# How to Calculate the Beta of a Portfolio to a Factor

### 1. Introduction

Autumn 2020 was largely impacted by two major risk events: the US elections and the Covid-19 vaccine trial results. Both scenarios acted as a catalyst for a violent equity factor reversal.

Accurately measuring risk exposures to a specific factor and understanding the limitations of the risk metrics used is one of the major challenges of risk management. In this article, we present different ways to measure risk exposure to the general markets and to a specific index with a focus on:

• Defining beta exposure;
• Discussing quality control toolkits;
• Illustrating the beta exposure concept with simple practical examples;
• Reviewing some important considerations when using beta.

A traditional risk measure employed in the asset-management industry is the market value exposure. While useful, it does not accurately explain the portfolio’s behaviour as the market moves.

### 2. Beta Exposure

A traditional risk measure employed in the asset-management industry is the market value exposure, which represents the notional exposure in percentage allocation of the fund. This measure is not subjective. While useful, it does not accurately explain the portfolio’s behaviour as the market moves. Understanding and accurately quantifying this behaviour is captured by a risk metric that is called the beta (‘β’).

The beta metric for a portfolio with respect to a pre-defined index, called X, captures the sensitivity of the fund to X. Basically, the fund’s beta to X tries to capture how much money the fund makes as X goes up (or down) by a specified amount. For complex and large portfolios, this calculation is far from simple, because to be accurate, one needs to capture precisely how all positions move with respect to each other as X moves by a pre-defined shock.

X can represent anything one needs to calculate the exposure to, such as the S&P 500 Index, the GBPUSD currency cross or any other index. Likewise, X can relate to more complex indices, such as a Growth versus Value index, or its sector-neutral version. Whatever the index, accurately computing a portfolio’s beta to it is a powerful risk management tool.

### 3. Types of Beta

There are two types of beta measures:

• Ex-post beta focuses on the actual, realised and observed return of a fund and compares it, after its realisation, to X;
• Ex-ante beta tries to capture the expectation of a portfolio change upon a move in X – the ex-ante measure is based only on the current positions and not historical holdings.

Generally, ex-ante measures will input current positions in the calculation and simulate hypothetical profit and loss returns assuming constant positions. The ex-ante measure estimates forward risk by using the current position and assumes that past market conditions such as returns, correlations and volatilities are a good representation of future market returns.

Both ex-ante and ex-post are important measures and tend to differ. For a specific date, their main difference lies in that the ex-ante will attempt to describe the current portfolio behaviour, while the ex-post attempts to capture the historical portfolio behaviour. For example, an ex-post measure will capture the historical change in positions between that date and the start of the lookback period, whilst an ex-ante does not.

More details on the technical computation of ex-post and ex-ante beta exposures are provided in the appendix.

### 4. Quality Control Toolkit

Beta is usually computed using simple linear regression statistical tools. As statistical tools, the results have limitations and are valid in a specific domain of application. There are at least four risk management tools that can be used to control the quality of the output:

• P-value: gives a sense of whether the regression is meaningful;
• R-squared: gives a sense of how much of the risk is explained;
• Analysis of errors: gives a sense of the domain validation1;
• Robustness: when dealing with non-normal data, a robust way to compute the beta.

There are further details in the appendix about the quality control toolkit and its technical aspects.

Ex-post beta is useful for investors as it can determine how the fund has historically behaved versus specific factors.

### 5. Illustrations

#### 5.1. An Example of Ex-Post Beta

As discussed above, ex-post beta is useful for investors as it can determine how the fund has historically behaved versus specific factors such as the market, Momentum2, Value, interest rates, gold or oil. Below, we show the ex-post beta of a long-only fund that perfectly replicates the returns of the FTSE 250 Index to GBPUSD. In this case, we define the ex-post beta as the solution of the simple linear regression using a least square method approach. For our purpose, we used a 6-month lookback coupled with daily returns as parameters.

