IEMS 326 Notes03Bonds

IEMS 326 Notes03Bonds - Bonds and More on Cashflow...

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Bonds and More on Cashflow Sequences IEMS 326 Lecture 3
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Inflation and Annuities 3-year annuity: get C in 1 year, C in 2 years, C in 3 years. What about C in 1 year, C(1+g) in 2 years, C(1+g) 2 in 3 years? Or an “inflating perpetuity” paying in every year n the amount C(1+g) n-1 ?
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Inflation and Annuities We had used the formula x + x 2 + x 3 + … = 1 / ( (1/x) – 1) to find the PV of a perpetuity: The PV of getting $1 in n years is 1/(1+r) n . x = 1/(1+r) so perpetuity PV=1/((1+r)-1)=1/r. The PV of (1+g) n in n years is (1+g) n /(1+r) n . So x = (1+g)/(1+r), PV of inflating perpetuity is 1 / ( (1+r)/(1+g) - 1) = (1+g) / (r-g).
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Inflation and Annuities If the payment in 1 year is C, the PV of the inflating perpetuity is C / (r-g). For an n-year inflating annuity: if we wait n years, the next payment will be C(1+g) n . The value of the inflating perpetuity then will be C(1+g) n / (r-g). Discount back to the present: (C(1+g) n / (r-g)) / (1+r) n . Subtract: (C / (r-g)) (1 – ((1+g)/(1+r)) n ).
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The PV of an annuity of n payments of C at periods 1, …, n is (C/r)(1 – (1+r) -n ). The PV of an “annuity due” i.e. an annuity of n payments of C at periods 0, …, n-1 is C + (C/r)(1 – (1+r) -(n-1) ), or equivalently, (C/r)(1 – (1+r) -n )(1+r) What if the first payment is in k periods? PV =
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IEMS 326 Notes03Bonds - Bonds and More on Cashflow...

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