ah right ... cumulative discount factors are a short cut when discounting annuity cash flows ...

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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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ah right ... cumulative discount factors are a short cut when discounting annuity cash flows ...

Common shares are not an annuity. You are discounting uncertain future cash flows. Just investing based on filings isn't enuf

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www.wikihow.com/Calculate-Annuity-Payments

https://www.accountingcoach.com/present-value-of-a-single-amount/explanation/3

www.investopedia.com/walkthrough/corporate./3/./present-value-discounting.aspx

https://www.accountingcoach.com/present-value-of-an-ordinary-annuity/./3

www.investopedia.com/walkthrough/corporate./discounted-cash-flow/annuities.aspx

www.investopedia.com/articles/03/101503.asp