The machine is worth 0 at this point what is the rate

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The machine is worth 0 at this point. What is the rate of return of this investment? Timeline or I PV = $355,000 I FV = $400,000 I n = 4 I r = ? 8 FV n = PV 0 ! (1 + r ) n 400 = 355 ! (1 + r ) 4 400 355 " # $ % & ' 1 4 = 1 + r r = 400 355 " # $ % & ' 1 4 ( 1 = 3.0286% Or in general: r = FV n PV 0 ! " # $ % & 1 n ' 1 = FV n PV 0 n ' 1
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Example: Solving for the Number of Periods n I You just met the girl of your dreams. I You currently have $10,000 in your Engagement Ring Fund, which earns 12% per year. I The engagement ring you want costs $20,000. I How long do you have to wait before you can pop the question? What info do we have? I PV = $10,000 I FV = $20,000 I n = ? I r = 12% FV n = PV 0 ! (1 + r ) n => 20,000 = 10,000*(1 + 0.12) n => 2 = (1.12) n " (1.12) n = 2 => ln(1.12) n = ln(2) => n *ln(1.12) = ln(2) => n = ln(2) ln(1.12) => n = 6.1163 Or in general: n = ln FV n PV 0 ( ) ln(1 + r ) 9
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Aside: The Rule of 72 again I According to the this rule, it would have taken you years to double your money at an annual rate of interest of 12%. 6 12 72 72 = = r 10
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Summary n n r PV FV ) 1 ( * + = n n n n r FV r FV PV + = + = ) 1 ( * ) 1 ( 11 ) 1 ln( ln r PV FV n + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = n PV FV r
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Compounding Frequencies: APRs vs. EARs 12
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13 Compounding Frequencies I So far we assumed (implicitly) that compounding occurred once per period. I What happens if interest is compounded more frequently than once a year? I Example: (Note: this is not how banks state returns) A) Bank A offers an effective annual interest rate of r Yr = 12% B) Bank B offers an effective semi-annual rate of r semi = 6%
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