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The Time Value of Money Compounding and Discounting Single Sums and Annuities 1999, Prentice Hall, Inc.

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Notes: n Although it is easiest to use your financial calculator to solve time value problems, you MUST understand what you are doing. This will require a lot of practice to eliminate mistakes. n Understanding the concept of Time Value of Money NOW is extremely important because all the remaining chapters will require TVM concept application.

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Notes: n In your Test, you will be REQUIRED to show both your financial calculator solution and Mathematical solution (either using the formula or the Financial Tables) n Only in multiple choice questions will you not be required to show your calculation. Therefore, you can use the financial calculator alone. You must however, make sure that you know how to use your calculator properly; otherwise you will easily make mistakes with the use of a financial calculator.

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Using your Financial Calculators (TI BAII Plus and Sharp EL-733A) n For the first time that you are using your calculator, perform the following: –TI BAII Plus users, Set Payment Frequency and Compounding Frequency to 1 (press 2 nd, press P/Y, press 1, press ENTER, press down arrow, press 1, press ENTER, press 2 nd, press QUIT) –Sharp EL-733A users, Set to FIN Mode if not yet in FIN Mode ( press 2 nd F, press Mode). You should see FIN on the Display. n Set Decimal to 4 places –TI BAII Plus users, press 2 nd, press Format, press 4, press ENTER, press 2 nd, press QUIT. –Sharp EL-733A users, press 2 nd F, press TAB, press 4.

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Using your Financial Calculators (TI BAII Plus and Sharp EL-733A) n To start each calculation, –TI BAII Plus users, press CE/C, press 2 nd, press CLR TVM, press 2 nd, press CLR Work. BAII Plus has a continuous memory. Turning-off the calculator does not erase what was previously stored in its memory, although turning it on again resets the display to zero. Therefore, it is extremely important to clear memory before each calculation. –Sharp EL-733A users, press 2 nd F, press CA. n To erase the previously entered number, –TI BAII Plus users, simply press CE/C. –Sharp EL-733A users, simply press C CE. n Enter Outflow Value as negative. To enter it as negative enter the value/s, press +/-. DO NOT use the minus sign key.

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Using your Financial Calculators (TI BAII Plus and Sharp EL-733A) n The order in which data (PV, n, I, etc) are entered does not matter. n To compute for the result, press CPT for TI BAII Plus users, press COMP for Sharp EL-733A users. Then press whatever variable you are computing for (PV. FV, etc) n To perform calculations involving annuity dues, payment must be set to the begin mode. –TI BAII Plus users, press 2 nd, press BGN, press 2 nd,press SET, press 2 nd, press QUIT. –Sharp EL-733A users, press BGN. n It is important to reset the mode back to END. Most payment problems are made at the end of each year (ordinary annuities).

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We know that receiving $1 today is worth more than $1 in the future. This is due to the time value of money. The 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 interest cost, we can: ? n Translate $1 today into its equivalent in the future (COMPOUNDING). Today Future

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If we can MEASURE this interest 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 – single sum

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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: I/Y = i = 6 N = n = 5 I/Y = i = 6 N = n = 5 PV = -100 PV = -100 FV = $133.82 FV = $133.82 0 5 0 5 PV = -100 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 5 years? Calculator Solution: Calculator Solution: I/Y = i = 6 I/Y = i = 6 N = n = 5 PV = -100 N = 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 (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 0 5 PV = -100 FV = 133. 82

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Calculator Solution: Calculator Solution: I/Y = i = 1.5 I/Y = i = 1.5 N = n = 20 PV = -100 N = n = 20 PV = -100 FV = $134.68 FV = $134.68 0 20 0 20 PV = -100 FV = 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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Calculator Solution: Calculator Solution: I/Y = i = 1.5 I/Y = i = 1.5 N = n = 20 PV = -100 N = 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 (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 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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Present Value – single sum

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Calculator Solution: Calculator Solution: I/Y = i =6 I/Y = i =6 N = n = 5 FV = 100 N = n = 5 FV = 100 PV = -74.73 PV = -74.73 0 5 0 5 PV = FV = 100 Present Value - single sums If you will receive $100 5 years from now, what is the PV of that $100 if the interest rate is 6%?

