In this article, Youssef Louraoui (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of *beta*, one of the most fundamental concepts in the financial industry, which is heavily used in asset pricing and risk assessment.

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 beta represents the sensitivity of an asset 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.

The beta for a portfolio in respect to a predefined index, called by the letter X, indicates the fund’s sensitivity to X. Essentially, the fund’s beta to X attempts to capture the amount of money made when X increases (or decreases) by a specified amount. This computation is far from straightforward for complicated and large portfolios, as it needs exactly capturing how all positions move in respect to one another as X changes due to a predefined shock. X is usually modelled as an index, such as the S&P 500 index or any other index. Similarly, X can refer to more complex indices such as a Growth versus Value index.

Graphically, the beta represents the slope of the straight line through a regression of data points between the asset returns in comparison to the market returns. It is a traditional risk measure used in the asset management industry. In order to give a more insightful explanation, a regression analysis has been performed by downloading a 2-year time series of APPL stock and the S&P500 to see how the stock behaved in relation to the global market fluctuations. Figure 1 depicts the regression between APPL stock and the S&P500 index, where we can see a beta that is near zero (b = 0.3508), which indicates that the stock price didn’t fluctuate along with the market index.

Figure 1. Apple regression over the S&P500 index.

Source: Computation by the author (data source: Thomson Reuters).

## Mathematical derivation of Beta

In this model, the beta (β) parameter is a key parameter and is defined as:

where *Cov*(*r*,*r _{m}*) represents the covariance of the asset return with the overall market return, and σ(

*r*)

_{m}^{2}is the variance of market return.

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 in 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 in 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.

## Estimation of the beta of an asset

You can download an Excel file with data for Apple stock price and the S&P500 index (used as a representation of the market).

## Use of beta

William Sharpe, John Lintner, and Jan Mossin separately developed another 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.

Where :

- E(r) represents the expected return of the asset
- r
_{f}the risk-free rate - β a measure of the risk of the asset
- E(r
_{m}) the expected return of the market - E[r
_{m}– r_{f}] represents the market risk premium.

## Why should I be interested in this post?

If you are a business school or university undergraduate or graduate student, this content will help you in broadening your knowledge of finance.

## Relevance to the SimTrade Certificate

The SimTrade Certificate enables students to the enhance their comprehension on finance.

### About theory

- By taking the Investment strategy By taking the SimTrade course, you will know more about how investors can use various strategies to invest in order to trade in the market.

### About practice

- By launching the Market Maker simulation By launching the series of Market maker simulations, you can extend your learning about financial markets and trading approaches.

## Related posts on the SimTrade blog

▶ Louraoui Y. Systematic and specific risks

▶ Louraoui Y. Portfolio

▶ Louraoui Y. Alpha

▶ Walia J.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).