Here's the problem.
We want to split our investments between various asset classes, say Large Cap and Small Cap and ...
>Or Growth or Value or Equity or Income or ...
Pay attention.
We choose an allocation between various assets so that our Risk
is minimized.
>Your Risk is in green ... why?
So as to distinguish it from Risk = Volatility or
Standard Deviation (SD).
(See Standard Deviation and Risk)
>You say a Frontier. I always thought a frontier was ... like, in the Wild Wild West ...
The socalled Efficient Frontier of Modern Portfolio Theory deals with minimizing
Risk = Standard Deviation (or Volatility).
(See Modern Portfolio Theory.)
Anyway, if we define Risk to mean the
Probability of a Loss,
then we stare at the Cumulative Probability Distribution for a particular asset and determine
the probability that the Return is negative. For example, suppose the Cumulative Distribution ...
>Remind me. What's this Cumulative stuff?
Check out Distributions.
Here's a picture
In this case, there's a 20% probability of achieving a negative return.
>For a particular asset, right?
Yes. In this example I've actually assumed a Normal distribution which is defined by the Mean
and Standard Deviation (or Volatility) of the asset.
 Fig. 1

Anyway, here's the problem:
 Assume a distribution for each asset class. (Example: Normal, Lognormal or whatever)
 Pick fractions of your portfolio which are to be devoted to each asset.
(Example: three assets with fractions x, y and z where x+y+z=1.)
 Assuming annual (or monthly or daily or whatever) rebalancing (to maintain the ratios
x:y:z), determine the Cumulative Distribution for the mix of assets
(hence the probability of a loss, or the Risk, as per Fig. 1).
 Vary the fractions x, y and z so as to minimize this Risk.
>I assume that x=0.3 means 30% of your portfolio is devoted to the first asset.
Yes. Anyway, it means that we must determine the Cumulative Distribution for the mix of assets
... which, of course, will depend upon the distributions for each
... and the fractions x, y and z.
>If I knew Risk for each, why wouldn't
I just put everything into the asset class with the least Risk? Say x = 1.0?
You mean Risk.
Actually, you may be able to reduce the Risk if one component
is negatively correlated. But be patient. We'll see.
Fig. 2

>I assume the Mean Return and Standard Deviation for each asset
will affect their Risk?
You mean Risk. Yes. For a Normal Distribution, Fig. 2 is
a chart showing how the Risk depends upon the Mean and SD.
For example, with Mean=10% and SD=15% then Risk=25%.
>You mean Risk.

Instead of generating an elaborate mathematical analysis ...
>Good!
... we'll use a spreadsheet where we can enter parameters for a couple of assets ...
>Like Mean, Standard Deviation?
Yes. Pay attention. Here's a spreadsheet where we withdraw a certain percentage of a
portfolio each year, splitting our assets between, say, Stocks and Bonds.
The spreadsheet (notyetfinished!) will look like this:
If y'all want to play with the spreadsheet (such as it is), then RIGHTclick on the
picture, above, and Save Target file ...
If you play with this spreadsheet, you can generate enuff data
to justify the infamous 60/40 stock/bond split
... still in progress (I think) ...
