2Time Value Topics Future value Present value Rates of return Amortization
3Determinants of Intrinsic Value: The Present Value Equation Net operatingprofit after taxesRequired investmentsin operating capital−Free cash flow(FCF)=FCF1FCF2FCF∞...Value =(1 + WACC)1(1 + WACC)2(1 + WACC)∞Weighted averagecost of capital(WACC)For value box in Ch 4 time value FM13.Cost of debtCost of equity
4Why is timing important? You are asked to choose from the following options:1. Receive $1 million today2. Receive $1 million 10 years from nowWould you choose 1 or 2?
5Money has time valueMost people prefer to receive it sooner rather than later because they place a higher value on the cash received earlier.
6Time value of money: Practical relevance ExamplesRetirement planMortgage paymentPricing a financial securitiesHelping your company to decide which project to undertake
7Time lines show timing of cash flows. CF0CF1CF3CF2123I%Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
8Time line for a $100 lump sum due at the end of Year 2. 12 YearI%
9Time line for an ordinary annuity of $100 for 3 years 123I%
11Preparing BAII Plus for use Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’.Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’.Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.
12FV of an initial $100 after 3 years (I = 10%) 12310%Finding FVs (moving to the righton a time line) is called compounding.100
30Spreadsheet Solution Use the RATE function: = RATE(N, PMT, PV, FV)
31ExercisesSuppose you deposit $150 in an account today and the interest rate is 6 percent p.a.. How much will you have in the account at the end of 33 years?You deposited $15,000 in an account 22 years ago and now the account has $50,000 in it. What was the annual rate of return that you received on this investment?You currently have $38,000 in an account that has been paying 5.75 percent p.a.. You remember that you had opened this account quite some years ago with an initial deposit of $19,000. You forget when the initial deposit was made. How many years (in fractions) ago did you make the initial deposit?
32Perpetuity 1Perpetuity: a stream of equal cash flows ( C ) that occur at the end of each period and go on forever.PV of perpetuity =
33Perpetuity 2 We use the idea of a perpetuity to determine the value of A preferred stockA perpetual debt
34Perpetuity questionsSuppose the value of a perpetuity is $38,900 and the discount rate is 12 percent p.a.. What must be the annual cash flow from this perpetuity?Verify that C = $4,668.An asset that generates $890 per year forever is priced at $6,000. What is the required rate of return?Verify that r = %
35Ordinary AnnuityOrdinary annuity: a cash flow stream where a fixed amount is received at the end of every period for a fixed number of periods.
36What’s the FV of a 3-year ordinary annuity of $100 at 10%? 12310%110121FV = 331
37Financial Calculator Solution 331.00NI/YRPMTFVPVINPUTSOUTPUTHave payments but no lump sum PV, so enter 0 for present value.
38Spreadsheet Solution Use the FV function: = FV(I, N, PMT, PV)
39What’s the PV of this ordinary annuity? 10012310%90.9182.6475.13= PV
40Financial Calculator Solution INPUTSNI/YRPVPMTFVOUTPUTHave payments but no lump sum FV, so enter 0 for future value.
41Spreadsheet Solution Use the PV function: = PV(I, N, PMT, FV)
42Annuity, find FVYou open an account today with $20,000 and at the end of each of the next 15 years, you deposit $2,500 in it. At the end of 15 years, what will be the balance in the account if the interest rate is 7 percent p.a.?PV=-20000, PMT=-2500, N=15, I/Y=7, FV=?
43Annuity, find I/YYou lend your friend $100,000. He will pay you $12,000 per year for the ten years and a balloon payment at t = 10 of $50,000. What is the interest rate that you are charging your friend?PV=-100,000, FV=50,000, PMT=12,000, N = 10, I/Y=?
44Annuity, find PMTNext year, you will start to make 35 deposits of $3,000 per year in your Individual Retirement Account (so you will contribute from t=1 to t=35). With the money accumulated at t=35, you will then buy a retirement annuity of 20 years with equal yearly payments from a life insurance company (payments from t=36 to t=55). If the annual rate of return over the entire period is 8%, what will be the annual payment of the annuity?
45Annuity DueAnnuity due: a cash flow stream where a fixed amount is received at the beginning of every period for a fixed number of periods.
46Ordinary Annuity vs. Annuity Due PMT123I%Annuity Due
48a relationship between ordinary annuity and annuity due? PV of annuity due= (PV of ordinary annuity) x (1 + r)FV of annuity due= (FV of ordinary annuity) x (1 + r)
49Find the FV and PV if the annuity were an annuity due. 10012310%
50PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due:= (PV of ordinary annuity) (1+I)= ($248.69) ( ) = $273.56FV of annuity due:= (FV of ordinary annuity) (1+I)= ($331.00) ( ) = $364.10
51PV of Annuity Due: Switch from “End” to “Begin” NI/YRPVPMTFVINPUTSOUTPUTBEGIN Mode
52FV of Annuity Due: Switch from “End” to “Begin” NI/YRPVPMTFVINPUTSOUTPUTBEGIN Mode
53Excel Function for Annuities Due Change the formula to:=PV(0.10,3,-100,0,1)The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:=FV(0.10,3,-100,0,1)
54What is the PV of this uneven cash flow stream? 10013002310%-50490.91247.93225.39-34.15= PV
57In-class group project You will need to pay for your son’s private school tuition (first grade through 12th grade) a sum of $8,000 per year for Years 1 through 5, $10,000 per year for Years 6 through 8, and $12,500 per year for Years 9 through 12. Assume that all payments are made at the beginning of the year, that is, tuition for Year 1 is paid now (i.e., at t = 0), tuition for Year 2 is paid one year from now, and so on. In addition to the tuition payments you expect to incur graduation expenses of $2,500 at the end of Year 12. If a bank account can provide a certain 10 percent p.a. rate of return, how much money do you need to deposit today to be able to pay for the above expenses?
