Volatility of returns indicates different relative riskiness than Beta
Hi everyone,
I'm trying to compare the riskiness of two bond markets, using BAML indices as proxies. In one approach, I am calculating annualized volatility of total returns and in the second approach, I am calculating the Beta of one index relative to the other.
...however I got some results that don't seem to make much sense:
Index 1 has an annualized volatility of returns that is greater than Index II, but in the Beta calculation, Index I has a Beta of below 1 relative to Index II (indicating Index I is less risky than Index II).
In both cases, I am calculating the volatility of total returns.
For the annualized volatility calculation, I have verified my figures using a Bloomberg function, so I know my methodology is correct. And on the Beta calculation, I have seen some in the market mention similar results.
I was wondering if anyone could see a reason for annualized volatility of total returns to give a different indication of the riskiness of two markets than a Beta calculation?
Thank you very much for any help and advice in advance!
Martin
Your volatility measure is a TOTAL calculation, while your beta calculation is a RELATIVE calculation. So you can get a vol number for any set of returns, but you can't get a beta unless you are comparing it to something. Hence, your comparison of beta of index 1 vs 2 only indicates how 1 and 2 move in relation to each other, which tells you nothing about how they compare against each other. So your interpretation of the beta is not right - the only thing you can tell from your beta calculation is that index 1 and index 2 are not perfectly correlated and therefore may have some 'diversification' benefit if you were to invest in both.
Are you looking just for which investment on a standalone basis is riskier? If so, volatility is really all you need...
Thank you very much for your reply.
In the Beta calculation, I always assumed that a value above 1 means that the volatility of returns for Index 1 is greater than the volatility of returns for Index 2.
A Beta of Abs volatility of Index 2, then I don't see how a Beta between the two should indicate Index 1 is less volatile than Index 2. Unless the period of calculation is the problem: I have calculated Beta by taking the covariance of Index 1 and Index 2 over the variance of Index 2, with the calculations carried out over a 51 datapoints of weekly data.
I'm not a super mathematician to prove to you that this could happen, but consider this: Another more straightforward for formula for beta is:
beta = correl (1,2) * stdev (1)/ stdev(2).
So if the correlation between 1 and 2 is 0.5 and stdev (1)/stdev(2) is 1.2 you will get a beta that is under 1 (I'm not sure if it's mathematically possible to have that, but let's run with it....)
It's completely possible. You can check your calculations, but I don't think the relationship between higher absolute vol and higher beta has to hold. If someone knows differently, my apologies for being confusing.
Thank you very much for your explanation.
Are volatility and beta relevant measures? If both are indices, they are both likely diversified from individual credit risk, so your main concern is interest rate risk measured by duration and convexity, and the debt rating breakdown each index . If they are different in terms of duration/optionality and credit quality holdings, these will tell you what is "riskier", which is easier than trying to calculate metrics that may or may not be meaningful.
Thank you for your reply. My intention was to try to show the how Index 1 moves in relation to Index 2, because Index 2 is larger and its influence has traditionally extended to Index 1. I am therefore considering Index 2 the "Market" index due to its relative size to Index 1, and its common investor base. The measure may not be relevant in the strictest sense.
But I agree with you that measures such as duration, convexity and credit rating breakdown are also crucial to understand what is going on in the two markets.
Thank you again for your kind response.
Makes more sense, maybe R^2 is potentially a metric that can be used?
In mathematical terms, your question boils down to whether cov(i,ii) can be smaller than var(ii) while var(i)>var(ii).
cov(i,ii) = corr(i,ii)stdev(i)stdev(ii). Clearly, cov(i,ii) can be smaller than var(ii) (aka stdev(ii)^2) if corr(i,ii) is small enough, even though stdev(i)*stdev(ii) > var(ii) given that var(i)>var(ii).
More intuitively, the volatility of an asset's returns has 2 components -- idiosyncratic/uncorrelated and market/systemic i.e. beta (in this case index 2 is your mkt). So even if overall volatility is high, if most of it is attributable to the former, the beta can still be small. Also, your measure of beta is merely telling you about how both indices move relative to each other, not how risky they are in absolute.
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