2 Time Value Topics Future value Present value Rates of return Amortization
3 Time Value Basic Concepts Time linesFuture value / Present value of lump sumFV / PV of annuityPerpetuitiesUneven CF streamCompounding periodsNominal / Effective / Periodic ratesAmortizationRates of return
4 Determinants 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.Market interest ratesFirm’s debt/equity mixCost of debtCost of equityMarket risk aversionFirm’s business risk
5 Time 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.
6 Time line for a $100 lump sum due at the end of Year 2. 12 YearI%
7 Time line for an ordinary annuity of $100 for 3 years 123I%
12 After 2 years FV2 = FV1(1+I) = PV(1 + I)(1+I) = PV(1+I)2 = $100(1.10)2 = $121.00
13 After 3 years FV3 = FV2(1+I)=PV(1 + I)2(1+I) = PV(1+I)3 = $100(1.10)3 = $133.10In general,FVN = PV(1 + I)N
14 Four Ways to Find FVsStep-by-step approach using time line (as shown in Slides 7-10).Solve the equation with a regular calculator (formula approach).Use a financial calculator.Use a spreadsheet.
15 Financial calculator: HP10BII Adjust display brightness: hold down ON and push + or –.Set number of decimal places to display: Orange Shift key, then DISP key (in orange), then desired decimal places (e.g., 3).To temporarily show all digits, hit Orange Shift key, then DISP, then =.
16 HP10BII (Continued)To permanently show all digits, hit ORANGE shift, then DISP, then . (period key).Set decimal mode: Hit ORANGE shift, then ./, key. Note: many non-US countries reverse the US use of decimals and commas when writing a number.
17 HP10BII: Set Time Value Parameters To set END (for cash flows occurring at the end of the year), hit ORANGE shift key, then BEG/END.To set 1 payment per period, hit 1, then ORANGE shift key, then P/YR.
18 Financial Calculator Solution Financial calculators solve this equation:PV (1+I)N = FVNThere are 4 variables (PV, I, N, FV). If 3 are known, calculator solves for 4th.
19 Here’s the setup to find FV N I/YR PV PMT FV133.10INPUTSOUTPUTClearing automatically (shift Clear-All) sets everything to 0, but for safety enter PMT = 0.
21 After 4 years, but different compounding per year Semi-annualQuarterlyPV = $100N = 4 yrs x 2 = 8 periodsi = 10% / 2 = 5% per periodFV = ? =PV = $100N = 4 yrs x 4 = 16 periodsi = 10% / 4 = 2.5% per periodFV = ? =
22 Spreadsheet SolutionUse FV function: see spreadsheet in Ch04 Mini Case.xls= FV(I, N, PMT, PV)= FV(0.10, 3, 0, -100) =
23 Discounting $$ Money needed today to accumulate x$ value in future Solve for Present Value (PV)Mathematical process (divide)
24 What’s the PV of $110 due in 1 year if I/YR = 10%? Finding PVs is discounting, it’s reverse of compounding.110123PV = ?
25 Solve FVN = PV(1 + I )N for PV 1101PV=1.10PV= $110
26 What’s the PV of $110 due in 1 year if I/YR = 10%? Annual CompoundingSemi-annuallyFV = $110N = 1 yri = 10%PV = ? =FV = $110N = 1 yr x 2 = 2 periodsi = 10% / 2 = 5.0% per periodFV = ? =
27 What’s the PV of $100 due in 3 years if I/YR = 10%? Finding PVs is discounting, and it’s the reverse of compounding.100123PV = ?
28 Solve FVN = PV(1 + I )N for PV 31PV=$1001.10= $100(0.7513) = $75.13
29 Financial Calculator Solution N I/YR PV PMT FV-75.13INPUTSOUTPUTEither PV or FV must be negative. HerePV = Put in $75.13 today, takeout $100 after 3 years.
30 Spreadsheet SolutionUse PV function: see spreadsheet in Ch04 Mini Case.xls= PV(I, N, PMT, FV)= PV(0.10, 3, 0, 100) =
31 Cash Flow signs Investing $ today Borrowing $ today Outlay (invest) $ today in present to earn greater return in the future.Earn interest (revenue), plus principalPV = <->FV = +Take in (borrow) $ today in present to use now, then repay with interest in the future.Pay interest (expense), plus principalPV = +FV = <->
32 Periods or Interest Rate unknown Solve for NSolve for iInvest $100 today earning 10% & need $ How long will it takeDeposit $100 today. You need $ in 4 years. What’s the annual interest rate if the money is compounded quarterly?
