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- 1. Topic 3 Time Value of Money
- 2. Learning Objectives Define the time value of money. Explain the significance of time value of money in financial management. Define the meaning of compounding and discounting. Calculate the future value and present value. Calculate future and present value, ordinary annuity or annuity due. Define the meaning of perpetuity and how to calculate it.
- 3. Generally, receiving $1 today is worth more than $1 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. Today Future
- 4. If we can measure this opportunity cost, we can: Translate $1 today into its equivalent in the future (compounding). Translate $1 in the future into its equivalent today (discounting). ? Today Future Today ? Future
- 5. Significance of the time value of money Time value of money is important in understanding financial management. It should be considered for making financial decisions. It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities.
- 6. Simple Interest Interest is earned only on principal. Example: Compute simple interest on $100 invested at 6% per year for three years. 1st year interest is $6.00 2nd yearinterest is $6.00 3rd year interest is $6.00 Total interest earned: $18.00
- 7. Compound Interest Compounding is when 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). Is the amount a sum will grow to in a certain number of years when compounded at a specific rate. Compounding : process of determining the Future Value (FV) of cash flow. Compounded amount = Future Value (beginning amount plus interest earned. )
- 8. Compound Interest Example: Compute compound interest on $100 invested at 6% for three years with annual compounding. 1st year interest is $6.00 Principal now is $106.00 2nd year interest is $6.36 Principal now is $112.36 3rd year interest is $6.74 Principal now is $119.11 Total interest earned: $19.10
- 9. Future Value Future Value is the amount a sum will grow to in a certain number of years when compounded at a specific rate. Two ways to calculate Future Value (FV): by using Manual Formula or Using Table. Manual Formula Table FVn = PV (1 + r)n FVn = PV (FVIFi,n)n Where : FVn = the future of the investment at the end of “n” years r = the annual interest (or discount) rate n = number of years PV= the present value, or original amount invested at the beginning of the first year FVIF=Futurevalueinterestfactororthecompoundsum$1
- 10. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .06, 1 ) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)1 = $106 0 1 PV = -100 FV = ???
- 11. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .06, 5 ) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)5 = $133.82 0 5 PV = -100 FV = ???
- 12. Compound Interest With Non-annual Periods Non-annual periods : not annual compounding but occur semiannually, quarterly, monthly or daily… If semiannually compounding : FV = PV (1 + i/2)n x 2 or FVn= PV (FVIFi/2,nx2) If quarterly compounding : FV = PV (1 + i/4)n x 4 or FVn= PV (FVIFi/4,nx4) If monthly compounding : FV = PV (1 + i/12)n x 12 or FVn= PV (FVIFi/12,nx12) If daily compounding : FV = PV (1 + i/365)n x 365 or FVn= PV (FVIFi/365,nx365)
- 13. Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .015, 20 ) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = $134.68 0 20 PV = -100 FV = 134.68 Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?
- 14. Example: If you invest RM10,000 in a bank where it will earn 6% interest compounded annually. How much will it be worth at the end of a) 1 year and b) 5 years Compounded for 1 year FV1 = RM10,000 (1 + 0.06)1 FV1 = RM10,000 (FVIF 6%,1 ) = RM10,000 (1.06)1 = RM10,000 (1.0600) = RM10,600 = RM10,600 Compounded for 5 years FV5 = RM10,000 (1 + 0.06)5 FV1 = RM10,000 (FVIF 6%,5 ) = RM10,000 (1.06)5 = RM10,000 (1.3382) = RM13,380 = RM13,382
- 15. Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .005, 60 ) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.005)60 = $134.89 0 60 PV = -100 FV = 134.89 Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
- 16. 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 Manual Formula Table PVn = FV/ (1 + r)n PVn = FV (PVIFi,n)n Where : FVn = the future of the investment at the end of “n” years r = the annual interest (or discount) rate n = number of years PV= the present value, or original amount invested at the beginning of the first year PVIF=Present Value Interest Factor or the discount sum$1
- 17. Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF .06, 1 ) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)1 = $94.34 PV = ??? FV = 100 0 1 Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?
