January 5, 2010
The horizontal axis shows the percentile of the outcome and the vertical axis show each possible outcome. The 10th percentile return for the S&P500 (SPY) is a return of -19%; there is a projected 10% chance that the S&P500 will lose 19% or more over the next twelve months. The projected 1% tail for the S&P500 is -44%, and the worst one-year return for the S&P500 in the period from 1926-2005 reported by D’Antonio is -43.4%. The Monte Carlo results account for the fact that the holder of the collar receives the dividends from the position in SPY.
The lower percentiles for the collar strategy are flat – we cannot have a one-year return worse than -10% because this is where the put options kick in. The high percentiles are similarly truncated due to the call that we sold in creating the collar. We get all the gains on the S&P500 up to 20%, but no more. Between where the put and call kick in, we will track the S&P500. If the S&P500 ends up anywhere between -10% and +20% for the year, our returns with the collar strategy will be indistinguishable from the S&P500 – and that is easily seen on the chart above. The impacts of extreme events (good and bad) have been removed from the portfolio.
The following graph shows the returns for a portfolio that is 50% invested in SPY and 50% invested in AGG, overlaid on the previous chart.
The collar portfolio and the 50% SPY/50% AGG portfolios have the same standard deviation of returns, but the other components of risk in these two strategies are quite different. The standard deviation measures of the average size of swings in a portfolio, but tells you nothing about the shape of the distribution of portfolio returns. If returns are normally distributed (which D’Antonio verified for the one-year returns on the S&P500), the returns for the collar strategy are normally distributed with the extreme tails cut off – the distribution is more square-shaped. In statistical terms, the kurtosis is much lower for the collar. (For statistically-schooled readers, you will recognize that the standard deviation is one moment of the distribution. The collar changes the other moments – especially the kurtosis, which is a measure of the ‘fatness’ of the tails of the distribution.)
Once you layer options into the portfolio, simply looking at the average return and the standard deviation of returns no longer tells the full story – you have to look at the tails. If things get really bad in the equity markets, you can lose considerably more with the 50/50 SPY/AGG portfolio than with the collar. With the 50/50 portfolio, the projected 1% tail is -22% – more than twice the maximum loss with the collar strategy, even though both strategies have almost identical standard deviations.
Why would an investor prefer the 50/50 portfolio vs. the collar portfolio? Given all of the uncertainties in estimating risk and return, let’s assume the expected returns of both of the strategies are the same. The collar is more sensitive to movements in the S&P500 than the 50/50 portfolio, as long as the S&P500 stays within -10% and +20%. If the S&P500 returns -8%, the collar will also return -8%, but the 50/50 portfolio will tend to do better because the 50% in bonds will temper the loss in equities. The collar is preferable if the market moves down dramatically (the collar wins at the 10th percentile), but the 50/50 portfolio, by keeping the upside in equities, outperforms at the 90th percentile and above.
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