33 Periods or Interest Rate unknown Solve for NSolve for iInvest $100 today earning 10% & need $ How long will it takeDeposit $100 today. You need $ in 4 years. What’s the annual interest rate if the money is compounded quarterly?
34 Finding the Time to Double 20%21?-1FV = PV(1 + I)NContinued on next slide
35 Time to Double (Continued) N LN(1.2) = LN(2)N = LN(2)/LN(1.2)N = 0.693/0.182 = 3.8
36 Financial Calculator Solution N I/YR PV PMT FV3.8INPUTSOUTPUT
37 Spreadsheet SolutionUse NPER function: see spreadsheet in Ch04 Mini Case.xls= NPER(I, PMT, PV, FV)= NPER(0.10, 0, -1, 2) = 3.8
41 Ordinary Annuity vs. Annuity Due Series of equal payments made at fixed intervals or specified number of periods.Ordinary endAnnuity beg
42 Ordinary Annuity vs. Annuity Due $100=PMT$100123I%Annuity Due
43 What’s the FV of a 3-year ordinary annuity of $100 at 10%? 12310%110121FV = 331
44 FV Annuity FormulaThe future value of an annuity with N periods and an interest rate of I can be found with the following formula:= PMT(1+I)N-1I= $100(1+0.10)3-10.10= $331
45 Financial Calculator Formula for Annuities Financial calculators solve this equation:FVN + PV(1+I)N + PMT(1+I)N-1I= 0There are 5 variables (PV, PMT, N, I, FV. If 4 are known, calculator solves for 5th. Pay attention to inflows & outflows (signs).
46 Financial Calculator Solution 331.00NI/YRPMTFVPVINPUTSOUTPUTHave payments but no lump sum PV, so enter 0 for present value.
47 Spreadsheet Solution Use FV function: see spreadsheet. = FV(I, N, PMT, PV)= FV(0.10, 3, -100, 0) =
48 What’s the PV of this ordinary annuity? 10012310%90.9182.6475.13= PV
49 PV Annuity FormulaThe present value of an annuity with N periods and an interest rate of I can be found with the following formula:= PMT1I−I (1+I)N= $1000.10.1(1+0.1)3= $248.69
50 Financial Calculator Solution INPUTSNI/YRPVPMTFVOUTPUTHave payments but no lump sum FV, so enter 0 for future value.
51 Spreadsheet Solution Use PV function: see spreadsheet. = PV(I, N, PMT, FV)= PV(0.10, 3, 100, 0) =
52 Find the FV and PV if the annuity were an annuity due. 10012310%
53 PV 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
54 PV of Annuity Due: Switch from “End” to “Begin” NI/YRPVPMTFVINPUTSOUTPUTBEGIN Mode
55 FV of Annuity Due: Switch from “End” to “Begin” NI/YRPVPMTFVINPUTSOUTPUTBEGIN Mode
56 Excel 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)
57 Retirement problem for you ScenarioSolutionWant to retire in 35 yearsDeposit (invest) $2500 year into an S&P 500 Index fund (which returns 12.1% annually)How much will you have to retire on in 35 years?How much cash did you have to outlay in total to accumulate that much?Pmt = $2500N= 35i = 12.1%FV = ? = $1,104,853$2500/yr x 35 yrs = $87,500 total cash outlay
58 Retirement problem for your friend the slacker ScenarioSolutionWant to retire with you in 35 years, but is ski bum & fails to save his 1st 15 yearsDeposit (invest) $2500 year into an S&P 500 Index fund (which returns 12.1% annually)How much will you have to retire on in 35 years?How much cash did you have to outlay in total to accumulate that much?Pmt = $2500N= 20i = 12.1%FV = ? = $182,231$2500/yr x 20 yrs = $50,000 total cash outlay$1,104,853 vs. $182,231
59 What is the PV of this uneven cash flow stream? 10013002310%-50490.91247.93225.39-34.15= PV
60 Financial calculator: HP10BII Clear all: Orange Shift key, then C All key (in orange).Enter number, then hit the CFj key.Repeat for all cash flows, in order.To find NPV: Enter interest rate (I/YR). Then Orange Shift key, then NPV key (in orange).
61 Financial calculator: HP10BII (more) To see current cash flow in list, hit RCL CFj CFjTo see previous CF, hit RCL CFj –To see subsequent CF, hit RCL CFj +To see CF 0-9, hit RCL CFj 1 (to see CF 1). To see CF 10-14, hit RCL CFj . (period) 1 (to see CF 11).
