3 more general present value of an annuity 1 m

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More General: Present Value of an Annuity (1) M 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. M 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 -----|----------|----------|----------|----------|--------------- PV = ? A A A A Cash Flows 4
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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: ) 2 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 3 2 1 * + + + + + + = + r A r A r A A PV r 5 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 4 3 2 1 * + + + + + + + = r A r A r A r A PV
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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 * (1 + r ) PV * = A + A (1 + r ) ! 1 + A (1 + r ) ! 2 + A (1 + r ) ! 3 (2) 6 PV * = A (1 + r ) ! 1 + A (1 + r ) ! 2 + A (1 + r ) ! 3 + A (1 + r ) ! 4 " # $ % (1) 4 * * ) 1 ( ) 1 ( + = + => r A A PV PV r
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More General: Present Value of an Annuity (4) ! PV * + rPV * " PV * = A " A (1 + r ) " 4 7 4 * * ) 1 ( ) 1 ( + = + r A A PV PV r ( ) 4 * ) 1 ( 1 + = r A rPV ( ) r r A PV 4 * ) 1 ( 1 + =
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Even More General: Present Value of an Annuity M This calculation works not only for n = 4 but for any finite n! M So to wrap it up – and that s the only thing you really need to know: <DRUMROLLS> M For an interest rate ‘r’, the present value of an annuity with n level cash flows ‘A’ that start at the end of the first period is equal to: ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + = n n n r r A r r A r r A PV 1 1 1 1 1 1 ) 1 ( 1 8
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