Lecture Note
University
California State UniversityCourse
FIN 430 | International Financial ManagementPages
10
Academic year
2023
Jesenia Cuellar
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FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation Future and Present Values of Multiple Cash Flows Saving Up Revisited Based on this example, there are two ways to calculate future values for multiple cash flows: (1) compound the accumulated balance forward one year at a time or (2) calculate the future value of each cash flow first and then add these up. Both give the same answer. so you can do it either way. To illustrate the two different ways of calculating future values, consider the future values, consider the future value of $2,000 invested at the end of each of the next five years. The current balance is zero, and the rate is 10 percent. We first draw a time line. Time line for $2,000 per year for five years 0 1 2 3 4 5 Time (years) $2,000 $2,000 $2,000 $2,000 $2,000 On the time line, notice that nothing happens until the end of the first year when we make the first $2,000 investment. This first $2,000 earns interest for the next four (not five) years. Also, notice that the last $2,000 is invested at the end of the fifth year, so it earns no interest at all. Illustrated below are the calculations involved if we compound the investment one period at a time. As illustrated, the future value is $12,210.20 Future value created by compounding forward one period at a time 0 1 2 3 4 5 Time (years) - Beginning amount $0 $ 0 $2,200 $4,620 $7,282 $10,210.20 + Additions 2,000 2,000 2,000 x1.1 x1.1 x11 2,000 x1.1 x1.1 2,000.00 Ending amount $2,000 $4,200 $6,620 $9,282 $12,210.20 The figure below goes through the same calculations, but it uses the second technique. Naturally, the answer is the same. Future value calculated by compounding each cash flow separately
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation 0 1 2 3 4 5 Time (years) $2,000 $2,000 $2,000 $2,000 $ 2,000 x1.1 2,200.00 x1.12 2,420.00 x1.1 2,662.00 x1.14 2,928.20 $12,210.20 How To Calculate Present Values with Multiple Future Cash Flows Using a Financial Calculator To calculate the present value of multiple cash flows with a financial calculator, we will discount the individual cash flows one at a time using the same technique we used in our previous chapter, so this is not really new. There is a shortcut, however, that we can show you. We will use the numbers in the example to illustrate. To begin, of course, we first remember to clear out the calculator! Next from the example below, the first cash flow is $200 to be received in one-year and the discount rate is 12 percent, so we do the following: Enter 12 200 N I/Y PMT PV FV Solve for -178.57 Now, you can write down this answer to save it, but that's inefficient. All calculators have a memory where you can store numbers. Why not save it there? Doing so cuts way down on mistakes because you don't have to write down and/ or rekey numbers, and it's much faster. Next, we value the second cash flow. We need to change N to 2 and FV to 400. As long as we haven't changed anything else, we don't have to reenter I/Y or clear out the calculator, so we have: Enter 2 400 N I/V PMT PV FV Solve for -318.88 You save this number by adding it to the one you saved in our first calculations, and SO on, for the remaining two calculations As we will see in a later chapter, some financial calculators will let you enter all of the future cash flows at once, but we'll discuss that subject when we get to it. A Note on Cash Flow Timing
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation In working present and future value problems, cash flow timing is critically important. In almost all such calculations, it is implicitly assumed that the cash flows occur at the end of each period. In fact, all the formulas we have discussed, all the numbers in a standard present value or future value table, and, very importantly, all the preset (or default) settings on a financial calculator or spreadsheet assume that cash flows occur at the end of each period. Unless you are very explicitly told otherwise, you always should assume that this is what is meant. Valuing Level Cash Flows: Annuities and Perpetuities More generally, a series of constant, or level, cash flows that occur at the end of each period for some fixed number of periods is called an ordinary annuity: or. more correctly, the cash flows are said to be in ordinary annuity form. Present Value for Annuity Cash Flows It turns out that the present value of an annuity of C dollars per period for I periods when the rate of return, or interest rate, is r is given by: Annuity present value = Cx ( Present value factor 1-fraterly The term in parentheses on the first line is sometimes called the present value interest factor for annuities and abbreviated PVIFA (r, 1). The expression for the annuity present value may look a little complicated, but it isn't difficult to use. Notice that the term in square brackets on the second line, 1/(1 1+r)^t, is the same present value factor we've been calculating. In the preceding example, the interest rate is 10 percent and there are three years involved The usual present value factor is thus: Present value factor=1/1.1^3=1/1.33 = 751315 How Much Can You Afford? The loan payments are in ordinary annuity form, SO the annuity present value factor is: Annuity PV factor = (1 - Present value factor) /r = 1 (1/1.01) /01 = (1 .62026)/.01 37.9740 With this factor, we can calculate the present value of the 48 payments of $632 each as: Present value = $632 X 37.9740 = $24,000 Therefore, $24,000 is about what you can afford to borrow and repay.