- 18. Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF .06, 5 ) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)5 = $74.73 Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = ??? FV = 100
- 19. Mathematical Solution: PV = FV (PVIF i, n ) PV = 1000 (PVIF .07, 15 ) (use PVIF table, or) PV = FV / (1 + i)n PV = 1000 / (1.07)15 = $362.45 Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? 0 15 PV = -362.45 FV = 1000
- 20. Finding i 1. At what annual rate would the following have to be invested; $500 to grow to RM1183.70 in 10 years. FVn = PV (FVIF i,n ) 1183.70 = 500 (FVIF i,10 ) 1183.70/500 = (FVIF i,10 ) 2.3674 = (FVIF i,10 ) refer to FVIF table i = 9% 2. If you sold land for $11,439 that you bought 5 years ago for $5,000, what is your annual rate of return? FV = PV (FVIF i, n ) 11,439 = 5,000 (FVIF ?, 5 ) 11,439/ 5,000= (FVIF ?, 5 ) 2.3866 = (FVIF ?, 5 ) i = .18
- 21. Finding n 1. How many years will the following investment takes? $100 to grow to $672.75 if invested at 10% compounded annually FVn = PV (FVIF i,n ) 672.75 = 100 (FVIF 10%,n ) 672.75/100 = (FVIF 10%,n ) 6.7272 = (FVIF 10%,n ) refer to FVIF table n = 20 years 2. Suppose you placed $100 in an account that pays 9% interest, compounded annually. How long will it take for your account to grow to $514? FV = PV (1 + i)n 514 = 100 (1+ .09)N 514/100 = (FVIF 9%,n ) 5.14 = (FVIF 9%,n ) refer to FVIF table n = 19 years
- 22. Hint for single sum problems: In every single sum present value and future value problem, there are four variables: FV, PV, i and n. When doing problems, you will be given three variables and you will solve for the fourth variable. Keeping this in mind makes solving time value problems much easier!
- 23. Compounding and Discounting Cash Flow Streams 0 1 2 3 4
- 24. Two types of annuity: ordinary annuity and annuity due. ordinary annuity: a sequence of equal cash flows, occurring at the end of each period. Annuity due: annuity payment occurs at the beginning of the period rather than at the end of the period. 0 1 2 3 4 Annuities
- 25. Mathematical Solution: FVA = PMT (FVIFA i, n ) FVA = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or) FVA = PMT (1 + i)n - 1 i FVA = 1,000 (1.08)3 - 1 = $3246.40 .08 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
- 26. Mathematical Solution: PVA = PMT (PVIFA i, n ) PVA = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or) 1 PVA = PMT 1 - (1 + i)n i 1 PV A= 1000 1 - (1.08 )3 = $2,577.10 .08 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
- 27. Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. You can think of a perpetuity as an annuity that goes on forever.
- 28. PMT i PV = So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity
- 29. What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PMT $10,000 i .08 = $125,000 PV = =
- 30. Ordinary Annuity vs. Annuity Due $1000 $1000 $1000 4 5 6 7 8
- 31. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7 PV in END Mode FV in END Mode
- 32. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8 PV in BEGIN Mode FV in BEGIN Mode
- 33. Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: The Future Value (at 3) is $3,246.40. The Present Value (at 0) is $2,577.10. 0 1 2 3 1000 1000 1000
- 34. What about this annuity? Same 3-year time line, Same 3 $1000 cash flows, but The cash flows occur at the beginning of each year, rather than at the end of each year. This is an “annuity due.” 0 1 2 3 1000 1000 1000
- 35. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FVA due = PMT (FVIFA i, n ) (1 + i) FVA due = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or) FVA due = PMT (1 + i)n - 1 i FVA due = 1,000 (1.08)3 - 1 = $3,506.11 .08 (1 + i) (1.08)
- 36. Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PVA due = PMT (PVIFA i, n ) (1 + i) PVA due = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or) 1 PVA due = PMT 1 - (1 + i)n i 1 PVA due = 1000 1 - (1.08 )3 = $2,783.26 .08 (1 + i) (1.08)
- 37. Annual Percentage Yield (APY) Which is the better loan: 8% compounded annually, or 7.85% compounded quarterly? We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! We need to calculate the APY. Note: APY can be called as the Effective Annual rate (EAR)
- 38. Annual Percentage Yield (APY) Find the APY for the quarterly loan: The quarterly loan is more expensive than the 8% loan with annual compounding! APY = ( 1 + ) m - 1quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
- 39. Practice Problems
- 40. 1. To what amount will the following investments accumulate? a. $4,000 invested for 11 years at 9% compounded annually b. $8,000 invested for 10 years at 8% compounded annually
- 41. 2. How many years will the following take? a. $550 to grow to $1,043.90 if invested at 6% compounded annually b. $40 to grow to $88.44 if invested at 12% compounded annually
- 42. 3. At what annual rate would the following have to be invested? a. $550 to grow to $1,898.60 in 13 years b. $275 to grow to $406.18 in 8 years
- 43. 4. What is the present value of the following annuities? a. $3,000 a year for 10 years discounted back to the present at 8% b. $50 a year for 3 years discounted back to the present at 3% a. PV = $3,000 (PVIFAr,t) PV = $3,000 (PVIFA8%,10) PV = $3,000 (6.7101) PV = $20,130 b. PV = PMT (PVIFAr,t) PV = $50 (PVIFA3%,3) PV = $50 (2.8286) PV = $141.43
- 44. To pay for your child’s education, you wish to have accumulated $25,000 at the end of 15 years. To do this, you plan on depositing an equal amount in the bank at the end of each year. If the bank is willing to pay 7% compounded annually, how much must you deposit each year to obtain your goal? FVA = PMT (FVIFA i, n ) $25,000 = PMT (PVIFA .07, 15 ) $25,000 = PMT (25.129) Thus, PMT = $994.87

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