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Among other things, this site provides a home for our "The Geometry of Risk and Reward" paper. The paper is a 599 KB PDF document, optimized for printing. It can be read with Adobe's Acrobat Reader. It is the support for everything in this exposition.
The paper is rather technical. We suggest that you browse through this web site to get the idea of what we are about—before downloading the paper.
The symbol SOLNG stands for "Sharpe-optimal long portfolio." We will be dealing with two distinct SOLNG portfolios.
The ex ante SOLNG portfolio is a dynamic portfolio drawn from a limited universe of mutual funds. The graphic on the title page shows the cummulative performance, since the beginning of 1996, of the ex ante SOLNG portfolio corresponding to four Vanguard index funds.
On the other hand, the ex post SOLNG portfolio is a mathematical construct. Start with a certain number of risky funds and three years of quarterly returns. With these data one can compute the ex post Sharpe ratio of any formal, long portfolio in the risky funds. The ex post SOLNG portfolio (for the period under consideration) is the formal, long portfolio with maximal Sharpe ratio. Generally this portfolio is uniquely determined by the data.
(Think of a formal portfolio as a list of percentages of each fund in the portfolio. The quarterly returns of the portfolio are sums of the same proportions of the quarterly returns of the individual funds. These returns do not correspond, in any obvious way, to the returns of an actual portfolio of funds.)
The relationship between the ex ante and ex post SOLNG portfolios is quite simple. The proportions of the individual funds in the ex ante portfolio at the beginning of any quarter are precisely the proportions of the funds in the ex post portfolio corresponding to the previous 12 quarters. If quarterly return data are available very shortly after the end of each quarter, the ex ante SOLNG portfolio can be realized as an effective investment strategy. Indeed this strategy could have been quite profitable over the past six-plus years—as our title page graphic indicates.