Risk-Efficient Portfolios; Estimation Error In Essence

(1)

School of Education, Culture and Communication Division of Applied Mathematics

MASTER THESIS IN MATHEMATICS / APPLIED

MATHEMATICS

RISK-EFFICIENT PORTFOLIOS;

ESTIMATION ERROR IN ESSENCE

AUTHOR:

DANIEL KOFI ADOBAH-OTCHEY

Masterarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

(2)

School of Education, Culture and Communication

Division of Applied Mathematics

Master Thesis In Mathematics / Applied Mathematics

Date:

2016-06-10

Project name:

Risk-Efficient Portfolios; Estimation Error In Essence

Author:

Daniel Kofi Adobah-Otchey

Supervisors:

Lars Pettersson, Senior Lecturer Professor Anatoliy Malyarenko

Reviewer:

Richard Bonner, Senior Lecturer

Examiner:

Linus Carlsson, Senior Lecturer

Comprising:

(3)

Abstract

This thesis primarily looks at estimation error problems and other related issues arising in connection with portfolio optimization. With some available assets, a portfolio program or optimizer seeks to distribute a fixed amount of capital among these available assets to optimize some cost function. In this regard, Markowitz portfolio selection basis defines the variance of the portfolio return to being that of the portfolio risk and tries to find an allocation that reduces or minimizes the risk subject to a target mean or expected return. Should the mean return vector and the covariance matrix of returns for the underlying assets be known, the Markowitz problem is said to have a closed-form solution.

In practice, however, an estimation is made from historical data for unknown expected returns and the covariance matrix of the returns, and this brings into the domain several problems such as estimation problems and renders the Markowitz theory impracticable in real-life portfolio applications. Estimators necessary to remedy these problems would be made bare to show how possible it is to tackle such issues.

In the concept demonstration sections, the analysis starts with the price data of 40 stocks and the S&P index. The efficient frontier is introduced and used to show how the estimators take effect.

Finally, implementation is made possible using the R Programming Language to demonstrate the necessary concepts with the conclusion presented at the end.

(4)

Acknowledgements

Special appreciation goes to my family for the encouragement and making it possible to fulfill this Master programme in Financial Engineering.

I am also grateful to my supervisors, Lars Pettersson and Professor Anatoliy Malyarenko, for their continuous and prompt guidance during the writing of this thesis. It is also in their teach-ings that I gained an inspiration to pursue this topic. Their comments and teachteach-ings were of much significance.

(5)

List of Figures

2.7.1 Boxplots illustration . . . 14

2.7.2 Skewness illustration of boxplots . . . 14

3.0.1 Scatter plot and Distributions of stock GS: price and log returns . . . 17

3.1.1 Boxplots of monthly log returns . . . 18

4.2.1 Varying curves as correlation changes . . . 24

4.2.2 Varying returns for n assets . . . 24

4.4.1 Minimum-variance . . . 26

4.7.1 Risk-Return plots for the stocks under consideration . . . 29

4.8.1 Mean-Variance Efficient Frontier, QP (Short Selling Allowed) . . . 30

4.9.1 Mean-Variance Efficient Frontier, QP (Short Selling Not Allowed) . . . 32

4.10.1Tangency Portfolio; Sharpe Ratio in play . . . 33

(8)

List of Tables

3.0.1 Listed Stocks under consideration . . . 16

3.1.1 Monthly log returns summary statistics . . . 19

3.1.1 Monthly log returns summary statistics (continuation) . . . 20

(9)

Chapter 1 Introduction

1.1 Background

Discerning investors are always keen with regards to portfolio risk and return. A good intu-ition of efficient portfolio structure is vital for optimally managing the investment benefits of portfolios. Effective portfolio organization lessens the impact of risk while building up re-turns. For discerning investors, portfolio efficiency holds much importance in cognizance to the estimation of risk and return of assets.

Investors acknowledge the investment benefits that comes with the diversification of effi-cient portfolio of assets. Managing portfolio risk to the best level is an essential component in this regard.

