AnnuityandPerpetuityFormula - 1 Annuities and Perpetuities:...

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1 Annuities and Perpetuities: The goal of these notes is to derive all perpetuity and annuity formulas. It turns out that we only need to know one calculus formula: T X j =1 x j 1 = 1 x T 1 x (1) The proof of this calculus formula is at the end of this document (for those should be really excited now! A growing annuity has the following cash ±ows: t = 0 t = 1 t = 2 t = 3 ... T 0 1 1 + g (1 + g ) 2 ... (1 + g ) T 1 (2) Then, we can conveniently write this as: (The P sign just indicates summation) PV = T X t =1 (1 + g ) t 1 (1 + r ) t (3) First trick: Multiply and divide by (1 + r ) : PV = T X t =1 (1 + g ) t 1 (1 + r ) t 1 + r 1 + r = 1 1 + r T X t =1 1 + g 1 + r ± t 1 Second trick: De²ne x ± 1+ g 1+ r so that: PV = 1 1 + r T X t =1 x t 1 (4) We can now apply the formula in equation 1, i.e: P T t =1 x t 1 = 1
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AnnuityandPerpetuityFormula - 1 Annuities and Perpetuities:...

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