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Chapter 5 Future Value Present Value Annuities Different compounding Periods Adjusting for frequent compounding Effective Annual Rate (EAR) 5-1 You want to buy a computer and a friend offers you a $1000. Would you prefer use the money now. Later (after year for example). The answer to that question depends on: Inflation rate. Deferred consumption. Forgone investment opportunity Uncertainty (Risk) 5-2 There are several application for the TVM from which both individuals and firms benefit, such as: Planning for retirement, Valuing businesses or any asset (including stocks and bonds), Setting up loan payment schedules Making corporate decisions regarding investing in new plants and equipments. The rest of this book and course heavily depends on your understanding of the concepts of TVM and your proficiency in doing its calculations. 5-3 0 CF0 I% 1 2 3 CF1 CF2 CF3 Help visualize what is happening in a particular problem. Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period. 5-4 Finding the future value (FV) or compounding): The amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate. Finding the present value (PV): The value today of a future cash flow or series of cash flows when discounted at a given interest. Compounding : is the process to determine the FV of a cash flow or series of payments. (multiplying) Discounting : is the reverse of compounding. The process of determining the PV of a cash flow or series of payments (dividing) 5-5 1. $100 lump sum (single payments) due in 2 years 0 I% 1 2 2. 3-year $100 ordinary annuity 0 I% 100 1 2 3 100 100 100 3. 3-year $100 annuity due 0 100 I% 1 2 100 100 3 Annuity: A series of equal payments at fixed intervals for a specified # of periods 5-6 Examples of obligations that uses annuities: Auto, student, mortgage loans However, many financial decisions involve non constant (not equal) payments: Dividend on common stocks. Investment in capital equipment 4. Uneven cash flow stream (payments are not equal) 0 -50 I% 1 2 3 100 75 -50 5-7 4. Perpetuities (annuity that has payments that go forever) 0 I% 1 100 2 100 ∞ 100 5-8 Compounding interest rates is when interest is earned on interest. 0 5% 100 1 105 = 100(1.05) Interest = $5 Amount = $100 2 110.25 = 100(1.05)2 Interest = $5.25 Amount= $105 3 115.76 = 100(1.05)3 Interest = $5.5125 Amount= $110.25 Thus, FV of annuity due > FV of ordinary annuity Simple interests: interest is not earned on interest FV = PV + PV (i)(N) = 100 + 100(0.05)(3) = 115 5-9 + relation between FV and interest rates + relation between FV and N 5-10 (-) relation between PV and interest rates (-) relation between PV and N 5-11 7-12 So far we are assuming that interest is compounded yearly (annual compounding). However, there are many situations where interest is due 2,4, 12, 26, 52, 365 times a year. In general, bonds pay interest semiannually. Most mortgages, student, and auto loans require payments to be monthly. 5-13 A CD that offers a state rate of 10% compounded annually is different from a CD that offers a state rate of 10% compounded semiannually. The 10% is called the nominal rate (INOM), quoted, stated, or annual percentage rate (APR) since it ignores compounding effects. It is the rate that is stated by banks, credit card companies, and auto, student, and mortgage loans. Periodic rate (IPER): amount of interest charged each period, e.g. annually, monthly, quarterly, daily, and/or continuously. IPER = INOM/M, where M is the number of compounding periods per year. M = 4 for quarterly, M = 12 for monthly , and M = continuous compounding 5-14 We can go on compounding every hour, minute, and second continuous compounding FV5 PV (e ) 100(e IN 0.10( 5 ) ) $164.87 5-15 Thus, if $1,649 is due in 10 years, and if the appropriate continuous discount rate, is 5%, then the present value of this future payment is $1,000: PV10 FV IN FV (e ) IN (e ) 1,649(e 0.05(10) ) $1000.1690 5-16 You have $100 and an investment horizon of 3 year and have 2 choices: CD that offers a state rate of 10% annually CD that offers a state rate of 10% semiannually. The first choice will offer you a FV of 0 10% 1 2 100 3 133.10 Annually: FV3 = $100(1.10)3 = $133.10 5-17 As for the second choice (semiannually compounding): There must be 2 main adjustments: ▪ Covert the stated interests to periodic rate ▪ Convert the number of year into number of periods. 0 0 100 5% 1 1 2 3 2 4 5 3 6 134.01 Semiannually: FV6 = $100(1.05)6 = $134.01 5-18 Thus, the FV of a lump sum will be larger if compounded is more often, holding the stated I% constant Because interest is earned on interest more often. Will the PV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant? Why? 5-19 PV of a lump sum will be lower when interest rate is discounted more frequently. This is because interest is discounted sooner and thus there will be more discounting. PV of 100 at 10% annually for 3 year is PV of 100 at 10% semiannually for 3 year is 5-20 7-21 In general, different compounding is used by different investments. However, we cannot compare between these investments until we put them on a common basis. We cannot compare a CD that offers 10% annually with that that offers it semiannually or quarterly use the Effective Annual Rate (EAR) (EAR or EFF%): the annual rate of interest actually (truly)being earned, accounting for compounding. 5-22 EFF% for 10% semiannual interest EFF%= (1 + INOM/M)M – 1 = (1 + 0.10/2)2 – 1 = 10.25% Excel: =EFFECT(nominal_rate,npery) =EFFECT(.10,2) Should be indifferent between receiving 10.25% annual interest and receiving 10% interest, compounded semiannually. 5-23 Nominal Rate Effective Annual Rate when compounded Yearly Semiannually Quarterly Monthly Daily Continuously 1% 1% 1.0025% 1.0038% 1.0046% 1.0050% 1.0050% 2% 2% 2.0100% 2.0151% 2.0184% 2.0201% 2.0201% 3% 3% 3.0225% 3.0339% 3.0416% 3.0453% 3.0455% 4% 4% 4.0400% 4.0604% 4.0742% 4.0808% 4.0811% 5% 5% 5.0625% 5.0945% 5.1162% 5.1267% 5.1271% 6% 6% 6.0900% 6.1364% 6.1678% 6.1831% 6.1837% 8% 8% 8.1600% 8.2432% 8.3000% 8.3278% 8.3287% 10% 10% 10.2500% 10.3813% 10.4713% 10.5156% 10.5171% 15% 15% 15.5625% 15.8650% 16.0755% 16.1798% 16.1834% 25% 25% 26.5625% 27.4429% 28.0732% 28.3916% 28.4025% INOM: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. IPER: Used in calculations and shown on time lines. If M = 1 INOM = IPER = EAR = [1+(Inom/1]. EAR: Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods. For example: interest rate of 10% is compounded semiannually, but payments of annuity are occurring annually. 5-25 I nom (1 EAR) 1 M M 1 M I (ARP) M (1 EAR) -1 nom 265-26 Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? APR 12 (1 .12) 1/12 1 .113865515 or 11.39% 27

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