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Chapter 5

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The Time Value of Money

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Simple Interest n Interest is earned on principal n $100 invested at 6% per year n 1 st yearinterest is $6.00 n 2 nd yearinterest is $6.00 n 3 rd year interest is $6.00 n Total interest earned:$18

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Compound Interest n Interest is earned on previously earned interest n $100 invested at 6% with annual compounding n 1 st yearinterest is $6.00Principal is $106 n 2 nd yearinterest is $6.36Principal is $112.36 n 3 rd yearinterest is $6.74Principal is $119.11 n Total interest earned: $19.10

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We know that 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

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If we can measure this opportunity cost, we can: n Translate $1 today into its equivalent in the future (compounding). n Translate $1 in the future into its equivalent today (discounting). ? Today Future Today ? Future

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Future Value

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Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? 0 1 PV = FV =

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Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Calculator Solution: Calculator Solution: P/Y = 1I = 6 P/Y = 1I = 6 N = 1 PV = -100 N = 1 PV = -100 FV = $106 FV = $106 0 1 0 1 PV = -100 FV = 106

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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 (1 + i) n FV = 100 (1.06) 1 = $106 0 1 0 1 PV = -100 FV = 106

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Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: Calculator Solution: P/Y = 1I = 6 P/Y = 1I = 6 N = 5 PV = -100 N = 5 PV = -100 FV = $133.82 FV = $133.82 0 5 0 5 PV = -100 FV = 133. 82

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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 (1 + i) n FV = 100 (1.06) 5 = $ 133.82 0 5 0 5 PV = -100 FV = 133. 82

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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?

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0 ? PV = FV =

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Calculator Solution: Calculator Solution: P/Y = 4I = 6 P/Y = 4I = 6 N = 20 PV = -100 N = 20 PV = -100 FV = $134.68 FV = $134.68 0 20 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?

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Mathematical Solution: FV = PV (1 + i/m) m x n FV = 100 (1.015) 20 = $134.68 0 20 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?

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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?

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0 ? PV = FV =

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Calculator Solution: Calculator Solution: P/Y = 12I = 6 P/Y = 12I = 6 N = 60 PV = -100 N = 60 PV = -100 FV = $134.89 FV = $134.89 0 60 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?

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Mathematical Solution: FV = PV (1 + i/m) m x n FV = 100 (1.005) 60 = $134.89 0 60 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?

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Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding, after 100 years?

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0 ? PV = FV =

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Present Value

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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%? 0 ? PV = FV =

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Calculator Solution: Calculator Solution: P/Y = 1I = 6 P/Y = 1I = 6 N = 1 FV = 100 N = 1 FV = 100 PV = -94.34 PV = -94.34 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%? PV = -94. 34 FV = 100 0 1 0 1

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Mathematical Solution: PV = FV / (1 + i) n PV = 100 / (1.06) 1 = $94.34 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%? PV = -94. 34 FV = 100 0 1 0 1

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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%?

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0 ? PV = FV =

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Calculator Solution: Calculator Solution: P/Y = 1I = 6 P/Y = 1I = 6 N = 5 FV = 100 N = 5 FV = 100 PV = -74.73 PV = -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 0 5 PV = -74. 73 FV = 100

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Mathematical Solution: 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 0 5 PV = -74. 73 FV = 100

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Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%?

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0 15 0 15 PV = FV =

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Calculator Solution: Calculator Solution: P/Y = 1I = 7 P/Y = 1I = 7 N = 15 FV = 1,000 N = 15 FV = 1,000 PV = -362.45 PV = -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 0 15 PV = -362. 45 FV = 1000

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Mathematical Solution: 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 0 15 PV = -362. 45 FV = 1000

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Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

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0 5 0 5 PV = FV =

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Calculator Solution: Calculator Solution: P/Y = 1N = 5 P/Y = 1N = 5 PV = -5,000 FV = 11,933 PV = -5,000 FV = 11,933 I = 19% I = 19% 0 5 0 5 PV = -5000 FV = 11,933 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

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Mathematical Solution: PV = FV / (1 + i) n PV = FV / (1 + i) n 5,000 = 11,933 / (1+ i) 5 5,000 = 11,933 / (1+ i) 5.419 = ((1/ (1+i) 5 ).419 = ((1/ (1+i) 5 ) 2.3866 = (1+i) 5 2.3866 = (1+i) 5 (2.3866) 1/5 = (1+i) i =.19 (2.3866) 1/5 = (1+i) i =.19 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

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Hint for single sum problems: n In every single sum future value and present value problem, there are 4 variables: n FV, PV, i, and n n When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. n Keeping this in mind makes “time value” problems much easier!

