Earning 65% (50% ex-futures) in 4 years - Statistical Analysis
In this post I will analyze some statistical properties of the series of returns on my portfolio.
While the return earned by holding a share of stock is often assumed to have a normal (Gaussian) distribution, a stock option is a non-linear asset. Even under the simplest Black-Scholes framework, the return earned by holding a single option has a very complicated distribution. Then, of course, the distribution of returns of an actively traded portfolio is yet another completely different subject.
In what follows I present statistical properties of the series of daily returns for
- My entire portfolio
- My portfolio, without the contribution of trades in futures.
- The S&P 500 index, for which each raw daily return will be incremented by 1.79%/250, the assumed contribution from a 1.79% per annum dividend yield, divided over approximately 250 trading days per year.
A return over a time interval with the starting value S[i] and an ending value S[i+1] is defined as log(S[i+1]/S[i]), where “log” is the natural logarithm. So, these are series of “continuously compounded” returns. The convenience of this definition, as compared to using simple returns (S[i+1]-S[i]) / S[i] is that the former is additive when applied to consecutive time periods. For example, the (continuously compounded) annual return for a given year is simply 250 times the average daily return shown in the table for this year.
For simplicity, I focus on the average and the standard deviation of a daily return. The Sharpe ratio is defined as the ratio of average to standard deviation. Therefore, for a risk-averse speculator, a “better” portfolio will have a higher Sharpe ratio.
| Full | Ex-Futures | S&P 500 | |
| Average | 0.214% | 0.065% | 0.087% |
| St.Dev. | 0.77% | 1.22% | 1.08% |
| Sharpe Ratio | 0.278 | 0.054 | 0.080 |
| Full | Ex-Futures | S&P 500 | |
| Average | 0.039% | -0.020% | 0.041% |
| St.Dev. | 1.55% | 1.19% | 0.70% |
| Sharpe Ratio | 0.025 | -0.017 | 0.059 |
| Full | Ex-Futures | S&P 500 | |
| Average | -0.057% | 0.049% | 0.019% |
| St.Dev. | 1.66% | 1.24% | 0.65% |
| Sharpe Ratio | -0.034 | 0.039 | 0.029 |
| Full | Ex-Futures | S&P 500 | |
| Average | 0.107% | 0.066% | 0.058% |
| St.Dev. | 1.33% | 0.82% | 0.64% |
| Sharpe Ratio | 0.080 | 0.080 | 0.090 |
| Full | Ex-Futures | S&P 500 | |
| Average | 0.059% | 0.040% | 0.051% |
| St.Dev. | 1.42% | 1.14% | 0.79% |
| Sharpe Ratio | 0.041 | 0.035 | 0.065 |
Why are the averages in the above tables so much smaller than standard deviations? Consider the return for S&P 500 over a single trading day. The index will either go up, or it will go down. For the sake of example, it might move up or down by 1% - in which case the standard deviation will be exactly 1%. However, the average of signed returns would be much smaller. If equally-sized up- and down-moves were equally likely, the average would in fact be zero, despite the fact that the series will have a significant standard deviation.
The other way of looking at the relative magnitudes of average and standard deviation of a daily return is by realizing that, at least for S&P 500, one could annualize them, by multiplying the average by 250 (the number of trading days in a year), and the standard deviation by about 16 (approximately the square root of 250). Taking the numbers from the last table, covering the entire period from the day the trading began in 2002 to November 24, 2006, the annualized average return for S&P 500 is 0.051%*250=12.75%, whereas the annualized standard deviation is 0.79%*16=12.64%. So, the mean grew relative to the standard deviation. As the time horizon gets longer, the likely total returns on the index are dominated by the mean (which scales linearly with time), while the standard deviation, scaling as the square root of time, increases in absolute terms but declines relative to the mean total return. This behavior is nothing but the central limit theorem of statistics.
The scaling argument just applied to explain the order of magnitude of average and standard deviation of daily returns of S&P 500 cannot be applied to returns on my portfolio, because it is actively traded. However, one can still compare them to the respective numbers for the index. If I am able to attain a higher mean return, this is good. If the same could be attained while keeping the standard deviation lower than that of the index, this would be still better. This is a high hurdle to overcome. I believe that the change over time of numbers in the above tables shows an improvement. For example, in the latest period, year 2006 to-date (to November 24), the mean return on the portfolio was higher than that of the index and while the “noise” (the standard deviation) of the portfolio was also higher, its Sharpe ratio at 0.08 was only a little lower than the value of 0.09 for the index.