M timeline 1 0 1 2 3 4 periods 1000 1000 1000 1000

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M Timeline: -1 0 1 2 3 4 Periods --|---------|----------|----------|----------|----------|--------------- 1,000 1,000 1,000 1,000 Cash Flows PV = ? M Correct answer is d) One way to show this is to solve for PV -1 first 85 . 486 , 3 $ ) 1 . 1 ( * 87 . 169 , 3 $ 87 . 169 , 3 $ 1 . 0 ) 1 . 1 ( 1 * 000 , 1 $ ) 1 ( * ) 1 ( 1 * 0 1 4 1 1 0 1 = = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = PV PV PV r PV PV r r A PV n 12
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iClicker Example: PV of an Annuity Due (3) M Timeline: -1 0 1 2 3 4 Periods --|---------|----------|----------|----------|----------|--------------- 1,000 1,000 1,000 1,000 Cash Flows PV = ? M Alternatively: 85 . 486 , 3 $ 85 . 486 , 2 $ 000 , 1 $ 1 . 0 ) 1 . 1 ( 1 * 000 , 1 $ 000 , 1 $ 1 . 1 000 , 1 $ 1 . 1 000 , 1 $ 1 . 1 000 , 1 $ 000 , 1 $ 0 3 0 3 2 0 = + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = + + + = PV PV PV 13
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Future Value of an Annuity M Consider an n-period annuity, where payments of A are made at the end of each period. The periodic effective interest rate is r. Then the future value of the annuity is given by: FV n = PV 0 (1 + r ) n ! FV n = A 1 " 1 (1 + r ) n r # $ % % % % & ' ( ( ( ( (1 + r ) n ! FV n = A (1 + r ) n " (1 + r ) n (1 + r ) n # $ % & ' ( r # $ % % % % & ' ( ( ( ( ! FV n = A (1 + r ) n " 1 r # $ % & ' ( = A r 1 + r ( ) n " 1 ( ) 14
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Future Value of an Annuity: Car Buying Example (Cont d) M How much money would you have at the end of four years if you invested the $500 per month instead? Assume that you earn 1% interest per month on your invested money. We are looking for FV 48 . Alternatively: FV 48 = PV*(1+r) 48 = $18,986.98*(1.01) 48 = $30,611.30 ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = r r A FV n n 1 1 15 30 . 611 , 30 $ 01 . 0 1 ) 01 . 1 ( * 500 48 48 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = FV
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iClicker Example: FV of an Annuity Due (1) M Recall the example from before, but assume your grandparents give your sister $1,000 at the beginning of each year in college (starting now) for four years. What is the value of this annuity exactly four years from now? As before, the interest rate is 10% per year.
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