In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) explains the concept of beta, one of the most fundamental concepts in the financial industry, which is heavily used in asset management to assess the risk of assets and portfolios.
This article is structured as follows: we introduce the concept of beta in asset management. Next, we present the mathematical foundations of the concept. We finish with an interpretation of beta values for risk analysis.
Introduction
The (market) beta represents the sensitivity of an individual asset or a portfolio to the fluctuations of the market. This risk measure helps investors to predict the movements of their assets according to the movements of the market overall. It measures the asset risk in comparison with the systematic risk inherent to the market.
In practice, the beta for a portfolio (fund) in respect to the market M represented by a predefined index (the S&P 500 index for example) indicates the fund’s sensitivity to the index. Essentially, the fund’s beta to the index attempts to capture the amount of money made (or lost) when the index increases (or decreases) by a specified amount.
Graphically, the beta represents the slope of the straight line through a regression of data points between the asset return in comparison to the market return for different time periods. It is a traditional risk measure used in the asset management industry. To give a more insightful explanation, a regression analysis has been performed using data for the Apple stock (APPL) and the S&P500 index to see how the stock behaves in relation to the market fluctuations (monthly data for the period July 2018 – June 2020). Figure 1 depicts the regression between Apple stock and the S&P500 index (excess) returns. The estimated beta is between zero and one (beta = 0.3508), which indicates that the stock price fluctuates less than the market index.
Figure 1. Linear regression of the Apple stock return on the S&P500 index return.
Source: Computation by the author (data source: Thomson Reuters).
Mathematical derivation of Beta
Use of beta
William Sharpe, John Lintner, and Jan Mossin separately developed key capital markets theory as a result of Markowitz’s previous works: the Capital Asset Pricing Model (CAPM). The CAPM was a huge evolutionary step forward in capital market equilibrium theory since it enabled investors to appropriately value assets in terms of systematic risk, defined as the market risk which cannot be neutralized by the effect of diversification.
The CAPM expresses the expected return of an asset a function of the risk-free rate, the beta of the asset, and the expected return of the market. The main result of the CAPM is a simple mathematical formula that links the expected return of an asset to these different components. For an asset i, it is given by:
Where:
- E(ri) represents the expected return of asset i
- rf the risk-free rate
- βi the measure of the risk of asset i
- E(rm) the expected return of the market
- E(rm)- rf the market risk premium.
The risk premium for asset i is equal to βi(E(rm)- rf), that is the beta of asset i, βi, multiplied by the risk premium for the market, E(rm)- rf.
In this model, the beta (β) parameter is a key parameter and is defined as:
Where:
- Cov(ri, rm) represents the covariance of the return of asset i with the return of the market
- σ2(rm) the variance of the return of the market.
Excel file to compute the beta
You can download below an Excel file with data for Apple stock returns and the S&P500 index returns (used as a representation of the market). This Excel file computes the beta of apple with the S&P500 index.
Interpretation of the beta
Beta helps investors to explain how the asset moves compared to the market. More specifically, we can consider the following cases for beta values:
- β = 1 indicates a fluctuation between the asset and its benchmark, thus the asset tends to move at a similar rate than the market fluctuations. A passive ETF replicating an index will present a beta close to 1 with its associated index.
- 0 < β < 1 indicates that the asset moves at a slower rate than market fluctuations. Defensive stocks, stocks that deliver consistent returns without regarding the market state like P&G or Coca Cola in the US, tend to have a beta with the market lower than 1.
- β > 1 indicates a more aggressive effect of amplification between the asset price movements with the market movements. Call options tend to have higher betas than their underlying asset.
- β = 0 indicates that the asset or portfolio is uncorrelated to the market. Govies, or sovereign debt bonds, tend to have a beta-neutral exposure to the market.
- β < 0 indicates an inverse effect of market fluctuation impact in the asset volatility. In this sense, the asset would behave inversely in terms of volatility compared to the market movements. Put options and Gold typically tend to have negative betas.
Why should I be interested in this post?
If you are a business school or university student, this post will help you to understand the fundamentals of investment.
Related posts on the SimTrade blog
▶ Youssef LOURAOUI Systematic and specific risks
▶ Youssef LOURAOUI Portfolio
▶ Youssef LOURAOUI Alpha
▶ Jayati WALIA Capital Asset Pricing Model (CAPM)
Useful resources
Academic research
Fama, Eugene F. 1965. The Behavior of Stock Market Prices.Journal of Business 37: January 1965, 34-105.
Fama, Eugene F. 1967. Risk, Return, and General Equilibrium in a Stable Paretian Market. Chicago, IL: University of Chicago.Unpublished manuscript.
Fama, Eugene F. 1968. Risk, Return, and Equilibrium: Some Clarifying Comments. Journal of Finance, (23), 29-40.
Lintner, J. 1965a. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1): 13-37.
Lintner, J. 1965b. Security Prices, Risk and Maximal Gains from Diversification. The Journal of Finance 20(4): 587-615.
Mangram, M.E., 2013. A simplified perspective of the Markowitz Portfolio Theory. Global Journal of Business Research, 7(1): 59-70.
Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.
Mossin, J. 1966. Equilibrium in a Capital Asset Market. Econometrica 34(4): 768-783.
Sharpe, W.F. 1963. A Simplified Model for Portfolio Analysis. Management Science 9(2): 277-293.
Sharpe, W.F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance 19(3): 425-442.
Business analysis
JP Morgan Asset Management, 2021. Glossary of investment terms: Beta
Man Institute, 2021. How to calculate the Beta of a portfolio to a factor
Nasdaq, 2021. Beta
About the author
The article was written in November 2021 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).