (found on the Tools menu) to calculate the IRR:

14 Chapter As shown here, the IRR is. percent: 0 A B C D E F IRR?.% Year USING A LOAN TABLE TO FIND THE IRR Principal at beginning of year Division of cash flow between investment income and return of principal Cash flow at end of year Income Principal, <-- =C-D , =$B$*B =B-E Of course, we could have simplified life by just using the IRR function: 0 IRR A B C D Direct calculation of IRR Year Cash flow 0 -, % <-- =IRR(B:B)

15 Basic Financial Calculations.. Excel s Rate Function Excel s Rate function computes the IRR of a series of constant future payments. In the following example, we pay $,000 today for an annual payment of $00 for the next 0 years. Rate shows that the IRR is.0 percent: A B C USING EXCEL'S RATE FUNCTION TO COMPUTE THE IRR Initial investment,000 Periodic cash flow 00 Number of payments 0 IRR.0% <-- =RATE(B,B,-B) Note: Rate works much like PMT and PV, discussed elsewhere in this chapter; it requires a sign change between the initial investment and the periodic cash flow (note that we have used B in cell B). It also has switches to allow for payments that start today and payments that start one period from now (not shown in the example).. Multiple Internal Rates of Return Sometimes a series of cash flows has more than one IRR. In the next example we can tell that the cash flows in cells B : B have two IRRs, since the NPV graph crosses the x-axis twice:

16 Chapter A B C D E F G H I Discount rate % NPV -. <-- =NPV(B,B:B)+B DATA TABLE Discount rate NPV Year Cash flow -. Table header, <-- =B 0-0% % % % %. - %. %.0 Net present value.00 Identifying the two IRRs First IRR.% <-- =IRR(B:B,0) Second IRR.% <-- =IRR(B:B,0.) MULTIPLE INTERNAL RATES OF RETURN Two IRRs % % 0% % 0% % 0% % 0% Discount rate %.0 %. % -0. 0% -. % -.0 % -. % -0. Note: For a discussion of how to create data tables in Excel see Chapter. Excel s IRR function allows us to add an extra argument that will help us find both IRRs. Instead of writing = IRR(B : B), we write = IRR(B : B,guess). The argument guess is a starting point for the algorithm that Excel uses to find the IRR; by adjusting the guess, we can identify both the IRRs. Cells B0 and B give an illustration. There are two things to note about this procedure: The argument guess merely has to be close to the IRR; it is not unique. For example by setting the guesses equal to 0. and 0., we will still get the same IRRs: 0 A B C D Identifying the two IRRs First IRR.% <-- =IRR(B:B,0.) Second IRR.% <-- =IRR(B:B,0.)

17 Basic Financial Calculations In order to identify the number and the approximate value of the IRRs, it helps greatly to graph (as we did above) the NPV of the investment as a function of various discount rates. The internal rates of return are then the points where the graph crosses the x-axis, and the approximate location of these points should be used as the guesses in the IRR function. From a purely technical point of view, a set of cash flows can have multiple IRRs only if it has at least two changes of sign. Many typical cash flows have only one change of sign. Consider, for example, the cash flows from purchasing a bond having a 0 percent coupon, a face value of $,000, and eight more years to maturity. If the current market price of the bond is $00, then the stream of cash flows changes signs only once (from negative in year 0 to positive in years ). Thus there is only one IRR: 0 A B C D E F G H I J K BOND CASH FLOWS: NPV CROSSES x-axis ONLY ONCE, SO THERE IS ONLY ONE IRR Year Cash flow Data table: Effect of 0-00 discount rate on NPV 00, <-- =NPV(E,B:B)+B, table header 00 0%, %.0 NPV of Bond Cash Flows 00 % % % % % %. 00 IRR.% <-- =IRR(B:B) % % % % 0% % 0% 0% -. NPV -00 Discount rate. Flat Payment Schedules Another common problem is to compute a flat repayment for a loan. For example, you take a loan for $0,000 at an interest rate of percent per year. The bank wants you to make a series of payments that will pay off the loan and the interest over six years. We can use Excel s PMT function to determine how much each annual payment should be:. If you don t put in a guess (as we did in the example), Excel defaults to a guess of 0. Thus, in this case, IRR(B : B) will return. percent.

