*Loss coherence applied to random variables; loss coherence matrix compared to correlation matrix; loss coherence for asset classes and trading strategies; some surprises…*

In the previous post, I created a way to measure the tendency of two investments to lose money at the same and called it *loss coherence.* Loss coherence (“LC”) is roughly analogous to correlation, except that it measures a very specific property of how two investments relate to each other.

In this post, we’ll look some actual values of loss coherence and compare them to correlation.

### Loss Coherence of Random Variables

Because loss coherence is roughly analogous to correlation, it should have a mean value of zero when it is calculated for two random variables. I would also think it should be somewhat correlated to correlation. Chart 1 shows the result of generating two psuedo-random variables one thousand times and comparing the correlation with the loss coherence.

Note the following:

- As expected, for random variables, loss coherence is related to correlation. The correlation coefficient is .
- The median value of loss coherence for these 1,000 tests is zero.
- For these random variables, loss coherence varies approximately 4X faster than correlation and has a range of nearly -1 to +1, whereas correlation has a range of approximately -0.4 to +0.4.
- Loss coherence is quantized, and this is particularly obvious at values near zero. The quantization is due to the discrete nature of the binomial distribution.

No surprises here – so far, so good.

### Loss Coherence of Common Asset Classes

Let’s compare the loss coherence and correlation between several asset classes. The asset classes are:

- International equities (MSCI World Index)
- US equities (S&P 500 TRI)
- CTAs (SocGen CTA Index)
- Trend-following CTAs (SocGen Trend Sub-index)
- Hedge Funds (Barclay HF Index)
- REITs (Nareit US Real Estate Index)
- Bonds (continuation prices of Ten-Year Note futures)

We’re using monthly returns from Jan-2000 to Dec-2017, except for Ten-Year Notes, which starts in Feb-2001.

First, the correlation table:

MSCI World Index | S&P 500 TRI | SG CTA Index | SG CTA Trend Sub-Index | Barclay HF Index | Nareit US Real Estate Index | Ten-Year Notes | |
---|---|---|---|---|---|---|---|

MSCI World Index | 1.00 | ||||||

S&P 500 TRI | 0.97 | 1.00 | |||||

SG CTA Index | -0.10 | -0.14 | 1.00 | ||||

SG CTA Trend Sub-Index | -0.09 | -0.14 | 0.97 | 1.00 | |||

Barclay HF Index | 0.84 | 0.77 | 0.02 | 0.02 | 1.00 | ||

Nareit US Real Estate Index | 0.63 | 0.61 | 0.00 | 0.02 | 0.52 | 1.00 | |

Ten-Year Notes | -0.32 | -0.36 | 0.27 | 0.28 | -0.29 | -0.07 | 1.00 |

Now, the loss coherence table:

Loss Coherence | MSCI World Index | S&P 500 TRI | SG CTA Index | SG CTA Trend Sub-Index | Barclay HF Index | Nareit US Real Estate Index | Ten-Year Notes |
---|---|---|---|---|---|---|---|

MSCI World Index | 1.00 | ||||||

S&P 500 TRI | 1.00 | 1.00 | |||||

SG CTA Index | 0.63 | 0.43 | 1.00 | ||||

SG CTA Trend Sub-Index | 0.90 | 0.78 | 1.00 | 1.00 | |||

Barclay HF Index | 1.00 | 1.00 | 0.31 | 0.74 | 1.00 | ||

Nareit US Real Estate Index | 1.00 | 1.00 | 0.67 | 0.90 | 0.99 | 1.00 | |

Ten-Year Notes | -0.55 | -0.89 | 0.90 | 0.93 | -0.61 | 0.96 | 1.00 |

There are some surprises here. I’ve highlighted two places where the correlation and loss coherence are notably different.

One of the primary selling points of trend-following CTAs is their non-correlation to stocks. Looking at the correlation and loss coherence of the S&P 500 TRI and the SG CTA Trend Sub-Index, the correlation is -0.14. Highly uncorrelated, as expected. When we look at loss coherence, however, the LC is 0.78. So even though the two asset classes are uncorrelated, they both lose money in the same months at a far greater frequency than would be expected if the two classes were independent.

The same issue occurs with REITs and bonds. While the correlation is -0.07, the LC is 0.96. Once again, two uncorrelated return streams tend to lose money in the same months at a far greater rate than if the two returns streams were truly unrelated.

Let’s dig a little deeper into this second case. There are 203 months in the sample, 84 negative months for Bonds, and 74 negative months for REITs. So the probability of a lose/lose month is:

(1)

and the expected number of months when both investments lose money is:

(2)

The actual number of months when both investments lose money is 41, so there are 11 more lose/lose months than expected. This may not seem like a lot. Assuming the two investments are random and independent, however, we can use the binomial distribution to calculate the probability of this happening. *The probability of having 41 or more months where both investments lose money is only 0.03.*

### Summary

The remarkable thing about cases like Bonds and REITs is not that the loss coherence is so high. The remarkable thing is that the loss coherence is so vastly different than the correlation coefficient.

One would expect that uncorrelated investments would tend to lose money at different times and that the frequency of periods when both investments lose money would be roughly the same as the binomial distribution suggests. In particular cases, however, we find this not to be the case.

Can loss coherence help build better portfolios? More research is needed to answer this question. It appears, though, that it can reveal some relationships between investments that are not apparent from looking at correlations.