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation Annuity Present Values The loan payments are in ordinary annuity form, so the annuity present value factor is: Annuity PV factor = (1 - Present value factor) /r = 1 (1/1.01) /.01 = (1 .62026)/.01 = 37.9740 With this factor, we can calculate the present value of the 48 payments of $632 each as: Present value = $632 x 37.9740 = $24,000 Therefore, $24,000 is about what you can afford to borrow and repay Annuity Present Values To find annuity present values with a financial calculator, we need to use the PMT key (you were probably wondering what it was for). Compared to finding the present value of a single amount, there are two important differences. First, we enter the annuity cash flow using the PMT key, and second, we don't enter anything for the future value, FV. So, for example, the problem we have been examining is a three-year, $500 annuity. If the discount rate is 10 percent, we need to do the following (after clearing out the calculator!): Enter 3 10 500 N I/Y PMT PV FV Solve for -1,243.43 Finding the Payment Suppose you wish to start a business that specializes in the latest health food trend, frozen yak milk. To produce and market your product, the Yankee Doodle Dandy, you need to borrow $100,000. Because it strikes you as unlikely that this particular fad will be long-lived, you propose to pay off the loan quickly by making five equal annual payments. If the interest rate is 18 percent, what will the payments be? In this case, we know that the present value is $100,000. The interest rate is 18 percent, and there are five years to make payments. The payments are all equal, so we need to find the relevant annuity factor and solve for the unknown cash flow: Annuity present value = $100,000 = Cx (1 - Present value factor)/r $100,000 = Cx(1-1/1.18*).18 = - = CX(1-.4371).18 - = Cx 3.1272 C = $100,000/3.1272 = $31,977.78 Therefore, you'll make five payments of just under $32,000 each.
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation Annuity Payments Using a spreadsheet to work the same problem goes like this: A B C D E F 1 2 Using a spreadsheet to find annuity payments 3 4 What is the annuity payment If the present value is $100,000. the Interest rate is 18 percent, and 5 there are 5 periods? We need to solve for the unknown payment in an annuity, so we use the 6 formula PMT(rate, nper, pv, M 7 8 Annuity present value: $100,000 9 Number of payments: 5 10 Discount rate: 18 11 12 Annuity payment: ($31,977.78) 13 14 The formula entered in cell B12 is =PMT(B10, 89, 88): notice that the 15 payment Is negative because it is an outflow to us. Finding The Number of Payments You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can afford to make only the minimum payment of $20 per month. The Interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000? What we have here us an annuity of $20 per month at 1.5 percent per month for some unknown length of time. The present value is $1,000 (the amount you owe today). WE need to do a little algebra (or else use a financial calculator): $1,000 = $20 x (1 - Present value factor) /015 ($1,000/20) x 015 = 1 - Present value factor Present value factor = 25 = 1/(1+r)' 1.015 = 1/25=4 At this point, the problem boils down to asking the following question: How long does it take for your money to quadruple at 1.5 percent per month? Based on our previous chapter, the answer is about 93 months: 1.015^93 = 4 It will take you about 93/12 = 7.75 years at this rate. Future Value For Annuities In general, the future value factor for an annuity is given by: Annuity FV factor = (Future value factor - 1)/r [(1+r) - 1]/n
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation A Note On Annuities Due Remember that with an ordinary annuity. the cash flows occur at the end of each period. When you take out a loan with monthly payments, for example, the first loan payment normally occurs one month after you get the loan. However, when you lease an apartment, he first lease is usually due immediately. The second payment is due at the beginning of the second month, and so on A lease is an example of an annuity due. An annuity due is an annuity for which the cash flows occur at the beginning of each period. Almost any type of arrangement in which we have to prepay the same amount each period is an annuity due. There are several different ways to calculate the value of an annuity due. With a financial calculator, you switch it into "due" or "beginning" mode. It is very important to remember to switch it back when you are finished! Another way to calculate the present value of an annuity due can be illustrated with a time line. Suppose an annuity due has five payments of $400 each, and the relevant discount rate is 10 percent. The time line looks like this: 0 1 2 3 4 5 $400 $400 $400 $400 $400 Notice how the cash flows here are the same as those for a four-year ordinary annuity, except that there is an extra $400 at Time 0. For practice, verify that the present value of a four-year $400 ordinary annuity at 10 percent is $1,267.95. If we add on the extra $400, we get $1,667.95, which is the present