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Unformatted text preview: x = 1+ g 1+ r to get: PV = 1 1 + r 1 & & 1+ g 1+ r ± T 1 & 1+ g 1+ r = 1 & & 1+ g 1+ r ± T (1 + r ) h 1 & 1+ g 1+ r i = 1 & & 1+ g 1+ r ± T 1 + r & (1 + g ) = 1 & & 1+ g 1+ r ± T r & g (9) That±s it!! From here, we can easily derive all formulas: T & period Annuity (no problem, set g = 0) 1 & ( 1 1+ r ) T r T & period growing annuinty (we solved for this!) 1 & ( 1+ g 1+ r ) T r & g Growing Perpetuity (let T ! 1 ), works only if g < r , why?? 1 r & g Regular Annuity 1 r Proof. T X j =1 x j & 1 = T & 1 X j =0 x j = ² 1 + x + x 2 + x T & 1 ³ Multiply both sides by (1 & x ) and you obtain: (1 & x ) T X j =1 x j & 1 = (1 & x ) ² 1 + x + x 2 + x T & 1 ³ = ² 1 + x + x 2 + x T & 1 ³ & x ² 1 + x + x 2 + x T & 1 ³ = ² 1 + x + x 2 + x T & 1 ³ & ² x + x 2 + ::: + x T ³ = 1 & x T Hence: (1 & x ) T X j =1 x j & 1 = 1 & x T or T X j =1 x j & 1 = 1 & x T 1 & x 2...
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 Fall '09
 ODEAN
 Time Value Of Money, Annuity, Perpetuity, Expression, Xt

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