© 2004 by Nelson, a division of Thomson Canada Limited Contemporary Financial Management Chapter 4: Time Value of Money

© 2004 by Nelson, a division of Thomson Canada Limited 2 Introduction ● This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.

© 2004 by Nelson, a division of Thomson Canada Limited 3 Payment of Interest ● Interest is the cost of money ● Interest may be calculated as: ● Simple interest ● Compound interest

© 2004 by Nelson, a division of Thomson Canada Limited 4 Simple Interest ● Interest paid only on the initial principal Example: $1,000 is invested to earn 6% per year, simple interest. -$1,000 $

© 2004 by Nelson, a division of Thomson Canada Limited 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. -$1,000 $60.00 $63.60 $

© 2004 by Nelson, a division of Thomson Canada Limited 6 Future Value ● The value of an investment at a point in the future, given some rate of return. 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 Simple Interest Compound Interest

© 2004 by Nelson, a division of Thomson Canada Limited 7 Future Value: Simple Interest Example: You invest $1,000 for three years at 6% simple interest per year. -$1, %

© 2004 by Nelson, a division of Thomson Canada Limited 8 Future Value: Compound Interest Example: You invest $1,000 for three years at 6%, compounded annually. -$1, %

© 2004 by Nelson, a division of Thomson Canada Limited 9 Future Value: Compound Interest ● Future values can be calculated using a table method, whereby “future value interest factors” (FVIF) are provided. ● See Table 4.1 (page 135), where: FV = future value PV = present value FVIF = future value interest factor i = interest rate n = number of periods

© 2004 by Nelson, a division of Thomson Canada Limited 10 Future Value: Compound Interest Example: You invest $1,000 for three years at 6% compounded annually.

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 12 Present Value Example: You will receive $1,000 in three years. If the discount rate is 6%, what is the present value? $1, %

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 14 Present Value Example: What is the present value of $1,000 to be received in three years, given a discount rate of 6%?

© 2004 by Nelson, a division of Thomson Canada Limited 15 A Note of Caution ● Note that the algebraic solution to the present value problem gave an answer of ● The table method gave an answer of $840. Caution: Tables provide approximate answers only. If more accuracy is required, use algebra!

© 2004 by Nelson, a division of Thomson Canada Limited 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.

© 2004 by Nelson, a division of Thomson Canada Limited 17 Annuities ● Ordinary annuity: cash flows occur at the end of each period Example: 3-year, $100 ordinary annuity $

© 2004 by Nelson, a division of Thomson Canada Limited 18 Annuities ● Annuity Due: cash flows occur at the beginning of each period Example: 3-year, $100 annuity due $

© 2004 by Nelson, a division of Thomson Canada Limited 19 Difference Between Annuity Types $ $100 Ordinary Annuity Annuity Due

© 2004 by Nelson, a division of Thomson Canada Limited 20 Annuities: Future Value ● Future value of an annuity - sum of the future values of all individual cash flows. $ FV FV of Annuity

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 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%?

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 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%?

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 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?

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 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?

© 2004 by Nelson, a division of Thomson Canada Limited 29 Annuities: Present Value ● The present value of an annuity is the sum of the present values of all individual cash flows. $ PV PV of Annuity

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 31 Annuities: Present Value – Algebra Example: What is the present value of a three year, $100 ordinary annuity, given a discount rate of 6%?

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 33 Annuities: Present Value – Algebra Example: What is the present value of a three year, $100 annuity due, given a discount rate of 6%?

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 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?

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 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?

© 2004 by Nelson, a division of Thomson Canada Limited 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 AnnuityAnnuity Due

© 2004 by Nelson, a division of Thomson Canada Limited 39 Other Uses of Annuity Formulas ● Loan Amortization and Capital Recovery Problems: calculating the payments necessary to pay off, or amortize, a loan.

© 2004 by Nelson, a division of Thomson Canada Limited 40 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 $

© 2004 by Nelson, a division of Thomson Canada Limited 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).

© 2004 by Nelson, a division of Thomson Canada Limited 42 Perpetuities Example: What is the present value of a $100 perpetuity, given a discount rate of 8% compounded annually?

© 2004 by Nelson, a division of Thomson Canada Limited 43 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 -$1,000 $60.00 $ % $ % 12%

© 2004 by Nelson, a division of Thomson Canada Limited 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

© 2004 by Nelson, a division of Thomson Canada Limited 45 More Frequent Compounding Example: What is a $1,000 investment worth in five years if it earns 8% interest, compounded quarterly?

© 2004 by Nelson, a division of Thomson Canada Limited 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?

© 2004 by Nelson, a division of Thomson Canada Limited 47 Impact of Compounding Frequency $1,000 Invested at Different 10% Nominal Rates for One Year

© 2004 by Nelson, a division of Thomson Canada Limited 48 Effective Annual Rate (EAR) ● The annually compounded interest rate that is identical to some nominal rate, compounded “m” times per year.

© 2004 by Nelson, a division of Thomson Canada Limited 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?

© 2004 by Nelson, a division of Thomson Canada Limited 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