Geometry of Risk and Reward 3


Let no one unskilled in geometry enter (inscription over the door of Plato's Academy)

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and at once it is something entirely different.

— Johann Wolfgang von Goethe, Maxims and Reflections

Geometry

We are not economists or statisticians. Our bent is for geometry and numerical analysis. Conveniently the theory of portfolio choice, as espoused by Markowitz and Sharpe, has a natural geometric framework.

Risky funds live as points in a Euclidean "risk space" with the benchmark money market fund at its origin. Portfolios of risky funds span a "portfolio flat." The vector from the origin to a point on the portfolio flat is the "risk vector," or just the "risk," of that portfolio, its norm or length is the "scalar risk," and its square length is the variance of the rewards of the portfolio.

We require two axioms of our geometry:

Axiom 1. The portfolio flat does not pass through the origin. (No portfolio of risky funds is as riskless as money market.)
Axiom 2. Expected reward is a linear function of (vector) risk.

Singular value manipulation of data allows these axioms to hold in very general circumstances.

In view of Axiom 2 the expected reward of any portfolio is the inner product of its risk (vector) with some fixed vector in risk space. We call this fixed vector the Sharpe vector. The line through the origin in the direction of the Sharpe vector is the Sharpe axis.

By definition, the Sharpe ratio of a portfolio is the ratio of its expected reward to its scalar risk. It follows that the Sharpe ratio is a scaled cosine of the angle between the fund's risk vector and the Sharpe axis. The scale factor the is length of the Sharpe vector. Portfolios with the same Sharpe ratio lie on a right circular cone about the Sharpe axis and sweep out a connected component of a conic section on the portfolio flat.

The power of geometry

The power of geometry is that you can "see" what is going on. Here is an example in context.

Suppose you are interested in three mutual funds, A, B, and C, and, using the past 12 quarters of returns, you compute the following statistics.

You plan to invest a certain amount of money in these funds, and you want to earn the highest reward possible per unit of risk taken. How should you apportion your money to achieve such a Sharpe-optimal portfolio?

Who knows what the exact proportions of the funds should be, but certain things seem clear. Fund A, having the highest reward and highest Sharpe ratio, should certainly be included in the optimal portfolio. And fund C, which is not even expected to beat money market over the next quarter, is probably not a candidate or, in any case, should take up a very small percentage of the optimal portfolio.

fund	expected reward	scalar risk	Sharpe ratio
A	28.0	16.0	1.75
B	20.0	12.5	1.60
C	0.0	2.5	0.00

A bit of geometry shows how wrong such intuition can be. The picture on the right represents the given data. Expected reward levels are marked on the Sharpe axis. The other numbers are components of the risks of funds B and C.

Risk space is three-dimensional here. M represents the money market origin. We assume that the line BC meets the Sharpe axis at point S, and fund A does not lie on the MBSC plane.

The indicated right circular cone about the Sharpe axis represents all points with Sharpe ratio 1.60. B is on the cone: it has Sharpe ratio exactly 1.60. A is inside the cone: its Sharpe ratio is higher than 1.60. And of course C, with Sharpe ratio 0.00, is outside the cone. (The Sharpe ratio of M, the vertex of the cone, being formally 0/0, is undefined.)

The plane through A,B,C, and S is the portfolio flat in this example. Since S lies right on the Sharpe axis, the cosine of the angle of its risk vector with the Sharpe axis is maximal. Thus S, with Sharpe ratio 2.00, is the portfolio we are looking for. But S = 25% B + 75% C; it has no A and is mostly C. So much for intuition!

This is a hypothetical example, but it is not completely unrealistic. A bond fund C may have high negative correlations with stock funds A and B. (The correlation of C with B is -7.5/12.5 = -0.6 in this example.) Bond funds are typically less risky than stock funds, as in the example, and may not have positive expected reward, as in the example.