Deceived By Correlations: A Quant Conundrum [View article]
@LeftBanker: The risk of using correlation has nothing to do with the assumption that the time-series are GBM -- that was just for convenience to take independent normal random variables and translate them into asset prices.
This risk is if the random-variables have meaningfully different means from 0. On a daily measurement level, most assets have near-zero mean. On longer horizons, however, their means can differ in a meaningful manner.
Consider monthly returns for SPY and UNG from 1/30/2009 to 1/30/2012. SPY has a positive monthly mean log-return and UNG has a negative monthly mean log-return; a naive correlation estimate is at -2.6%; an adjusted correlation estimates puts the two at -15.5%. While some quick math tells us that the latter calculation is only significant at an 80% level, this fact often get's swept under the rug. Toss these two numbers into a mean-variance optimization and we can end up with dramatically different results.
The goal here was to simply demonstrate that the naive use of correlation can lead to dramatically different results than what we expect; by assuming mean-returns, at any timeframe, are zero can help bring the number back in line with our expectations. Not a parlor trick -- just math.
Deceived By Correlations: A Quant Conundrum [View article]
@Business Economics Analyst: Indeed, correlation prediction is very difficult. However, this is a problem we see time and time again: people assume that negatively correlated assets must have divergent trends and that positively correlated assets move in the same direction. By the definition of correlation, this need not be so. Given that many people utilize correlation as the basis for their diversification decisions, it is important to highlight how the strict statistical measure may deviate from their intuition -- and a quick and dirty method for correcting for it: namely, assume "means" are zero.
Deceived By Correlations: A Quant Conundrum [View article]
@change is the only constant: It is true that the independence of random variables necessarily implies zero correlation between those variables, though zero correlation does not necessarily imply independence of random variables.
What we are trying to point out here is that the "trend" component is ignored because the variation occurs around the mean. Therefore, any measures of joint-deviation are going to be about the "noise" and not the underlying trends of the return distributions.
By assuming a zero mean, our "trend" component gets measured as noise -- and therefore positive (or negative) correlations emerge.
Deceived By Correlations: A Quant Conundrum [View article]
@HeWho: My apologies for not being clearer in this scenario. Our "intuition" is that these assets are negatively correlated (as their trends are in opposite directions) -- but the statistical measure says they are independent.
@Princeton Ivy Management LLC, thanks for the comments and question. You've asked as very broad question, so I'm not going to go into a lot of detail around the issue; however I can provide you some starting points to formulate your own answers visa vie your own analysis.
Remember the yield curve represents many aspects of investor expectations: one is a maturity premium that investors require to hold longer dated instruments, another is the inflation premium investors demand to hold assets paying constant streams of income (that aren't inflation adjusted). If you're able to more clearly isolate, given the scenario you're interested, *why* the changes in rates have taken place (inflationary expectations increase, term premiums increase, increase in supply of long-dated instruments, etc.) then that's a good place to begin your analysis.
@briandp32, thanks for your comment, great question. The links to the data provide you with all the necessary tools do an analysis to test your hypothesis. No one would argue the Fed has an arduous task of unloading a massive balance sheet onto a bond market concerned about your very question. One could reasonably conjecture that greater misalignment between expectations and outcomes implies (1) greater rate volatility and (2) greater changes in rates. However, I would still expect the basic premise of "lower credit quality to depreciate less than high credit quality during rising interest rate environments" to hold -- all else equal.
It's worth noting that there were several periods used in this analysis where consecutive cumulative changes are close to 2% (see the x-axis of the scatter plots from the first graph series), which could be construed as information release grossly misaligning with expectations. So although the analysis isn't entirely based upon "big surprises," one could argue there are a couple data points in the sample where that is the case.
@the social scientist: Thanks for your comment. Cumulative changes in bond prices are plotted against cumulative changes in interest rates. You are correct, autocorrelation is introduced in the process. This makes tests of statistical significance and explained variation meaningless, which is why these values aren't provided or included in the post. It does not however, prevent legitimate illustration of sensitivity towards some factor, which is what the scatter plots illustrate.
I'm not sure where you're getting the "conclusion" that junk bonds are uncorrelated to high quality. Junk bonds are highly correlated to high quality, but they tend to fall less than high yield during rising interest rate environments as credit spreads compress (the conclusion of the article). The statement about interest rate risk and credit risk moving inversely is not meant to imply that fixed income of differing credit quality are not correlated -- it is a statement about correlation of underlying risk factors.
A quote from Swedroe, "while junk bonds have a relatively low positive correlation to equities on average, that correlation has a nasty tendency to dramatically increase at exactly the wrong time -- when equity risks show up." http://cbsn.ws/Xo9A1w
So, if you foresee large equity declines in the near future, you're exactly right that you can expect a similar behavior between High Yield and Equities. However, if look at 6 month rolling correlations between HYG (High Yield Bonds) and IWV (Russell 3000) since October 2007, the correlation has moved between 0.20 and 0.89, so your logic (as Swedroe would agree) should be applied only under certain scenarios.
