Chapter 5: Discounted Cash Flow Valuation

FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow ValuationFuture and Present Values of Multiple Cash FlowsSaving Up RevisitedBased on this example, there are two ways to calculate future values for multiple cashflows: (1) compound the accumulated balance forward one year at a time or (2) calculatethe 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 futurevalues, consider the future value of $2,000 invested at the end of each of the next fiveyears.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 years012345Time(years)$2,000$2,000$2,000$2,000$2,000On the time line, notice that nothing happens until the end of the first year when we makethe 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 periodat a time. As illustrated, the future value is $12,210.20Future value created by compounding forward one period at a time012345Time(years)-Beginning amount$0$0$2,200$4,620$7,282$10,210.20+Additions2,0002,0002,000x1.1x1.1x112,000x1.1x1.12,000.00Ending amount$2,000$4,200$6,620$9,282$12,210.20The 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 FinanceChapter 5: Discounted Cash Flow Valuation012345Time(years)$2,000$2,000$2,000$2,000$ 2,000x1.12,200.00x1.122,420.00x1.12,662.00x1.142,928.20$12,210.20How To Calculate Present Values with Multiple Future Cash Flows Using a Financial CalculatorTo calculate the present value of multiple cash flows with a financial calculator, we willdiscount the individual cash flows one at a time using the same technique we used in ourprevious chapter, so this is not really new.There is a shortcut, however, that we can show you. We will use the numbers in theexample to illustrate.To begin, of course, we first remember to clear out the calculator! Next from the examplebelow, the first cash flow is $200 to be received in one-year and the discount rate is 12percent, so we do the following:Enter12200NI/YPMTPVFVSolve for-178.57Now, you can write down this answer to save it, but that's inefficient. All calculatorshave a memory where you can store numbers. Why not save it there? Doing so cuts waydown on mistakes because you don't have to write down and/ or rekey numbers, and it'smuch faster.Next, we value the second cash flow. We need to change N to 2 and FV to 400. As longas we haven't changed anything else, we don't have to reenter I/Y or clear out thecalculator, so we have:Enter2400NI/VPMTPVFVSolve for-318.88You save this number by adding it to the one you saved in our first calculations, and SOon, for the remaining two calculationsAs we will see in a later chapter, some financial calculators will let you enter all of thefuture cash flows at once, but we'll discuss that subject when we get to it.A Note on Cash Flow Timing FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow ValuationIn working present and future value problems, cash flow timing is critically important. Inalmost all such calculations, it is implicitly assumed that the cash flows occur at the endof each period.In fact, all the formulas we have discussed, all the numbers in a standard present value orfuture value table, and, very importantly, all the preset (or default) settings on a financialcalculator 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 whatis meant.Valuing Level Cash Flows: Annuities and PerpetuitiesMore generally, a series of constant, or level, cash flows that occur at the end of eachperiod for some fixed number of periods is called an ordinary annuity: or. morecorrectly, the cash flows are said to be in ordinary annuity form.Present Value for Annuity Cash FlowsIt turns out that the present value of an annuity of C dollars per period for I periods whenthe rate of return, or interest rate, is r is given by:Annuity present value = Cx ( Present valuefactor1-fraterlyThe term in parentheses on the first line is sometimes called the present value interestfactor for annuities and abbreviated PVIFA (r, 1).The expression for the annuity present value may look a little complicated, but it isn'tdifficult to use. Notice that the term in square brackets on the second line, 1/(1 1+r)^t, isthe same present value factor we've been calculating.In the preceding example, the interest rate is 10 percent and there are three yearsinvolved The usual present value factor is thus:Present value factor=1/1.1^3=1/1.33 = 751315How 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)/.0137.9740With this factor, we can calculate the present value of the 48 payments of $632 each as:Present value = $632 X 37.9740 = $24,000Therefore, $24,000 is about what you can afford to borrow and repay. FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow ValuationAnnuity Present ValuesThe 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=37.9740With this factor, we can calculate the present value of the 48 payments of $632 each as:Present value = $632 x 37.9740 = $24,000Therefore, $24,000 is about what you can afford to borrow and repayAnnuity Present ValuesTo 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 importantdifferences.First, we enter the annuity cash flow using the PMT key, and second, we don't enteranything for the future value, FV.So, for example, the problem we have been examining is a three-year, $500 annuity. Ifthe discount rate is 10 percent, we need to do the following (after clearing out thecalculator!):Enter310500NI/YPMTPVFVSolve for-1,243.43Finding the PaymentSuppose you wish to start a business that specializes in the latest health food