DISCOUNTED CASH FLOW VALUATIONChapter 4Copyright 2014 McGraw-Hill Education.All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

KEY CONCEPTS AND SKILLSAppreciate the significance of compound vs. simple interestDescribe and compute the future value and/or present value of a single cash flow or series of cash flowsDefine and calculate the return on an investmentRecognize and compute the impact of compounding periods on the true return of stated interest ratesDevelop facility with a financial calculator and/or spreadsheet to solve time value problemsComprehend and calculate time value metrics for perpetuities and annuitiesFamiliarization with loan types and amortization

4-*

CHAPTER OUTLINE4.1 Valuation: The One-Period Case4.2 The Multiperiod Case4.3 Compounding Periods4.4 Simplifications4.5 Loan Types and Loan Amortization4.6 What Is a Firm Worth?4-*

THE ESSENTIAL PREMISEA dollar today is more valuable than a dollar to be received in the future

Why?A dollar today is more valuable because:It can be invested to make more dollarsIt can be immediately consumedThere is no doubt about its receipt

4-*

AN IMPORTANT COROLLARYIf you know your required rate of return and the length of time before cash is harvested, you can calculate some critical metrics:

The value today of a payment to be received in the futureThis measure is called a Present ValueThe value in the future of a sum invested todayThis measure is called a Future Value

Present and Future Values can be calculated over single and multiple periods

4-*

4.1 THE ONE-PERIOD CASEIf you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500.

$500 would be interest (=$10,000 .05)

$10,000 is the principal repayment (=$10,000 1)

$10,500 is the total due. It can be calculated as:

$10,500 = $10,000(1.05)

The total amount due at the end of the investment is call the Future Value (FV). 4-*

FUTURE VALUEIn the one-period case, the formula for FV can be written as:FV = C0(1 + r)

Where C0 is cash flow today (time zero), and r is the appropriate interest rate.4-*

PRESENT VALUEPresent Value is todays value of a sum to be received in the future given a specific rate of interest and time horizon.Suppose you were promised $10,000 due in one year when interest rates are 5-percent. Your investment be worth $9,523.81 in todays dollars. 4-*The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is the Present Value (PV).Note that $10,000 = $9,523.81(1.05).

PRESENT VALUEIn the one-period case, the formula for PV can be written as:

4-*Where C1 is cash flow at date 1, and r is the appropriate interest rate.

NET PRESENT VALUEThe Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.

Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%.

Should you buy?4-*

NET PRESENT VALUE4-*The present value of the cash inflow is greaterthan the cost. In other words, the Net PresentValue is positive, so the investment should be purchased.

NET PRESENT VALUEIn the one-period case, the formula for NPV can be written as:NPV = Cost + PV4-*If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FV would be less than the $10,000 the investment promised, and we would be worse off in FV terms :

$9,500(1.05) = $9,975 < $10,000

4.2 THE MULTIPERIOD CASEThe previous examples considered only one period of time.It is possible to compute PV & FV for multiple periods of time.Doing so requires discrimination between simple and compound interest4-*

SIMPLE INTERESTSuppose an investor has $1,000 to invest at an interest rate of 9%In one year, they will have earned $90 [=1000 x (1+.09)]If they take the $90 earned out of the capital market and reinvest the $1,000 capital at 9% they will again earn $90 in a years time.This process could continue ad infinitum with the annual earnings never contributing further to the investors wealthWith simple interest, therefore, valuing future or present values never has to extend beyond the single period case4-*

COMPOUND INTERESTSuppose an investor has $1,000 to invest at an interest rate of 9%In one year, they will have earned $90 (=1000 x .09)Now suppose that the investor reinvests both the original $1,000 capital AND $90 earnings for another year at 9%In one year they will have earned $98.10, an amount $8.10 greater than the previous year.In this example, the interest in the second year is higher than the first because it is paid on the initial capital AND prior earnings. In other words, the earnings also earn interest or are compoundedCompounding may not seem very compelling the early years of an investment. But, we will see that it is a very powerful long term force4-*

MULTIPERIOD FUTURE VALUEThe general formula for the future value of an investment over many periods can be written as:FV = C0(1 + r)TWhere C0 is cash flow at date 0,r is the appropriate interest rate, andT is the number of periods over which the cash is invested.4-*

EXAMPLE: MULTIPERIOD FUTURE VALUESuppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years.What will the dividend be in five years?

