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- 1. The Time Value of Money TOPIC 3
- 2. Learning Objectives1. Define the time value of money2. The significance of time value of money in financial management3. Define and understand the conceptual and calculation of future and present value in cash flows4. Define the meaning of compounding and discounting5. Work with annuities and perpetuities WRMAS 2
- 3. TIME VALUE OF MONEY Basic Principle : A dollar received today is worth more than a dollar received in the future.• This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.• Example Invest RM1 today at a 6% annual interest rate. At the end of the year you will get $1.06. SO You can say: 1. The future value of RM1 today is $1.06 given a 6% interest rate a year. OR WE CAN SAY 2. The present value of the $1.06 you expect to receive in one year is only $1 today. WRMAS 3
- 4. Translate $1 today into its equivalent in the future (compounding) – Future Value Today Future ? Translate $1 in the future into its equivalent today (discounting)- Present Value Today Future ? WRMAS 4
- 5. SIGNIFICANCE OF TIME VALUE OF MONEY• This concept is so important in understanding financial management.• We must take this time value of money into consideration when we are making financial decisions.• It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities. WRMAS 5
- 6. COMPOUND INTEREST AND FUTURE VALUE• Future value is the value at a given future date of an amount placed on deposit today and earning interest at a specified rate.• Compound interest is interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum (that includes the principal and interest earned so far). (Process of determining FV)• Principal is the amount of money on which interest is paid. WRMAS 5-6
- 7. Simple Interest• Interest is earned only on principal.• Example: Compute simple interest on $100 invested at 6% per year for 3 years. – 1st year interest is $6.00 – 2nd year interest is $6.00 – 3rd year interest is $6.00 – Total interest earned: $18.005-7
- 8. Compound Interest and Future ValueExample:Compute compound interest on $100 invested at 6% for 3years with annual compounding. 1st year interest is $6.00 Principal is $106.00 2nd year interest is $6.36 Principal is $112.36 3rd year interest is $6.74 Principal is $119.11 Total interest earned: $19.10 WRMAS 8
- 9. The Equation for Future Value• We use the following notation for the various inputs: – FVn = future value at the end of period n – PV = initial principal, or present value – r = annual rate of interest paid. (Note: On financial calculators, I is typically used to represent this rate.) – n = number of periods (typically years) that the money is left on deposit OR FVn = PV (1+r)n FVn = PV (FVIFr,n)WRMAS 9
- 10. Future Value ExampleExample: What will be the FV of $100 in 2 years at interest rate of 6%? Manually Table FV2= $100 (1+.06)2 FV2= PV(FVIF6%,2) = $100 (1.06)2 = $100 (1.1236) = $112.36 SAME ANSWER! = $112.36ExerciseJane Farber places $800 in a savings account paying 6% interestcompounded annually. She wants to know how much money will bein the account at the end of 5 years. WRMAS 10
- 11. Future ValueChanging I, N and PV• Future Value can be increased or decreased by changing: Increasing number of years of compounding (n) Increasing the interest or discount rate (i) Increasing the original investment (PV) WRMAS 11
- 12. FV- Changing i, n, and PVExercise(a) You deposit $500 in a bank for 2 years. What is the FV at 2%? What is the FV if you change interest rate to 6%? FV at 2% = 500 (1.0404) = ? i ( OR ) FV ? FV at 6% = 500 (1.1236) = ?(b) Continue same example but change time to 10 years. What is the FV now? n ( OR ) FV ? FV at 6% = 500 (1.7908) = ?(c) Continue same example but change contribution to $1500. What is the FV now? PV ( OR ) FV ? FV at 6%, year 10 = 1500 (1.7908) = ? WRMAS 12
- 13. Future Value RelationshipWe can increase the FV by:1. Increasing the number of years for which money is invested; and/or2. Investing at a higher interest rate.WRMAS 13