18 Chapter Notice that we have put a minus sign in the space labeled Pv Excel s nomenclature for the initial loan principal. As discussed in footnote, if we do not do so, Excel returns a negative payment (a minor irritant). You can confirm that the answer of $,0. is correct by creating a loan table: 0 A B C D E F G FLAT PAYMENT SCHEDULES Loan principal 0,000 Interest rate % Loan term <-- Number of years over which loan is repaid Annual payment,0. <-- =PMT(B,B,-B) Split payment into: Year Principal at beginning of year Payment at end of year Interest Return of principal 0,000.00, ,.,0.0,0. 0.,.,0.,0..,00. =C-F,0.0,0..0,.,.,0..,.,0.,0.., =$B$*C =D-E

19 Basic Financial Calculations The zero in cell C indicates that the loan is fully repaid over its term of six years. You can easily confirm that the present value of the payments over the six years is the initial principal of $0,000.. Future Values and Applications We start with a triviality. Suppose you deposit $,000 in an account today, leaving it there for 0 years. Suppose the account draws annual interest of 0 percent. How much will you have at the end of 0 years? The answer, as shown in the following spreadsheet, is $,.: 0 A B C D E Interest 0% Year Account balance, beginning of year Interest earned during year Total in account, end year, ,00.00 <-- =C+B, ,0.00 <-- =C+B, ,.00,.00.0,.0,.0.,0. =$B$*B,0..0,.,..,.,..,.,..,. 0,..,.,. =D A simpler way SIMPLE FUTURE VALUE,. <-- =B*(+B)^0 As cell C shows, you don t need all these complicated calculations: The future value of $,000 in 0 years at 0 percent per year is given by FV =, 000 ( + 0% ) 0 =,.

20 0 Chapter Now consider the following, slightly more complicated, problem: Again, you intend to open a savings account. Your initial deposit of $,000 today will be followed by a similar deposit at the beginning of years,,. If the account earns 0 percent per year, how much will you have in the account at the start of year 0? 0 A B C D E F Interest 0% Annual deposit,000 <-- Made today and at beginning of each of next years Number of deposits 0 Year Future value FUTURE VALUE WITH ANNUAL DEPOSITS Account balance, beginning of year Deposit at beginning of year Interest earned during year Total in account, end year 0.00, ,00.00 <-- =D+C+B,00.00, ,0.00 <-- =D+C+B,0.00,000.00,.00,.00,000.0,0.0,0.0,000 0.,.,.,000.,.,.,000. 0,. 0,.,000,.,.,.,000,.,. 0,.,000,.,.,. <-- =FV(B,B,-B,,) =E =$B$*(B+C) This problem is easily modeled in Excel: Thus the answer is that we will have $,. in the account at the end of year 0. This same answer can be represented as a formula that sums the future values of each deposit: Total at beginning of year 0 =, 000 ( + 0% ) +, 000 ( + 0% ) , 000 ( + 0% ) 0 =, 000 ( + 0% ) t t= 0 An Excel Function Note from cell B that Excel has a function FV that gives this sum. The dialog box brought up by FV is the following:

21 Basic Financial Calculations We note three things about this function: For positive deposits FV returns a negative number (look back at footnote ). This is an irritating property of this function that it shares with PV and PMT. To avoid negative numbers, we have put the Pmt in as,000. The line Pv in the dialog box refers to a situation where the account has some initial value other than 0 when the series of deposits is made. In this example, this space has been left blank, indicating that the initial account value is zero. As noted in the picture, Type (either or 0) refers to whether the deposit is made at the beginning or the end of each period (in our example the former is the case).. A Pension Problem Complicating the Future-Value Problem A typical exercise is the following: You are currently years old and intend to retire at age 0. To make your retirement easier, you intend to start a retirement account:

