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6-1 CHAPTER 5 Time Value of Money Read Chapter 6 (Ch. 5 in the 4 th edition) Future value Present value Rates of return Amortization

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6-2 Time Value of Money Problems Use a financial calculator Bring your calculator to class Will need on exams We will not use the tables

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6-3 Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 i% 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.

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6-4 A. (1) a. Time line for a $100 lump sum due at the end of Year 2. 100 012 Year i %

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6-5 A. (1) b. Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

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6-6 A. (1) c. Time line for uneven CFs -$50 at t=0 and $100, $75, and $50 at the end of Years 1 through 3. 100 50 75 0123 i% -50

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6-7 What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? 0123 10% 100 Finding FVs is Compounding.

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6-8 After 1 year: FV 1 =PV + I 1 =PV + PV (i) =PV(1 + i) =$100 (1.10) =$110.00. After 2 years: FV 2 =PV(1 + i) 2 =$100 (1.10) 2 =$121.00.

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6-9 After 3 years: FV 3 =PV(1 + i) 3 =100 (1.10) 3 =$133.10. In general, FV n =PV (1 + i) n

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6-10 Three ways to find FVs: 1.‘Solve’ the Equation with a Scientific Calculator 2.Use Tables (the book describes this but not for use in this class) 3. Use a Financial Calculator 4. Spreadsheet (has built-in formulas) -- won’t work on exams

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6-11 310-100 0 NI/YR PV PMTFV 133.10 INPUTS OUTPUT Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Check your calculator. Set: P/YR = 1 and END (“BEGIN” should not show on the display)

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6-12 What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0123 10% PV = ?

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6-13 Financial Calculator Solution: 310 0100 NI/YR PV PMTFV -75.13 INPUTS OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.

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6-14 If sales grow at 20% per year, how long before sales double? Solve for n: FV n = 1(1 + i) n ; In our case 2 = (1.20) n. Take the log of both sides: ln(2) = n ln(1.2) n = ln(2)/ln(1.2)=.693…/0.1823.. =3.8017

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6-15 20 -1 0 2 NI/YR PV PMTFV 3.8 INPUTS OUTPUT Graphical Illustration: 0 1234 1 2 FV 3.8 Year Financial calculator solution

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6-16 What’s the difference between an ordinary annuity and an annuity due?

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6-17 Ordinary vs. Annuity Due PMT 0123 i% PMT 0123 i% PMT

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6-18 What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

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6-19 310 0 -100 331.00 INPUTS OUTPUT NI/YRPVPMTFV Financial Calculator Solution: If you enter PMT of 100, you get FV of -331. Get used to the fact that you have to figure out the sign.

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6-20 What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV

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6-21 310 100 0 -248.69 INPUTS OUTPUT NI/YRPVPMTFV Have payments but no lump sum FV, so enter 0 for future value. Financial Calculator Solution:

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6-22 Technical Aside: Your calculator really is assuming a NPV equation, with PV as a time zero cash flow as follows: When you use the top row of calculator keys, the calculator assumes NPV=0 and solves for one variable.

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6-23 Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

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6-24 310 100 0 -273.55 INPUTS OUTPUT NI/YRPVPMTFV Switch from “End” to “Begin”. Then enter variables to find PVA 3 = $273.55. Then enter PV = 0 and press FV to find FV = $364.10.

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6-25 Alternative: n The first payment is in the present and thus has a PV of 100. n The next two payments comprise a two period ordinary annuity -- use the formula with n=2, PMT=100, and i=.10. n Sum the above two for the present value. n If you already have the PV, multiply by To get FV

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6-26 Perpetuities n A perpetuity is a stream of regular payments that goes on forever An infinite annuity n Future value of a perpetuity Makes no sense because there is no end point n Present value of a perpetuity A diminishing series of numbers Each payment’s present value is smaller than the one before

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6-27 Perpetuities—Example Example You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below. PV N PMT I/Y 250 999 5 2 0FV Answer Q:The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for? A:Convert the k to a quarterly k and plug the values into the equation.

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6-28 What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

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6-29 n Input in “CFLO” register ( CF j ) : CF 0 =0 CF 1 =100 CF 2 =300 CF 3 =300 CF 4 =-50 n Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

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6-30 What’s Project L’s NPV? 108060 0123 10% Project L: -100.00 9.09 49.59 60.11 18.79 = NPV L

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6-31 Calculator Solution: Enter in CFLO for L: -100 10 60 80 10 CF 0 CF 1 NPV CF 2 CF 3 i = 18.78 = NPV L

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6-32 TI Calculators BA-35 doesn’t appear to do uneven cash flows (NPV and IRR) BA II PLUS CF CF 0 = -100 Enter C01= 10 Enter F01= 1.00 C02= 60 Enter F02= 1.00 C03= 80 Enter F03= 1.00 NPV I=10 Enter CPT NPV= 18.78 IRR CPT IRR= 18.13

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6-33 The Sinking Fund Problem n Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the bonds’ lives Principal is repaid at maturity in a lump sum –A sinking fund provides cash to pay off a bond’s principal at maturity Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem

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6-34 The Sinking Fund Problem –Example Q:The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside? A:The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator. PMT N FV I/Y 407,768.35 20 15,000,000 6 0 PV Answer Example

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6-35 What interest rate would cause $100 to grow to $125.97 in 3 years? 3-100 0 125.97 INPUTS OUTPUT NI/YRPVFV PMT 8% $100 (1 + i ) 3 = $125.97.

