Chapter 5 Time Value of Money Future Value Present Value Annuities Different compounding Periods Adjusting for frequent compounding Effective Annual Rate (EAR)
The Concept of TVM • 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)
Application of TVM • 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.
Time Lines • 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. 0 1 2 3 I% CF0 CF1 CF2 CF3
Important Terminology • 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)
2. 3-year $100 ordinary annuity 3. 3-year $100 annuity due 2 1 3 0 I% 0 1 2 3 I% 100 100 100 100 100 100 Different Time Lines 1. $100 lump sum (single payments) due in 2 years I% 0 1 2 100 Annuity: A series of equal payments at fixed intervals for a specified # of periods
4. Uneven cash flow stream (payments are not equal) 0 1 2 3 I% -50 100 75 -50 Different Time Lines • 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. Perpetuities (annuity that has payments that go forever) 0 1 2 I% 100 100 100 Different Time Lines ∞
0 1 2 3 5% 100 105 110.25 How Compounding and Discounting Works • 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) = 115 115.76 = 100(1.05)3 Interest = $5.5125 Amount= $110.25 = 100(1.05) Interest = $5 Amount = $100 = 100(1.05)2 Interest = $5.25 Amount= $105
A Graphic view of the compounding process (FV) (lump sum) + relation between FV and interest rates + relation between FV and N
A Graphic view of the discounting process (PV) (lump sum) (-) relation between PV and interest rates (-) relation between PV and N
Different Compounding Periods • 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.
Different Compounding Periods • 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
Different Compounding Periods • We can go on compounding every hour, minute, and second continuous compounding
Continuous Discounting • 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:
0 1 2 3 10% 100 133.10 How to adjust for frequent compounding? Annually: FV3 = $100(1.10)3 = $133.10 • 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 1 2 3 0 1 2 3 4 5 6 5% 100 134.01 How to adjust for frequent compounding? • 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. Semiannually: FV6 = $100(1.05)6 = $134.01
Differences in FVs when compounding is frequent • 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?
Will the PV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant? • 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
Classifications of Interest Rates • 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.
Example • 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.
When is each rate used? • 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.
Example 2 • 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?