62 CF & HP Input in “CFLO” register: Enter I/YR = 10, then press NPV button to get NPV = (Here NPV = PV.)
64 What’s PV of this 3-yr, $100 per yr CF Stream, 10%=I, semi-annual compounding? 12310%= PV
65 Nominal 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)
66 Periodic 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 = %.
67 The 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?
68 The 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.
69 FV Formula with Different Compounding Periods INOMFVN = PVMM N
70 $100 at a 12% nominal rate with semiannual compounding for 5 years INOMFVN = PVMM N0.12FV5S = $22x5= $100(1.06)10 = $179.08
71 FV 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(Mon.)= $100(1.01)60= $181.67FV(Daily)= $100(1+(0.12/365))(5x365)= $182.19
72 Nominal vs. Effective Rates (APR vs. EAR or Eff) $100 today, 10% nominal rate compounded annually vs. semi-annually.
73 Effective Annual Rate (EAR = EFF%) The EAR is the annual rate that causes PV to grow to the same FV as under multi-period compounding.
74 Effective 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.
75 Comparing 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.
76 EFF% for a nominal rate of 12%, compounded semiannually INOMM= − 10.122= (1.06)= = 12.36%.
77 Finding EFF with HP10BIIType in nominal rate, then Orange Shift key, then NOM% key (in orange).Type in number of periods, then Orange Shift key, then P/YR key (in orange).To find effective rate, hit Orange Shift key, then EFF% key (in orange).
78 EAR (or EFF%) for a Nominal Rate of 12% (APR) EARAnnual = 12%.EARQ = 2 p/yr = %.EARM = 12 p/yr = %.EARD(365) = 365 p/yr = %.
79 Can effective rate ever be equal to nominal? Yes, but only if annual compounding is used, i.e., if p/yr = 1.If p/yr > 1, EFF% will always be greater than nominal.
80 When is each rate used? INOM: Written into contracts, quoted by banks and brokers. Not used in calculations or shownon time lines, unless annual compounding.
81 When is each rate used? (Continued) IPER:Used in calculations, shown on time lines.If INOM has semi-annual compounding, then periodic rate is IPER = INOM/2
82 When 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.
83 Fractional Time Periods On January 1 you deposit $100 in an account that pays a nominal interest rate of %, with daily compounding (365 days).How much will you have on October 1, or after 9 months (273 days)? (Days given.)
84 Convert interest to daily rate IPER = %/365= % per day12273%-100FV=?
91 2nd Method: Treat as an annuity, use financial calculator Find the EFF% (EAR) for the quoted rate:10 shift Nom; 2 sift p/yr; EFF% = ?
92 Use EAR = 10.25% as the annual rate in calculator. INPUTSNI/YRPVPMTFVOUTPUT331.80
93 What’s the PV of this stream? 10015%2390.7082.2774.62247.59
94 Comparing Investments You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a % nominal rate, with 365 daily compounding. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.Should you buy it?
95 Daily time lineIPER = % per day.365456 days……-8501,000
96 Three solution methods 1. Greatest future wealth: FV2. Greatest wealth today: PV3. Highest rate of return: EFF%
97 1. Greatest Future Wealth Find FV of $850 left in bank for15 months and compare withnote’s FV = $1,000.FVBank = $850( )456= $ in bank.Buy the note: $1,000 > $
98 Calculator Solution to FV 924.97INPUTSOUTPUTNI/YRPVFVIPER = INOM/M= /365= per day.PMT
99 2. Greatest Present Wealth Find PV of note, and comparewith its $850 cost:PV = $1,000/( )456= $918.95Buy the note: $ > $850
100 Financial Calculator Solution INPUTSOUTPUTNI/YRPVFVPMT/365 =PV of note is greater than its $850 cost, so buy the note. Raises your wealth.
101 3. Rate of ReturnFind the EFF% on note and compare with 7.0% bank pays, which is your opportunity cost of capital:FVN = PV(1 + I)N$1,000 = $850(1 + I)456Now we must solve for I.
103 Using interest conversion P/YR = 365NOM% = (365) = 13.01EFF% = 13.89Since 13.89% > 7.0% opportunity cost,buy the note.
104 AmortizationConstruct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.
105 Step 1: Find the required payments. PMT12310%-1,000INPUTSOUTPUTNI/YRPVFV402.11
106 Step 2: Find interest charge for Year 1. INTt = Beg balt (I)INT1 = $1,000(0.10) = $100
107 Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT= $ $100= $302.11
108 Step 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.
109 Amortization Table YEAR BEG BAL PMT INT PRIN PMT END BAL 1 $1,000 $402 $100$302$698269840270332366337TOT1,206.34206.341,000
110 Interest declines because outstanding balance declines.
111 Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!Financial calculators (and spreadsheets) are great for setting up amortization tables.