Misc Stuff

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:

write the annual percentage return as a decimal (by dividing by 100) example: 22% becomes return = 0.22

then "add 1" (so 1+return = 1.22)

then you multiply your initial portfolio by (1+return) example: multiply by 1.22

For negative returns ... same thing:

Write -10% as a decimal (so -10% becomes return = -0.10)
then "add 1" (so it's now 1+return = 0.90)
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, ... r_N,
(expressed as decimals by dividing the percentage returns by 100),
the Annualized Return is:
CAGR = [ (1+r₁)(1+r₂)...(1+r_N) ]^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 ] %

Excel Tips:

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

Simpler Return Calculations:

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

Determine a bunch of returns:

1999	13.26%
2000	-3.34%
2001	-5.49%
2002	-13.09%
to May 31, 2003	9.80%

Multiply together like so:
(1+0.1326)(1-0.0334)(1-0.0549)(1-0.1309)(1+0.0980) = 0.9874

That gives 0.9874 as the increase over 4*12+5 = 53 months or 53/12 = 4.417 years
so, to get the "annualized" return, we need to determine R so that (1+R)^4.417 = 0.9874 so 1+R = 0.9874^1/4.417

That means we need to calculate a root, namely: 0.9874^1/4.417

So here's how you can calculate the n^th root of a (positive) number A, namely A^1/n:

Take a guess and call it X₀. (Example: X₀ = 1.)
Calculate X₁ = [(n-1)X₀ⁿ + A]/[n X₀^n-1]
Calculate X₂ = [(n-1)X₁ⁿ + A]/[n X₁^n-1]
Calculate X₃ = [(n-1)X₂ⁿ + A]/[n X₂^n-1]
Continue until the numbers don't hardly change none

Example 1:
For n = 4 and A = 0.9874 and we want 0.9874^1/4 :

X₀ = 1
X₁ = [3 X₀⁴ + 0.9874]/[4 X₀³] = 0.996850
X₂ = [3 X₁⁴ + 0.9874]/[4 X₁³] = 0.996835
X₃ = [3 X₂⁴ + 0.9874]/[4 X₂³] = 0.996835
We conclude that 0.9874^1/4 = 0.996835

Example 2:
For n = 5 and A = 0.9874 and we want 0.9874^1/5 :

X₀ = 1
X₁ = [4 X₀⁵ + 0.9874]/[5 X₀⁴] = 0.997480
X₂ = [4 X₁⁵ + 0.9874]/[5 X₁⁴] = 0.997467
X₃ = [4 X₂⁵ + 0.9874]/[5 X₂⁴] = 0.997467
We conclude that 0.9874^1/5 = 0.997467

Example 3:
For n = 4.417 and A = 0.9874 and we want 0.9874^1/4.417 :

Do Example 1 and get 0.996835 for n = 4.
Do Example 2 and get 0.997467 for n = 5.
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 = 0.9874^1/4.417 = 0.9971
so R = -0.0029 or -0.29% annualized return.

About Return Distributions:

Suppose you have a bunch of returns (or any other numbers!)
in cells A2 to A20 and bins from B2 to B13 ... as seen here

Select cells from D2 to D14 with your mouse
... one more cell than the number of bins !
Type: =FREQUENCY(A2:A20,B2:B13)
Press three keys: Ctrl+Shift+Enter (as noted above).
Cells D2 to D14 will give the number of returns which lie in the various bins:
R<= -50%, -50% < R <= -40%, -40% < R <= -30%, ... and finally R> 60%
(You may want to put appropriate labels on a plot as in column C
so the chart has these labels on the horizontal axis.)

About Arithmetic and Geometric Means:

Q: The annual returns of my portfolio over two years are P and Q (P = 0.12 means 12%).
What will my portfolio be in two years?

A: After one year, $A changes to A(1+P). After two years it's A(1+P)(1+Q) = A G₁G₂
where G₁ = 1+P and G₁ = 1+Q are the annual Gain Factors.

Q: The amount A G₁G₂ is equivalent to achieving what "constant" annual return?

A: At a "constant" annual return of R, $A becomes (after two years) A(1+R)².
And A(1+R)² equals A G₁G₂ provided (1+R)² = G₁G₂ or 1+R = SQRT[G₁G₂]
and SQRT[G₁G₂] is the Geometric Mean of the Gain Factors G₁ and G₂.
That means our annual Gain Factor would be GM, the Geometric Mean of G₁ and G₂.

Q: What would I have if I got the Arithmetic Mean?

A: The Arithmetic Mean of P and Q is (P+Q)/2 and two years of this return would yield:
A[1+(P+Q)/2]² where we note that [1+(P+Q)/2] = (1/2)[(1+P) + (1+Q)] = (1/2)[G₁+G₂].
Hence A[1+(P+Q)/2]² = A[(1/2)(G₁+G₂)]².
That means our annual Gain Factor would be AM, the Arithmetic Mean of G₁ and G₂.

Q: How do the two compare?

A: Using R gives $A(1+R)² = $A[GM]² where 1+R = GM is the Geometric Mean of Gain Factors G₁ and G₂.
Using the Average gives $A[1+(P+Q)/2]² = $A[AM]² where AM is the Arithmetic Mean of G₁ and G₂.

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):

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

To see the difference, click here for an online spreadsheet.