value of this annuity due. There is an even easier way to calculate the present or future value of an annuity due. If we assume that cash flows occur at the end of each period when they really occur at the beginning, then we discount each one by one period too many. We could fix this by multiplying our answer by (1 + r), where r is the discount rate. In fact, the relationship between the value of an annuity due and an ordinary annuity with the same number of payment is: Annuity due value = Ordinary annuity value X (1 + r) This works for both present and future values, so calculating the value of an annuity due involves two steps: (1) calculate the present or future value as though it were an ordinary annuity and (2) multiply your answer by Perpetuities An important special case of an annuity arises when the level stream of cash flows continues forever. Such an asset is called a perpetuity because the cash flows are perpetual. The present value of a perpetuity is: Perpetuity PV = C/r
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation For example, an investment offers a perpetual cash flow of $500 every year. The return you require on such an investment is 8 percent. What Is the value of this investment? The value of this perpetuity is: Perpetuity PV = C/r = $500/.08 = $6,250 This concludes our discussion of valuing investments with multiple cash flows. Preferred Stock Preferred stock (or preference stock) is an important example of a perpetuity. When a corporation sells preferred stock, the buyer is promised a fixed cash dividend every period (usually every quarter) forever. This dividend must be paid before any dividend must be paid to regular stockholders, hence the term preferred. Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell? The issue that is already out has a present value of $40 and a cash flow of $1 every quarter forever. Because this is a perpetuity: Present value = $40 = $1 (1/r) r = .025. or 2.5% To be competitive, the new Fellini issue also will have to offer 2.5 percent per quarter; so, if the present value is to be $100, the dividend must be such that: Present value = $100 = C (1/025) C = $2.50 (per quarter) Comparing Rates: The Effect of Compounding This subject causes a fair amount of confusion because-rates are quoted in many different ways. Sometimes the way a rate is quoted is the result of tradition, and sometimes it's the result of legislation. Unfortunately, at times, rates are quoted in deliberately deceptive ways to mislead borrowers and investors. Effective Annual Rates and Compounding As our example illustrates, 10 percent compounded semiannually is actually equivalent to 10.25 percent per year. Put another way, we would be indifferent between 10 percent compounded semiannually and 10.25 percent compounded annually. Any time we have compounding during the year, we need to be concerned about what the rate really is.
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation In our example, the 10 precent is called a stated, or quoted, interest rate. Other names are used as well. The 10.25 percent, which is actually the rate that you will earn, is called the effective annual rate (EAR). To compare different investments or interest rates, we will always need to convert to effective rates. Some general procedures for doing this are discussed next. Calculating and Comparing Effective Annual Rates This example illustrates two things. First, the highest quoted rate is not necessarily the best. Second, compounding during the year can lead to a significant difference between the quoted rate and the effective rate. Remember that the effective rate is what you get or what you pay. If you look at our examples, you see that we computed the EARs in three steps. We first divided the quoted rate by the number of times that the interest is compounded. We then added 1 to the result and raised it to the power of the number of times the interest is compounded. Finally, we subtracted the 1. If we let m be the number of times the interest is compounded during the year, these steps can be summarized as: EAR = +Quoted rate/m)^ m - Suppose you were offered 12 percent compounded monthly. EARs and APRs Sometimes it's not altogether clear whether a rate is an effective annual rate or not. A case in point concerns what is called the annual percentage rate (APR) on a loan. Truth-in-lending laws in the United States require that lenders disclose an APR on virtually all consumer loans. This rate must be displayed on a loan in a prominent and unambiguous way. Given that an APR must be calculated and displayed, an obvious question arises: Is an APR an effective annual rate? Put another way: If a bank quotes a car loan at 12 percent APR, is the consumer actually paying 12 percent interest? Surprisingly, the answer is no. There is some confusion over this point, which we discuss next. The confusion over APRs arises because lenders are required by law to compute the APR in a particular way. By law, the APR is equal to the interest rate per period multiplied by the number of periods in a year. The EAR on such a loan is thus: EAR = (1+ APR/12)¹2 - 1 = 1.01 ¹2 1 = .126825, or 12.6825% EARs, APRs, Financial Calculators, and Spreadsheets With a spreadsheet, we can easily do these conversions To convert a quoted rate (or an APR) to an effective rate in Excel, for example, use the formula EF ECT (nominal rate,