trend, frozenyak 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 theinterest 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 findthe 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 3.1272C = $100,000/3.1272 = $31,977.78Therefore, you'll make five payments of just under $32,000 each. FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow ValuationAnnuity PaymentsUsing a spreadsheet to work the same problem goes like this:ABCDEF12Using a spreadsheet to find annuity payments34What is the annuity payment If the present value is $100,000. the Interest rate is 18 percent, and5there are 5 periods? We need to solve for the unknown payment in an annuity, so we use the6formula PMT(rate, nper, pv, M78Annuity present value:$100,0009Number of payments:510Discount rate:181112Annuity payment:($31,977.78)1314The formula entered in cell B12 is =PMT(B10, 89, 88): notice that the15payment Is negative because it is an outflow to us.Finding The Number of PaymentsYou ran a little short on your spring break vacation, so you put $1,000 on your creditcard. 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 topay off the $1,000?What we have here us an annuity of $20 per month at 1.5 percent per month for someunknown length of time. The present value is $1,000 (the amount you owe today). WEneed 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 factorPresent value factor = 25 = 1/(1+r)'1.015 = 1/25=4At this point, the problem boils down to asking the following question: How long does ittake for your money to quadruple at 1.5 percent per month? Based on our previouschapter, the answer is about 93 months:1.015^93 = 4It will take you about 93/12 = 7.75 years at this rate.Future Value For AnnuitiesIn 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 FinanceChapter 5: Discounted Cash Flow ValuationA Note On Annuities DueRemember 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 paymentnormally occurs one month after you get the loan. However, when you lease anapartment, 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 anexample of an annuity due.An annuity due is an annuity for which the cash flows occur at the beginning of eachperiod. Almost any type of arrangement in which we have to prepay the same amounteach period is an annuity due.There are several different ways to calculate the value of an annuity due. With a financialcalculator, 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 atime line. Suppose an annuity due has five payments of $400 each, and the relevantdiscount rate is 10 percent. The time line looks like this:012345$400$400$400$400$400Notice 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 10percent is $1,267.95.If we add on the extra $400, we get $1,667.95, which is the present value of this annuitydue.There is an even easier way to calculate the present or future value of an annuity due. Ifwe assume that cash flows occur at the end of each period when they really occur at thebeginning, 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 withthe 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 dueinvolves two steps: (1) calculate the present or future value as though it were an ordinaryannuity and (2) multiply your answer byPerpetuitiesAn important special case of an annuity arises when the level stream of cash flowscontinues forever. Such an asset is called a perpetuity because the cash flows areperpetual.The present value of a perpetuity is:Perpetuity PV = C/r FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow ValuationFor example, an investment offers a perpetual cash flow of $500 every year. The returnyou require on such an investment is 8 percent. What Is the value of this investment? Thevalue of this perpetuity is:Perpetuity PV = C/r = $500/.08 = $6,250This concludes our discussion of valuing investments with multiple cash flows.Preferred StockPreferred stock (or preference stock) is an important example of a perpetuity. When acorporation sells preferred stock, the buyer is promised a fixed cash dividend everyperiod (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 shareand offers a dividend of $1 every quarter. What dividend will Fellini have to offer if thepreferred stock is going to sell?The issue that is already out has a present value of $40 and a cash flow of $1 everyquarter 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 CompoundingThis subject causes a fair amount of confusion because-rates are quoted in many differentways. Sometimes the way a rate is quoted is the result of tradition, and sometimes it's theresult of legislation.Unfortunately, at times, rates are quoted in deliberately deceptive ways to misleadborrowers and investors.Effective Annual Rates and CompoundingAs our example illustrates, 10 percent compounded semiannually is actually equivalent to10.25 percent per year.Put another way, we would be indifferent between 10 percent compounded semiannuallyand 10.25 percent compounded annually.Any time we have compounding during the year, we need to be concerned about what therate really is. FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow ValuationIn our example, the 10 precent is called a stated, or quoted, interest rate. Other namesare used as well.The 10.25 percent, which is actually the rate that you will earn, is called the effectiveannual rate (EAR).To compare different investments or interest rates, we will always need to convert toeffective rates. Some general procedures for doing this are discussed next.Calculating