Markowitz (1959, 1987), known for contributing immensely to Modern Portfolio Theory, gives the vintage definition of portfolio selection. Markowitz says a portfolio is efficient if its expected return is maximum for some given level of variance or if there is a minimum variance for some given level of expected return of all portfolios from a given set of securities. His theory offers a solution for a risk-averse investor to the problem of portfolio choice: the optimal portfolios are those that have the lowest risk for a certain return according to a rational investor’s thought. Such portfolios are defined to be mean-variance efficient. Mean-Variance Efficiency (MVE) is a useful framework for defining portfolio optimality and building optimal portfolios and asset allocations.

The model of Markowitz, which in theory is of absolute excellence, relies on returns, vari-ances, and covariance of returns to find the optimal portfolio setting. In practice, the problem revolves around the uncertainty of the parameters, as true parameters that optimize the alloca-tion are unknown to the investor and must be estimated. Such a situaalloca-tion brings to the fact in how well we can reduce or alleviate these estimation errors that arises from parameter estima-tion to a level or point where we can prudently know or say the portfolio under consideraestima-tion is efficient to give out the best output for decisions to be taken. This thesis primarily looks at estimation error problems and other related issues arising in connection with portfolio optim-ization. With some available assets, a portfolio program or optimizer seeks to distribute a fixed amount of capital among these available assets to optimize some cost function. In this regard, Markowitz portfolio selection basis defines the variance of the portfolio return to being that of the portfolio risk and tries to find an allocation that reduces or minimizes the risk subject to a target mean or expected return. Should the mean return vector and the covariance matrix of re-turns for the underlying assets be known, the Markowitz problem is said to have a closed-form

(10)

solution.

In practice, however, an estimation is made from historical data for unknown expected re-turns and the covariance matrix of the rere-turns, and this brings into the domain several problems such as estimation problems and renders the Markowitz theory impracticable in real-life port-folio applications.

1.2 Current Discussions and Findings

How can one invest wealth acquired? Portfolio theory provides a solution to this based on two principles:

• Maximizing the expected return and • Minimizing the risk of the portfolio.

These two principles are somewhat at odds considering that riskier assets have higher ex-pected returns, as investors demand a reward for bearing greater risk. The difference between the expected return of a risky asset and the risk-free rate of return is called the risk premium. Without risk premiums, not every investor would be willing to invest in risky assets.

Markowitz (1959) Mean-Variance (MV) optimization has been, and probably still is, the standard for efficient portfolio construction for over 50 years. Almost every portfolio program or optimizers for asset allocation is based on some form of variation of the Markowitz process. In modern finance, Markowitz method is theoretically necessary. However, the MV optim-ization is sensitive to the uncertainty which comes with the estimates of the expected return and the variance-covariance matrix of the returns and characteristically results in an ambiguous portfolio optimality and poor out-of-sample1 performance. Tests done have demonstrated that equal weighting subjugates unbounded MV optimized portfolios, and they essentially have no practical investment value. DeMiguel et al. (2007) in their literature tested the performance of 14 models, and this, as stated earlier, was regarding unbounded MV portfolio choice with estimation error in context. They found out that, as in Jobson and Korkie (1981), that none of the models seem to have steady improvements over equal weighting and has in practice, made the MV optimization to be used primarily as a convenient frame of reference for imposing provisional or specific constraints and providing a scientific guise for marketing purposes.

The forerunners of the estimate parameter uncertainty with a statistical perspective of the MV optimization include Frost and Savarino (1986, 1988), Jorion (1986, 1992) and Jobson and Korkie (1980, 1981). Interestedly, the majority of traditional research focused in this area has been on utility maximization or unbounded in-sample MV optimization while overlooking the uncertainty of estimates or out-of-sample performance. Examples of such research include the Black and Litterman (1992) technique. While an in-sample investigative context may be manageable and suitable, the unbounded MV optimization, without considering estimation er-ror, leads most of the time to distorted or irrelevant trading and constraint policies that may harmfully affect loads of dollars or any currency of relevance under any portfolio management.

