Timeline 15 typical annuities o most consumer loans

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Timeline:
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15 Typical Annuities O Most consumer loans Car loans Loans for appliances Credit card loans (if you do not add to the negative balance). O Student loans O Mortgages O In all these cases, you borrow a certain amount today (the PV of the annuity), which you then will repay by making equal periodic payments (the cash flows associated with the annuity) for a fixed number of periods.
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16 Example: Present Value of an Annuity (1) O Your younger sister is starting college today. Your grandparents promise to give her $1,000 at the end of each year while she goes to college (maximum four years). How much money do your grandparents have to set aside today to make sure they can pay her as promised, assuming the annual interest rate is 10%?
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17 Example: Present Value of an Annuity (2) 0 1 2 3 4 Periods -----|----------|----------|----------|----------|--------------- PV = ? 1,000 1,000 1,000 1,000 Cash Flows
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18 Example: Present Value of an Annuity (3) Interpretation: O This means your grandparents have to put aside $ today to be able to pay your sister $1,000 at the end of each of the next four years (assuming an effective annual interest rate of 10%). 0 1 2 3 4 Periods -----|--------------|-------------|--------------|-------------|--------- $1,000 $1,000 $1,000 $1,000 Cash Flows
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19 More General: Present Value of an Annuity (1) O Discounting each cash flow individually can get cumbersome for longer annuities (think of a 30-year mortgage with monthly payments). Fortunately, we can simplify things quite a bit. O Consider the following 4-period annuity. We want to know the present value of these cash flows at *. The periodic interest rate is r . * Periods -----|----------|----------|----------|----------|--------------- A A A A Cash Flows
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20 More General: Present Value of an Annuity (2) * Periods -----|----------|----------|----------|----------|--------------- PV * = ? A A A A Cash Flow We can multiply both sides of this equation by (1+r). We can now subtract equation (1) from equation (2) to get:
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21 More General: Present Value of an Annuity (3) * Periods -----|----------|----------|----------|----------|--------------- PV * = ? A A A A Cash Flow ___________________________________________________ Now we have to solve this last equation for PV *
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22 More General: Present Value of an Annuity (4)
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23 Even More General: Present Value of an Annuity O This calculation works not only for n = 4 but for any finite n!
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