1. Future Value Present Value Annuities Different compounding Periods Adjusting for frequent compounding Effective Annual Rate (EAR) Chapter 5 5-1

2.  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

3.  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

4.  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 CF 0 CF 1 CF 3 CF 2 0123 I%

5.  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

6. 5-6 100 0 1 2 3 I% 2. 3-year $100 ordinary annuity 100 012 I% 1. $100 lump sum (single payments) due in 2 years 100 0123 I% 3. 3-year $100 annuity due Annuity: A series of equal payments at fixed intervals for a specified # of periods

7.  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 5-7 100 -50 75 0123 I% -50 4. Uneven cash flow stream (payments are not equal)

8. 5-8 100 012 I% 4. Perpetuities (annuity that has payments that go forever) ∞

9.  Compounding interest rates is when interest is earned on interest.  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 5-9 115.76 105110.25 0123 5% 100 = 100(1.05) Interest = $5 Amount = $100 = 100(1.05) 2 Interest = $5.25 Amount= $105 = 100(1.05) 3 Interest = $5.5125 Amount= $110.25

10. + relation between FV and interest rates + relation between FV and N 5-10

11. (-) relation between PV and interest rates (-) relation between PV and N 5-11

12. 7-12

13.  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

14.  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 (I NOM ), 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 (I PER ): amount of interest charged each period, e.g. annually, monthly, quarterly, daily, and/or continuously.  I PER = I NOM /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

15.  We can go on compounding every hour, minute, and second  continuous compounding 5-15

16.  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: 5-16

17.  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 5-17 Annually: FV 3 = $100(1.10) 3 = $133.10 0123 10% 100133.10

18.  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. 5-18 Semiannually: FV 6 = $100(1.05) 6 = $134.01 0123 5% 456 134.01 123 0 100

19.  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

20.  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

21. 7-21

22.  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

23.  EFF% for 10% semiannual interest  EFF%= (1 + I NOM /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

24. NominalEffective Annual Rate when compounded RateYearly Semiannually QuarterlyMonthlyDaily Continuously 1% 1.0025%1.0038%1.0046%1.0050% 2% 2.0100%2.0151%2.0184%2.0201% 3% 3.0225%3.0339%3.0416%3.0453%3.0455% 4% 4.0400%4.0604%4.0742%4.0808%4.0811% 5% 5.0625%5.0945%5.1162%5.1267%5.1271% 6% 6.0900%6.1364%6.1678%6.1831%6.1837% 8% 8.1600%8.2432%8.3000%8.3278%8.3287% 10% 10.2500%10.3813%10.4713%10.5156%10.5171% 15% 15.5625%15.8650%16.0755%16.1798%16.1834% 25% 26.5625%27.4429%28.0732%28.3916%28.4025%

25.  I NOM : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.  I PER : Used in calculations and shown on time lines.  If M = 1  I NOM = I PER = 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

26. 26 5-26

27. 27  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?