1_{In-sample and out-of-sample terms are typically used in optimization or fitting methods. Running an approach} over a particular data set (say over the last three years) is said to be done in-sample. That is results are evaluated over the sample used to fit it and this should technically give the best possible result. If evaluated over, say next two months, then it is done out-of-sample, then evaluation was made in a time period contrary to the approach it was optimized on. In other words, in-sample refers to data available and out-of-sample is data one wants to

(11)

As mentioned earlier, academic research has mostly concentrated on unbounded in-sample expected utility maximization when it comes to estimation error. An early reference is Bawa et al. (1979). More recent explorations include Jorion (1986) and DeMiguel et al. (2007). However, portfolio optimizers typically prefer MV efficiency for defining portfolio optimality as investors feel more at ease estimating asset returns and risks than parameters related to a utility function. Levy and Markowitz (1979) have shown that portfolios on the MV efficient frontier are regularly good approximations of portfolios that tend to maximize expected utility or many utility functions and return generating processes in practice.

1.3 Practical hindrances and its implication/effect

With every ‘perceived perfect’ model comes with it some limitations. Markowitz’s theory has no adverse impact on the investment outcome. Markowitz model gives the right channel to make investments if and only if the specific correct inputs are known and used. The more severe problem arising is estimation error or the parameter uncertainty that comes with the optimization inputs needed. Risk-return estimates are highly very much uncertain practically, and mostly, sensitivity to the changes in inputs of portfolio optimization leads to ambiguity in portfolio optimality.

Practically, expected returns and covariance matrix of the returns are unknown and as such estimated from historical data. This introduces three well-known problems:

• Financial data is characteristically non-stationary2 _{over extended periods of time. This}

characteristic limits the amount of data that can be essentially used to estimate the mean and covariance of the returns. Conversely, however, the sample covariance has numerous parameters, and this requires large amounts of data to estimate. Considering a portfolio that includes 1000 stocks, then the sample covariance has approximately 500,000 para-meters needed to be estimated. This, as a result, requires alternative estimators of the covariance.

• The estimated means and covariance tend to have a significant influence on the optimal portfolio weights. This is to say, a small change of the estimates may lead to extreme changes in the portfolio weights. These portfolio weights computation are made possible by replacing the mean vector and the covariance matrix of returns with their estimates. • The optimal portfolio is likely to increase a significant amount of estimation errors in

certain directions. This comes from the fact that should the variance of an asset be sig-nificantly underestimated, it appears to be small, and the optimal portfolio assigns a significant weight to it. Similarly, a considerable weight will be allocated if the mean return of an asset appears to be large as a result of being significantly overestimated and results in the risk of the estimated optimal portfolio being typically under-predicted and return, over-predicted.

2_{The financial data here being non-stationary means the mean and variance vary with time with the underlying} probability distribution. Stationarity, on the other hand, means the distribution does not change with time.

(12)

1.4 Thesis Objective

Estimating the expected returns and the covariance matrix of stock returns has always been one of the tackiest points. Estimating these parameters is not a new issue. There is lots of literature which discuss them. Some of this literature include:

• That of Stein (1955) and in this article, Stein presents that the accepted sample statistics are not suitable for multivariate problems.

• Furthermore, a series of papers also describes the issue in detail. These papers include that of Michaud (1989) and Barry (1974).

• Empirical tests done by Chopra and Ziemba (1993) also shows that return estimate errors are predominant in risk estimate errors.

• Also, Monte Carlo simulations are used to estimate the enormity of the problem by Jorion (1992) and Broadie (1993).

Notwithstanding this, the standard statistical method of finding these parameters is usually gathering a history of past stock returns and computing their sample covariance matrix and mean. Unfortunately, this process creates problems that have been well covered by Jobson and Korkie (1980). When the number of stocks under study is enormous, in relation to the number of historical return observations available, the sample covariance matrix is estimated with much error and this is usually the case. This implies the most extreme coefficients in the matrix that have been estimated is likely to take on extreme values; not that it is correct rather because they contain an extreme amount of error. Customarily, any MV program or optimizer or software will latch onto these extreme values and place its biggest stakes on those coefficients which are the most extremely unreliable. Michaud (1989) has called this occurrence “error maximization.”

The key parameters in the Mean-Variance Portfolio (MVP) model are the mean vector and the covariance matrix of the returns of securities in the portfolio under consideration. These parameters are assumed to be known. However, in a practical sense, they need to be estimated from the observed market or financial data. Large historical returns of data are required to get quality statistical estimates for the key parameters. However, as stated in Section 1.3, financial data is characteristically non-stationary over long periods of time and as such only a limited amount of recent historical data is relevant. Long periods of time here could be a time period of 30 years or more and a shorter span, between 10 to 15 years. The estimated covariance matrix and the mean return vectors are the key inputs of the optimization program.