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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 0 5 0 5 PV = -74. 73 FV = 100 Present Value - single sums If you will receive $100 5 years from now, what is the PV of that $100 if the interest rate is 6%?

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Calculator Solution: Calculator Solution: N = n = 5 N = n = 5 PV = -5,000 FV = 11,933 PV = -5,000 FV = 11,933 I/Y = i =19% I/Y = i =19% 0 5 0 5 PV = -5,000 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 (PVIF i, n ) PV = FV (PVIF i, n ) 5,000 = 11,933 (PVIF ?, 5 ) 5,000 = 11,933 (PVIF ?, 5 ) 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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Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 PV = FV =

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Calculator Solution: n FV = 500 n I/Y = i = 0.8PV = -100 n N = n = 202 months Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 ? 0 ? PV = -100 FV = 500

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Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i) n 100 = 500 / (1+.008) N 5 = (1.008) N ln 5 = ln (1.008) N ln 5 = N ln (1.008) 1.60944 =.007968 N N = 202 months

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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. 01 234

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Examples of Annuities: n If you buy a bond, you will receive equal 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 at the end of the next 3 years, at 8%, how much would you have after 3 years? 0 1 2 3

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

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Calculator Solution: Calculator Solution: I/Y= i = 8N = n = 3 I/Y= i = 8N = n = 3 PMT = -1,000 PMT = -1,000 FV = $3,246.40 FV = $3,246.40 Future Value - annuity If you invest $1,000 at the end of the next 3 years, 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 (FVIFA i, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) Future Value - annuity If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?

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Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i Future Value - annuity If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?

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Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 1,000 (FVIFA.08, 3 ) (use FVIFA table, or) 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 at the end of the next 3 years, 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 discount rate is 8%? 0 1 2 3

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Calculator Solution: Calculator Solution: I/Y = i = 8N = n = 3 I/Y = i = 8N = n = 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 discount rate is 8%?

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Calculator Solution: Calculator Solution: I/Y = i = 8N = n = 3 I/Y = i = 8N = n = 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 discount rate is 8%?

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Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%?

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Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) 1 1 PV = PMT 1 - (1 + i) n i Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%?

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Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) 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 discount rate is 8%?

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Ordinary Annuity vs. Annuity Due

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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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What about this annuity? n Same 3-year time line, n Same 3 $1000 cash flows, but n The cash flows occur at the beginning of each year, rather than at the end of each year. n This is an “annuity due.” 0 1 2 3 1000 1000 1000 1000 1000 1000

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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? 0 1 2 3

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Calculator Solution: Calculator Solution: Mode = BEGIN I/Y = i = 8 Mode = BEGIN I/Y = i = 8 N = n = 3 PMT = -1,000 N = n = 3 PMT = -1,000 FV = $3,506.11 FV = $3,506.11 0 1 2 3 -1000 -1000 -1000 -1000 -1000 -1000 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?

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0 1 2 3 -1000 -1000 -1000 -1000 -1000 -1000 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? Calculator Solution: Calculator Solution: Mode = BEGIN I/Y = i = 8 Mode = BEGIN I/Y = i = 8 N = n = 3 PMT = -1,000 N = n = 3 PMT = -1,000 FV = $3,506.11 FV = $3,506.11

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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: FV = PMT (FVIFA i, n ) (1 + i) FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or)

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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: FV = PMT (FVIFA i, n ) (1 + i) FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + i) n - 1 FV = PMT (1 + i) n - 1 i (1 + i)

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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: FV = PMT (FVIFA i, n ) (1 + i) FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + i) n - 1 FV = PMT (1 + i) n - 1 i FV = 1,000 (1.08) 3 - 1 = $3,506.11 FV = 1,000 (1.08) 3 - 1 = $3,506.11.08.08 (1 + i) (1.08)

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Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if the discount rate is 8%? 0 1 2 3

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Calculator Solution: Calculator Solution: Mode = BEGIN I/Y = i = 8 Mode = BEGIN I/Y = i = 8 N = n= 3 PMT = 1,000 N = n= 3 PMT = 1,000 PV = $2,783.26 PV = $2,783.26 0 1 2 3 1000 1000 1000 1000 1000 1000 Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if the discount rate is 8%?