and Comparing Effective Annual RatesThis example illustrates two things. First, the highest quoted rate is not necessarily thebest. Second, compounding during the year can lead to a significant difference betweenthe quoted rate and the effective rate. Remember that the effective rate is what you get orwhat you pay.If you look at our examples, you see that we computed the EARs in three steps. We firstdivided 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 theinterest is compounded.Finally, we subtracted the 1. If we let m be the number of times the interest iscompounded during the year, these steps can be summarized as:EAR = +Quoted rate/m)^ m -Suppose you were offered 12 percent compounded monthly.EARs and APRsSometimes it's not altogether clear whether a rate is an effective annual rate or not. Acase 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 onvirtually all consumer loans. This rate must be displayed on a loan in a prominent andunambiguous way.Given that an APR must be calculated and displayed, an obvious question arises: Is anAPR an effective annual rate? Put another way: If a bank quotes a car loan at 12 percentAPR, 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 APRin a particular way.By law, the APR is equal to the interest rate per period multiplied by the number ofperiods 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 SpreadsheetsWith a spreadsheet, we can easily do these conversions To convert a quoted rate (or anAPR) to an effective rate in Excel, for example, use the formula EF ECT (nominal rate, FIN3403: Corporate FinanceChapter 5: Discounted Cash Flow Valuationinpery), where nominal rate is the quoted rate or APR and inpery is the number ofcompounding periods per year. Similarly, to convert an EAR to a quoted rate, useNOMINAL (effect rate, inpery), where effect rate is the EAR.Loan Types and Loan AmortizationWhenever a lender extends a loan, some provision will be made for repayment of theprinciple (the original loan amount).Pure Discount LoansThe pure discount loan is the simplest form of loan. With such a loan, the borrowerreceives money today and repays a single lump sum at some time in the future.Interest-Only LoansA second type of loan has a repayment plan that calls for the borrower to pay interesteach period and to repay the entire principle (the original loan amount) at some point inthe future.Such loans are called interest-only loans. Notice that if there is just one period, a purediscount 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 borrowerwould 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 everyyear for the next 50 years and then repay the principal. In the extreme, the borrower paysthe interest every period forever and never repays any principal. As we discussed earlierin the chapter, the result is a perpetuity.Most corporate bonds have the general form of an interest-only loan. Because we will beconsidering bonds in some detail in the next chapter, we defer a further discussion ofthem for now.Amortized LoansWith a pure discount or interest-only loan, the principle is repaid all at once. Analternative is an amortized loan, with which the lender may require the borrower to repayparts of the loan amount over time. The process of paying off a loan by making regularprincipal reductions is called amortizing the loan.A-simple way of amortizing a loan is to have the borrower pay the interest each periodplus 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 eachyear and to reduce the loan balance each year by $1,000. Because the loan amountdeclines 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. Thereason 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 FinanceChapter 5: Discounted Cash Flow ValuationFor example, the interest in the first year will be $5,000 x .09 = $450. The total paymentwill 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, andthe total payment is $1,360. We can calculate the total payment in each of the remainingyears by preparing an amortization schedule as follows:Year Beginning Balance Total Payment Interest Paid Principal Paid Ending Balance1$5,000$1,450$ 450$1,000$4,00024,0001,3603601,0003,00033,0001,2702701,0002,00042,0001,1801801,0001,00051,0001,090901,0000Totals$6,350$1,350$5,000Probably the most common way. of amortizing a loan is to have the borrower make asingle, fixed payment every. period. Almost all consumer loans (such as car loans) andmortgages work this way.Loan Amortization Using A SpreadsheetLoan amortization is a very common spreadsheet application. To illustrate, we will set upthe problem that we have just examined, a five-year, $5,000, 9 percent loan with constantpayments. Our spreadsheet looks like this:ABCoFGH12Using spreadsheet to amortize loan34Loan amount$5,0005Interest ete096Loan terms&7Loan payment$1.285.4BNote payment is calculated using PMTPate rest or M9Amortization table1011YearBeginningTotalInterestPrincipalEnding12BalancePaymentPaidPaidBalance13166,000.00$1,285.44$450.00$835.46$4,664.541424364.541,285.44374.00910.662,202.001523,263.001,285.46292.95992.612,261.271642,261.271,286.44203551,081.951773.32175179.321,295.46106541,179.32.0018Totals$6,427.90$1,427.31$5,000.001920Formulas in the amortiation tabler2122YearBeginningTotalInterestPrincipalEnding23BalancePaymentPaidPaidBalance241=+04-$057-013-813252-+0-$057263w-014$057DIS-E15274*+015-$0$7DIG-EW285e+016-$0$7+SOSSCIT-CI2-FO2930Note: totals in the amortization table are calculated using the SUM formula31