The objective is to look into some procedures or estimators which tend to reduce or more likely, correct the estimation errors and the problems that comes with it to provide a much better framework for better investment output. There are several approaches to remedy these issues. However, in this project, two methods are considered. One particular approach has been proposed to remedy the statistical instability issues that arise from poor statistical estimates of the key parameters and this involves the use of Monte Carlo simulations or Resampling (bootstrap) to find a range of optimal portfolios or in other words using resampling (bootstrap) together with optimization to deal with the bias in the weight of the efficient portfolio. Another approach includes using shrinkage estimates of the covariance matrix. This method seeks to strive a compromise between the instability of the sample covariance estimator and the biases introduced by model-based estimators.

(13)

Chapter 2 Reviewing Fundamental Concepts

Under this chapter, we look at some concepts needed involving diverse mathematical and fin-ancial notions.

2.1 Returns

The aim of investing is to make a profit. The revenue from investing, or the loss, negative rev-enue in this case, is dependent upon price change and the amounts of assets held. All Investors are interested in earnings that are high in relation to the amount of their initial investments made. Returns are dependent on time such as days or months and independent of other units such as dollars or euros. Other types of returns are discussed next.

2.2 Net Returns

Assuming Pt is the price of an asset at time t and dividends are not considered, the net return

from time t_{− 1 to time t is}

R_t =Pt− Pt−1 P_t₋₁ =

P_t P_t₋₁− 1 The above equation also represents one-period returns.

The numerator Pt− Pt₋₁is the revenue under consideration. The denominator Pt₋₁is the initial

investment at the start of the holding period.

A simple mathematical arrangement of the net return stated above gives us the gross return. The gross return for k period is

1+ Rt(k) = Pt P_t_−k = P_t Pt₋₁ P_t₋₁ Pt₋₂ ... Pt−k+1 P_t_−k = (1 + Rt)(1 + Rt₋₁)...(1 + Rt−k+1)

(14)

2.3 Multi-period returns

Multi-period returns, an extension of one-period returns, can be shown as follows. Using the gross return over k periods

1+ Rt(k) = Pt P_t_−k = k₋₁

∏

i=0 (1 + Rt_−i)

2.4 Log Returns

Continuously compounded returns are known as log returns. Rt denotes log returns in this

instance and defined

r_t= log(1 + Rt) = log Pt P_t₋₁ = log(Pt)− log(Pt−1) = pt− pt₋₁

where the log price is pt = log(Pt)

If Rt is small, log returns rt become approximately equal to net returns Rt. That is log(1 + Rt)≈

R_t as the time step ∆t approaches 0. An advantage of using log returns is with its simplicity when multi-period returns are considered. A k-period log return is just the summation of the single-period log returns. The k-period log return here becomes

rt(k) = log(1 + Rt(k))

= log (1 + Rt)(1 + Rt₋₁)...(1 + Rt−k+1)

= log(1 + Rt) + (1 + Rt₋₁) + ... + (1 + Rt−k+1)

= rt+ rt−1+ ... + rt−k+1

2.5 Adjusting for Dividends

Many stocks, listed or not, disburse dividends to shareholders and is accounted for when returns are computed. Equally, bonds do pay interest. Let Dt represent dividends or interest. If Dt is

paid between t_{− 1 and t, the net return at time t is}

R_t= Pt+ Dt P_t₋₁ − 1

(15)

2.6 Random Walk Synopsis

Stock prices play a pivotal role in this project and as such a key knowledge in its behaviour is of much consequence. An aspect not to overlook is that of its random nature and what comes to mind quickly is the random walk hypothesis. The random walk hypothesis or theory states that changes in stock prices are independent of each other and have the same distribution, and presupposes that a stock price’s movement or trend in the past cannot be used to predict its future movement.

Most Mathematical Finance work assumes stock prices follow a log-normal geometric ran-dom walk or geometric Brownian motion. Is this assumption true? There are two assumptions made on the log-normal geometric random walk1:

• Log returns are normally distributed N(µ,σ2_).

