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:

**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 r_{1} and r_{2} respectively and that inflation is i.

At the end of the year the assets have changed to A(1+r_{1}) and B(1+r_{2}) so your portfolio is now worth
A(1+r_{1}) + B(1+r_{2}).

Aah, but inflation reduces the buying power of your portfolio, so in __real__ terms, it's only worth
[A(1+r_{1}) + B(1+r_{2})] / (1+i).

The increase (in real terms) is from (A+B) to [A(1+r_{1}) + B(1+r_{2})] / (1+i) and
(dividing the latter by the former) that's a gain of:

[A/(A+B)](1+r_{1})/ (1+i) + [B/(A+B)](1+r_{2})/ (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+r_{1})/ (1+i) + **y** (1+r_{2})/ (1+i) - 1 =
[**x** + **y** + **x** r_{1} + **y** r_{2}] / (1+i) - 1 =
[1 + **x** r_{1} + **y** r_{2}] / (1+i) - 1 =
(**x r**_{1} + y r_{2} - i) / (1+i).

Note that this is the same as the formula above except that the portfolio return **r** is replaced by **x** r_{1} + **y** r_{2}.

Note that the formula can also be written:
**x (r**_{1} - i) / (1+i) + y (r_{2} - 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: