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Qinglei Dai, FEUNL, 2006 Finance I Sept 26

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Qinglei Dai, FEUNL, 2006 Topics Covered Compounding period Stated interest rate Effective interest rate Continuous compounding Simplifications Perpetuity Growing perpetuity Annuity Growing annuity

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Qinglei Dai, FEUNL, 2006 Compounding Periods Compounding may occur more frequently than once in a year Eg. An interest rate of 12% compounded monthly, quarterly, semiannually, and annually

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Qinglei Dai, FEUNL, 2006 AnnuallySemiannuallyQuarterlyMonthly 1 month 3 months 6 months 1 year

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Qinglei Dai, FEUNL, 2006 Compounding an investment m times a year provides end-of-year wealth Stated annual interest rate, Effective annual interest rate Compounding Periods

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Qinglei Dai, FEUNL, 2006 Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

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Qinglei Dai, FEUNL, 2006 Compounding Periods Compounding over T years E.g. Investing $5,000 at a stated annual interest rate of 12% per year, compounded quarterly, for 5 years. At the end of five years the wealth should be ……..

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Qinglei Dai, FEUNL, 2006 Continuous compounding Compounding Periods

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Qinglei Dai, FEUNL, 2006 Simplifications Perpetuity: a constant stream of cash flows without end

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Qinglei Dai, FEUNL, 2006 Simplifications Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

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Qinglei Dai, FEUNL, 2006 Simplifications Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

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Qinglei Dai, FEUNL, 2006 Simplifications Growing perpetuity: a stream of cash flows that grow at a constant rate without end

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Qinglei Dai, FEUNL, 2006 Example: A business brings in €100,000 after costs next year. The cash flow is expected to grow at 5% per year. The appropriate discount rate is 11%. What is the PV of the business? Simplifications

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Qinglei Dai, FEUNL, 2006 Example: A business brings in €100,000 after costs. The cash flow grows at 5% per year. The appropriate discount rate is 11%. What is the PV of the business? Simplifications

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Qinglei Dai, FEUNL, 2006 Three important points: 1. 2. 3. Simplifications

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Qinglei Dai, FEUNL, 2006 Simplifications Annuity:

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Qinglei Dai, FEUNL, 2006 Example: you have won a lottery that will pay you €100,000 per year for 5 years. The interest rate is 8%. What is the present value of your gain? Simplifications

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Qinglei Dai, FEUNL, 2006 typedescriptioncalculation Normal annuity The 1st payment will start one year from now Delayed annuity The 1st payment will start three years from now Advanced annuity The 1st payment starts today Infrequent annuity The 1st payment starts in 2 years. The annuity is payable once in two years.

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Qinglei Dai, FEUNL, 2006 Growing Annuity: Simplifications

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Qinglei Dai, FEUNL, 2006 Example: You have been offered a job at x a year. You expect the income to grow at y% a year until retirement in z years. Given an interest rate of r, what is the PV of your lifetime salary?

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