- 14. FV- Finding I and nExample 1At what annual rate would the following have to be invested ; $500to grow to $1183.70 in 10 years.1183.70 = 500 (FVIFi,,10)1183.70/500 = FVIFi,102.3674 = FVIFi,10 (refer to FVIF table)2.3674 = 9%Example 2How many years will the following take? $100 to grow to $672.75 ifinvested at 10% compounded annually.$672.75 = $100 (FVIF10%,n)672.75/100 = FVIF10%,n6.7275 = FVIF10%,n (refer to FVIF table)6.7275 = 20 years WRMAS 14
- 15. Exercise-Finding i and na) How many years will the following take : i. $100 to grow to $298.60 if invested at 20% compounded annually ii. $550 to grow to $1,044.05 if invested at 6% compounded annuallyb) At what annual rate would the following have to be invested : i. $200 to grow to $497.65 in 5 years ii. $180 to grow to $485.93 in 6 years WRMAS 15
- 16. DISCOUNT INTEREST AND PRESENT VALUE Present value reflects the current value of a future payment or receipt. How much do I have to invest today to have some amount in the future? Finding Present Values (PVs) = discounting Example You need RM400 to buy textbook next year. Earn 7% on your money. How much do you have to put today? WRMAS 16
- 17. PRESENT VALUE Formula of Present Value (PV): FVn PV = or PV = FVn (PVIFi,n) (1+i )nWhere;FVn = the future value of the investment at the end of n yearsn = number of years until payment is receivedi = the interest ratePV = the present value of the future sum of moneyFVIF = Future value interest factor or the compound sum $1[ 1/(1+i)n ] is also known as discounting factor WRMAS 17
- 18. Present ValueExample : What is the PV of $800 to be received 10 years from today if our discount rate is 10%.Manually PV = 800/(1.10)10 = $308.43Table SAME ANSWER! PV = $800 (PVIF 10%,10yrs) = $800 (0.3855) = $308.40 WRMAS 18
- 19. Present Value ExerciseExercise 1 (finding PV)Pam Valenti wishes to find the present value of $1,700 that will bereceived 8 years from now. Pam’s opportunity cost is 8%.Exercise 2 (changing i)Find the PV of $10,000 to be received 10 years from today if ourdiscount rate is:a) 5% b) 10% c) 20% i ( OR ) PV ?Exercise 3 (finding n)How many years will it take for your initial investment of RM7,752to grow to RM20,000 with a 9% interest ? n ( OR ) PV ? WRMAS 19
- 20. Present Value RelationshipPV is lower if:1. Time period is longer; and/or2. Interest rate is higher. WRMAS 20
- 21. ANNUITYAn annuity is a series of equal payments for a specified numbersof years. These cash flows can be inflows of returns earned oninvestments or outflows of funds invested to earn future returns.There are 2 types of annuities*: - An ordinary annuity is an annuity for which the cash flow occurs at the end of each period (much more frequently in finance) - An annuity due is an annuity for which the cash flow occurs at the beginning of each period.Note: An annuity due will always be greater than an ordinaryannuity because interest will compound for an additional period. WRMAS 21
- 22. Ordinary Annuity-PVa) Present Value of Annuity (PVA)• Pensions, insurance obligations, and interest owed on bonds are all annuities. To compare these three types of investments we need to know the present value (PV) of each. Formula: PVAn = PMT [1-(1+i)-n] PVAn = PMT (PVIFAi,n) i or WRMAS 22
- 23. Ordinary Annuity-FVb) Future Value of Annuity (FVA)• Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow. Formula FVAn = PMT (1+ i)n -1 or FVAn = PMT (FVIFAi,n) i WRMAS 23
- 24. FV of Annuity: Changing PMT, N & r1. What will $5,000 deposited annually for 50 years be worth at 7%? – FV= $2,032,644 – Contribution = 250,000 (= 5000*50)2. Change PMT = $6,000 for 50 years at 7% – FV = 2,439,173 – Contribution= $300,000 (= 6000*50)3. Change time = 60 years, $6,000 at 7% – FV = $4,881,122 – Contribution = 360,000 (= 6000*60)4. Change r = 9%, 60 years, $6,000 – FV = $11,668,753 – Contribution = $360,000 (= 6000*60) 24
- 25. ANNUITY DUE• Remember-Annuity due is ordinary annuities in which all payments have been shifted forward by one time period. a) Future Value of Annuity Due (FVAD): FVADn = PMT (FVIFAi,n) (1+i) b) Present Value of Annuity Due (PVAD) formula: PVADn = PMT (PVIFAi,n) (1+i) WRMAS 25