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6-36 Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semi-annually, quarterly, or daily--interest is earned on interest more often.

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6-37 0123 10% 0123 5% 456 134.01 100 133.10 123 0 100 Semi-annually: Annually: FV 3 = 100(1.10) 3 = 133.10. FV 6/2 = 100(1.05) 6 = 134.01.

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6-38 We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. The literal rate applied each period EAR= EFF% = effective annual rate.

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6-39 i Nom is stated in contracts. Periods per year (m) must also be given. Sometimes (incorrectly) referred to as the “simple” interest rate. Examples: 8%, Daily interest (365 days) 8%; Quarterly

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6-40 Periodic rate = i Per = i Nom /m, where m is periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: i per = 8/4 = 2% 8% daily (365): i per = 8/365 = 0.021918%

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6-41 Effective Annual Rate (EAR = EFF%): The annual rate which cause PV to grow to the same FV as under multiperiod compounding. Example: EFF% for 10%, semiannual: FV=(1 + i nom /m) m =(1.05) 2 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually: (1.1025) 1 = 1.1025 (1.05) 2 = 1.1025

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6-42 Comparing Financial Investments An 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.

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6-43 How do we find EFF% for a nominal rate of 10%, compounded semi-annually?

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6-44 EAR = EFF% of 10% EAR Annual =10%. EAR Q = (1 + 0.10/4) 4 - 1=10.38%. EAR M = (1 + 0.10/12) 12 - 1=10.47%. EAR D = (1 + 0.10/360) 360 - 1= 10.5155572%.

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6-45 Can 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.

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6-46 When is each rate used? i nom : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

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6-47 i per : Used in calculations, shown on time lines. If i nom has annual compounding, then i per = i nom /1 = i nom.

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6-48 EAR = EFF%: Used to compare returns on investments with different payments per year and in advertising of deposit interest rates. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.)

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6-49 FV of $100 after 3 years under 10% semi-annual compounding? Quarterly? = $100(1.05) 6 = $134.01 FV 3Q = $100(1.025) 12 = $134.49

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6-50 What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually? 01 100 23 5% 456 100 6-month periods

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6-51 Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

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6-52 1st method: Compound each CF 01 100 23 5% 456 100100.00 110.25 121.55 331.80 FVA 3 = 100(1.05) 4 + 100(1.05) 2 + 100 = 331.80

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6-53 What’s the PV of this stream? 0 100 1 5% 23 100 90.70 82.27 74.62 247.59 Years 2 )05.1(100 4 )05.1(100 6 )05.1(100

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6-54 Second Method: use your financial calculator! Follow these two steps: a.Find the EAR for the quoted rate : This is the i per for a period of one year. Use in formula (or calculator) with the period equal to a year.

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6-55 0 100 1 10.25% 23 100 Time line

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6-56 3 10.25 0 -100 INPUTS OUTPUT NI/YRPVFVPMT 331.80 b. Calculator inputs

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6-57 NIPVPMTFV 10 1000 58 0 7 -5001000 15 -7501001000 2408/12 -100,000 0 501010010 Calculator Workout: fill in the blanks

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6-58 0.75 10 - 100 0 ? =107.41 INPUTS OUTPUT NI/YRPVPMTFV Fractional Time Periods 00.250.500.75 10% - 100 1.00 FV = ? Example: $100 deposited in a bank at 10% interest for 0.75 of the year

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6-59 AMORTIZATION Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

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6-60 This is what an amortization schedule looks like. Amortization Table BeginningEnding PrincipalTotalInterestPrincipal PeriodBalancePayment Balance 1$1,000.00$402.11$100.00$302.11$697.89 2 $402.11$69.79$332.33$365.56 3 $402.11$36.56$365.56$0.00

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6-61 Step 1: Find the required payment. PMT 0123 10% -1000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV PMT 402.11

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6-62 Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = 1000(0.10)=$100. Step 3: Find repayment of principal in Year 1. Repmt.= PMT - INT = 402.11 - 100 = $302.11.

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6-63 Step 4: Find ending balance after Year 1. End bal=Beg bal - Repmt =1000 - 302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

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6-64 Amortization Table BeginningEnding PrincipalTotalInterestPrincipal PeriodBalancePayment Balance 1$1,000.00$402.11$100.00$302.11$697.89 2 $402.11$69.79$332.33$365.56 3 $402.11$36.56$365.56$0.00 Interest declines. Tax Implications.

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6-65 Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.

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6-66 Amortized Loans—Example Example PMT N PV I/Y 293.75 48 10,000 1.5 0 FV Answer This can also be calculated using the PVA formula of PVA = PMT[PVFA k, n ] with an n of 48 and a k of 1.5%, resulting in $10,000 = PMT[34.0426] = $293.75. Q:Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment? A:Adjust your interest rate and number of periods for monthly compounding and input the following keystrokes into your calculator.

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6-67 Amortized Loans—Example PV N FV I/Y 15,053.75 36 0 1 500 PMT Answer Example This can also be calculated using the PVA formula of PVA = PMT[PVFA k, n ] with an n of 36 and a k of 1%, resulting in PVA = $500[30.1075] = $15,053.75. Q:Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car? A:Adjust your k and n for monthly compounding and input the following calculator keystrokes.

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6-68 Loan Amortization Schedules—Example Example Q:Develop an amortization schedule for the loan demonstrated in Example 5.12. Note that the Interest portion of the payment is decreasing while the Principal portion is increasing.

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