I am reading Henk Tijms' "Understanding probability" and have become a bit confused. A man deposits $100,000 in a pension fund for 20 years with an average growth of 14%. Assuming a fixed rate of growth of 14%, the book tells me that the equation below, solved for x, will give me the amount each year so that the initial investment capital, which must remain in the fund for 20 years, remains undisturbed (1+r)[itex]^{20}A - [/itex][itex]\sum[/itex][itex]^{19}_{k=0}[/itex](1+r)[itex]^{k}[/itex]x =0 This gives $15098. Doesn't this amount breach the condition of keeping the $100,000 undisturbed after the first year? Aren't we down to $98902 after the first withrawl of $15098? Secondly, could someone please explain to me how the equation works? It seems to be the undisturbed total in the fund after 20 years of investment at 14% minus the sum of all the growth multiples for 1 years investment, 2 years investment .... 20 years investment multilplied by x (the amount he can "safely" withdraw). I can't really get my head round it. In case I have misinterpreted the book here is the page I am referring to: http://books.google.co.uk/books?id=...ntleman would like to place $100,000"&f=false
I think I see the problem. The equation is based on equating the expected gain from leaving A invested compound, withdrawing nothing, to the sum expected to be achieved by investing x each year at 14% for 20 years (the investment being made at the end of each year, not the beginning). What the author overlooked is that it is the gain on A that should be used, A[(1+r)^{20}-1], not A(1+r)^{20}. If you plug that in you get x = 14000.
I realise that the last link I posted might have been a bit broken, this might work better: http://books.google.co.uk/books?id=Ua-_5Ga4QF8C&pg=PA151&lpg=PA151&dq="a+retired+gentleman+would+like+to+place+$100,000"&source=bl&ots=jLf6YRt9TS&sig=B4i7k4PePVuGoKF4_RbaS3U6eXc&hl=en&sa=X&ei=7JqrUKz1CMKt0QWrEQ&ved=0CB0Q6AEwAA#v=onepage&q="a retired gentleman would like to place %24100%2C000"&f=false It's page 151, paragraph 2 of the book I presumed that leaving the $100,000 "undisturbed" meant that the amount should at no point in the 20 years fall below $100,000 but then this could be the root of my misunderstanding.
That's the root of your misunderstanding. The idea is that there must be enough to make that fixed annual payment throughout that 20 year span. If the 20th payment brings the balance down to exactly zero, that's OK. In fact, that is what this $15098 payment does. (Even better, $15098.60; this leaves the balance after 20 annual payments at one penny.) What happens if, for example, the interest rate is 8.3% for the first ten years, 19.7% for the last ten? That's still an average rate of 14%. However, a fixed annual payment of $15098 would draw the balance down to almost zero in ten years. You'd get a paltry $127.40 in the 11th year, nothing thereafter.
Thanks for that, i now understand the basic premise but I still do not know how you would arrive at this equation. Where does it come from? Why does it work?
Hi DH, I agree that the model you describe leads to the equation in the text, but it is clearly inconsistent with the wording, accurately quoted in the OP: "the initial investment capital, which must remain in the fund for 20 years, will not be disturbed". It is an error in the book. However, I doubt it matters for what follows there. Appleton, how you arrive at the equation is going to depend on which consistent pair of model and equation we choose. Based on the words, the equation is very obvious: just take 14% of $100,000 out each year. So presumably DH's model is the one intended: the capital should be almost zero after 20 years. This is exactly like a house mortgage, with the investor as the bank. Here's one way to get it: Suppose at the end of each year, the investor withdraws x and invests it somewhere else at the same interest rate. At the end of 20 years, the first x withdrawn will have been reinvested for 19 years, the second for 18, and so on. The final x (which emptied the original investment) was invested for no years. The total in the new investment is now x(1+r)^{19}+x(1+r)^{18}+..+x. Summing the geometric series gives the RHS of the equation. But this must be equivalent to having left the entire original investment untouched, no withdrawals, for 20 years. That gives the LHS.