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Contemporary Financial Management Chapter 4: Time Value of Money © 2004 by Nelson, a division of Thomson Canada Limited 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 2 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 3 Simple Interest ● Interest paid only on the initial principal Example: $1,000 is invested to earn 6% per year, simple interest. 0 1 2 3 -$1,000 $60 $60 $60 © 2004 by Nelson, a division of Thomson Canada Limited 4 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 1 2 3 -$1,000 $60.00 $63.60 $67.42 © 2004 by Nelson, a division of Thomson Canada Limited 5 Future Value ● The value of an investment at a point in the future, given some rate of return. Simple Interest Compound Interest FVn = PV0+(PV0 i n) FVn = PV0(1 + i)n 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 © 2004 by Nelson, a division of Thomson Canada Limited 6 Future Value: Simple Interest Example: You invest $1,000 for three years at 6% simple interest per year. 6% 0 1 6% 2 6% 3 -$1,000 FV3 = PV0+(PV0 i n) = $1,000 $1,000 0.06 3 = $1,180.00 © 2004 by Nelson, a division of Thomson Canada Limited 7 Future Value: Compound Interest Example: You invest $1,000 for three years at 6%, compounded annually. 0 6% 1 6% 2 6% 3 -$1,000 FV3 = PV0 (1 + i)n = $1,000 1 0.06 3 = $1,191.02 © 2004 by Nelson, a division of Thomson Canada Limited 8 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) FVn = PV0(FVIFi,n ), where: FVIFi,n = 1+i n 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 9 Future Value: Compound Interest Example: You invest $1,000 for three years at 6% compounded annually. Table 4.1 Excerpt: FVIFs for $1 End of Period (n) 5% 6% 8% 2 3 4 1.102 1.158 1.216 1.124 1.191 1.262 1.166 1.260 1.360 FV3 = PV0 (FVIF6%,3 ) =$1,000(1.191) =$1,191.00 © 2004 by Nelson, a division of Thomson Canada Limited 10 Present Value ● What a future sum of money is worth today, given a particular interest (or discount) rate. PV0 FVn 1+i n FV = future value PV = present value i = interest (or discount) rate n = number of periods © 2004 by Nelson, a division of Thomson Canada Limited 11 Present Value Example: You will receive $1,000 in three years. If the discount rate is 6%, what is the present value? 0 6% 1 6% 2 6% 3 $1,000 PV0 FV3 1+i n $1,000 1 0.06 © 2004 by Nelson, a division of Thomson Canada Limited 3 $839.62 12 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) PV0 = FVn(PVIFi,n ), where: PVIFi,n = 1 1+i n 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 13 Present Value Example: What is the present value of $1,000 to be received in three years, given a discount rate of 6%? Table 4.2 Excerpt: PVIFs for $1 End of Period (n) 5% 6% 8% 2 3 4 0.907 0.864 0.823 0.890 0.840 0.792 0.857 0.794 0.735 PV0 = FV3(FVIF6%,3 ) =$1,000(0.840) =$840.00 © 2004 by Nelson, a division of Thomson Canada Limited 14 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! © 2004 by Nelson, a division of Thomson Canada Limited 15 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 16 Annuities ● Ordinary annuity: cash flows occur at the end of each period Example: 3-year, $100 ordinary annuity 0 1 2 3 $100 $100 $100 © 2004 by Nelson, a division of Thomson Canada Limited 17 Annuities ● Annuity Due: cash flows occur at the beginning of each period Example: 3-year, $100 annuity due 0 1 2 $100 $100 $100 © 2004 by Nelson, a division of Thomson Canada Limited 3 18 Difference Between Annuity Types Ordinary Annuity 0 1 2 3 $100 $100 $100 Annuity Due 0 1 2 3 $100 $100 $100 $100 © 2004 by Nelson, a division of Thomson Canada Limited 19 Annuities: Future Value ● Future value of an annuity - sum of the future values of all individual cash flows. 0 1 2 3 $100 $100 $100 FV FV FV FV of Annuity © 2004 by Nelson, a division of Thomson Canada Limited 20 Annuities: Future Value – Algebra ● Future value of an ordinary annuity 1+in -1 FVOrdinary= PMT i 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 21 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%? 1+in -1 FVOrdinary= PMT i Annuity 1.06 3 1 100 . 06 $318.36 © 2004 by Nelson, a division of Thomson Canada Limited 22 Annuities: Future Value – Algebra ● Future value of an annuity due: 1+in -1 FVAnnuity= PMT 1 + i i 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 23 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%? 1+in -1 FVAnnuity= PMT 1+i i Due 1.06 3 1 100 1.06 . 