Visualizing Increased Correlation During Market Crashes [View article]
Brad,
We appreciate the comment. The example we used in the article was just that, an example, and not a recommendation by any means. Your recommended portfolio of SPLV, DBA and TLH is an interesting one. In particular, we like the use of SPLV in order to take advantage of the, well-documented at this point, premium offered by low volatility equities. However, this portfolio still has a drawdown of ~22% during the credit crisis.
While this is indeed a large improvement in risk-adjusted returns, it may still be highly damaging depending on the type and purpose account. The main point we meant to highlight is that for investor's that need more return than that offered by a very conservative portfolio, but that are also highly sensitive to drawdown (think retiree's with a relatively long retirement horizon remaining) tactical management is one of the only answers that is cost-effective (assuming you pick the right manager).
Understanding The Perils Of Mean-Variance Optimization [View article]
xinge2010: The peril of unconstrained MVO derives from the risk that low eigenvalue (low variance) principal components (principal portfolios) may have high relative expected excess returns, which will lead to the low variance principal portfolios dominating the resulting portfolio.
Consider the case where the majority of principal portfolios have an expected excess return to variance ratio of ~0.5, but the lowest variance portfolio has an expected excess return to variance ratio of 10. When it is leveraged, it will become the single dominating "factor" in the portfolio.
The risk here is that Random Matrix Theory tells us that the principal components with the lowest variance (eigenvalues) are likely dominated by sampling noise, meaning that they are meaningless portfolios. From one rebalance period to the next, not only will the portfolio exhibit instability, but expected excess return and variance estimates may be fairly meaningless.
Understanding The Perils Of Mean-Variance Optimization [View article]
A quick math correction: The scaling factor in the SPY & AGG example should be 7.11, AGG's gamma should be 0.2844, and the optimal weights are 21.95% for SPY and 78.04% for AGG.
The Importance Of Benchmarking In Strategy Evaluation [View article]
One problem with R-squared is that it is going to rely very much on the time period that you choose for the calculation. Say you had a product that tactically allocated between SPY and AGG such that the product was either 100% SPY, 100% AGG, or 50%/50% SPY/AGG. I would argue that a static 50/50 portfolio would be a decent starting point for a benchmark since this is the "base" allocation around which the product rotates. However, if the product is 100% AGG for a significant period of time, the R-squared to this 50/50 benchmark may be low. R-squared is one criteria to evaluate the choice of a benchmark, but it should be used in conjunction with other quantitative and qualitative measures.
As a point of reference, the R-squared between EFA and the strategy in the article is 0.95, even though I think EFA is not the right benchmark.
The Importance Of Benchmarking In Strategy Evaluation [View article]
It can be more difficult to find a benchmark in more specialized circumstances. Oftentimes, it means creating your own customized benchmark. For example, an investor using a proprietary momentum model may compare his or her results to the results of using a more simplified momentum model (perhaps a 200 day moving average).
The Equity Growth Value Paradigm, Part I: What It Is Not [View article]
All returns used are total returns and incorporate the effects of dividends, splits and price movements. The data can be found on Kenneth French's website.
Most of the simple rules of thumb in the market can indeed be shot down. However, there are rules that are able to be more dynamic and truly add value for the investor. The trick is knowing how to distinguish between the former and the latter.
Deceived By Correlations: A Quant Conundrum [View article]
The risk of using correlation has nothing to do with the assumption that the time-series are GBM -- that was just for convenience to take independent normal random variables and translate them into asset prices.
This risk is if the random-variables have meaningfully different means from 0. On a daily measurement level, most assets have near-zero mean. On longer horizons, however, their means can differ in a meaningful manner.
Consider monthly returns for SPY and UNG from 1/30/2009 to 1/30/2012. SPY has a positive monthly mean log-return and UNG has a negative monthly mean log-return; a naive correlation estimate is at -2.6%; an adjusted correlation estimates puts the two at -15.5%. While some quick math tells us that the latter calculation is only significant at an 80% level, this fact often get's swept under the rug. Toss these two numbers into a mean-variance optimization and we can end up with dramatically different results.
The goal here was to simply demonstrate that the naive use of correlation can lead to dramatically different results than what we expect; by assuming mean-returns, at any timeframe, are zero can help bring the number back in line with our expectations. Not a parlor trick -- just math.
Deceived By Correlations: A Quant Conundrum [View article]
Indeed, correlation prediction is very difficult. However, this is a problem we see time and time again: people assume that negatively correlated assets must have divergent trends and that positively correlated assets move in the same direction. By the definition of correlation, this need not be so.
