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Contemporary Financial Management

Contemporary Financial Management. Chapter 4: Time Value of Money. Introduction. This chapter introduces the concepts and skills necessary to understand the time value of money and its applications. Payment of Interest. Interest is the cost of money Interest may be calculated as:

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Contemporary Financial Management

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  1. Contemporary Financial Management Chapter 4: Time Value of Money

  2. Introduction • This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.

  3. Payment of Interest • Interest is the cost of money • Interest may be calculated as: • Simple interest • Compound interest

  4. Simple Interest • Interest paid only on the initial principal Example: $1,000 is invested to earn 6% per year, simple interest. 0 3 2 1 $60 -$1,000 $60 $60

  5. Compound Interest • Interest paid on both the initial principal and on interest that has been paid & reinvested. Example: $1,000 invested to earn 6% per year, compounded annually. 0 3 2 1 $60.00 -$1,000 $67.42 $63.60

  6. The value of an investment at a point in the future, given some rate of return. Simple Interest Compound Interest Future Value FV = future value PV = present value i = interest rate n = number of periods FV = future value PV = present value i = interest rate n = number of periods

  7. Example: You invest $1,000 for three years at 6% simple interest per year. 6% 6% 6% Future Value: Simple Interest 0 3 2 1 -$1,000 FV=?

  8. Example: You invest $1,000 for three years at 6%, compounded annually. 6% 6% 6% Future Value: Compound Interest 0 3 2 1 -$1,000 FV=?

  9. Future values can be calculated using a table method, whereby “future value interest factors” (FVIF) are provided. See Table 4.1 (page 135) Future Value: Compound Interest , where: FV = future value PV = present value FVIF = future value interest factor i = interest rate n = number of periods

  10. Example: You invest $1,000 for three years at 6% compounded annually. Future Value: Compound Interest

  11. Present Value • What a future sum of money is worth today, given a particular interest (or discount) rate. FV = future value PV = present value i = interest (or discount) rate n = number of periods

  12. 6% 6% 6% 0 3 2 1 $1,000 PV=? Present Value Example: You will receive $1,000 in three years. If the discount rate is 6%, what is the present value?

  13. Present Value • Present values can be calculated using a table method, whereby “present value interest factors” (PVIF) are provided. • See Table 4.2 (page 139) , where: FV = future value PV = present value PVIF = present value interest factor i = interest rate n = number of periods

  14. Example: What is the present value of $1,000 to be received in three years, given a discount rate of 6%? Present Value

  15. A Note of Caution • Note that the algebraic solution to the present value problem gave an answer of 839.62 • The table method gave an answer of $840. Caution: Tables provide approximate answers only. If more accuracy is required, use algebra!

  16. Annuities • The payment or receipt of an equal cash flow per period, for a specified number of periods. Examples: mortgages, car leases, retirement income.

  17. Annuities • Ordinary annuity: cash flows occur at the end of each period Example: 3-year, $100 ordinary annuity 0 3 2 1 $100 $100 $100

  18. Annuities • Annuity Due: cash flows occur at the beginning of each period Example: 3-year, $100 annuity due 0 3 2 1 $100 $100 $100

  19. Difference Between Annuity Types Ordinary Annuity 0 3 2 1 $100 $100 $100 Annuity Due 0 3 2 1 $100 $100 $100 $100

  20. Annuities: Future Value • Future value of an annuity - sum of the future values of all individual cash flows. 0 3 2 1 $100 $100 $100 FV FV FV FV of Annuity

  21. Annuities: Future Value – Algebra • Future value of an ordinary annuity FV = future value of the annuity PMT = equal periodic cash flow i = the (annually compounded) interest rate n = number of years

  22. Annuities: Future Value Example: What is the future value of a three year ordinary annuity with a cash flow of $100 per year, earning 6%?

  23. Annuities: Future Value – Algebra • Future value of an annuity due: FV = future value of the annuity PMT = equal periodic cash flow i = the (annually compounded) interest rate n = number of years

  24. Annuities: Future Value – Algebra Example: What is the future value of a three year annuity due with a cash flow of $100 per year, earning 6%?

  25. Annuities: Future Value – Table • The future value of an ordinary annuity can be calculated using Table 4.3 (p. 145), where “future value of an ordinary annuity interest factors” (FVIFA) are provided. , where: PMT = equal periodic cash flow i = the (annually compounded) interest rate n = number of periods FVAN = future value (ordinary annuity) FVIFA = future value interest factor

  26. Ordinary Annuity: Future Value Example: What is the future value of a 3-year $100 ordinary annuity if the cash flows are invested at 6%, compounded annually?

