Morningstar
So there I was, browsing various discussion forums and finding that I didn't understand certain arguments on the difference between
investing some fixed annual amount in an asset that provides >Creeping senility?
where >I assume you can prove that?
Then we can write:
Anyway, if you were to invest in an asset with a constant annual return, say R, then the 1-year, 2-year, 3-year etc.
gains would be (1+R), (1+R) ^{2}, (1+R)^{3} ... so the number of units you'd hold (after N years) would be (using [1]):
>So when is U2 greater than U1? That's the question, right?
We can get an exact formula for that sum in U2, however, to get a quick-and-dirty answer, we'll assume that the U2-sum goes on forever
Let's look at:
>And the portfolio values?
>How about a magic formula?
- The final value of Portfolio #1 is [final price][number of units] = [P
_{0}G_{N}]* [(A/P_{0})(1 +**gMS**] = AG_{N}(1 +**gMS**) - Similarly, for Portfolio #2 we'd get [p
_{0}(1+R)^{N}]*[(A/p_{0})(1 + 1/R)] = A(1+R)^{N}(1 + 1/R) - Portfolio #2 would be larger provided:
(1+R)^{N}(1+1/R) > G_{N}(1 +**gMS**) or (1+R) > G_{N}^{1/N}{(1+**gMS**)/(1+1/R)}^{1/N} Note that G_{N}^{1/N}is the annualized Gain Factor - For N large (that's our "long haul") {(1+
**gMS**)/(1+1/R)}^{1/N}would be close to "1". (For example, if (1 +**gMS**)/(1 + 1/R) = (1+10)/(1+1/.05) = 0.52 and N = 40 then (0.52)^(1/40) = 0.98) - That'd give the following result:
**A constant return greater than the annualized return of a volatile asset will provide a larger (eventual) portfolio**
For a one-time, initial investment, your final portfolio will depend only upon the final asset price
... and remember, the annualized return does, indeed, ignore all intermediate prices. Aah, but for a sequence of annual investments, you'd expect your final portofolio to depend upon these intermediate prices.
What is suggested here is that (approximately, for the long haul) the comparison depends only upon that annualized return.
Nothing, it's just interesting, don't you think? >No, and besides ... where's the spreadsheet that'll give Figure 3? Here.
withdrawing, not investing?
Mmm ... good question. Now A is the annual withdrawal amount.
If we assume you begin with $ A (and _{0}A units) and withdraw $A each year (starting at the _{0}/ P_{0}end of the first year), then
(for Portfolio #1) the asset prices at the end of each year would be P _{0}G_{1}, P_{0}G_{2}, etc. etc.
and the units sold each year would be A/(P _{0}/G_{1}), A/(P_{0}/G_{2}), , etc. etc.
... that's Dollars/UnitPrice
and your number of units held (after N years) would be A _{0}/P_{0} - (A/P_{0})(1/G_{1} + 1/G_{2} + ... + 1/G_{N}) = A_{0}/P_{0} - (A/P_{0})gMS
so (multiplying by the final asset price of P _{0}G_{N}) your final Portfolio #1 would be:
where we've set
For Portfolio #2 (the "constant return" portfolio) we get the corresponding result by putting
That gives a final Portfolio #2:
>That's with "long haul" approximations, right?
Okay, now the burning question:
Portfolio #2 (with constant return >So we plot the left side >The dot gives ... what?
I might point out that the values of |