Miscellaneous Stuff
suggested by discussions at morningstar.com and investorhub.com

If your portfolio goes up 22% in a year, then \$1.00 grows to \$1.22 in that year ... since \$1.00*(1.22) = \$1.22
If, the next year, the return is 73%, you multiply your starting portfolio by 1.73
Your \$1.22 portfolio grows to \$2.11 since \$1.22*(1.73) = \$2.11
In general, you:

1. write the annual percentage return as a decimal (by dividing by 100)   example: 22% becomes return = 0.22
2. then "add 1" (so 1+return = 1.22)
3. then you multiply your initial portfolio by (1+return)   example: multiply by 1.22

For negative returns ... same thing:

1. Write -10% as a decimal (so -10% becomes return = -0.10)
2. then "add 1" (so it's now 1+return = 0.90)
3. then you multiply your initial portfolio by 1+return = 0.90
Hence, for annual returns of 22%, 73%, 21%, 45% and -10% a \$15.00 initial portfolio would grow to
\$15.00*(1.22)(1.73)(1.21)(1.45)(0.90) = \$49.99
That's a gain of \$34.99, since you started with \$15.00, so that's a total gain of \$34.99/\$15.00 = 2.33 or 233%

What's the Equivalent "Annualized" Return
... or Compound Annual Growth Rate (CAGR) ?

That five year sequence of annual returns is equivalent to what "fixed" annual return, over five years?
Suppose this "fixed" return is r (so, if r turns out to be 0.123, that means 12.3%)
Then an initial portfolio of \$15, over these five years, would grow to
\$15*(1+r)(1+r)(1+r)(1+r)(1+r) = \$15*(1+r)5
which is supposed to be equivalent to
\$15.00*(1.22)(1.73)(1.21)(1.45)(0.90)
That makes (1+r)5 = (1.22)(1.73)(1.21)(1.45)(0.90)
or r = [(1.22)(1.73)(1.21)(1.45)(0.90)]1/5 - 1 = 0.272 or 27.2% per year.
and that gives the prescription for finding the CAGR from a sequence of annual returns, namely:
 For N annual returns of r1, r2, ... rN, (expressed as decimals by dividing the percentage returns by 100), the Annualized Return is: CAGR = [ (1+r1)(1+r2)...(1+rN) ]1/N - 1
 Note: In an MS Excel spreadsheet, if cells A1 to A25 contain 25 returns, then:      =AVERAGE(A1:A25) gives the Average (or Arithmetic Mean)) of these Returns. To get the Annualized Return do this: Type:      =GEOMEAN(1+A1:A25)-1 but don't hit the Enter key ... yet!! Hold down the Ctrl and Shift keys and NOW hit the Enter key. It changes to {=GEOMEAN(1+A1:A25)-1} giving the Annualized Return.

Example: for the five annual returns given earlier, we'd get

Note: The Annualized Return is not to be confused with the Average Return \$#^@!&#
The Average and Annualized returns would be the same only if all annual returns were the same!
Example: for five returns all equal to R, the average return would equal R. The above prescription would then say:
CAGR = [ (1+R)(1+R)(1+R)(1+R)(1+R) ]1/5 - 1 = 1+R - 1 = R

Note:

 The CAGR (or Annualized Return or GeoMean) is the answer to the question: What constant annual return would make my portfolio go from \$A to \$B in n years?
The answer can be obtained from the individual annual returns (using the GeoMean)
or (more simply) as   100 [ (B/A)1/n - 1 ] %

Above, we noted the slick trick of using =GEOMEAN(1+A1:A25)-1 and Ctrl / Shift / Enter to get the Annualized Return (or geometric mean).
This ritual involves adding 1 to every cell. There are some other similar tricks, like the following (each using Ctrl+Shift+Enter).
In each case there's some arithmetic applied before the calculation:

=PRODUCT(1+A1:A25,1+A1:A25) where "1" is added before the PRODUCT is calculated.
=PRODUCT(1+2*A1:A25,1+2*A1:A25) where each number is multiplied by "2" then added to "1" before the PRODUCT is calculated.
=SUM(A1:A25/100) where each number is divided by 100 before SUMming.
=SUMPRODUCT(A1:A25-A26,A1:A25-A26)   where the number in cell A26 is subtracted from every number in A1 to A25 before calculating the SUMPRODUCT.
=SUMPRODUCT(A1:A25-AVERAGE(A1:A25),A1:A25-AVERAGE(A1:A25))   where the AVERAGE is subtracted from each number.
This would give the sum of squares of the deviations from the Mean.
=SUMPRODUCT(A1:A25-A26,A1:A25-A26,N(A1:A25<A26))   where A26 is subtracted from each number and the SUMPRODUCT is calculated for those numbers less than A26.
If A26 is the Mean, then this would give the sum of squares of the deviations from the Mean for those numbers less than the Mean.
=AVERAGE(IF(B1:B25>B14,A1:A25))   where A26 is some date and column B has dates associated with each number in A1 to A25.
The AVERAGE of A1 to A25 is calculated for those numbers with dates less than A26.
=SUM(A1:A25*B1:B25)   where each number in A1:A25 is first multiplied by the corresponding number in B1:B25 ... then the SUM of products is calculated.
This is the same as =SUMPRODUCT(A1:A25,B1:B25)
There are (probably!) other neat tricks ... but I don't know 'em

If you don't have access to Excel (hence the GEOMEAN function) or to a calculator that'll extract roots, you can try this:

So here's how you can calculate the nth root of a (positive) number A, namely A1/n:
1. Take a guess and call it X0. (Example: X0 = 1.)
2. Calculate X1 = [(n-1)X0n + A]/[n X0n-1]
3. Calculate X2 = [(n-1)X1n + A]/[n X1n-1]
4. Calculate X3 = [(n-1)X2n + A]/[n X2n-1]
5. Continue until the numbers don't hardly change none
Example 1:
For n = 4 and A = 0.9874 and we want 0.98741/4 :
1. X0 = 1
2. X1 = [3 X04 + 0.9874]/[4 X03] = 0.996850
3. X2 = [3 X14 + 0.9874]/[4 X13] = 0.996835
4. X3 = [3 X24 + 0.9874]/[4 X23] = 0.996835
5. We conclude that 0.98741/4 = 0.996835
Example 2:
For n = 5 and A = 0.9874 and we want 0.98741/5 :
1. X0 = 1
2. X1 = [4 X05 + 0.9874]/[5 X04] = 0.997480
3. X2 = [4 X15 + 0.9874]/[5 X14] = 0.997467
4. X3 = [4 X25 + 0.9874]/[5 X24] = 0.997467
5. We conclude that 0.98741/5 = 0.997467
Example 3:
For n = 4.417 and A = 0.9874 and we want 0.98741/4.417 :
1. Do Example 1 and get 0.996835 for n = 4.
2. Do Example 2 and get 0.997467 for n = 5.
3. For n = 4.417 calculate the approximate answer by interpolating between 4 and 5:
0.996835 + (4.417 - 4)(0.997467 - 0.996835)
and you'll get 0.997099 whereas the "correct" answer is 0.997133

That makes 1+R = 0.98741/4.417 = 0.9971
so R = -0.0029 or -0.29% annualized return.

Conclusion?
The Geometric Mean of Gain Factors gives the correct, final portfolio value
... not the Arithmetic Mean
... and the Geometric Mean is smaller than the Arithmetic Mean!
See
Average vs Annualized.

Note:
There are (at least) two approximations to the Annualized Return if you only know the Mean (or Average) and Standard Deviation (or Volatility):

1. AnnualizedReturn = AverageReturn - (1/2) Volatility2
2. AnnualizedReturn = SQRT[ (1+AverageReturn)2 - Volatility2 ] - 1

Conclusion?
At an annual inflation rate of I (I = 0.03 means 3%), the "buying power" of today's \$A is [A/(1+I)N] in N years.
Just divide by (1+Inflation) ... for each year

Your portfolio is currently worth \$A.
A particular item is currently worth \$P.
Currently, you can buy A/P items.
In N years you may or may not be able to buy that many items; it'll depend upon your portfolio gain and inflation:

Suppose that your \$A investment grows at a rate r (r = 0.08 meaning 8%) and inflation is i (i = 0.03 meaning 3%).

Conclusion?
At an annual portfolio gain of r the "REAL", inflation-adjusted return is R = [(1+r)/(1+i)] - 1 = (r - i)/(1 + i).

Note: For small inflation (that is, i is small), one often puts R = r - i which is an approximation to the "REAL" rate of return.

REAL Return ... with several assets:

Now suppose that you have a portfolio with two assets worth \$A and \$B, so your portfolio is worth A + B.
Suppose, further, that the two assets have annual returns of r1 and r2 respectively and that inflation is i.
At the end of the year the assets have changed to A(1+r1) and B(1+r2) so your portfolio is now worth A(1+r1) + B(1+r2).

Aah, but inflation reduces the buying power of your portfolio, so in real terms, it's only worth [A(1+r1) + B(1+r2)] / (1+i).
The increase (in real terms) is from (A+B) to [A(1+r1) + B(1+r2)] / (1+i) and (dividing the latter by the former) that's a gain of:

[A/(A+B)](1+r1)/ (1+i) + [B/(A+B)](1+r2)/ (1+i).

Moral?
If your portfolio has a fraction x devoted to asset #1 and y = 1 - x devoted to asset #2, then your "real" return is:

x (1+r1)/ (1+i) + y (1+r2)/ (1+i) - 1 = [x + y + x r1 + y r2] / (1+i) - 1 = [1 + x r1 + y r2] / (1+i) - 1 = (x r1 + y r2 - i) / (1+i).

Note that this is the same as the formula above except that the portfolio return r is replaced by x r1 + y r2.
Note that the formula can also be written: x (r1 - i) / (1+i) + y (r2 - i) / (1+i)
so it's the weighted sum of the inflation-adjusted return for each component.

This ritual extends to a portfolio with a jillion assets ... just use the "weighted" return
For example, if 60% is devoted asset #1, 30% to asset #2 and 10% to asset #3, then you'd have a real return of:

Conclusion?
For Portfolio fractions x1, x2, x3 ...   devoted to assets with returns of R1, R2, R3 ...
the Portfolio return is x1R1 + x2R2 + x3R3 + ...
For Annual Rebalancing:
This is the way to get the annual return (each year).

Note:
If R1, R2, R3 ... are the Average Annual Returns for each asset (over umpteen years)
then x1R1 + x2R2 + x3R3 + ... will give the Average Annual Portfolio Return (or Arithmetic Mean Return)
... provided you do "Annual Rebalancing".

This gives the Rule of Thumb:
Save for retirement so that your Portfolio is 25x the income you'd want from your investments, at retirement.

You can (of course) stick in your own numbers or use a calculator ...

If you think that, because your expenses would be less,
you can live on, say, 75% of your salary (after you retire),
and your current salary increases at some annual rate,
you can try this (quick-and-dirty) calculator
to see how much of your salary you should invest ... until you retire.