  27. Annuity Due: Future Value • Calculated using Table 4.3 (p. 145), where FVIFAs are found. Ordinary annuity formula is adjusted to reflect one extra period of interest. , where: PMT = equal periodic cash flow i = the (annually compounded) interest rate n = number of periods FVAND = future value (annuity due) FVIFA = future value interest factor

  28. Annuity Due: Future Value Example: What is the future value of a 3-year $100 annuity due if the cash flows are invested at 6% compounded annually?

  29. Annuities: Present Value • The present value of an annuity is the sum of the present values of all individual cash flows. 0 3 2 1 PV PV PV PV of Annuity $100 $100 $100

  30. Annuities: Present Value – Algebra • Present value of an ordinary annuity PV = present value of the annuity PMT = equal periodic cash flow i = the (annually compounded) interest or discount rate n = number of years

  31. Annuities: Present Value – Algebra Example: What is the present value of a three year, $100 ordinary annuity, given a discount rate of 6%?

  32. Annuities: Present Value – Algebra • Present value of an annuity due: PV = present value of the annuity PMT = equal periodic cash flow i = the (annually compounded) interest or discount rate n = number of years

  33. Annuities: Present Value – Algebra Example: What is the present value of a three year, $100 annuity due, given a discount rate of 6%?

  34. Annuities: Present Value – Table • The present value of an ordinary annuity can be calculated using Table 4.4 (p. 149), where “present value of an ordinary annuity interest factors” (PVIFA) are found. , where: PMT = cash flow i = the (annually compounded) interest or discount rate n = number of periods PVAN = present value (ordinary annuity) PVIFA = present value interest factor

  35. Annuities: Present Value – Table Example: What is the present value of a 3-year $100 ordinary annuity if current interest rates are 6% compounded annually?

  36. Annuities: Present Value – Table • Calculated using Table 4.4 (p. 149), where PVIFAs are found. Present value of ordinary annuity formula is modified to account for one less period of interest. PMT = cash flow i = the (annually compounded) interest or discount rate n = number of periods PVAND = present value (annuity due) PVIFA = present value interest factor

  37. Annuities: Present Value – Table Example: What is the present value of a 3-year $100 annuity due if current interest rates are 6% compounded annually?

  38. Other Uses of Annuity Formulas • Sinking Fund Problems: calculating the annuity payment that must be received or invested each year to produce a future value. Ordinary Annuity Annuity Due

  39. Other Uses of Annuity Formulas • Loan Amortization and Capital Recovery Problems: calculating the payments necessary to pay off, or amortize, a loan.

  40. 4 0 3 2 1 $60 $60 $60 $60 Perpetuities • Financial instrument that pays an equal cash flow per period into the indefinite future (i.e. to infinity). Example: dividend stream on common and preferred stock

  41. Perpetuities • Present value of a perpetuity equals the sum of the present values of each cash flow. • Equal to a simple function of the cash flow (PMT) and interest rate (i).

  42. Perpetuities Example: What is the present value of a $100 perpetuity, given a discount rate of 8% compounded annually?

  43. 6% 6% More Frequent Compounding • Nominal Interest Rate: the annual percentage interest rate, often referred to as the Annual Percentage Rate (APR). Example: 12% compounded semi-annually 12% 6% 0.5 0 1.5 1 $60.00 -$1,000 $67.42 $63.60

  44. More Frequent Compounding • Increased interest payment frequency requires future and present value formulas to be adjusted to account for the number of compounding periods per year (m). Future Value Present Value

  45. More Frequent Compounding Example: What is a $1,000 investment worth in five years if it earns 8% interest, compounded quarterly?

  46. More Frequent Compounding Example: How much do you have to invest today in order to have $10,000 in 20 years, if you can earn 10% interest, compounded monthly?

  47. Impact of Compounding Frequency $1,000 Invested at Different 10% Nominal Rates for One Year

  48. Effective Annual Rate (EAR) • The annually compounded interest rate that is identical to some nominal rate, compounded “m” times per year.

  49. Effective Annual Rate (EAR) • EAR provides a common basis for comparing investment alternatives. Example: Would you prefer an investment offering 6.12%, compounded quarterly or one offering 6.10%, compounded monthly?

  50. Major Points • The time value of money underlies the valuation of almost all real & financial assets • Present value – what something is worth today • Future value – the dollar value of something in the future • Investors should be indifferent between: • Receiving a present value today • Receiving a future value tomorrow • A lump sum today or in the future • An annuity

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