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Archives of Business Research – Vol. 10, No. 5

Publication Date: May 25, 2022

DOI:10.14738/abr.105.12442. Kantarelis, D. (2022). Risky Portfolio Efficient Frontier & Performance: Bayesian versus Frequentist. Archives of Business Research,

10(05). 211-228.

Services for Science and Education – United Kingdom

Risky Portfolio Efficient Frontier & Performance: Bayesian versus

Frequentist

Demetri Kantarelis, Ph.D.

Professor, Department of Economics, Finance & International Business

Grenon School of Business, Assumption University

500 Salisbury Street, Worcester, MA 01609, USA

ABSTRACT

The location of the Efficient Frontier in Expected Portfolio Return (Rp) / Portfolio

Standard Deviation (Sp) space depends on what kind of summary estimates are

utilized by the investor. By relying on Bayesian and Frequentist summary

measures, in conjunction with a 5-stock portfolio of non-volatility free stocks,

analysis in this paper proves that use of reliable Bayesian estimates (estimates that

are not sensitive to prior distributions) (a) causes the locus of the Efficient Frontier

to appear higher than, and to the right of, the one constructed on Frequentist

geometric measures, and (b) generates improved measures of portfolio

performance. Additionally, it is demonstrated that the decision of the investor to

rely on simple (arithmetic or sample) summary measures describing non-volatility- free assets, would cause the Efficient Frontier to appear above the Bayesian,

implying overoptimism. It is concluded that Bayesian estimates, that are not

sensitive to mistaken prior distributions, take the investor closer to truer

performance levels.

INTRODUCTION

As per collective practical experience, the Mean-Variance Optimization theory of Markowitz

(1952, 1959) suffers from several weaknesses. According to Haugh (2016), such weaknesses

include “estimation errors in the mean return vector and covariance matrix [...], the portfolio

weights tend to be extremely sensitive to very small changes in the expected returns [...], [and]

the presence of heavy tails in the return distributions can result in significant errors in

covariance estimates as well.” In efforts to deal with such weaknesses, many researchers have

established that Bayesian analysis is an effective alternative tool that can be used to deal with

parameter uncertainty in portfolio management so that we end up with results closer to the

truth. As stated by Avramov et al. (2010) in their abstract, “The Bayesian framework neatly

accounts for these uncertainties, whereas standard statistical models often ignore them.”

Bauder et al. (2018), show the advantages of the Bayesian approach over the Frequentist (or

sample) both theoretically and empirically and they conclude that “the constructed Bayesian

efficient frontier improves the overoptimism which is present in the sample efficient frontier.”

Researchers are concerned with the common questions faced by Bayesians such as proper prior

distributions, importance of historical data horizons, use of models other than simple mean- variance optimization (such as risk-based or characteristic-based pricing models) and

difficulties in data collection in association with these models. See among others, Bawa et al.

(1979), Aguilar et al. (2000), Barry (1974), Black et al. (1992), Bodnar et al. (2008, 2009, 2010,

2017), Brown (1976), Frost et al. (1986), Jorion (1986), Kan et al. (2007, 2008), Klein et al.

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(1976), MacKinlay et al. (2000), Pastor (2000), Rachev et al. (2008), Sekerke (2015),

Stambaugh (1997), Tu et al. (2010).

Based on simple mean-variance optimization, my aim in this paper is to derive a Bayesian

Efficient Frontier (BEF) and compare it to the Frequentist Efficient Frontier (FEF) for a 5-stock

portfolio. Additionally, I will use the Bayesian and Frequentist results to measure portfolio

performance relative to each other and to a market. Assuming that readers are familiar with

the Markowitz and Capital Asset Pricing Model (CAPM) models as well as the Sharpe, Treynor,

Jensen and M2 measures of portfolio performance, I will try to re-verify the assertion that the

BEF is a better tool to rely on for managing a portfolio. Overall, with the assistance of widely

available software statistical packages such as Excel and STATA, I hope to contribute to the

basic pragmatic pedagogy associated with portfolio efficient frontiers needed by students as

well as active investors.

I will proceed as follows: in Section 2, I will deal with the traditional FEF, CAPM, and

performance measures. In Section 3, I will deal with the BEF, Bayesian CAPM, and the

corresponding performance measures; the analysis will include prediction intervals. In Section

4, I will compare the Frequentist results to the Bayesian results and, thereafter, conclude in

Section 5.

Data for my 5-stock portfolio along with a risk-free asset and a market are described in Table

1.

Table 1: Data and Concepts

Time Horizon Daily rate of return: December 12, 2017 to October 29, 2021

n = number of observations = 219

Stocks

Eli Lilly & Co. (LLY)

Walmart Inc. (WMT)

Caterpillar Inc. (CAT)

Deere & Co. (DE)

National Fuel Gas (NFG)

Risk-free Asset rate (Rf) One-month US Treasury bill rate *

Market Rate (Rm) Value-weight return of all CRSP firms incorporated in the US *

(CRSP = Center for Research in Security Prices)

Fama / French Factors

SMB = return spread of small minus large stocks

HML = return spread of cheap minus expensive stocks

RMW = return spread of the most profitable firms minus the least

profitable

CMA = return spread of firms that invest conservatively minus

aggressively *

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Kantarelis, D. (2022). Risky Portfolio Efficient Frontier & Performance: Bayesian versus Frequentist. Archives of Business Research, 10(05). 211-228.