• Log returns are mutually independent.

With the sum of normal random variables also being normal, the normality of single-period log returns will imply normality of multiple-period log returns.

Once again, are the assumptions stated above true? The response is a “no.” Investigating the marginal distributions of numerous series of log returns shows that the return density has a bell shape, and this is somehow similar to that of normal densities. This concept will be portrayed in Chapter 3 using one of the stocks to give a much better visualization. However, the tails of the log return distributions are heavy-tailed in comparison to that of a normal distribution.

Heavy-tailed distributions play a fundamental role in finance and such as stock returns and other changes in market prices usually are heavy-tailed. This is to say that most extreme values occur in this regard and more likely than would be predicted. In finance applications, one needs to be apprehensive when the return or log return distribution is heavy-tailed. Such distributions have the possibility of an enormous negative return, which could, in an instance of consideration, entirely deplete a company’s capital reserves. Significant positive returns are also a matter of concern should short selling be done.

2.7 Box-plots

Box-plots, also known as box and whisker plots, are a useful graphical means for comparing data and can be displayed in a vertical or horizontal format. It is used to display patterns of quantitative data based on these measures: minimum, maximum, median (Q2), first quartile (Q1), third quartile (Q3). Based on this, the range and interquartile range (IQR) can be found which are also common measures showing the spread in data. Figure 2.7.1 shows the measures stated earlier. The difference between the third and first quartile gives the interquartile range (IQR) as seen in figure 2.7.1. The “whiskers” show the location of the maximum and minimum points. Some datasets will display, surprisingly, high maximums or surprisingly low minimums called outliers.

Boxplots can also provide information regarding the shape of data. Most likely being that the data is skewed to the right, left or it is symmetric (Figure 2.7.2).

1_{The process P}

t= P0exp(rt+ rt−1+ ... + r1) is a geometric or exponential random walk. In the case where r1,r2... are i.i.d. N(µ, σ2), then Ptbecomes log-normal geometric random walk having N(µ, σ2).

(16)

Outlier : More than 3/2 times of the upper quartile

Median Upper Quartile Maximum

Lower Quartile

Minimum

Outlier : Less than 3/2 times of the lower quartile

Figure 2.7.1: Boxplots illustration

Symmetric Positively Skewed

Negatively Skewed

(17)

Chapter 3 Data

The portfolio under consideration is based on ten (10) different industries listed and ranked on Forbes World’s biggest Public Companies for 2015. The focus is primarily on the United States of America (USA) industries. Concerning each industry from the list, four (4) compan-ies (ranked top 4) are chosen to represent a total of 40 stocks.

The selected industries are Gas and Oil Operations, Discount Stores, Software Programming and Computer Hardware, Industrial Conglomerates, Diversified Insurance, Investment Ser-vices, Pharmaceuticals, Telecommunication SerSer-vices, Major Banks and Household & Personal Care. Table 3.0.1 on page 16 shows the stocks under consideration in this portfolio.

The period under consideration is a 10-year period from 31st December 2005 to 31st December 2015. Historical stock prices were obtained from Yahoo! Finance (www.finance.yahoo. com) to which they are readily available. The benchmark in focus is the S&P 500 stock market index (with code ^GSPC) as this primarily focuses on stocks issued and traded on the American stock exchanges.

Also, the risk-free rate used is 0.03118_{≈ 3.12% from the US Treasury; a 10-year treasury by} month (http://www.multpl.com/10-year-treasury-rate/table/by-month). The rate was obtained by averaging the monthly risk-free rates for the period under review. With a 10-year period in focus, each stock has a lifespan of 120 monthly price data. The histor-ical stock price data is converted to monthly return (log returns). The Adjusted Closing Price is used and also, dividends have already been accounted. The values of concern here are the initial investment value and the ending investment value. Due to this calculation process, one observed value is lost making the data be of length T= 119.

Using Goldman Sachs (GS) as a demonstration, the distribution of both the prices and log returns with a scatter plot are shown in Figure 3.0.1 on page 17. The scatter plot gives a visu-alization of how random the prices behave. The histogram for the prices is also an indication that stock prices are indeed not normally distributed. In this regard, the distribution of the log returns appears to be bell-shaped. A normal curve is generated using the R programming language to support this. However, just as mentioned, normality assumption remains fairly reasonable in this context.