22 Chapter 0 0 At the beginning of each of years,,, (that is, starting today and at the beginning of each of the next four years), you intend to make a deposit into the retirement account. You think that the account will earn percent per year. After retirement at age 0, you anticipate living eight more years. At the beginning of each of these years you want to withdraw $0,000 from your retirement account. Your account balances will continue to earn percent. How much should you deposit annually in the account? The following spreadsheet fragment shows how easily you can go wrong in this kind of problem in this case, you ve calculated that in order to provide $0,000 per year for eight years, you need to contribute $0,000/ = $,000 in each of the first five years. As the spreadsheet shows, you ll end up with a lot of money at the end of eight years! (The reason you ve ignored the powerful effects of compound interest. If you set the interest rate in the spreadsheet equal to 0 percent, you ll see that you re right.) A B C D E F Interest % Annual deposit, Annual retirement withdrawal 0, Year A RETIREMENT PROBLEM Account balance, beginning of year Deposit at beginning of year Interest earned during year Total in account, end year =$B$*(C+B) 0.00,000.00,0.00,0.00 <-- =D+C+B,0.00,000.00,.0 0,.0 0,.0,000.00,.,.,.,000.00,0.,.,.,000.00,. 0,. 0,. -0,000.00,.,0.,0. -0,000.00,.,.,. -0, ,.,.0,.0-0,000.00,.0,0. 0,0. -0,000.00,00.,0.,0. -0,000.00,.,.,. -0,000.00,.,.,. -0,000.00,.,. Note: This problem has five deposits and eight annual withdrawals, all made at the beginning of the year. The beginning of year is the last year of the retirement plan; if the annual deposit is correctly computed, the balance at the beginning of year after the withdrawal should be zero.. Of course you re going to live much longer! And I wish you good health! The dimensions of this problem have been chosen to make it fit nicely on a page.

23 Basic Financial Calculations There are several ways to solve this problem. The first involves Excel s Solver. This can be found on the Tools menu. Clicking on the Solver makes a dialog box appear. In the following illustration we ve filled it in:. If the Solver does not appear on the Tools menu, then you have to load it. Go Tools Add-Ins and click Solver Add-In on the list of programs. Note that you could also use the Goal Seek tool to solve this problem. For simple problems such as this one, there is not much difference between the Solver and Goal Seek; the one (not inconsiderable) advantage of the Solver is that it remembers its previous arguments, so that if you bring it up again on the same spreadsheet, you can see what you did in the previous iteration. In later chapters we will illustrate problems that cannot be solved by Goal Seek and where the use of the Solver is a necessity. Solver and Goal Seek are compared in Chapter.

24 Chapter If we now click on the Solve box, we get the answer: 0 A B C D E F Interest % Annual deposit,. Annual retirement withdrawal 0, Year A RETIREMENT PROBLEM Account balance, beginning of year Deposit at beginning of year Interest earned during year Total in account, end year =$B$*(C+B) 0.00,.,0.,. <-- =D+C+B,.,.,.,0.,0.,.,.0 0,0. 0,0.,. 0,.,0.,0.,.,.,.0,.0-0,000.00,.,.,. -0,000.00,0.,.0,.0-0,000.00,.0,.,. -0,000.00,.0 0,. 0 0,. -0,000.00,.0,.,. -0,000.00,.,.,. -0,000.00,. 0, , ,

25 Basic Financial Calculations.. Solving the Retirement Problem Using Financial Formulas We can solve this problem in a more intelligent fashion if we understand the discounting process. The present value of the whole series of payments, discounted at percent, must be zero: Initial deposit 0, t = t t= 0 (. 0) t= (. 0), Initial deposit = t t t= (. 0) t= 0 (. 0) Both the numerator on the right-hand side as 0, 000 t and the denominator (. 0) t= (. 0) using Excel s PV function: 0, 000 (. 0) = t t= (. 0) can be calculated t t= 0 A Interest % Annual deposit, Annual retirement withdrawal 0, Numerator Denominator Annual deposit B A RETIREMENT PROBLEM Solution using formulas,. <-- =/(+B)^*PV(B,,-B). <-- =PV(B,,-,,),. <-- =B/B C. Continuous Compounding Suppose you deposit $,000 in a bank account that pays percent per year. At the end of the year you will have $,000 * (.0) = $.00. Now suppose that the bank interprets percent per year to mean that it pays you. percent interest twice a year. Thus after six months you ll have $,0, and after one year you will have. $ , + $, 00. =. By this logic, if you get paid interest n times per year, your accretion at the end of the year will be. $, n. As n increases, this amount gets larger, converging n

