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  • Discounted Cash Flow Valuation?
    The first part of your question a) is the annuity part where in you have a starting balance of say P and you withdraw X at the beginning of each one year period during which you earn 1.09 or R on the balance hence after the first year you have a balance of: F(1) = ( P - X ) * R After the second year you have: F(2) = ( F(1) - X ) * R .: F(2) = ( ( P - X ) * R - X ) * R F(2) = ( P * R - X * R - X ) * R F(2) = P * R^2 - X * ( R^2 + R ) After the third year you have: F(3) = ( F(2) - X ) * R .: F(3) = P * R^3 - X * ( R^3 + R^2 + R ) The general form is: F(n) = P * R^n - X * summation of the term R^k for k from 1 to n therefore applying the summation of a geometric sequence equation we have: F(n) = P * R^n - X * ( ( 1 - R^(n+1) ) / ( 1 - R ) - 1 ) At the end of 20 years, the balance would be 0 so we have: P * R^20 = X * ( ( 1 - R^21 ) / ( 1 - R ) - 1 ) solving for P we have: P = €25,000 * ( ( 1 - 1.09^21 ) / ( 1 - 1.09 ) - 1 ) / 1.09^20 P = €248,752.87 So you must have €248,752.87 saved up by your 60th birthday in order to provide the desired €25,000 per year income for 20 years. The second part of your question deals with compound saving up to P on your 60th birthday and hence we're interested in the balance on the day of the birthday so the balance on the first birthday (25th birthday) is: F(1) = X On the second it's: F(2) = F(1) * R + X .: F(2) = X * R + X F(2) = X * ( R + 1 ) On the third it's F(3) = F(2) * R + X .: F(3) = X * ( R^2 + R + 1 ) The general form is: F(n) = X * summation of the term R^k for k from 0 to n-1 Applying the summation of a geometric sequence equation, you have: F(n) = X * ( 1 - R^n ) / ( 1 - R ) We know that F(35) = P which we calculated in the first part of the problem so solving for X we have: X = P * ( 1 - R ) / ( 1 - R^35 ) X = €248,752.87 * ( 1 - 1.09 ) / ( 1 - 1.09^35 ) X = €1,153.18 So each deposit will have to be at least €1,153.18 to meet your goals.
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    discounting annuity cash flows