Given that many people utilize correlation as the basis for their diversification decisions, it is important to highlight how the strict statistical measure may deviate from their intuition -- and a quick and dirty method for correcting for it: namely, assume "means" are zero.
Deceived By Correlations: A Quant Conundrum [View article]
Thanks for sharing!
Deceived By Correlations: A Quant Conundrum [View article]
It is true that the independence of random variables necessarily implies zero correlation between those variables, though zero correlation does not necessarily imply independence of random variables.
What we are trying to point out here is that the "trend" component is ignored because the variation occurs around the mean. Therefore, any measures of joint-deviation are going to be about the "noise" and not the underlying trends of the return distributions.
By assuming a zero mean, our "trend" component gets measured as noise -- and therefore positive (or negative) correlations emerge.
Deceived By Correlations: A Quant Conundrum [View article]
My apologies for not being clearer in this scenario. Our "intuition" is that these assets are negatively correlated (as their trends are in opposite directions) -- but the statistical measure says they are independent.
Bond Bubbles And Credit Quality [View article]
Remember the yield curve represents many aspects of investor expectations: one is a maturity premium that investors require to hold longer dated instruments, another is the inflation premium investors demand to hold assets paying constant streams of income (that aren't inflation adjusted). If you're able to more clearly isolate, given the scenario you're interested, *why* the changes in rates have taken place (inflationary expectations increase, term premiums increase, increase in supply of long-dated instruments, etc.) then that's a good place to begin your analysis.
Happy (insight) hunting!
Bond Bubbles And Credit Quality [View article]
It's worth noting that there were several periods used in this analysis where consecutive cumulative changes are close to 2% (see the x-axis of the scatter plots from the first graph series), which could be construed as information release grossly misaligning with expectations. So although the analysis isn't entirely based upon "big surprises," one could argue there are a couple data points in the sample where that is the case.
Bond Bubbles And Credit Quality [View article]
I'm not sure where you're getting the "conclusion" that junk bonds are uncorrelated to high quality. Junk bonds are highly correlated to high quality, but they tend to fall less than high yield during rising interest rate environments as credit spreads compress (the conclusion of the article). The statement about interest rate risk and credit risk moving inversely is not meant to imply that fixed income of differing credit quality are not correlated -- it is a statement about correlation of underlying risk factors.
A quote from Swedroe, "while junk bonds have a relatively low positive correlation to equities on average, that correlation has a nasty tendency to dramatically increase at exactly the wrong time -- when equity risks show up." http://cbsn.ws/Xo9A1w
So, if you foresee large equity declines in the near future, you're exactly right that you can expect a similar behavior between High Yield and Equities. However, if look at 6 month rolling correlations between HYG (High Yield Bonds) and IWV (Russell 3000) since October 2007, the correlation has moved between 0.20 and 0.89, so your logic (as Swedroe would agree) should be applied only under certain scenarios.
Visualizing Increased Correlation During Market Crashes [View article]
We appreciate the comment. The example we used in the article was just that, an example, and not a recommendation by any means. Your recommended portfolio of SPLV, DBA and TLH is an interesting one. In particular, we like the use of SPLV in order to take advantage of the, well-documented at this point, premium offered by low volatility equities. However, this portfolio still has a drawdown of ~22% during the credit crisis.
While this is indeed a large improvement in risk-adjusted returns, it may still be highly damaging depending on the type and purpose account. The main point we meant to highlight is that for investor's that need more return than that offered by a very conservative portfolio, but that are also highly sensitive to drawdown (think retiree's with a relatively long retirement horizon remaining) tactical management is one of the only answers that is cost-effective (assuming you pick the right manager).
Understanding The Perils Of Mean-Variance Optimization [View article]
Consider the case where the majority of principal portfolios have an expected excess return to variance ratio of ~0.5, but the lowest variance portfolio has an expected excess return to variance ratio of 10. When it is leveraged, it will become the single dominating "factor" in the portfolio.
The risk here is that Random Matrix Theory tells us that the principal components with the lowest variance (eigenvalues) are likely dominated by sampling noise, meaning that they are meaningless portfolios. From one rebalance period to the next, not only will the portfolio exhibit instability, but expected excess return and variance estimates may be fairly meaningless.
Understanding The Perils Of Mean-Variance Optimization [View article]
The Importance Of Benchmarking In Strategy Evaluation [View article]
As a point of reference, the R-squared between EFA and the strategy in the article is 0.95, even though I think EFA is not the right benchmark.
The Importance Of Benchmarking In Strategy Evaluation [View article]
The Equity Growth Value Paradigm, Part I: What It Is Not [View article]
As Goes January, So Goes The Year? [View article]