Low (and high) volatility strategy effects

Does minimum variance act differently from low volatility?  Do either of them act like low beta?  What about high volatility versus high beta?

Inspiration

Falkenblog had a post investigating differences in results when using different strategies for low volatility investing.  Here we look not at a single portfolio of a given strategy over time, but a thousand.

Ingredients

The data are daily closing prices of almost all of the stocks in the S&P 500.  The final date in the data is 2012 February 24.

The in-sample period was 2006.  That is the period in which the variance matrix — and hence the volatilities — were estimated.  The betas relative to the S&P 500 were also estimated using 2006 data.

Figure 1: The volatility versus beta estimates from 2006. Figure 1 demonstrates that the belief that beta is about volatilty really is a myth.  Note that the stock with the largest volatility (RHT) has a beta of 0.97.

The portfolios were formed as of the start of 2007.

Constraints

All the portfolios were built with these three constraints:

1. long-only
2. 45 to 50 names in the portfolio
3. all assets have (starting) weights of no more than 4%

Six sets of random portfolios (1000 in each set) were generated.  The additional constraints were:

• vanilla: (no additional constraints)

• low variance: the portfolio variance is no more than 120% of the minimum attainable variance (as estimated from 2006) given the first three constraints

• low volatility: only stocks from the lowest quartile of volatilities (as estimated from 2006)

• low beta: only stocks from the lowest quartile of betas (as estimated from 2006)

• high volatility: only stocks from the highest quartile of volatilities (as estimated from 2006)

• high beta: only stocks from the highest quartile of betas (as estimated from 2006)

Results over time

Figures 2 through 7 show the valuations over time of the random portfolios (in blue).  These are buy-and-hold portfolios — there is no trading.

The black line shows the median valuation at each point in time.  The gold lines delimit the central 95% of valuations at each timepoint.  None of these three lines follows a particular portfolio.

Figure 2: The “vanilla” portfolio values.

Figure 3: The “low variance” portfolio values.

Figure 4: The “low volatility” portfolio values.

Figure 5: The “low beta” portfolio values.

Figure 6: The “high volatility” portfolio values.

Figure 7: The “high beta” portfolio values.

Caveat

These results have significant survival bias.

This is not selecting S&P 500 stocks in 2007.

This is selecting stocks in 2007 that will be in the S&P 500 in 2012.

Summary

Not forgetting the caveat above, what I see is:

• low volatility and low beta look quite similar
• low variance performed better during the period than either low volatility or low beta
• there are subtle — and perhaps superficial — differences between high volatility and high beta
• the vanilla (and high vol and beta) portfolios have an extraordinary spread in valuation by mid-2008; the spread then seriously collapses (Why?)
• In contrast 2011 August looks more like a fault line in geological strata

Appendix R

estimate betas

Estimating the betas is one line of R code:

sp5.beta06 <- coef(lm(sp5.ret[1:250,] ~ spxret[rownames(sp5.ret)[1:250]]))[2,]

generate random portfolios

require(PortfolioProbe)

Select the subset of stock ids that are desired:

bp03.betaq1 <- names(sp5.beta06)[sp5.beta06 < quantile(sp5.beta06, .25)]

Now use that in specifying the portfolios to create:

bpo3.rp.lowbeta <- random.portfolio(1000, prices=sp5.close[251,], gross=1e6, long.only=TRUE, port.size=c(45, 50), max.weight=.04, universe=bp03.betaq1)

valuation

It is just one command to get the matrix of valuations over time for each set of random portfolios:

bp03.rpval.lowbeta <- valuation(bp03.rp.lowbeta, sp5.close[-1:-250,], collapse=TRUE)

plot

The function used to do the plotting (x is the valuation matrix) is:

> pp.valplot
function (x, scale=1e6, ...)
{
        x <- x / scale
        pp.timeplot(x, ylab="Value", div="years",
               col="steelblue", ...)
        med <- apply(x, 1, median)
        ci <- apply(x, 1, function(y)
                quantile(y, c(.025, .975)))
        lines(med, col="black", lwd=2)
        matlines(t(ci), col="gold", lwd=2, lty=1)
        axis(4)
}

This uses the pp.timeplot function.

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