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Class 4 Jan 16th Completed

# The government promises to buy all the output of the

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The government promises to buy all the output of the machine in four years for \$400,000. – 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
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
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% C) Bank C offers an effective quarterly rate of r Q = 3%
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The government promises to buy all the output of the...

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