FV = C0(1 + r)T

$5.92 = $1.10(1.40)54-*

FUTURE VALUE AND COMPOUNDINGNotice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:

$5.92 > $1.10 + 5[$1.10.40] = $3.30

This is due to compounding.4-*

FUTURE VALUE AND COMPOUNDING4-*

MULTIPERIOD PRESENT VALUE4-*The general formula for the present value of an investment over many periods can be written as:PV = CT / (1 + r)TWhere CT is cash flow at date T,r is the appropriate interest rate, andT is the number of periods over which the cash is invested.

EXAMPLE: MULTIPERIOD PRESENT VALUE AND DISCOUNTINGHow much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?4-*$20,000PV

SOLVE FOR ANY VARIABLEExamples thus far have offered the time and interest rate and solved for PV or FVKeep in mind that there are four variables:PVFVTRIf you have any three you can solve for the fourthThe math can become cumbersomeFinancial calculators and spreadsheets are very helpful

4-*

EXAMPLE: SOLVING FOR TIMEIf we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?4-*

EXAMPLE: SOLVING FOR REQUIRED RETURNAssume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your childs education? 4-*About 21.15%.

CALCULATOR ORIENTATION: KEYSTexas Instruments BA-II PlusFV = future valuePV = present valueI/Y = periodic interest rateP/Y must equal 1 for the I/Y to be the periodic rateInterest is entered as a percent, not a decimalN = number of periodsRemember to clear the registers (CLR TVM) after each problemOther calculators are similar in format4-*

MULTIPLE CASH FLOWSConsider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows?If the issuer offers this investment for $1,500, should you purchase it?4-*

MULTIPLE CASH FLOWS4-*Present Value < Cost Do Not Purchase

VALUING UNEVEN CASH FLOWSFirst, set your calculator to 1 payment per year.Then, use the cash flow menu:4-*CF2CF1F2F1CF0120011,432.930400INPV12CF4CF3F4F316001800

4.3 COMPOUNDING PERIODSAll examples thus far have assumed annual compoundingInstances of other compounding schedules abound:Banks compound interest quarterly, monthly or dailyMortgage companies compound interest monthlyYet, almost all interest rates are expressed annuallyIf a rate is expressed annually, but compounded more frequently, then the effective rate is higher than the stated rateThis concept is called the Effective Annual Rate or EAR4-*

COMPUTING FV WITH MULTIPLE COMPOUNDING PERIODSCompounding an investment m times a year for T years provides for the future value of wealth:4-*

COMPOUNDING PERIODSFor example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to4-*

EFFECTIVE ANNUAL RATES OF INTERESTA reasonable question to ask in the above example is what is the effective annual rate of interest on that investment?4-*The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years:

EFFECTIVE ANNUAL RATES OF INTERESTSo, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually.4-*

EFFECTIVE ANNUAL RATES OF INTERESTFind the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.What we have is a loan with a monthly interest rate of 1%.This is equivalent to a loan with an annual interest rate of 19.56%.4-*

EAR ON A FINANCIAL CALCULATOR4-*Texas Instruments BAII Plus [] [C/Y=] 12 [ENTER]Sets 12 payments per year

CONTINUOUS COMPOUNDINGThe general formula for the future value of an investment compounded continuously over many periods can be written as:FV = C0erTWhere C0 is cash flow at date 0,r is the stated annual interest rate, T is the number of years, ande is a constant that is approximately equal to 2.718. ex is a key on your calculator.4-*

4.4 SIMPLIFICATIONSPerpetuityA constant stream of cash flows that lasts foreverGrowing perpetuityA stream of cash flows that grows at a constant rate foreverAnnuityA stream of constant cash flows that lasts for a fixed number of periodsGrowing annuityA stream of cash flows that grows at a constant rate for a fixed number of periods4-*

PERPETUITYA constant stream of cash flows that lasts forever4-*

PERPETUITY: EXAMPLEWhat is the value of a British consol that promises to pay 15 every year for ever? The interest rate is 10%.4-*

GROWING PERPETUITYA growing stream of cash flows that lasts forever4-*

GROWING PERPETUITY: EXAMPLEThe expected dividend next year is $1.30, and dividends are expected to grow at 5% forever.If the discount rate is 10%, what is the value of this promised dividend stream?4-*