26 Chapter (rather quickly, as you will soon see) to e 0.0, which in Excel is written as the function Exp. When n is infinite, we refer to this practice as continuous compounding. (By typing Exp() in a spreadsheet cell, you can see that e =.....) As you can see in the next display, $,000 continuously compounded for one year at percent grows to $,000 * e 0.0 = $,0. at the end of the year. Continuously compounded for t years, it will grow to $,000 * e 00 *t, where t need not be a whole number (for example, when t =., then the accumulation factor e 0.0 *. measures the growth of the initial investment at percent annually, continuously compounded for four years and three months). A MULTIPLE COMPOUNDING PERIODS Initial deposit,000 Interest rate % Number of compounding periods per year Interest per compounding period Accretion in one year Continuous compounding with Exp End-year accretion B C.00% <-- =B/B,00. <-- =B*(+B)^B,0. <-- =B*EXP(B) Effect of Multiple Compounding Periods,0.0,0.0,0.00,00.0,00.0,00.0,00.0,00.00 Number of compounding intervals, Compounding periods per year End-year accretion, <-- =$B$*(+$B$/A)^A,00. <-- =$B$*(+$B$/A)^A 0,0.0 0,0.0 0,0. 00,0. 0,0. 00,0. 00,0.

27 Basic Financial Calculations.. A Technical Note on the Graph The graph is an Excel XY (Scatter) chart; the x-axis in the chart has been set to be in logarithmic scale. This emphasizes the compounding process. The following picture shows the graph s x-axis marked and the relevant dialog box (right-click after marking the axis and go to Format Axis)... Back to Finance Continuous Discounting If the accretion factor for continuous compounding at interest r over t years is e rt, then the discount factor for the same period is e rt. Thus a cash flow C t occurring in year t and discounted at continuously compounded rate r will be worth C t e rt today, as follows:

28 Chapter 0 A B C D Interest % Year Cash flow Continously discounted PV 00. <-- =B*EXP(-$B$*A) <-- =B*EXP(-$B$*A) Present value CONTINUOUS DISCOUNTING,. <-- =SUM(C:C).. Calculating the Continuously Compounded Return from Price Data Suppose at time 0 you had $,000 in the bank and suppose that one year later you had $,00. What was your percentage return? Although the answer may appear obvious, it actually depends on the compounding method. If the bank paid interest only once a year, then the return would be 0 percent:, 00, 000 = 0 % However, if the bank paid interest twice a year, you would need to solve the following equation to calculate the return: /, 00, 000 r + r, 00 % = =, 000 =. The annual percentage return when interest is paid twice a year is therefore *.% =.0%. In general, if there are n compounding periods per year, you have to solve r n = / n, 00 and then multiply the result appropriately. If n, 000, 00 is very large, this converges to r = ln =. %:, 000

29 Basic Financial Calculations A B C CALCULATING RETURNS FROM PRICES Initial deposit,000 End-of-year value,00 Number of compounding periods Implied annual interest rate.0% <-- =((B/B)^(/B)-)*B Continuous return.% <-- =LN(B/B) 0 Implied annual interest rate with n compounding periods Number of compounding periods Rate.0% <-- =B, data table header 0.00%.0%.%.% 0.%,000.%.. Why Use Continuous Compounding? All this may see somewhat esoteric. However, continuous compounding/discounting is often used in financial calculations. In this book, it is used to calculate portfolio returns (Chapters ) and in practically all of the options calculations (Chapters ). There s another reason to use continuous compounding its ease of calculation. Suppose, for example, that your $,000 grew to $,00 in one year and nine months. What s the annualized rate of return? The easiest and most consistent way to find this answer is to calculate the continuously compounded annual return. Since year and months equals. years, this return is, 00, 000 exp[ r. ] =, 00 r = ln %., 000 =.