58Nominal rate (INOM)Stated in contracts, and quoted by banks and brokers.Not used in calculations or shown on time linesPeriods per year (M) must be given.Examples:8%; Quarterly8%, Daily interest (365 days)
59Periodic rate (IPER )IPER = INOM/M, where M is number of compounding periods per year. M = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.Used in calculations, shown on time lines.Examples:8% quarterly: IPER = 8%/4 = 2%.8% daily (365): IPER = 8%/365 = %.
60The Impact of Compounding Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant?Why?
61The Impact of Compounding (Answer) LARGER!If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.
62Six-months / semiannual Common examplesCompounding periodCompounding frequencySix-months / semiannual2Quarter4Month12Day365
63When frequency of compounding is more than once a year ‘n’ = number of years‘m’ = frequency of compounding per year‘r’ = nominal rate
64Effective Annual Rate (EAR = EFF%) The EAR is the annual rate that causes PV to grow to the same FV as under multi-period compounding.
65Effective Annual Rate Example Example: Invest $1 for one year at 12%, semiannual:FV = PV(1 + INOM/M)MFV = $1 (1.06)2 = $EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.
66$100 at a 12% nominal rate with semiannual compounding for 5 years INOMFVN = PVMM N0.12FV5S = $22x5= $100(1.06)10 = $179.08
67FV of $100 at a 12% nominal rate for 5 years with different compounding FV(Ann.)= $100(1.12)5= $176.23FV(Semi.)= $100(1.06)10= $179.08FV(Quar.)= $100(1.03)20= $180.61FV(Daily)= $100(1+(0.12/365))(5x365)= $182.19How to solve with financial calculator?
68Comparing RatesAn investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.Banks say “interest paid daily.” Same as compounded daily.
69EFF% for a nominal rate of 12%, compounded semiannually INOMM= − 10.122= (1.06)= = 12.36%.
70EAR (or EFF%) for a Nominal Rate of of 12% EARAnnual = 12%.EARQ = ( /4)4 - 1 = %.EARM = ( /12) = %.EARD(365) = ( /365) = %.
71Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if M = 1.If M > 1, EFF% will always be greater than the nominal rate.
72When is each rate used? INOM: Written into contracts, quoted by banks and brokers. Not used in calculations or shownon time lines.
73When is each rate used? (Continued) IPER:Used in calculations, shown on time lines.If INOM has annual compounding,then IPER = INOM/1 = INOM.
74When is each rate used? (Continued) EAR (or EFF%): Used to compare returns on investments with different payments per year.Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.
75Annuity with semiannual compounding You would like to accumulate $16,500 over the next 8 years. How much must you deposit every six months, starting six months from now, given a 4 percent per annum rate with semiannual compounding?
76Loan AmortizationAmortization is the process of separating a payment into interest payment and repayment of principal.Amortization schedule is a table that shows how each payment is split into principal repayment and interest payment.
77Amortization Example 1Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.
78Step 1: Find the required payments. PMT12310%-1,000INPUTSOUTPUTNI/YRPVFV402.11
79Step 2: Find interest charge for Year 1. INTt = Beg balt (I)INT1 = $1,000(0.10) = $100
80Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT= $ $100= $302.11
81Step 4: Find ending balance after Year 1. End bal = Beg bal - Repmt= $1,000 - $ = $697.89Repeat these steps for Years 2 and 3to complete the amortization table.
82Amortization Table YEAR BEG BAL PMT INT PRIN PMT END BAL 1 $1,000 $402 $100$302$698269840270332366337TOT1,206.34206.341,000
83Interest declines because outstanding balance declines.
84Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!
85Example of loan amortization 1 You have borrowed $8,000 from a bank and have promised to repay the loan in five equal yearly payments. The first payment is at the end of the first year. The interest rate is 10 percent. Draw up the amortization schedule for this loan.
86Example of loan amortization 2 1) Compute periodic payment.PV=8000, N=5, I/Y=10, FV=0, PMT=?Verify that PMT = -2,110.38
87Example of loan amortization 3 Suppose we want to work out the remaining balance immediately after the 2nd payment.Press [2ND], [AMORT] to activate the Amortization worksheet in BA II Plus.Press P1=2, [ENTER], ,Press P2=2, [ENTER], ,You will see BAL=5,248.20P1 = starting point in a range of payments, the first payment of interestP2 = ending point in a range of payments, the last payment of interest
88Example of loan amortization 4 Press again and you see the portion of the year 2 payment going towards repaying principal, PRN = -1,441.42Press again and you see the portion of year 2 payment going towards interest,INT =To get out of the Amortization schedule, press [2ND], Quit.
89Verify the amortization schedule YearBeg.BalancePaymentInterestPrincipalEnd.8,000.0012,110.38800.001,310.386,689.622668.961,441.425,248.203524.821,585.563,662.644366.261,744.121,918.535191.850.00
90Non-matching rates and periods What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?
91Time line for non-matching rates and periods 1100235%456 6-mos.periods
92Non-matching rates and periods Payments occur annually, but compounding occurs each 6 months.So we can’t use normal annuity valuation techniques.