ANNUITYA constant stream of cash flows with a fixed maturity4-*

ANNUITY: EXAMPLEIf you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?4-*

4-*What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

GROWING ANNUITYA growing stream of cash flows with a fixed maturity4-*

GROWING ANNUITY: EXAMPLEA defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?4-*

4.5 LOAN TYPES AND LOAN AMORTIZATIONPure Discount Loans are the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time.Interest-Only Loans require an interest payment each period, with full principal due at maturity.Amortized Loans require repayment of principal over time, in addition to required interest.4-*

PURE DISCOUNT LOANSTreasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?PV = 10,000 / 1.07 = 9,345.794-*

INTEREST-ONLY LOANSConsider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.What would the stream of cash flows be?Years 1 4: Interest payments of .07(10,000) = 700Year 5: Interest + principal = 10,700This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later.4-*

AMORTIZED LOAN WITH FIXED PRINCIPAL PAYMENTConsider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.Click on the Excel icon to see the amortization table4-*

Sheet1

YearBeginning BalanceInterest PaymentPrincipal PaymentTotal PaymentEnding Balance

150,0004,0005,0009,00045,000

245,0003,6005,0008,60040,000

340,0003,2005,0008,20035,000

435,0002,8005,0007,80030,000

530,0002,4005,0007,40025,000

625,0002,0005,0007,00020,000

720,0001,6005,0006,60015,000

815,0001,2005,0006,20010,000

910,0008005,0005,8005,000

105,0004005,0005,4000

AMORTIZED LOAN WITH FIXED PAYMENTEach payment covers the interest expense plus reduces principalConsider a 4 year loan with annual payments. The interest rate is 8% ,and the principal amount is $5,000.What is the annual payment?4 N8 I/Y5,000 PVCPT PMT = -1,509.60Click on the Excel icon to see the amortization table4-*

Sheet1

YearBeginning BalanceTotal PaymentInterest PaidPrincipal PaidEnding Balance

15,000.001,509.60400.001,109.603,890.40

23,890.401,509.60311.231,198.372,692.03

32,692.031,509.60215.361,294.241,397.79

41,397.791,509.60111.821,397.780.01

Totals6,038.401,038.414,999.99

Note: The ending balance of .01 is due to rounding. The last payment would actually be 1,509.61.

4.6 WHAT IS A FIRM WORTH?Conceptually, a firm should be worth the present value of the firms cash flows.The tricky part is determining the size, timing and risk of those cash flows.4-*

A NOTE ABOUT TECHNIQUEYou can solve time value problems in any of four ways:Math (Formulae given above)Tables (See Appendix A)Financial CalculatorSpreadsheet SoftwareFinancial calculators and spreadsheet software are the most common methods now.

4-*

QUICK QUIZHow is the future value of a single cash flow computed?How is the present value of a series of cash flows computed.What is the Net Present Value of an investment?What is an EAR, and how is it computed?What is a perpetuity? An annuity?Contrast interest-only loans to amortized loans.4-*

**It may be good at this point to discuss the difficulty of calculating time periods and interest rates, particularly without the help of a financial calculator.*I am providing information on the Texas Instruments BA-II Plus other calculators are similar. I choose this calculator since it is one that is allowable for use in taking the CFA exam. If you recommend or require a specific calculator other than this one, you may want to make the appropriate changes.

Note: the more information students have to remember to enter, the more likely they are to make a mistake. For this reason, I normally tell my students to set P/Y = 1 and leave it that way. Then I teach them to work on a period basis, which is consistent with using the formulas. If you want them to use the P/Y function, remind them that they will need to set it every time they work a new problem and that CLR TVM does not affect P/Y.

If students are having difficulty getting the correct answer, make sure they have done the following:Set decimal places to floating point (2nd Format, Dec = 9 enter) or show 4 to 5 decimal places if using and HPDouble check and make sure P/Y = 1Make sure to clear the TVM registers after finishing a problem (or before starting a problem). It is important to point out that CLR TVM clears the FV, PV, N, I/Y and PMT registers. C/CE and CLR Work DO NOT affect the TVM keys.**It is important to note to students that in this example the year 1 cash flow was given. If the current dividend were $1.30, then we would need to multiply it by one plus the growth rate to estimate the year 1 cash flow.*Remind students that the value of an investment is the present value of expected future cash flows.

1 N; 10,000 FV; 7 I/Y; CPT PV = -9,345.79