reSampled Frontier ... and portfolio allocation
motivated by e-mail from Levi F.

an Old Frontier

Once upon a time (1952) Markowitz introduced Modern Portfolio Theory and the so-called Efficient Frontier.
(He shared a Nobel prize for this work.)

>Efficient frontier? Huh?
That Efficient Frontier stuff goes like this:

Our portfolio contains N assets, with Mean Returns r1, r2, ...
We allocate fractions of our portolio to these N assets and we want to select these fractions, namely x1, x2, ... so that:

  1. x1 + x2 + ... + xN = 1
    The fractions add to "1" since 100% of our portfolio is devoted to these N components
  2. The Volatility (or Standard Deviation) of your portfolio, SD is prescribed
    That is, pick your volatility tolerance (according to your genetic makeup)!  
    The Volatility is sometimes called "Risk" (for no logical reason) ... so you're picking your Risk level.
    This Volatility (or Standard Deviation) will depend upon the returns of the N assets as well as your allocations x1, x2 etc.
  3. Determine the allocations so that your Average (or "expected") Return, namely R = x1r1 + x2r2 + ... + xNrN,   is a maximum.

>And how do you do all that?
You pick a set of fractions devoted to the N assets ...
>You pick x1, x2 etc.?
Yes, then you see what R-value that gives.
Then you pick another set of x's ...
>Which add up to 1, right?
Yes. That's #1, above. For example, you might have x1=0.60 and x2=0.25 and x3=0.15 meaning (for three assets) 60% + 25% + 15%.
Anyway, you see what R you get. Then you pick another ...
>Again and again until you get the largest R? That's a lot of work, eh?
A computer can do all the work, but I don't want to talk about the Efficient Frontier.
Instead, I want to ...
>Wait! What's that frontier look like?
The typical picture assocated with the Frontier is shown here
along with a selected Standard Deviation and the maximum R-value.

You can see the relationship to the Sharpe Ratio.
>You have something against Sharpe?
Who me? Actually, the big problem is: We look at the historical data for our assets, extract the Means and Variances and Covariances (and any other stuff we may need), determine our allocations and assume that all this will conveniently stay fixed for the future.

>Then we can relax, eh?
Yes ... and that ain't good!

Note that you could also prescribe the Return you'd like (that's R) and vary the allocations until you get the minimum SD.
(That'd be the minimum volatility and, if you believe that volatility = "Risk", then it's the minimum risk portfolio which provides that prescribed return.)

Or, you could just find the point which gives the minimum ... uh, "Risk".
See the chart (using a sample 3-asset portfolio)  
(Not surprisingly, it's got lots of bonds.)

One problem with this stuff (besides the fact that we rely heavily upon historical data) is that the Frontier moves about when we make small changes to the data.
>And the "optimal" allocations?
Yes, they move about too.

>So what do you intend to do?
Talk about a sampled Frontier ... or, better, a reSampled Frontier  

>Beg pardon?
It's an idea promoted by Michaud (1989) and Jorian (1992).

a New Frontier
Suppose we have a bunch of historical returns for our assets. Then we can calculate things like their Mean, Variance (which is Volatility2) and Covariance. That'd give us a set of fixed numbers we can use to construct our Efficient Frontier. But these numbers are actually estimates of future Means and Variances ... and future is not that fixed, eh?

Here's a way to introduce some randomness into future values:
  1. Take random samples of the set of historical returns.
    For example, if we have a set of returns, say 10.2%, 8.5%, 6.7% and 9.4% we could select a random sample of these four returns as 8.5%, 8.5%, 9.4% and 10.2% where, after making our selection, we put that return back into the set ... else the selection of "random" returns would be the same as the original set!
  2. Use this random sample to calculate an Efficient Frontier.
  3. Repeat steps 1 and 2 a jillion times ... each time generating a Frontier.
  4. Stare in awe (and trepidation) at the variation in those jillion Frontiers

>And the variation in associated allocations, eh?

Taking a set of historical returns and sampling from that set (again and again) is a neat way to get randomness from a fixed set of returns.
You never have to worry about assuming some kind of distribution (normal? lognormal?) or "fat tails".
I used it myself, in sensible withdrawals

To see the effect of changing the returns (for four assets with prescribed Means and Volatilities),
you can try this neat spreadsheet by Michael Kishinevsky.
  1. Click here.
  2. Select four (monthly) Means and Standard Deviations and an (annualized) Risk-free Rate.
  3. Press F9 to get a bunch of random returns for each asset (selected from four normal distributions with the prescribed Means and SDs).
  4. Look affectionately at the Efficient Frontier.
  5. Repeat step 3 (and 4).
We noted above that one can prescribe the Return and get the minimum Volatility (for that Return)
... hence an Efficient Frontier as in Figure 1A.

However, we can also plot the Volatility against the return, like Figure 1B (which is easier to interpret )
Let's consider a 3-asset Portfolio:

  • The Portfolio has a certain allocation of resources. (Example: 50% + 30% + 20% )
  • That gives a particular Mean Return and Volatility.
  • We modify the allocations and reduce the Volatility to a minimum
    while maintaining the Return. (Example: 35% + 37% +28% gives Min. Volatility)
    (It's got the same return with a smaller volatility.)
  • Clearly we prefer the blue portfolio to our original red portfolio.
  • We repeat this ritual for a bunch of Returns (each time minimizing the Volatility).
    This gives the Efficient Frontier curve.

>Did you say that one can prescribe the Return? Can I ask for 100%? Can I ask for ...?
Within reason ... say between the minimum and maximum asset returns.

Figure 1A

Figure 1B

For just three assets you can try this spreadsheet:

Click here to try it out, or
RIGHT-click on the picture and Save Target to download a copy.

>How do I get a feel for what the appropriate allocations might be?
  Or how that allocation might change as the returns are reSampled?
  Or if I start with different assets and different returns ... or?

Yes, well ... you might want to play the FRONTIER GAME  

>A game? How useful is that?
On a scale from 1 to 10, it's a 3, alongside Solitaire
... but it does illustrate the effect of reSampling the same set of returns and how the "optimal" allocation can change quite dramatically.

Some References:   in PDF format   in PDF format
a book   by Michaud on Efficient Asset Management
Stuff from Invesco   in PDF format