FTSE250_returnst = β * GBPUSD_returnst + ε

Figure 1 shows the beta (illustrated by the grey line) of the regression of historical FTSE 250 returns on GBPUSD returns using a least square method. We overrode the beta to zero (‘adjusted’, illustrated by the yellow line in Figure 1) whenever it was not statistically significant.3 This was done to emphasise the danger of naively using the beta when it is not statistically significant based on its P-value and R-squared.

Source: Bloomberg, Man Group; as of 19 October 2020.

The ex-post beta of the FTSE 250 Index depends on the past levels of the FTSE 250, which in turn is affected by participants’ beliefs – beliefs impact actions, which impacts supply and demand, which impacts observed market moves, which therefore impacts beta.

In addition, investors’ belief is a complicated function which is difficult to reduce to a simpler form. However, it is reasonable to assume it depends on:

• What other market participants do;
• Market fundamentals and their interpretation;
• Price action which can reinforce pre-existing behaviour;
• Newsflow.

None of these inputs are static, with investors updating their investment theses through time. As they change their beliefs, and therefore their actions, beta will also change over time.

Ultimately, supply and demand for both the British pound and the FTSE 250 will decide how they evolve in relation to each other. We still try to simplify this complex reality into three regimes, below:

1. GBPUSD is non-significant (i.e. p-value is less than 5%): In this case, there is no point trying to explain the exposure of FTSE 250 with respect to GBPUSD since there is no significant relationship between the two in that regime. Any price relationship between the two of them is non-significant, and chances are that beta is in fact zero. This is the case for example from December 2017 to December 2018 (illustrated by Zone D in Figure 1). If an investor is exposed to this kind of beta, it is certainly worth asking if this should be hedged or not;

2. Beta > 0 and is significant: In this case, the FTSE 250 and GBPUSD, on average, tend to move in the same direction. We can intuitively think of two different market environments which could support positive beta:

• Investment cycles where foreign investors are attracted by mid-cap UK shares could support this kind of regime. Indeed, in such an environment, the foreign demand for GBP to buy UK stocks can be high – all else equal. This seems to be the case before October 2012 (Zone A in figure 1);
• A large exogenous market shock could also support this kind of regime, as is demonstrated by the large spikes in 2016 (Zone B) and 2020 (Zone E). This was especially evident during Brexit- and Covid-related market events, with daily correlation between the FTSE 250 and GBP starting to increase i.e. the FTSE 250 increased as GBP increased, and vice versa. In the case of Covid-19, and to some extend Brexit, the US dollar acted as a safe haven for investors to reduce their exposure to UK risk. As more and more of those types of days kick in and are included in the lookback period, the beta consistently increases and eventually becomes significant. In this regime, the British pound and the FTSE 250 tend to move in a pair. Another way to describe this regime is to recognise that upon a large systemic shock, correlations go to one.
• It is worth diving into the Brexit phase to understand the dynamic in detail. Between 23 June 2016 and 27 June 2016, when the results of the Brexit referendum were announced, the FTSE 250 sold off by 5% and GBPUSD by 10%. Those three days materially increased the ex-post beta from a nonsignificant level to a significant level and explains the Brexit spike observed in Zone B. Subsequently, between 27 June 2016 and 30 December 2016, the relationship reversed and the FTSE 250 rallied by 20% while cable continued to sell off by -7%. This 6-month period eventually led to a material reduction in beta (Point Z). Furthermore, once the Brexit vote was out of the lookback period, the adjusted beta dropped to zero since it was not significant at this point (Zone C)4;

3. Beta < 0 and is significant: In this case, the FTSE 250 and GBPUSD tend to move in opposite directions. One potential explanation is that the FTSE 250 is made up of some large international companies which tend to improve their revenues when the British pound depreciates. This is a simple argument which justifies that if investors focus on this type of investment thesis, then we should not be surprised to get negative ex-post beta (Zone C).

#### 5.2. An Example of Ex-Ante Beta

To compute the ex-ante beta, we use current positions for a specific date and keep it unchanged in the calculation (see the appendix for further details on the computation). In this case, we focus on a portfolio which tracks the FTSE 250 Index (that we will call π) and calculate its ex-ante beta with respect to GBPUSD in Figure 2.