- 26. Earlier, we examined this “ordinary” annuity: 500 500 500 0 1 2 3 …….……5Using an interest rate of 5%, we find that:• The FVA (at 3) is $2,818.50• The PVA (at 0) is $2,106.00HOW ABOUT ANNUITY DUE?• FVAD5 (annuity due) = PMT{[(1 + r)n – 1]/r}* (1 + r) = 500(5.637)(1.06) = $2,987.61• PVAD0 = $2,818.80 WRMAS 26
- 27. Annuity ExerciseExercise 1Fran Abrams wishes to determine how much money she will have atthe end of 5 years if he chooses annuity A that earns 7% annuallyand deposit $1,000 per year.Exercise 1Branden Co., a small producer of plastic toys, wants to determine themost it should pay to purchase a particular annuity. The annuityconsists a cash flows of $700 at the end of each year for 5 years. Therequired return is 8%.Exercise 3Determine the answers for exercise 1 and 2 on annuity due. WRMAS 27
- 28. FV and PV With Non-annual PeriodsNon-annual periods : not annual compounding but occur semiannually, quarterly, monthly… – r = stated rate/# of compounding periods – N = # of years * # of compounding periods in a year• If semiannually compounding : FV = PV (1 + i/2)m x 2 or FVn = PV (FVIFi/2,nx2)• If quarterly compounding : FV = PV (1 + i/4)m x 4 or FVn = PV (FVIFi/4,nx4)• If monthly compounding : FV = PV (1 + i/12)m x 12 or FVn = PV (FVIFi/12,nx12)How about PV? WRMAS 28
- 29. Compound Interest With Non-annual PeriodsExample 1: If you deposit $100 in an account earning 6% with semiannually compounding, how much would you have in the account after 5 years?Manually TableFV5 = PV (1 + i/2)m x 2 FV5 = PV (FVIFi/2, nx2 ) = 100 (1 + 0.03 )10 = 100 (FVIF 3%,10) = 100 (1.3439) = $134.39 = 100 (1.3439) = $134.39Example 2: If you deposit $1,000 in an account earning 12% with quarterly compounding, how much would you have in the account after 5 years?Manually TableFV5 = PV (1 + i/4)m x 4 FV5 = PV (FVIFi/4, nx4 ) = 1000 (1 + 0.03)20 = 1000 (FVIF 3%,20) = 1000 (1.8061) = $1806.11 = 1000 (1.8061) =$1806.11 WRMAS 29
- 30. Exercise-Non AnnualExercise 1How much would you have today, if RM1,000 is being discounted at18% semiannually for 10 years.Exercise 2Calculate the PV of a sum of money, if RM40,000 is discounted backquarterly at 24% per annum for 10 years.Exercise 3Paul makes a single deposit today of $400. The deposit will be investedat an interest rate of 12% per year compounded monthly. What will bethe value of Paul’s account at the end of 2 years?Exercise 4Consider a 10-year mutual fund in which payments of $100 are made atthe beginning of each month. What is the amount today if the annualrate of interest is 5%? WRMAS 30
- 31. Quoted Vs. Effective Rate• We cannot compare rates with different compounding periods. 5% compounded annually is not the same as 5% compounded quarterly.• To make the rates comparable, we must calculate their equivalent rate at some common compounding period by using effective annual rate (EAR).• In general, the effective rate > quoted rate whenever compounding occurs more than once per year.WRMAS 31
- 32. Quoted Vs. Effective RateExample 1RM1 invested at 1% per month will grow to RM1.126825(=RM1.00(1.01)12) in 1 year. Thus even though the interest rate maybe quoted as 12% compounded monthly, the EAR is: EAR = (1 + .12/12)12 – 1 = 12.6825%Example 2Fred Moreno wishes to find the effective annual rate associatedwith an 8% quoted rate (r = 0.08) when interest is compounded (1)annually (m = 1); (2) semiannually (m = 2); and (3) quarterly (m = 4). WRMAS 32
- 33. PERPETUITY• A perpetuity is an annuity that continues forever.• The present value of a perpetuity is PV = PP i PV = present value of the perpetuity PP = constant dollar amount provided by the perpetuity i = annuity interest (or discount rate)ExampleWhat is the present value of $2,000 perpetuity discounted back tothe present at 10% interest rate?= 2000/.10 = $20,000 WRMAS 33
- 34. Perpetuity ExerciseExerciseWhat is the Present Value of the following :- A $100 perpetuity discounted back to the present at 12%- A $95 perpetuity discounted back to the present at 5% i ( OR ) P? $ ( OR ) PV ? WRMAS 34
- 35. WRMAS 35

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