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Calculator Solution: Calculator Solution: Mode = BEGIN I/Y = i = 8 Mode = BEGIN I/Y = i = 8 N = n= 3 PMT = 1,000 N = n= 3 PMT = 1,000 PV = $2,783.26 PV = $2,783.26 0 1 2 3 1000 1000 1000 1000 1000 1000 Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if the discount rate is 8%?

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Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or)

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Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) 1 1 PV = PMT 1 - (1 + i) n i (1 + i)

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Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) PV = 1,000 (PVIFA.08, 3 ) (1.08) (use PVIFA table, or) 1 1 PV = PMT 1 - (1 + i) n i 1 PV = 1000 1 - (1.08 ) 3 = $2,783.26.08.08 (1 + i) (1.08)

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Other Cash Flow Patterns 0123

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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 01 234 -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. Uneven Cash Flows 01 234 -10,000 2,000 4,000 6,000 7,000

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Uneven Cash Flows n Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 01 234 -10,000 2,000 4,000 6,000 7,000

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Uneven Cash Flows 01 234 -10,000 2,000 4,000 6,000 7,000 n Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

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Uneven Cash Flows n Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 01 234 -10,000 2,000 4,000 6,000 7,000

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Uneven Cash Flows n Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 01 234 -10,000 2,000 4,000 6,000 7,000

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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 01 234 -10,000 2,000 4,000 6,000 7,000

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Example 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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Example012345678 0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40 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.” 012345678 0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

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How to solve: 1) Discount each cash flow back to time 0 separately. Or, Or, 012345678 0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

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2) Find the PV of the annuity: PV 3: End mode; I/YR = i = 20; PMT = 40,000; N = n = 5 PV 3 = $119,624 012345678 0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

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119,624 012345678

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Then discount this single sum back to time 0. PV: End mode; I/YR = i = 20; N = n = 3; FV = 119,624; Solve: PV = $69,226 119,624 012345678 0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

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119,624 69,226 012345678

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119,624 69,226 n The PV of the cash flow stream is $69,226. 012345678 0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

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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, N = n = 360 PMT = -400 I/Y = i = 1 FV = $1,397,985.65 01 23... 360 400 400 400 400

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table)

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = PMT (1 + i) n - 1 FV = PMT (1 + i) n - 1 i

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = PMT (1 + i) n - 1 FV = PMT (1 + i) n - 1 i FV = 400 (1.01) 360 - 1 = $1,397,985.65 FV = 400 (1.01) 360 - 1 = $1,397,985.65.01.01

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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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01 23... 360 ? ? ? ?

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n Using your calculator, N = n = 360 N = n = 360 I/Y = i = 0.5833 PV = $100,000 PMT = -$665.30 01 23... 360 ? ? ? ? ? ? ? ?

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House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n ) PV = PMT (PVIFA i, n ) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table)

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House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n ) PV = PMT (PVIFA i, n ) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 1 1 PV = PMT 1 - (1 + i) n i

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House Payment Example Mathematical Solution: PV = PMT (PVIFA i, n ) PV = PMT (PVIFA i, n ) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 100,000 = PMT (PVIFA.07, 360 ) (can’t use PVIFA table) 1 1 PV = PMT 1 - (1 + i) n i 1 100,000 = PMT 1 - (1.005833 ) 360 PMT=$665.30.005833.005833

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Multiple Cash Flows –Future Value Example

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