Understanding how the beta behaves when changing parameters is a critical aspect of risk management.

### 6. Comparison of Ex-Post and Ex-Ante Beta

The results in Figure 2 show the ex-ante and ex-post beta exposures of the FTSE 250 to GBPUSD. The yellow line represents the ex-ante beta while the blue line is the ex-post beta of the same portfolio. It appears that the two betas are relatively similar except during shock periods (Brexit in Zone A and Covid in Zone B) for which the expost beta is larger than the ex-ante beta. This suggests that periods of large stress for which GBPUSD dropped, the FTSE 250 fell more than portfolio π.5

Source: Bloomberg, Man Group; as of 31 December 2020.

### 7. Important Considerations

Understanding how the beta behaves when changing parameters is a critical aspect of risk management. Besides ex-ante or ex-post decisions, the beta is also sensitive to different parameters. Practitioners should be aware of potential limitations related to the usage of the betas. Listed below are important questions we think makes sense to focus on when assessing beta:

1. Should the beta be computed ex-ante or ex-post?;
2. Is the beta significant and can the null hypothesis be rejected?;
3. Should indices be decomposed or not?;
4. How to treat options? Does it result in a risk approximation?;
5. Which lookback period should be taken and why?;
6. Should a weighting be applied in the most recent days?;
7. Does the lookback period represent the best expectation of future market regime?;
8. How does the beta change when the index is updated for a given theme (pure versus non-pure)?;
9. How does the beta change when using daily returns versus 3-day or 5-day returns?;
10. Do overlapping returns be used? How does it impact the results?;
11. Do robust regressions be used? How does it impact the results?;
12. How much of the P&L is explained by the beta (R-squared)?;
13. For which market shock in standard deviation terms is the beta an acceptable representation of the portfolio behaviour? At which point does the beta not accurate anymore?;
14. What information does backtesting give? Does the beta explain the actual P&L and, if not, why not?;
15. How to deal with autocorrelation and how to deal with lookback be dealt with?

Whichever beta one decides to use, the most important aspect of risk management is to have a good understanding of the assumptions underlying the calculation of beta and the regimes for which it works.

### 8. Conclusion

There are many subtleties to consider when applying a beta analysis to a portfolio. One important distinction that needs to be made is between ex-post and ex-ante. The ex-post beta of a fund to an index is particularly useful for an investor because it tries to capture how the fund has performed historically versus the index. The ex-ante beta is particularly useful for portfolio managers because it tries to capture how the current position would behave versus the index. Whichever beta one decides to use, the most important aspect of risk management is to have a good understanding of the assumptions underlying the calculation of beta and the regimes for which it works.

### 9. Appendix

#### 9.1. Beta Computation

##### 9.1.1. Ex-Post Beta

From a computational perspective, one could define the ex-post beta as the solution of the simple linear regression below using a least square method approach.

historical_portfolio_returnst = β * X_returnst + ε

##### 9.1.2. Ex-Ante Beta

From a computational perspective, a simple way to compute an ex-ante beta is to compute the risk of every asset in the portfolio to X. Once you have the covariance between X and every asset in the portfolio, you can then define the ex-ante beta to X as:

β = ∑nk=1 wk * Cov(X,sk) * h

where:

i) sk represents returns of stock k
ii) wk represents the current weight kept constant of the stock k
iii) h is the pre-defined given shock to compute the β to the factor X

A mathematically equivalent way to rewrite the above equation consists in computing the beta as a simple linear regression, using hypothetical portfolio returns and assuming constant weights. This is the method which was used in our illustration for portfolio π. In this approach, one would solve for the equation below using a least square method approach:

hypothetical_simulated_returnst = β * X_returnst + ε

where:

i) X_returns represents the daily returns of factor X
ii) hypothetical_simulated_returns represents the returns of a hypothetical portoflio with constant holding

#### 9.2. Quality Control Toolkit

##### 9.2.1. P-Value

In statistics, the p-value represents the probability of obtaining results at least as extreme as the observed results assuming that the null hypothesis6 is correct. A smaller p-value means that there is stronger evidence in favour of the alternative hypothesis. Practitioners usually set the p-value threshold at 5% below which one could fairly reject the null hypothesis. We then say the test is significant.

