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I have two time series of daily return calculated as $\frac{Price_{t}}{Price_{t-1}} -1$. One is the daily returns of a portfolio, the other the daily returns of the index (MSCI World). Period is 2020 YTD (so during COVID19 crash). There are 78 daily returns in each time series.

The portfolio (long only, no leverage) under performed the index of about 3% over the period. The first intuition is that Beta was above 1 since it under performs in a down market.

Trying to tie the actual portfolio return back to CAPM, portfolio daily returns are regressed on MSCI world daily returns, using Excel data analysis add-in. This gives the following:

Multiple R: 89.9%
R square: 80.8%
Standard error: 1.2%
F significance: 0%
Intercept: -0.05% (0.13% standard error, -0.4 t-stat, 0.7 p-value)
X Variable 1 (Beta): 80.69% (4.5% standard error, 17.9 t-stat, 0 p-value)

If: Portfolio return $= \alpha + \beta \times$ Market return then

Portfolio return $= -0.05\% + 80.69\% \times -10\%$ (index return)

which is outperforming the index, while in practice I am under performing.

Why is Beta so low?

Daily returns and regression output are available (Excel file) to download on this link: https://easyupload.io/uds211

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  • $\begingroup$ Well, the questin that arises in your question is how can CAPM have such a high $R^2$, since the vast literateure gives disapointing performance to the estimated model? Are you sure tha ypur single factor model si right? Furthermore, what ypu are estimating is the portfolio excess return over the market excess return and not simply the returns. Can you provide some of your data image? $\endgroup$ – Hunger Learn May 31 at 11:12
  • $\begingroup$ Sure, I have added the daily returns and regression output in the link in my question. It is in an Excel file to be downloaded. The "model" is a simple regression, I am trying to measure my realised Beta, and the one I get from the regression seems incredibly low so I wonder what's wrong. $\endgroup$ – tweedi May 31 at 14:44
  • $\begingroup$ Are they excess returns? Namely, have you subtructed the risk free from both types of returns? $\endgroup$ – Hunger Learn May 31 at 15:10
  • $\begingroup$ Try using weekly returns, as pointed out by @TimWilding. I wouldn't use daily returns for calculating the beta to MSCI World. $\endgroup$ – RWP - Down by the Bay Jun 1 at 14:33
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You are right to be sceptical of the beta of an international portfolio when it is calculated using daily returns. Beta estimates are often low for international portfolios because stock market returns are asynchronous. For example, Tokyo and the New York Stock Exchange have very different trading hours. Portfolios constructed with a tilt towards either country are likely to have very different daily returns. Using daily returns with the different market closes reduces the correlation between the portfolio and the benchmark. This, in turn, reduces the beta.

A simple way to check whether this is affecting your estimate of beta is to add lead and lagged versions of the MSCI as independent variables to your regression. Summing the individual betas gives a better estimate of the overall beta. See, for example, "Estimating Betas From NonSynchronous Data" by Scholes and Williams, 1977 (https://www.sciencedirect.com/science/article/abs/pii/0304405X77900411).

I did this quickly for your portfolio in Excel and came up with an overall beta for your portfolio of approx. 1.09. Almost all of the increase in beta comes from the correlation between the portfolio and the lagged MSCI returns.

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The -0.05% intercept (typo assumed) is a daily rate, so for 78 days, that is a cumulative alpha of around -3.9%. But your 80% beta to an index down 10% would have saved you around 2%. So on the numbers given, your portfolio should be about World -2%.

Another way of thinking about Beta is as Correlation times Relative Volatility. Since C (root of your R2) is higher than Beta, your lower beta is mostly a function of the fact that your portfolio was less volatile than World. In this case, lower vol simply provided no insurance against the price declines.

Before you get too disheartened by this, it is worth checking what is the geographical mix of your portfolio? Is it a truly global portfolio? And in what currency are you measuring the returns (in fact, also which of the two MSCI World indices are you using, the local or the dollar one)? It is actually very likely that all of your negative "alpha" (and maybe even more than that) is explicable by a single simple geographic/currency effect. If you own a lot of European or Asian stocks, thus is indeed probably the case in this case.

Even supposing your portfolio is majority US stocks, then ask yourself how much you have in Tech (or in Tech and Energy the other way). Tech has obviously been the big winner from all of this, trouncing non-Tech; with energy a total disaster (negative oil prices etc etc). Unless your portfolio is as completely depedent on FANG at al. as is the US market, then this kind of outcome really wouldnt look too out of place. In this scenario, there's simply a statistically significant positive Tech effect in the benchmark you're measuring yourself against. Don't play along with that, and you will lag as long as it continues to work.

Which is a HUGE problem for equity fund managers who might have latent fundamental suspicions about the valuations of these names. They will lag the index; and the longer this goes on, the greater the gap (allocation weights, as well as just performance) between them and index grows. So doing nothing effectively represents them doubling down on this view; but they are risk-constrained in how much "tracking error" (your portfolio standard error above) they are allowed tp run. So many end up forced to buy names at prices worse than they originally refused to buy as too rich. It's an infamous problem for many an institutional investor!

Hope this helps.

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I would also add that daily data are too noisy given what mentioned in previous answers.

Also, the time series isn’t long enough to make inference about portfolio performance.

In practice, regression analysis is not used as a way to evaluate the performance of a portfolio. It is only a thing in academia.

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  • $\begingroup$ I thought it is used to get the Beta and hedge this Beta. Is it not good practice to want to calculate the beta actually obtained (from the observed returns) and compare it to ex-ante Beta to see if the hedge has been accurate? $\endgroup$ – tweedi Jun 3 at 12:40
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    $\begingroup$ I would say for the purpose of comparing ex post and ex ante beta, yes, regressions can be useful. But you still need a longer time series to get accurate results. You are getting r-squared of .80 because in periods of down markets usually prices are highly correlated. However, for the purpose of evaluation portfolio performance (i.e. alpha and information ratios), regressions are not usually used in practice. $\endgroup$ – user28909 Jun 4 at 10:46
  • $\begingroup$ Thank you - very useful follow up. For the purpose of alpha calculation, in your experience, what method is used to do this without regression? $\endgroup$ – tweedi Jun 8 at 15:05
  • $\begingroup$ If I’m evaluating an investment manager, I calculate alpha by simply looking at the geometric return over multiple periods, say, 3 months, 1 year and 3 year vs the benchmark. I would also look at the performance vs the style benchmark (i.e. MSCI World growth or value; depending on the style of the investment manager. Rolling excess returns overtime to assess consistency. Information ratio to assess skill. Active share and tracking error to avoid closet indexers.... I hope this helps. $\endgroup$ – user28909 Jun 8 at 19:10

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