URL: http://dx.doi.org/10.14738/abr.105.12442

Portfolio (p) and

Market (m)

Performance Measures

Rp = Expected Portfolio Return

Sp = Portfolio St. Deviation

Sm = Market St. Deviation

βp = Portfolio Beta

Sharpe Portfolio Measure = (Rp - Rf)/Sp

Sharpe Market Measure = (Rm -Rf)/Sm

Treynor Portfolio Measure = (Rp - Rf)/ βp

Treynor Market Measure = (Rm – Rf)/1

Jensen Measure (Jensen’s alpha) = αp = Rp – [Rf + βp(Rm – Rf)]

Modigliani-squared Measure = M2 = (Sm/Sp)(Rp) + (1- Sm/Sp)Rf - Rm

Portfolio performs better than Market if Portfolio Measures exceed Market

Measures. High numerical values of Jensen and M2 imply good portfolio

performance (the higher the better.)

* Data and definitions:

Fama & French (1993, 2014, 2015)

French (2022)

<http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html>

Musarurwa, Rujeko, (2019), “Fama French Five Factor Asset Pricing Model”, in Quantisti.com

<https://blog.quantinsti.com/fama-french-five-factor-asset-pricing-model/>

FREQUENTIST EFFICIENT FRONTIER AND PORTFOLIO PERFORMANCE

Given that the stocks considered for the portfolio in this paper are not volatility-free, it is more

appropriate to utilize, as summary measures, geometric means, variances, and standard

deviations; BLOCK 1 of Table 2 displays such summary measures, based on 219 daily rates of

return.1 For the time horizon considered, the values of the Rf are all zero.

The same block displays estimated beta (β) values for each stock as well as the estimated

portfolio beta (βp). The empirical model used for estimation is the conventional, ordinary least

squares, 5-factor Capital Asset Pricing (CAPM) model, where t = time in days:

(Rit - Rft) = αi + βi(Rmt - Rft) + γiSMBt + δiHMLt + εiRMWt + ζiCMAt + eit (1)

The multiple regression results, for each stock, are reported in Appendix 1. The entire set and

subsets of the independent variables were used, but at the end I selected regressions in which

all independent variables considered were significant after correcting for serial correlation. The

results passed the F test but suffered from low R2 (especially the LLY stock) or low R"! (all other

stocks). I could not reject the H0 = 0 for the intercept coefficients, which is in support of the

CAPM. In conjunction with stocks LLY, WMT, DE and NFG, I could not reject the H0 ≤ 1 for the

beta, which is in some support of the CAPM. For stock CAT, I could not reject the H0 = 1 for its

beta, which is in full support of the CAPM. More details from the performed regressions are

reported in Appendix 1. My beta estimates are not far from those reported online [see Yahoo

1 For the computation of geometric summary measures, I used Excel:

=EXP(AVERAGE(LN(...)))-1

=EXP(VAR.S(LN(...)))-1

=EXP(STDEV.S(LN(...)))-1

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(2022)], but I trust my estimates more because they are based on data generously provided by

French (2022), data that has been recognized as more inclusive and thus more informative.

Table 2 displays results of the Equally Weighted Portfolio (BLOCK 2), Maximum Expected

Return Portfolio (BLOCK 3), Minimum Risk Portfolio (BLOCK 4), Tangency Portfolio (BLOCK 5),

and the Shorted Portfolio (BLOCK 6).

Expected Portfolio Return (Rp) and Portfolio Standard Deviation (Sp) are computed as

suggested by the traditional, widely known and used, Mean-Variance Model of Markowitz

(1952). 2

Using Excel, I computed Rp and Sp as follows: I calculated matrix X (stock rate or return minus

corresponding mean), a 219x5 matrix, and then I deposited it along with n in the functions box.

In turn, using the command =MMULT(TRANSPOSE(X),X/(n-1)) I calculated the below Variance- Covariance (VarCov) matrix which I also deposited in the functions box.

Frequentist Variance-Covariance matrix

LLY WMT CAT DE NFG

LLY 0.000362 3.79E-05 -7.1E-07 1.46E-05 1.98E-05

WMT 3.79E-05 0.000104 3.86E-05 5.12E-05 3.74E-05

CAT -7.1E-07 3.86E-05 0.000257 0.000225 9.78E-05

DE 1.46E-05 5.12E-05 0.000225 0.000339 9.96E-05

NFG 1.98E-05 3.74E-05 9.78E-05 9.96E-05 0.000189

Thereafter, I calculated Rp and Sp using the following commands:

Rp = SUMPRODUCT( ...weights... , ...means with dollar signs...)

Sp =SQRT(MMULT(MMULT(...weights... , VarCov),TRANSPOSE(...weights...)))

Finally, using Excel’s “Solver”, I derived, for each BLOCK in Table 2, the weights that optimize

Rp and Sp which I then used, in conjunction with the portfolio’s beta and the summary measures

of market and risk-free rates of return, to calculate the six Performance Measures, defined in

Table 13. (For the results in BLOCK 6, “Solver” was set to maximize the Sharpe Portfolio Measure

with leaving unchecked the box Make Unconstrained Variables Non-Negative.)

2 For a detailed description of the model, the interested reader may browse and / or download a version

of it available by the Hong Kong University of Science and Technology (2022).

3 The Investor can also use the Sortino Ratio [Sortino et al. (1994)] to measure portfolio performance.

Sortino modifies the Sharpe ratio by replacing Rf with a target rate of return and standard deviation

with downside deviation (negative returns as its risk measure) in the calculation. A good Sortino Ratio

is one with a score of 2 or above. Without a doubt, the Sortion measure, with a target rate of return,

would be better than the Sharpe measure, but because I do not plan to specify a target rate of return

for my portfolio, which portfolio performance measure is better than another is not my concern.