index.html [back] [next]

Pictures of two quarters

Pictures of the four Vanguard funds, on their 1999q2 and 1997q4 portfolio flats, are shown below. These are not fuzzy pictures. They are rigid, accurate, geometric realizations of 12 quarters of returns.

[1999q2]

1999q2 There are two components of risk on the portfolio flat, productive and incidental. Changes in the vertical, productive component of risk produce corresponding changes in expected reward. Changes in the in the horizontal, incidental component have no effect on expected reward at all. In addition to productive and incidental risk, all portfolios in this 1999q2 picture share 2.68 units of unseen, systemic risk. The fuchsia SYSTM portfolio has nothing but systemic risk. It is the point on the portfolio flat that is closest the money market origin of risk space.

The blue triangle represents the set of all long portfolios in the four funds. The lime curves are conic sections of constant Sharpe ratio, separated by a ratio increment of 0.25. The fuchsia circles are curves of constant scalar risk, separated by 5.0 units of risk.

The (ex post) SOLNG portfolio consists of 30.3% VIGRX and 69.7% VBMFX. SOLNG has a Sharpe ratio of 1.41 compared with 1.27 for VIGRX and 0.06 for VBMFX. This real-data situation has a lot in common with the hypothetical example of the previous section.

The constant Sharpe ratio conic through SOLNG is an ellipse of eccentricity 0.964. This ellipse is not shown. It is tangent to the VBMFX-VIGRX line at SOLNG. All the lime colored conics shown are hyperbolic. They have eccentricities greater than 1.

The SOLSH portfolio has the largest Sharpe ratio, 1.46, of any portfolio in the four funds. (SOLSH stands for "Sharpe-optimal long-short" portfolio.) It consists of 58.4% VIGRX, 64.5% VBMFX, and –22.9% VIVAX. (The VIVAX position is short.)

Geometrically, SOLSH is the point where the Sharpe axis of risk space cuts through the portfolio flat. This point is surrounded, on the portfolio flat, by constant Sharpe ratio ellipses of limiting eccentricity 0.934. The limiting eccentricity is the cosine of the angle between the Sharpe axis and the portfolio flat.


[1997q4]

1997q4 By 1999 bonds and stocks were marching to different drummers. They have been negatively correlated ever since (through 2001). This is a picture from a different era. Bonds and stocks were positively correlated throughout the 1996–1997 period. The picture is a case and point.

There is no reason to include the bond fund in the SOLNG portfolio now. This SOLNG portfolio consists of 56.9% VFINX and 43.1% VIVAX or, equivalently, 28.5% VIGRX and 71.5% VIVAX.

The SOLSH portfolio is off the map now. It has an expected reward of 45.3 compared with SOLNG's 22.1. Still, the Sharpe ratio of the Sharpe-optimal long-short portfolio is not much higher than the Sharpe ratio of SOLNG—2.39 compared with 2.33.


Geometry and reality

Twelve quarters of successive quarterly returns from a given fund make a point in 12-dimensional space. Four such fund points are the vertices of a tetrahedron. Such a tetrahedron is at most a three-dimensional object, no matter the dimension of its universe. And when the tetrahedron takes up almost no space at all in one of its three dimensions, it can be squashed to a planar quadrilateral without significant loss of information. This is what we see above, almost triangular quadrilaterals of funds lying in planar portfolio flats.

These pictures show the reality of the data. In fact the original quarterly return data for the four funds could be reconstructed from accurate drawings of the pictures with a loss of information of less than 0.5%. In addition to the pictures one would need an orthonormal frame of reference consisting of for each picture. (The systemic [scalar] risk and the ratio of reward to productive risk would also be convenient, though these numbers can obtained from the each picture as drawn.)

[home] [back] [next]
Last updated: 27-Mar-02
vic@norton.name