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The Time Value of Money Compounding and Discounting Cash Flow Streams 01 234

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Annuities n Annuity: a sequence of equal cash flows, occurring at the end of each period.

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01 234 Annuities

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Examples of Annuities: n If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. n If you borrow money to buy a house or a car, you will pay a stream of equal payments.

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Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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0 1 2 3

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Calculator Solution: Calculator Solution: P/Y = 1I = 8N = 3 P/Y = 1I = 8N = 3 PMT = -1,000 PMT = -1,000 FV = $3,246.40 FV = $3,246.40 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? 0 1 2 3 10001000 1000 10001000 1000

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Mathematical Solution: FV = PMT (1 + i) n - 1 i FV = 1,000 (1.08) 3 - 1 = $3246.40.08.08 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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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%? 0 1 2 3

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Calculator Solution: Calculator Solution: P/Y = 1I = 8N = 3 P/Y = 1I = 8N = 3 PMT = -1,000 PMT = -1,000 PV = $2,577.10 PV = $2,577.10 0 1 2 3 10001000 1000 10001000 1000 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%?

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Mathematical Solution: 1 1 PV = PMT 1 - (1 + i) n i 1 PV = 1000 1 - (1.08 ) 3 = $2,577.10.08.08 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%?

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Other Cash Flow Patterns 0123 The Time Value of Money

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Perpetuities n Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. n You can think of a perpetuity as an annuity that goes on forever.

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PMT i PV = n The PV of a perpetuity is very simple to find: Present Value of a Perpetuity

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What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

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PMT $10,000 PMT $10,000 i.08 i.08 PV = =

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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 PMT $10,000 i.08 i.08 = $125,000 PV = =

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Ordinary Annuity vs. Annuity Due $1000 $1000 $1000 4 5 6 7 8

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Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8

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Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7

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Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7 PVinENDModeFVinENDMode

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Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8

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Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8 PVinBEGINModeFVinBEGINMode

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Earlier, we examined this “ordinary” annuity: 0 1 2 3 10001000 1000 10001000 1000

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Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: n The Future Value (at 3) is $3,246.40. n The Present Value (at 0) is $2,577.10. 0 1 2 3 10001000 1000 10001000 1000

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n Is this an annuity? n How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate). Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000

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n Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows

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period CF PV (CF) period CF PV (CF) 0-10,000 -10,000.00 0-10,000 -10,000.00 1 2,000 1,818.18 1 2,000 1,818.18 2 4,000 3,305.79 2 4,000 3,305.79 3 6,000 4,507.89 3 6,000 4,507.89 4 7,000 4,781.09 4 7,000 4,781.09 PV of Cash Flow Stream: $ 4,412.95 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000

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Example 012345678012345678012345678012345678 $0 0 0 04040404040 n Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?

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n This type of cash flow sequence is often called a “deferred annuity.” 012345678012345678012345678012345678 $0 0 0 04040404040

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How to solve: 1) Discount each cash flow back to time 0 separately. Or, 012345678012345678012345678012345678 $0 0 0 04040404040

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2) Find the PV of the annuity: PV 3: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PV 3 = $119,624 012345678012345678012345678012345678 $0 0 0 04040404040

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119,624 012345678012345678012345678012345678

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Then discount this single sum back to time 0. PV: End mode; P/YR = 1; I = 20; N = 3; FV = 119,624; Solve: PV = $69,226 119,624 012345678012345678012345678012345678 $0 0 0 04040404040

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n The PV of the cash flow stream is $69,226. 69,226 012345678012345678012345678012345678 $0 0 0 04040404040 119,624

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Retirement Example n After graduation, you plan to invest $400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years?

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Retirement Example n After graduation, you plan to invest $400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years? 01 23... 360 400 400 400 400

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01 23... 360 400 400 400 400

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n Using your calculator, P/YR = 12 P/YR = 12 N = 360 PMT = -400 I%YR = 12 FV = $1,397,985.65 01 23... 360 400 400 400 400

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If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment? House Payment Example

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If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

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n Using your calculator, P/YR = 12 N = 360 N = 360 I%YR = 7 PV = $100,000 PMT = -$665.30 01 23... 360 ? ? ? ? ? ? ? ?

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1 IIS Chapter 5 - The Time Value of Money. 2 IIS The Time Value of Money Compounding and Discounting Single Sums.

1 IIS Chapter 5 - The Time Value of Money. 2 IIS The Time Value of Money Compounding and Discounting Single Sums.

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