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation inpery), where nominal rate is the quoted rate or APR and inpery is the number of compounding periods per year. Similarly, to convert an EAR to a quoted rate, use NOMINAL (effect rate, inpery), where effect rate is the EAR. Loan Types and Loan Amortization Whenever a lender extends a loan, some provision will be made for repayment of the principle (the original loan amount). Pure Discount Loans The pure discount loan is the simplest form of loan. With such a loan, the borrower receives money today and repays a single lump sum at some time in the future. Interest-Only Loans A second type of loan has a repayment plan that calls for the borrower to pay interest each period and to repay the entire principle (the original loan amount) at some point in the future. Such loans are called interest-only loans. Notice that if there is just one period, a pure discount loan and an interest-only loan are the same thing. For example, with a three-year, 10 percent, interest-only loan of $1,000, the borrower would pay $1,000 X .10 = $100 in interest at the end of the first and second years. A the end of the third year, the borrower would return the $1,000 along with another $100 in interest for that year. Similarly, a 50-year interest-only loan would call for the borrower to pay interest every year for the next 50 years and then repay the principal. In the extreme, the borrower pays the interest every period forever and never repays any principal. As we discussed earlier in the chapter, the result is a perpetuity. Most corporate bonds have the general form of an interest-only loan. Because we will be considering bonds in some detail in the next chapter, we defer a further discussion of them for now. Amortized Loans With a pure discount or interest-only loan, the principle is repaid all at once. An alternative is an amortized loan, with which the lender may require the borrower to repay parts of the loan amount over time. The process of paying off a loan by making regular principal reductions is called amortizing the loan. A-simple way of amortizing a loan is to have the borrower pay the interest each period plus some fixed amount. This approach is common with medium-term business loans. Suppose a business takes out a $5,000, five-year loan at 9 percent. The loan agreement calls for the borrower to pay the interest on the loan balance each year and to reduce the loan balance each year by $1,000. Because the loan amount declines by $1,000 each year, it is fully paid in five years. In this case we are considering, notice that the total payment will decline each year. The reason is that the loan balance goes down, resulting in a lower interest charge each year, while the $1,000 principal reduction is constant.
FIN3403: Corporate Finance Chapter 5: Discounted Cash Flow Valuation For example, the interest in the first year will be $5,000 x .09 = $450. The total payment will be $1,000 + 450 = $1,450. In the second year, the loan balance is $4,000, so the interest is $4,000 X .09 = $360, and the total payment is $1,360. We can calculate the total payment in each of the remaining years by preparing an amortization schedule as follows: Year Beginning Balance Total Payment Interest Paid Principal Paid Ending Balance 1 $5,000 $1,450 $ 450 $1,000 $4,000 2 4,000 1,360 360 1,000 3,000 3 3,000 1,270 270 1,000 2,000 4 2,000 1,180 180 1,000 1,000 5 1,000 1,090 90 1,000 0 Totals $6,350 $1,350 $5,000 Probably the most common way. of amortizing a loan is to have the borrower make a single, fixed payment every. period. Almost all consumer loans (such as car loans) and mortgages work this way. Loan Amortization Using A Spreadsheet Loan amortization is a very common spreadsheet application. To illustrate, we will set up the problem that we have just examined, a five-year, $5,000, 9 percent loan with constant payments. Our spreadsheet looks like this: A B C o F G H 1 2 Using spreadsheet to amortize loan 3 4 Loan amount $5,000 5 Interest ete 09 6 Loan terms & 7 Loan payment $1.285.4 B Note payment is calculated using PMTPate rest or M 9 Amortization table 10 11 Year Beginning Total Interest Principal Ending 12 Balance Payment Paid Paid Balance 13 1 66,000.00 $1,285.44 $450.00 $835.46 $4,664.54 14 2 4364.54 1,285.44 374.00 910.66 2,202.00 15 2 3,263.00 1,285.46 292.95 992.61 2,261.27 16 4 2,261.27 1,286.44 20355 1,081.95 1773.32 17 5 179.32 1,295.46 10654 1,179.32 .00 18 Totals $6,427.90 $1,427.31 $5,000.00 19 20 Formulas in the amortiation tabler 21 22 Year Beginning Total Interest Principal Ending 23 Balance Payment Paid Paid Balance 24 1 =+04 -$057 -013-813 25 2 -+0 -$057 26 3 w-014 $057 DIS-E15 27 4 *+015 -$0$7 DIG-EW 28 5 e+016 -$0$7 +SOSSCIT -CI2-FO 29 30 Note: totals in the amortization table are calculated using the SUM formula 31
Chapter 5: Discounted Cash Flow Valuation
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