While it is simple to know if a beta is significant at a specified quantile (‘μ’), by checking if if the p-value is less than μ, it is relatively difficult to know which quantile μ to use. Practitioners usually use μ = 5%. In addition, regime shifts can occur as we demonstrated in Figure 1, even though the beta exposure approach does not capture when regimes can change from non-significant to significant. In our opinion, caution is usually a good approach and taking time to ask whether the null hypothesis should be rejected is a good place to start. If in doubt, we recommend testing another similar hypothesis to be sure; for example, by changing slightly the index or the parameters or using weekly returns whilst correcting for autocorrelation.

Understanding if the beta is significant is a critical aspect of risk management. Unfortunately, our view is that this question is often viewed by practitioners as a mathematical consideration. From a statistical and practical standpoint, however, this technical detail matters. Indeed, if the beta is non-significant, then one can’t reject the null hypothesis, which means that beta is likely to be zero. It is then easy to understand that hedging a non-significant beta exposure potentially adds risk to a portfolio.

##### 9.2.2. R-Squared

R-squared is a statistical measure that represents the proportion of the variance for a dependent variable that is explained by an independent variable or variables in a regression. In simpler terms, it can be viewed as the percentage of risk explained by the factors used in the model input. A higher R-squared corresponds to more explanatory power of the factor used in the regression. When computing a single linear regression, the R-squared happens to be equal to the correlation squared.

##### 9.2.3. Analysis of Errors

The ordinary least square (‘OLS’) method is a statistical method for estimating the unknown parameters in a linear regression model. The OLS minimises the sum of the squares of the error ε i.e. the differences between the observed dependent variable (values of the variable being observed) in the given dataset and those predicted by the linear function.

It is customary to assume:

• Homoscedasticity: E[ε2i | X ] = σ2, which means that the error term has the same variance σ2; in each observation. When this requirement is violated, it is called heteroscedasticity. In this case, robust estimation techniques are recommended (see below);
• No autocorrelation: the errors are uncorrelated between observations: E[εi * εj | X ]=0 for i ≠ j.
##### 9.2.4. Robustness

Robust regression is a form of regression analysis designed to overcome some limitations of traditional linear regressions such as the OLS method. For instance, least square estimates are highly sensitive to outlier data points, such as multi-standard deviation moves. This is not normally a problem if the outlier is an extreme observation drawn from the tail of a normal distribution, but if the outlier results from nonnormal measurement error or some other violation of standard ordinary least squares assumptions, then it compromises the validity of the regression results if a non-robust regression technique is used.

Robust regression techniques include M-estimators (’M’ for ‘maximum likelihood-type’) popularised by Huber.7

1. When model developers build a model, they generally define a domain application. The domain application specifies how the model is supposed to be used by the model user. In particular, domain application describes for which input the output of the model can be trusted. Domain validation describes the set of inputs for which the model behaves adequately.
2. By Momentum we mean a signal defined by the performance of a long / short basket. The portfolio is long the stocks which performance is positive over a pre-defined time horizon, typically 12 months, and short the stocks which performance is negative over the same time horizon. This basket also rebalances over a different time horizon, typically every month. The performance of this basket is the Momentum signal.
3. When the p-value was over 5%.
4. Whilst the goal of this article is not to dig into the relationship between FX and equities during large FX shocks, it is nevertheless worth highlighting that on 15 January 2015, as the Swiss National Bank removed the 1.20 price peg from the EURCHF pair, a similar effect in the correlation between FX and equities was observed. On that day, correlation spiked as the EURCHF sold off by ~-2000 bps or ~-16.88% from 1.2010 to 0.9982 whilst SMI Index sold off by -8.67% from 9,198 to 8,400.
5. Ex-Ante Beta calculation assumes constant weights for portfolio π over the period.
6. The null hypothesis is the hypothesis that the beta is zero.
7. https://www.statsmodels.org/stable/rlm.html.