I have searched far and wide for a good definition of portfolio historical relative volatility. I started by looking at the standard mathematical definition of volatility and proceeded to research and analyze Beta, modified Sharpe, Sortino and Treynor ratios and the daily historical volatility that they measure for various mutual funds. After playing around with these and several other definitions from Modern and Post Modern Portfolio Theories I came to the realization that none of these measures, no matter how sophisticated and computationally intensive, gave me what I was really after. Having a background in applied mathematics, I went back to the drawing board and came up with my own (very simple to compute using a spreadsheet) measure that did not force me to make any market return assumptions and provided precisely what I was after. I call it simply Daily Portfolio Relative Volatility (DPRV). The definition could just as easily be applied to any other regular time period: minute, hour, week, month, quarter and etc. However, daily is just long enough to be practical and remove irrelevant randomness, while being an accurate psychological representation of portfolio volatility.

Of course, in order to compute volatility you must first define it. In my definition:
1) The least volatile way (trajectory) to get from the initial portfolio (or fund, or index) value to its final value is via a straight line. (This is a simple interest rate CD model and strictly speaking it is not correct, however it is a very good approximation to use for short time intervals with small changes between initial and final values. To get a more accurate measurement of volatility, we would have to use the compound interest rate CD model and fit to an exponential.)
2) Any deviation from the trajectory, no matter if it is negative or positive equally ads to the volatility. (While this definition can be argued, most everybody seems to agree with it.)
3) The most appropriate method for calculating the distance of a set of points from a trajectory is by computing the square root of the sum of the squares of the distances of those point from the trajectory and dividing the result by the number of data points. (Absolute value, another even power, or some other definition of distance could be used here just as well; however, I chose to stick to the most commonly used Euclidean distance.)

Using the above definitions, to calculate absolute volatility of any portfolio (or fund, or index) for a given period of time one must first compute a straight line between the initial and the final values of the portfolio and then take a square root of the sum of the squares of relative distances between such a line and the actual portfolio value on each of the intervening days between the initial and the final. This result is normalized by the number of intervening days in the calculation to produce absolute portfolio volatility.

Relative volatility can be measured with respect to any other portfolio (or fund, or index) by first measuring the absolute volatility of these in the same manner and on the same dates and then dividing the portfolio's in question volatility by its benchmark's volatility.

Of course, the greater the number of days used in this volatility calculation, the more reliable are the results. (Please note, however, that for longer periods of time, an exponential fit should be used instead of a straight line.)

For a CD that earns simple interest on a daily basis, this definition produces the expected result of 0 absolute volatility. Therefore, there would be no point in computing relative volatility of a portfolio with respect to a CD, as any other investment would be infinitely more volatile!

I tried applying this easy new volatility measure to my own portfolio's performance with the starting date of 12/11/07 (the date I started keeping track of its value on a daily basis) and ending today, 1/29/08. So far, using this measure, my stock portfolio appears to have 93.3% volatility of the S&P 500. (Using the more accurate exponential fit model would peg this relative volatility at 92.8%.) Equally important, my portfolio is down only 4% vs. S&P 500’s 7.8% drop over the same time period.