06 $337.46 © 2004 by Nelson, a division of Thomson Canada Limited 24 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. FVANn = PMT(FVIFAi,n ), where: 1 i n FVIFAi,n = 1 i 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 25 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? Table 4.3 Excerpt: FVIFA for $1 per period End of Period (n) 5% 6% 10% 2 3 4 2.050 3.152 4.310 2.060 3.184 4.375 2.100 3.310 4.641 FVANn = PMT(FVIFAi,n ) =$100 3.184 $318.40 © 2004 by Nelson, a division of Thomson Canada Limited 26 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. FVANDn = PMT FVIFAi,n 1 i , where: n 1 i 1 FVIFAi,n = i 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 27 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? Table 4.3 Excerpt: FVIFA for $1 per period End of Period (n) 5% 6% 10% 2 3 4 2.050 3.152 4.310 2.060 3.184 4.375 2.100 3.310 4.641 FVANDn = PMT FVIFAi,n 1 i $100 3.184(1.06) $337.50 © 2004 by Nelson, a division of Thomson Canada Limited 28 Annuities: Present Value ● The present value of an annuity is the sum of the present values of all individual cash flows. 0 1 2 3 PV PV PV PV of Annuity $100 $100 $100 © 2004 by Nelson, a division of Thomson Canada Limited 29 Annuities: Present Value – Algebra ● Present value of an ordinary annuity 1- 1+i-n PVOrdinary= PMT i 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 30 Annuities: Present Value – Algebra Example: What is the present value of a three year, $100 ordinary annuity, given a discount rate of 6%? 1- 1+i-n PVOrdinary=PMT i Annuity 1 - 1.06 -3 100 . 06 $267.30 © 2004 by Nelson, a division of Thomson Canada Limited 31 Annuities: Present Value – Algebra ● Present value of an annuity due: 1- 1+i-n PVAnnuity= PMT 1+i i 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 32 Annuities: Present Value – Algebra Example: What is the present value of a three year, $100 annuity due, given a discount rate of 6%? 1- 1+i-n PVAnnuity= PMT 1+i i Due 1 1.06 3 100 .06 $283.34 © 2004 by Nelson, a division of Thomson Canada Limited 1.06 33 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. PVAN0 = PMT(PVIFAi,n ), where: PVIFAi,n 1- 1+i-n = i 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 34 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? Table 4.4 Excerpt: PVIFA for $1 per period End of Period (n) 5% 6% 10% 2 3 4 1.859 2.723 3.546 1.833 2.673 3.465 1.736 2.487 3.170 PVAN0 = PMT(PVIFAi,n ) =$100 2.673 $267.30 © 2004 by Nelson, a division of Thomson Canada Limited 35 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. PVAND0 = PMT PVIFAi,n(1 i) PVIFAi,n 1- 1+in = i 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 36 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? Table 4.4 Excerpt: PVIFA for $1 per period End of Period (n) 5% 6% 10% 2 3 4 1.859 2.723 3.546 1.833 2.673 3.465 1.736 2.487 3.170 PVAND0 = PMT PVIFAi,n(1 i) $100 2.673 1.06 $283.34 © 2004 by Nelson, a division of Thomson Canada Limited 37 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 FVANn PMT= FVIFAi,n © 2004 by Nelson, a division of Thomson Canada Limited Annuity Due FVANn PMT= FVIFAi,n 1 i 38 Other Uses of Annuity Formulas ● Loan Amortization and Capital Recovery Problems: calculating the payments necessary to pay off, or amortize, a loan. PVAN0 PMT= PVIFAi,n © 2004 by Nelson, a division of Thomson Canada Limited 39 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 0 1 2 3 4 $60 $60 $60 $60 © 2004 by Nelson, a division of Thomson Canada Limited 40 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). PMT PVPER 0 n (1+i) t 1 © 2004 by Nelson, a division of Thomson Canada Limited PMT PVPER 0 i 41 Perpetuities Example: What is the present value of a $100 perpetuity, given a discount rate of 8% compounded annually? PMT $100 PVPER 0 $1,250.00 i 0.08 © 2004 by Nelson, a division of Thomson Canada Limited 42 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% 0 -$1,000 6% 0.5 6% $60.00 © 2004 by Nelson, a division of Thomson Canada Limited 1 $63.60 6% 1.5 $67.42 43 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 mn inom FVn PV0 1 m © 2004 by Nelson, a division of Thomson Canada Limited PV0 FVn mn inom 1+ m 44 More Frequent Compounding Example: What is a $1,000 investment worth in five years if it earns 8% interest, compounded quarterly? mn inom FVn PV0 1 m (4)(5) 0.08 $1,000 1 4 $1, 485.95 © 2004 by Nelson, a division of Thomson Canada Limited 45 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? PV0 FVn mn inom 1+ m $10,000 (12)(20) 0.10 1+ 12 © 2004 by Nelson, a division of Thomson Canada Limited $1,364.62 46 Impact of Compounding Frequency $1,000 Invested at Different 10% Nominal Rates for One Year $1,106 $1,105 $1,104 $1,103 $1,102 $1,101 $1,100 $1,099 $1,098 $1,097 Annual SemiAnnual Quarterly Monthly © 2004 by Nelson, a division of Thomson Canada Limited Daily 47 Effective Annual Rate (EAR) ● The annually compounded interest rate that is identical to some nominal rate, compounded “m” times per year. m ieff inom 1+ 1 m ieff effective annual rate inom nominal interest rate m = compounding frequency per year © 2004 by Nelson, a division of Thomson Canada Limited 48 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? m ieff inom 1+ 1 m m ieff 4 0.0612 1+ 1 4 6.262% © 2004 by Nelson, a division of Thomson Canada Limited inom 1+ 1 m 12 0.061 1+ 12 6.273% 1 49 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 © 2004 by Nelson, a division of Thomson Canada Limited 50