30 0 Chapter. Discounting Using Dated Cash Flows Most of the computations in this chapter consider cash flows that occur at fixed periodic intervals. Typically we look at cash flows that occur on dates 0,,..., n, where the period indicates an annual, semiannual, or other fixed interval. Two Excel functions, XIRR and XNPV, allow us to do computations on cash flows which occur on specific dates that need not be at even intervals. In the following example we compute the IRR of an investment of $,000 made on January 00 with payments on specific dates: A B C USING XIRR TO COMPUTE THE ANNUALIZED INTERNAL RATE OF RETURN Date Cash flow -Jan-0 -,000 -Mar-0 0 -Jul Oct-0 0 -Dec-0,000 IRR.% <-- =XIRR(B:B,A:A) The function XIRR outputs an annualized return. It works by computing the daily IRR and annualizing it, XIRR = ( + DailyIRR). XNPV computes the net present value of a series of cash flows occurring on specific dates:. If you do not see these functions, add them in by going to Tools Add-ins on the tool bar and checking Analysis ToolPak.

31 Basic Financial Calculations 0 A B C USING XNPV TO COMPUTE THE NET PRESENT VALUE Annual discount rate % Date Cash flow -Jan-0 -,000 -Mar Jul-0 -Oct-0 0 -Dec-0 00 Net present value.0 <-- =XNPV(B,B:B,A:A) Note that XNPV has a different syntax from NPV! XNPV requires all the cash flows, including the initial cash flow, whereas NPV assumes that the first cash flow occurs one period hence. Exercises. You are offered an asset costing $00 that has cash flows of $00 at the end of each of the next 0 years. a. If the appropriate discount rate for the asset is percent, should you purchase it? b. What is the IRR of the asset?. You just took a $0,000, five-year loan. Payments at the end of each year are flat (equal in every year) at an interest rate of percent. Calculate the appropriate loan table, showing the breakdown in each year between principal and interest.. You are offered an investment with the following conditions: The cost of the investment is $,000. The investment pays out a sum X at the end of the first year; this payout grows at the rate of 0 percent per year for years. If your discount rate is percent, calculate the smallest X that would entice you to purchase the asset. For example, as you can see in the following display, X = $00 is too small the NPV is negative.

32 Chapter 0 A B C Discount rate % Initial payment. NPV -. <-- =B+NPV(B,B:B) Year Cash flow < <-- =B*..00 <-- =B* The following cash-flow pattern has two IRRs. Use Excel to draw a graph of the NPV of these cash flows as a function of the discount rate. Then use the IRR function to identify the two IRRs. Would you invest in this project if the opportunity cost were 0 percent? 0 A B Year Cash flow ,000. In this exercise we solve iteratively for the internal rate of return. Consider an investment that costs 00 and has cash flows of 00, 00, 0,, in years (see cells A:B in the following spreadsheet). Setting up the loan table shows that 0 percent is greater than the IRR (since the return of principal at the end of year is less than the principal at the beginning of the year).

33 Basic Financial Calculations Setting the IRR? cell equal to percent shows that percent is less than the IRR, since the return of principal at the end of year is greater than the principal at the beginning of year : A B C D E F G H IRR? 0.00% Division of payment LOAN TABLE between: Year Cash flow Year Principal at beginning of year Payment at end of year Interest Principal <-- Should be zero for IRR By changing the IRR? cell, find the internal rate of return of the investment. A B C D E F G H IRR?.00% Division of payment LOAN TABLE between: Year Cash flow Year Principal at beginning of year Payment at end of year Interest Principal <-- Should be zero for IRR. An alternative definition of the IRR is the rate that makes the principal at the beginning of year equal to zero. In the preceding printout cell E gives the principal at the beginning of year. Using the Goal Seek function of Excel, find the rate that changes this figure to zero (the following picture shows how the screen should look).. In general, of course, the IRR is the rate of return that makes the principal in the year following the last payment equal to zero.

34 Chapter (Of course you should check your calculations by using the Excel IRR function.). Calculate the flat annual payment required to pay off a five-year loan of $00,000 bearing an interest rate of percent.. You have just taken a car loan of $,000. The loan is for months at an annual interest rate of percent (which the bank translates to a monthly rate of %/ =.%). The payments (to be made at the end of each of the next months) are all equal. a. Calculate the monthly payment on the loan. b. In a loan table calculate, for each month, the principal remaining on the loan at the beginning of the month and the split of that month s payment between interest and repayment of principal. c. Show that the principal at the beginning of each month is the present value of the remaining loan payments at the loan interest rate (use the PV function).. You are considering buying a car from a local auto dealer. The dealer offers you one of two payment options: You can pay $0,000 cash. The deferred payment plan : You can pay the dealer $,000 cash today and a payment of $,00 at the end of each of the next 0 months. As an alternative to the dealer financing, you have approached a local bank, which is willing to give you a car loan of $,000 at the rate of. percent per month. a. Assuming that. percent is the opportunity cost, calculate the present value of all the payments on the dealer s deferred payment plan. b. What is the effective interest rate being charged by the dealer? Do this calculation by preparing a spreadsheet like this (only part of the spreadsheet is shown you have to do this calculation for all 0 months):