(18)

Table 3.0.1: Listed Stocks under consideration Security Code/Ticker Symbol

Exxon XOM

ConocoPhilips COP

Valero Energy VLO

Chevron CVX

Wal-Mart Stores WMT Costco Wholesale COST Target Corporation TGT Dollar Tree DLTR Apple AAPL Hewlett-Packard HPQ Microsoft MSFT Oracle ORCL General Electric GE HoneyWell International HON United Technologies UTX

3M Company MMM

American International Group, Inc. AIG AllState Corporation ALL

Metlife MET

Hartford Financials HIG Berkshire Hathaway BRK-B Goldman Sachs GS Morgan Stanley MS Black Rock BLK Pfizer PFE Merck & Co MRK McKesson Corporation MCK Abbott Laboratories ABT

CenturyLink CTL

Verizon VZ

AT&T T

Level 3 Communications LVLT JP Morgan Chase JPM Wells Fargo and Company WFC

Citigroup C

Bank of America BAC Estee Lauder Companies EL Procter & Gamble PG Colgate-Palmolive CL Kimberly-Clark Corporation KMB S&P 500 Index ^GSPC

(19)

0

40

80

120

100

150

200 Scatter Plot of GS

Index

Pr

ice

0 5 10 100 150 200 Price Obser v ation Distribution of GS Price 0 2 4 6 −0.2 0.0 0.2 Log Returns Obser v ation

Distribution of GS Log Returns

(20)

3.1 Descriptive Statistics

The Table 3.1.1 shows the basic measures about descriptive statistics of the monthly log returns for the stocks under consideration.

A more subtle illustration is to use Boxplot to give an overview of the distribution of the data. Much has been discussed in Section 2.7. This will help display patterns of quantitative data such as the median, quartiles and interquartile range.

Figure 3.1.1 shows stocks such as LVLT, VLO and C have relatively wide dispersion (broader interquartile range), while stocks such as KMB, CL, PG and BRKB have relatively low level of dispersion (narrow interquartile range). Considerably the outliers to the right shows more of the log returns being positive in comparison to the negative values with most data values skewed to the right.

XOMCOP VLO CVX WMT COSTTGT DLTR AAPLHPQ MSFT ORCLGE HONUTX MMMAIG ALL METHIG BRKBGS MS BLK PFE MRK MCKABT CTLVZ T LVLTJPM WFCC BACEL PGCL KMB −1.0 −0.5 0.0 0.5 1.0 1.5

Boxplots of log returns for the assets

Assets

Log Retur

ns

(21)

Table 3.1.1: Monthly log returns summary statistics

XOM COP VLO CVX

Min. :-0.106920 Min. :-0.145599 Min. :-0.324672 Min. :-0.141612 1st Qu.:-0.042200 1st Qu.:-0.047742 1st Qu.:-0.080308 1st Qu.:-0.044386 Median :-0.002655 Median :-0.013036 Median :-0.024078 Median :-0.014079 Mean :-0.003856 Mean :-0.002583 Mean :-0.003289 Mean :-0.006348 3rd Qu.: 0.026363 3rd Qu.: 0.049375 3rd Qu.: 0.072155 3rd Qu.: 0.028855 Max. : 0.113967 Max. : 0.333028 Max. : 0.386828 Max. : 0.158971

WMT COST TGT DLTR

AAPL HPQ MSFT ORCL

Min. :-0.21326 Min. :-0.1987416 Min. :-0.222736 Min. :-0.206523 1st Qu.:-0.08028 1st Qu.:-0.0506441 1st Qu.:-0.051920 1st Qu.:-0.053947 Median :-0.02729 Median :-0.0129579 Median :-0.018394 Median :-0.018408 Mean :-0.01974 Mean : 0.0002379 Mean :-0.007615 Mean :-0.009552 3rd Qu.: 0.02596 3rd Qu.: 0.0576578 3rd Qu.: 0.036881 3rd Qu.: 0.041253 Max. : 0.39982 Max. : 0.3009437 Max. : 0.178358 Max. : 0.200671

