Trading Options

How to you know that you are earning good returns?

Earning 28% per year does sound good, compared to about 15.7% for the S&P 500 index and perhaps under 5% for a savings account. But then there is also the issue of noise in the growth of the portfolio. The value of a savings account just keeps growing and growing; there is no noise at all. Aside from a small chance of a bank’s bankruptcy and an FDIC intervention, it does not normally undergo any dramatic events. What about my portfolios and the S&P 500 index?

Investing in a single stock, or a broad-based index such as S&P 500, or a certain trading strategy or a fund manager brings about volatility in the path taken by the value of the account from the beginning of the year to its end. For instance, the highest daily return of S&P 500 in 2006 was 2.14% and the lowest return was -1.84%. So, a relevant question might be: what daily drop in the value of the portfolio would you be able to tolerate without responding to pain and going into less risky assets, such as bonds or even a savings account (which is essentially equivalent to investing in short-term money market instruments)?

The highest and lowest daily returns, for my entire portfolio in 2006, were approximately +4% and -4%, respectively. For the stock and option portfolio, they were roughly +5% and -5%, respectively. Both portfolios were “more noisy” than the S&P 500 index.

Designing an annual performance measure

In previous postings I have discussed a daily performance measure, an adaptation of the Sharpe Ratio concept, defined as the ratio of the average daily return to a standard deviation of the daily return. This measure does reward higher returns, as well as punishing noisier trading strategies, so it does reflect the risk-aversion of an educated speculator. However, most people are unaccustomed to thinking of average daily returns - people commonly think of annual returns instead. Looking at the sample standard deviation of a daily return is comparatively more useful. One would have to have a long - decades long - series of data to compute the standard deviation of annual returns. Such data do not exist for any fund manager. Using daily returns, and computing a sample standard deviation is hence the best substitute.

To create an easier understood performance measure, I will be looking to use an annual return.. As for a measure of noise, I will use the standard deviation of a daily return, scaled by the square root of the number of trading days in a year, assumed to be 250. The “Annualized Sharpe Ratio” is hence defined as (Annual Return - Risk-Free Rate) / (Standard Deviation of Daily Return * sqrt(Business Days Per Year)).

The above definition uses the textbook concept of a risk-free interest rate. It was not needed when the Sharpe Ratio was being computed on a daily scale. However, it does matter on an annual scale. If one wanted to leverage a position in the index, by borrowing cash and investing it in more than a 100%-weighted position in the index, one would be paying the borrowing cost and this is how an interest rate comes in. A realistic borrowing cost would likely be higher than the figures I assume, but this depends on the borrower. I assume this rate to be 5% for year 2006 and 3% for the entire trading history. These figures are chosen arbitrarily, but are probably in the right ballpark for typical short-term government yields, averaged over 2006 and over the last 4 years.

The conclusion from the above table is that the stock-and-option portfolio has outperformed the S&P 500 index in 2006, as judged by the Annualized Sharpe Ratio being 1.43 vs. 1.06. Less significantly (as futures trading is not meaningful and should not be considered in a fair comparison), the full portfolio has also outperformed the index: 1.16 vs. 1.06.

It would be very interesting to compare the Sharpe Ratios for many different portfolio managers and rank them accordingly. This would be a suitable way to rank my performance. However, the requisite information - namely, any sort of measure of the noise sustained by the portfolio as it attains a publicly disclosed annual return - is typically unavailable. So this comparison, while potentially very informative, cannot be done.

Annualized Sharpe Ratio for the entire trading history

The performance figures for my portfolio have improved over time. So, the figures given above for 2006 are better than figures computed for the entire trading history. To compute the latter, I define the
“Annualized Sharpe Ratio” as ((Business Days Per Year) * (Average Daily Return) - Risk-Free Rate) / (Standard Deviation of Daily Return * sqrt(Business Days Per Year)).

Compared to the definition of the Annualized Sharpe Ratio for 2006, this definition uses the number of business days in a year (assumed to be 250), multiplied by the average daily return, whereas the definition for year 2006 simply used the actual annual return figure for 2006. The results are:

Not very surprisingly, the results show that, considering the entire trading history, the full portfolio and also the stock-and-option part of the portfolio have both underperformed the S&P 500 index. This is just the other side of the coin. Although the performance improved in 2006, indeed, resulting in me outperforming the index, the prior years 2005 and 2004 were significantly noisier.