Portfolio Returns: Average, Annualized and their friends
Next: Predicting the future

You sit back, stare intently at your portfolio and its gains (losses?) over the past few years and ask yourself:

  1. What's my annual return?
  2. What might I expect for next year?
  3. If I had invested that extra $1000 a month earlier, how different would my return be?
  4. What if I withdraw some money then add it back, later ... how would that affect my return?
  5. What if ... ?
>So?
So we have to talk about how one calculates portfolio returns ... and there are several different ways to do that, each meaning something different.

For example:

  • You have a $1000 portfolio and your returns over the past 5 years are 2%, 5%, 6.95%, -3% and 8% and you neither add money nor withdraw money.
  • Then your portfolio will be 1000(1.02)(1.05)(1.0695)(0.97)(1.08) = $1200.
  • That's a 5-year gain factor of 1200 / 1000 = 1.20 so every $1.00 would have grown to $1.20.
  • That's a 5-year return of 0.20 or 20%.
  • That's like putting your $1000 in a bank at an annual interest of r and getting 1000 (1 + r)5 = 1200 after 5 years.
  • That'd make the bank interest equal to: r = (1200 / 1000)1/5 - 1 = 0.0371 or 3.71% per year.
That 3.7% would be your Annualized Return (or Compound Annual Growth Rate):
If R1, R2, R3 ... Rn are the last n annual returns and you neither add nor subtract from your portfolio, then
Annualized Return = CAGR = {(1+R1)(1+R2)... (1+Rn)}1/n - 1
is your annualized return (or CAGR) over n-years.

However, if you calculate the Average Return (for the example above) you'd get:
Average (or Mean) Return = (2 + 5 + 6.95 - 3 + 8)/5 = 3.79. That is, an average annual return of 3.79%.
If R1, R2, R3 ... Rn are the last n annual returns, then
Average (or Mean) Annual Return = (1/n) {R1 + R2 + R3 + ... + Rn}

So, if you'd like to guess at what next year's return might be, that Mean (historical) return is a good bet ... but don't count on it!

However, if you'd like to guess at what your gain factor might be over the next 5 years, you would NOT want to use: (1+Mean)5.
For the example, that'd turn $1000 into 1000(1.0379)5 = $1204, not $1200.
In fact, you'd use the CAGR: 1000(1 + CAGR)5. (See Poor Joe)

>And that'd tell me what to expect over the next 5 years?
No, it's just a guess ... don't count on it!
To know for sure, you'd need this.

Investing in a stock or mutual fund whose price varies dramatically from month to month (that's volatility, eh?) will reduce your CAGR.
In fact, a good approximation is:
CAGR = (Mean Return) - (1/2) (Volatility)2

Now, if you add money or withdraw money from your portfolio ... aah, that's a horse of a diff'runt hue.
Indeed, if you add or withdraw periodically (like once a month), there's IRR.
If you add or withdraw at random times, there's XIRR ... sometimes called ROI.
('course, ya gott watch our for XIRR bugs)
Both require sexy calculations which require a computer. There ain't no easy formula.

Further, some people calculate a return as:
[A]         newPrice / oldPrice - 1
Example: $53/$50 - 1 = 0.06 or 6%

Some prefer:
[B]         log(newPrice / oldPrice)
Example: log($53/$50) = 0.058 or 5.8%     (Remember to use natural logarithms, to the base e!)

>Use logs? Why?
If, over 4 years, the prices go from P1 to P2 to P3 to P4 to P5, the average return is either:
[A]         (1/4){ (P2/P1-1) + (P3/P2-1) + (P4/P3-1) + (P5/P4-1) }
or
[B]         (1/4){ log(P2/P1) + log(P3/P2) + log(P4/P3) + log(P5/P4) } = (1/4) log(P5/P1) = log[ (P5/P1)1/5] = log[1+CAGR].
Note that log(x) + log(y) = log(x*y). Clearly [B] will get math-types feeling warm all over.

Then there's Year To Date returns (YTD) and "real" Returns
(taking into consideration inflation, 'cause a 5% return in a year with 3% inflation is a "real" return of about 2%)

Then there's Time-weighted Returns and Dollar-weighted Returns and ...

>zzz
... and there's even Nepers !! (See Poor Joe)

>ZZZZ

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