Money In - Money Out:

If you put money into your portfolio, and take some out, then in and out etc. etc. then we're NOT talking about a buy-and-hold return. That makes the calculation of Annualized return more complicated, but there's an MS Excel command for this:
The XIRR command. It goes like this:

=XIRR(A1:A50,B1:B50,0.10)
where the range A1:A50 contains the cash flows
and the range B1:B50 contains the dates
and 0.10 is an initial guess (like 0.10 meaning 10%)
Here's an example
where you stick in the cash flows (negative if they're withdrawals) and the dates and end up with the current portfolio value (entered as a negative number, like -$44.100) and today's date. It's important to note that the (positive or negative) cash flows are new money invested (from your salary, for example) or withdrawals made (to your bank account, for example) ... and not internal activity within the portolio such as dividends or expenses due to commissions or fees etc.. This internal activity is ignored when calculating portfolio returns. (Their effect, of course, shows up in the final portfolio value!)
It's also important to note when an investment (or withdrawal) is made!
For example:

You invest $1K now.
In N months you invest another $1K.
In 12 months your portfolio is worth $2500.
You annualized return (or Compound Annual Growth Rate CAGR) depends upon N
... going from 25% to 50%.
The 25% number is (final portfolio)/(initial investment) - 1 = 2500/2000-1 = 0.25 or 25%.
The 50% number is (final portfolio)/(initial investment) - 1 = 1500/1000-1 = 0.50 or 50%
(where the $1K investment at N = 12 months is ignored).

Note 1: XIRR is an Excel add-in and may need to be installed.
Note 2: There may be more than one XIRR answer, depending upon your initial guess! (See XIRR.)
Note 3: You can use XIRR to calculate YTD Returns. (See XIRR-stuff.)

About inflation adjustments:

Q: $100 will buy how many $25 items?

A: It will buy [100]/25 = 4 items.

Q: If inflation is 3% per year, how many items will $100 buy next year?

A: Next year the $25 item will cost 25(1+.03) so $100 will buy 100/{25(1+.03)} = [100/(1+.03)] /25 items.

Q: If inflation is 3% per year, how many items will $100 buy in eight years?

A: In eight years the $25 item will cost 25(1+.03)⁸ and
$100 will buy 100/{25(1+.03)⁸} = [100/(1+.03)⁸]/25 items.

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

REAL Rate of Return:

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%).

Q: How many items, currently priced at $P, can you buy in N years?

A: In N years your $A portfolio will have grown to A(1+r)^N.
However, the $P items will then cost P(1+i)^N.
Hence you can buy [A(1+r)^N]/[P(1+i)^N] = A/P [(1+r)/(1+i)]^N items.
That means that, instead of A/P items, you can buy A/P [(1+r)/(1+i)]^N items (N years from now).
That means that the buying power of your portfolio has increased (or decreased) by a factor [(1+r)/(1+i)]^N over N years.
That's equivalent to an increase (or decrease) by a factor [(1+r)/(1+i)] per year.

Q: A change in buying power of [(1+r)/(1+i)] corresponds to what "REAL" portfolio growth (taking inflation into account)?

A: We need to find an equivalent, inflation-adjusted return R so that 1+ R = [(1+r)/(1+i)], hence the REAL rate of return is

R = [(1+r)/(1+i)] - 1.

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:

(0.6*r₁ + 0.3*r₂+ 0.1*r₃ - i) / (1+i)
or
0.6 (r₁ - i) / (1+i) + 0.3 (r₂ - i) / (1+i) + 0.1 (r₃ - i) / (1+i)

About Annual Rebalancing:

Q: My portfolio is 50% asset A, 30% asset B and 20% asset C (meaning fractions 0.50, 0.30 and 0.20).
The annual returns for each are P, Q and R (for a 12% return, P=0.12).
What is my portfolio return at the end of the year?

A: For a portfolio worth $1000
50%x1000 = $500 is in asset A which changes to 500*(1+P) = 0.50*1000*(1+P) after one year.
30%x1000 = $300 is in asset B which changes to 300*(1+Q) = 0.30*1000*(1+Q) after one year.
20%x1000 = $200 is in asset C which changes to 200*(1+R) = 0.20*1000*(1+R) after one year.

The final portfolio is then 0.50*1000*(1+P)+0.30*1000*(1+Q)+0.20*1000*(1+R)
which is the same as: 1000(1 + 0.50*P + 0.30*Q + 0.20*R)
which is the same as having an annual return of 0.50*P + 0.30*Q + 0.20*R.

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".

About Saving for Retirement:

Suppose that, were you to retire now, you'd need $50K per year from your investment portfolio.

Suppose, too, that you intend to retire in 30 years.

At 3% inflation, that $50K needs to be 50K(1+.03)³⁰ = $121K (to have the same buying power as $50K, today).

Suppose you think that, at retirement, you can safely withdraw 4% of your starting portfolio (this withdrawal increasing with inflation).

Then, the required $121K should be 4% of what Portfolio?

You'd want 0.04 x Portfolio = 121K so that Portfolio = 121K / 0.04

1/0.04 = 25 so you'd want 25 x 121K or 3025K or about a 3 million dollar Portfolio.

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.
Warning: the answers may be harmful to your health.

Years of Investing = (until retirement)
Salary Increases = % (that's inflation)
Return on Investments = % (while you're investing)
Percentage needed = %
(as a percentage of your current salary)
Withdrawal Rate = % at retirement

Percentage of Salary that you'll need to invest = %