Optimum Time Series Granularity in the Estimation of Financial Beta

Manuel G. Russon, Ph.D.St. John’s University

Qiaochu GengSt. John’s University

Abstract

In the world of finance and portfolio management, “beta” refers to the sensitivity of a security’s return to the sensitivity of the “market” portfolio and is an indication of the level of systematic risk, i.e. the amount of risk that a company’s equity shares with the entire market. Correct values for beta are crucial for portfolio managers, as the client contract almost always calls for a portfolio beta equal to 1.0. Typically, beta is estimated using Ordinary Least Squares with monthly granularity. This research considers that the ideal granularity might be something other than monthly. Betas for all companies in the SP500 are estimated with daily, monthly, quarterly and year granularity. Optimum granularity is determined to be quarterly.

I. Introduction

In the world of finance and portfolio management, “beta” refers to the sensitivity of a security’s return to the sensitivity of the “market” portfolio and is an indication of the level of systematic risk, i.e. the amount of risk that a company’s equity shares with the entire market.

Mathematically, beta is the regression slope in a linear regression of company rate of return onto the market rate of return. Eqn. 1, below, is the equation for beta and is referred to as the characteristic line.

rri = +*rrmkt (1)

Where rri - rate of return for company i

rrmkt - rate of return for market - alpha, intercept - beta, slope

The intercept, , is the expected return when the market return is equal to 0 and the slope, , is the percent change in the security for a one percent change in the market return, on average other things equal.

The conventional method to estimate the characteristic line alpha and beta is OLS, i.e. ordinary least squares using monthly observations. This research estimates beta for 4 levels of granularity, i.e. daily, monthly, quarterly, and yearly to find the optimum level of granularity.

Correct estimation of beta is important from an asset management, managerial finance and/or an investment banking perspective. From an asset management perspective, the following constraints are imposed upon portfolio managers:

1. Portfolio Turnover - usually limited to 100% per year2. Number of names in portfolio -usually required to be 50 – 1003. Tracking Error - usually limited to be +- 3% of index4. Weighted Ave. Beta - usually constrained to .98-1.02

All of these constraints are imposed to prevent excessive risk taking by the portfolio manager. In the case of item 4, incorrect betas can lead to unexpected volatility with concomitant issues in portfolio management regarding items 1-3. Institutional asset managers rely on betas provided by vendors such as Barra or Bloomberg. But even these vendors need to be sure their betas accurately reflect reality.

An individual retail investor can assess the expected volatility of the company’s equity relative to a universe benchmark by looking up a beta on Yahoo/Finance or other free source. In a more formal, institutional asset management context, portfolio manager of institutional clients, e.g. foundations, pension funds, mutual funds, etc. must satisfy a number of constraints in their portfolio management activities.

In a capital budgeting context, accurate betas are needed for estimation of cost of capital. An inaccurate beta generates inaccurate cost of capital, and this could lead to the incorrect acceptance or rejection of a capital project.

Figs. 1-4 display scatter-plots of rate of return for Caterpillar, Inc. vs. rate of return for SP500 for four levels of time series granularity with linear model plots overlaid. Beta is the slope of the linear model. The question to be addressed in this research is the appropriate level of granularity to discover the appropriate beta.

II. Methodology

End of day, month, quarter and year closing prices for all SP500 constituents as of 12/31/2015 were downloaded from Bloomberg for the period 12/31/1980-12/31/2016. As some companies are newer to the index, they have fewer data points than others. Returns will be calculated and beta coefficients estimated for each company for each time series granularity. Graphical (histograms and scatter plots) and analytical techniques (descriptive statistics, correlation and regression will be used to determine optimum granularity. The data will be estimated using S+.

Eqns. 2, 3, and 4 display the functional specification, population regression line and sample regression line in the estimation of beta. +

Eqn. 2 rri= f(rrmkt) (2)Eqn. 3 rri = +*rrmkt (3)Eqn. 4 rri = a+b*rrmkt (4)

III. Results

A histogram of all betas appears in Figs. 1-4, and pairs scatter-plots appear in Figs. 5. The beta results of all 500 companies are contained in Appendix 1.

Figs. 5-8 Histograms of Beta Coefficients

Table 1 displays beta coefficients for regressions from ten large companies, a subset of the result of time series regressions for all SP500 constituents.

Table 1 Beta Coefficients for 10 Companies

tkr betay betaq betam betad 15 aapl. 1.70 1.43 1.49 1.12 46 amzn. 1.74 1.25 1.49 1.30134 dd. 0.99 1.20 1.29 1.02153 duk. 0.76 0.53 0.47 0.56261 jnj. 0.34 0.35 0.42 0.52439 t. 0.63 0.52 0.59 0.78473 utx. 1.02 1.09 0.94 0.97490 wfc. 0.46 1.04 0.92 1.32496 wmt. 0.03 0.28 0.36 0.65506 xom. 0.43 0.70 0.57 0.85

Table 2 and 3 display descriptive statistics and correlation matrix for the betas contained in Appendix1.

Table 2 Descriptive Statistics for Beta Estimates of 500 Companies

mean med stdev skew kurt n betay 1.08 0.99 0.69 1.21 4.23 514betaq 1.07 1.01 0.52 0.73 0.77 514betam 1.04 1.01 0.48 0.40 -0.33 514betad 1.03 1.04 0.31 0.20 -0.24 514

Table 3 Corrrelation Matrix for Beta Estimates of 4 Granularities

betay betaq betam betad betay 1.00 0.71 0.70 0.59betaq 0.71 1.00 0.91 0.79betam 0.70 0.91 1.00 0.87betad 0.59 0.79 0.87 1.00

Fig. 9 displays scatterplot pairs of the beta estimates contained in Appendix 1.

Fig. 9 Scatterplot Pairs

A lot in the way of diagnostics is presented. The determination of which granularity must be made on the basis of these diagnostics. We find monthly beta to be most compelling on the basis that the mean and median are 1.04 and 1.01, both most near to 1.0.

Conclusion

This research evaluated beta estimates with four granularities to determine optimum granularity in the estimation of financial beta. Optimum granularity was determined to be monthly. Further research is needed for a more conclusive determination.