35 Basic Financial Calculations 0 D E F G Month Cash payment Payment under deferred payment plan Difference 0 0,000,000,000 <-- =E-F 0,00 -,00 <-- =E-F 0,00 -,00 0,00 -,00 0,00 -,00 0,00 -,00 0,00 -,00 0,00 -,00 0,00 -,00 H Now calculate the IRR of the numbers in column F; this is the monthly effective interest rate on the deferred payment plan. 0. You are considering a savings plan that calls for a deposit of $,000 at the end of each of the next five years. If the plan offers an interest rate of 0 percent, how much will you accumulate at the end of year? Do this calculation by completing the following spreadsheet. This spreadsheet does the calculation twice once using the FV function and once using a simple table that shows the accumulation at the beginning of each year. 0 A B C D Annual payment,000 Interest rate 0% Number of years Total value $,.0 <-- =FV(B,B,-B,,0) Year Accumulation at begining of year Payment at end of year Annual interest 0, ,000,000,00.00,00

36 Chapter. Redo the previous calculation, this time assuming that you make five deposits at the beginning of this year and the following four years. How much will you accumulate by the end of year?. A mutual fund has been advertising that, had you deposited $0 per month in the fund for the last 0 years, you would now have accumulated $,000. Assuming that these deposits were made at the beginning of each month for a period of 0 months, calculate the effective annual return fund investors got. Hint: Set up the following spreadsheet and then use Goal Seek. A B Monthly payment 0 Number of months 0 Effective monthly return? Accumulation C <-- =FV(B,B,-B,,) The effective annual return can then be calculated in one of two ways: ( + Monthly return) : This is the compound annual return, which is preferable, since it makes allowance for the reinvestment of each month s earnings. *Monthly return: This method is often used by banks.. You have just turned, and you intend to start saving for your retirement. Once you retire in 0 years (when you turn ), you would like to have an income of $00,000 per year for the next 0 years. Calculate how much you would have to save between now and age in order to finance your retirement income. Make the following assumptions: All savings draw compound interest of 0 percent per year. You make the first payment today and the last payment on the day you turn (0 payments). You make the first withdrawal when you turn and the last withdrawal when you turn (0 payments).. You currently have $,000 in the bank, in a savings account that draws percent interest. Your business needs $,000, and you are considering two options: (a) Use the money in your savings account or (b) borrow the money from the bank at percent, leaving the money in the savings account. Your financial analyst suggests that solution (b) is better. His logic: The sum of the interest paid on the percent loan is lower than the interest earned at the same time on the $,000 deposit. His calculations are illustrated in the following spreadsheet. Show that this logic is wrong. (If you think about it, it couldn t be preferable to take a percent loan when you are getting percent interest from the bank. However, the explanation may not be trivial.)

37 Basic Financial Calculations 0 A Interest earned % Interest paid % Initial deposit,000 Year Year B EXERCISE, financial analyst's calculations THE % LOAN Principal at beginning of year C Payment at end of year. Use XIRR to compute the internal rate of return for the following investment: D Interest paid E Repayment of principal,000.00,.,00.00,. <-- =C-D,.0,..,.0 Total interest paid,. Savings Account In savings account at beginning of year End-year interest earned In account at end of year,000.00,0.00,0.00,0.00,.0,.0 Interest earned,.0 F =PMT($B$,,-$B$) A B Date Cash flow 0-Jun-0 - -Feb-0 0 -Feb-0 0 -Feb-0 0 -Feb- 0 -Feb- 0 -Feb-,00. Use XNPV to value the following investment. Assume that the annual discount rate is %. 0 A B Date Cash flow 0-Jun Feb Feb Feb Feb- 00 -Feb- 00 -Feb- -,00. Identify the two internal rates of return of the investment in exercise.

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