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## CHAPTER 5 Time Value of Money

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**CHAPTER 5Time Value of Money**• Read Chapter 6 (Ch. 5 in the 4th edition) • Future value • Present value • Rates of return • Amortization**Time Value of Money Problems**• Use a financial calculator • Bring your calculator to class • Will need on exams • We will not use the tables**0**1 2 3 i% CF0 CF1 CF2 CF3 • Time lines show timing of cash flows. • Tick marksat ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.**A. (1) a. Time line for a $100 lump sum due at the end of**Year 2. 0 1 2 Year i % 100**0**1 2 3 i% 100 100 100 A. (1) b. Time line for anordinary annuity of $100 for3 years.**0**1 2 3 i% 100 75 50 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. -50**What’s the FV of an initial$100 after 3 years if i = 10%?**0 1 2 3 10% 100 FV = ? Finding FVs is Compounding.**After 1 year:**FV1 = PV + I1 = PV + PV (i) = PV(1 + i) = $100 (1.10) = $110.00. After 2 years: FV2 = PV(1 + i)2 = $100 (1.10)2 = $121.00.**After 3 years:**FV3 = PV(1 + i)3 = 100 (1.10)3 = $133.10. In general, FVn = PV (1 + i)n**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**Here’s the setup to find FV:**INPUTS 3 10 -100 0 N I/YR PV PMT FV 133.10 OUTPUT 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)**0**1 2 3 10% 100 What’s the PV of $100 duein 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. PV = ?**Financial Calculator Solution:**INPUTS 3 10 0 100 N I/YR PV PMT FV -75.13 OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.**If sales grow at 20% per year,how long before sales double?**Solve for n: FVn = 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**INPUTS**20 -1 0 2 N I/YR PV PMT FV 3.8 OUTPUT Financial calculator solution Graphical Illustration: FV 2 3.8 1 Year 0 1 2 3 4**What’s the differencebetween an ordinaryannuity and an**annuitydue?**0**1 2 3 i% PMT PMT PMT 0 1 2 3 i% PMT PMT Ordinary vs. Annuity Due PMT**What’s the FV of a 3-yearordinary annuity of $100 at10%?**0 1 2 3 10% 100 100 100 110 121 FV = 331**INPUTS**3 10 0 -100 331.00 N I/YR PV PMT FV OUTPUT 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.**0**1 2 3 10% 100 100 100 What’s the PV of this ordinaryannuity? 90.91 82.64 75.13 248.69 = PV**INPUTS**3 10 100 0 -248.69 OUTPUT Financial Calculator Solution: N I/YR PV PMT FV Have payments but no lump sum FV, so enter 0 for future value.**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.**0**1 2 3 10% 100 100 Find the FV and PV if theannuity were an annuity due. 100**Switch from “End” to “Begin”.**Then enter variables to find PVA3 = $273.55. INPUTS 3 10 100 0 -273.55 N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.**Alternative:**• The first payment is in the present and thus has a PV of 100. • The next two payments comprise a two period ordinary annuity -- use the formula with n=2, PMT=100, and i=.10. • Sum the above two for the present value. • If you already have the PV, multiply by To get FV**Perpetuities**• A perpetuity is a stream of regular payments that goes on forever An infinite annuity • Future value of a perpetuity Makes no sense because there is no end point • Present value of a perpetuity A diminishing series of numbers • Each payment’s present value is smaller than the one before**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. You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below. Example N 999 I/Y 2 PMT 5 FV 0 PV 250 Answer Perpetuities—Example**1**2 3 100 300 300 What is the PV of this uneven cashflow stream? 4 0 10% -50 90.91 247.93 225.39 -34.15 530.08 = PV**Input in “CFLO” register ( CFj ):**CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 • Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)**0**1 2 3 10% 10 60 80 What’s Project L’s NPV? Project L: -100.00 9.09 49.59 60.11 18.79 = NPVL**Calculator Solution:**Enter in CFLO for L: -100 10 60 80 10 CF0 CF1 CF2 CF3 i NPV = 18.78 = NPVL**TI Calculators**• BA-35 doesn’t appear to do uneven cash flows (NPV and IRR) BA II PLUS CF CF0=-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**The Sinking Fund Problem**• 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**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. N 20 Example I/Y 6 FV 15,000,000 0 PV 407,768.35 PMT Answer The Sinking Fund Problem –Example**INPUTS**3 -100 0 125.97 N I/YR PV FV PMT OUTPUT What interest rate wouldcause $100 to grow to $125.97 in 3 years? $100 (1 + i )3 = $125.97. 8%**Will the FV of a lump sum belarger or smaller if wecompound**more often, holdingthe 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.**0**1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. Semi-annually: 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 FV6/2 = 100(1.05)6 = 134.01.**We will deal with 3**different rates: iNom = nominal, or stated, or quoted, rate per year. iPer = periodic rate. The literal rate applied each period EAR = EFF% = effective annual rate.**iNom 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**Periodic rate = iPer = iNom/m, where m is periods per year.**m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. • Examples: 8% quarterly: iper = 8/4 = 2% 8% daily (365): iper = 8/365 = 0.021918%**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 + inom/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**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.**How do we find EFF% for a**nominal rate of 10%, compounded semi-annually?**EAR = EFF% of 10%**EARAnnual = 10%. EARQ = (1 + 0.10/4)4 - 1 = 10.38%. EARM = (1 + 0.10/12)12 - 1 = 10.47%. EARD = (1 + 0.10/360)360 - 1= 10.5155572%.**Can the effective rate ever beequal 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.**When is each rate used?**Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. inom:**Used in calculations,**shown on time lines. iper: If inom has annual compounding, then iper = inom/1 = inom.**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.)**FV of $100 after 3 yearsunder 10% semi-annualcompounding?**Quarterly? = $100(1.05)6 = $134.01 FV3Q = $100(1.025)12 = $134.49**4**5 6 0 1 2 3 5% 100 100 100 What’s the value at the endof Year 3 of the following CF stream if the quoted interestrate is 10%, compoundedsemi-annually? 6-month periods