GE HON UTX MMM

AIG ALL MET HIG

Min. :-1.238308 Min. :-0.25289 Min. :-0.267385 Min. :-0.664334 1st Qu.:-0.053425 1st Qu.:-0.04979 1st Qu.:-0.049065 1st Qu.:-0.058843 Median :-0.006521 Median :-0.01238 Median :-0.009633 Median :-0.006637 Mean : 0.023492 Mean :-0.00374 Mean :-0.001371 Mean : 0.003458 3rd Qu.: 0.061600 3rd Qu.: 0.02176 3rd Qu.: 0.040112 3rd Qu.: 0.049058 Max. : 1.854547 Max. : 0.55826 Max. : 0.522200 Max. : 1.379244

(22)

Table 3.1.1: Monthly log returns summary statistics (continuation)

BRKB GS MS BLK

Min. :-0.150972 Min. :-0.209179 Min. :-0.269748 Min. :-0.303221 1st Qu.:-0.033254 1st Qu.:-0.056160 1st Qu.:-0.057275 1st Qu.:-0.060408 Median :-0.008413 Median :-0.014027 Median :-0.009738 Median :-0.014579 Mean :-0.006821 Mean :-0.002978 Mean : 0.002783 Mean :-0.009823 3rd Qu.: 0.020685 3rd Qu.: 0.057065 3rd Qu.: 0.058787 3rd Qu.: 0.045728 Max. : 0.153370 Max. : 0.321775 Max. : 0.573923 Max. : 0.392643

PFE MRK MCK ABT

CTL VZ T LVLT

Min. :-0.127760 Min. :-0.131138 Min. :-0.094971 Min. :-0.4210296 1st Qu.:-0.046923 1st Qu.:-0.046229 1st Qu.:-0.042458 1st Qu.:-0.0830621 Median :-0.011266 Median :-0.008557 Median :-0.008960 Median :-0.0094075 Mean :-0.002562 Mean :-0.008133 Mean :-0.006684 Mean : 0.0002872 3rd Qu.: 0.034636 3rd Qu.: 0.019984 3rd Qu.: 0.021031 3rd Qu.: 0.0926748 Max. : 0.378147 Max. : 0.112063 Max. : 0.169175 Max. : 0.9444617

JPM WFC C BAC

Min. :-0.218127 Min. :-0.340177 Min. :-0.522754 Min. :-0.548887 1st Qu.:-0.059837 1st Qu.:-0.041993 1st Qu.:-0.056006 1st Qu.:-0.063759 Median :-0.013201 Median :-0.011004 Median :-0.005751 Median :-0.003731 Mean :-0.006301 Mean :-0.006891 Mean : 0.017219 Mean : 0.006357 3rd Qu.: 0.047505 3rd Qu.: 0.025661 3rd Qu.: 0.079302 3rd Qu.: 0.048026 Max. : 0.264597 Max. : 0.444550 Max. : 0.861482 Max. : 0.760721

EL PG CL KMB

(23)

Chapter 4 The Language of Portfolio

4.1 Risk - Return

Historical returns are summarized by expected value and standard deviation, usually referred to as risk in many finance applications. Risk here can be the standard deviation of an asset or portfolio under consideration. For an asset, its expected return, variance, and covariance are expressed in the following manner respectively:

Ri= N

∑

t=1 Rit N (4.1.0.1)

where Rit is the tth observed return on the ith asset , and N is the number of observed returns

on asset i. σi2= N

∑

t=1 (Rit− Ri)2 N_{− 1} (4.1.0.2) cov(Ri, Rj) = σi j = N

∑

t=1 (Rit− Ri)(Rjt− Rj) N_{− 1} (4.1.0.3)

Furthermore, the correlation coefficient between two assets or stocks is between -1 and 1 and is related as:

Corr(I, J) = ρ =cov(Ri, Rj) σiσj

= σi j σiσj

(4.1.0.4) A collection of assets or securities makes a portfolio and also basically a weighted average of the assets have the weights summing to one. The weights specify what portions of the total investment are needed for allocation among the assets. For a portfolio of assets, the portfolio is a linear combination of the individual assets in the portfolio. Let

w_irepresent the weight of asset i in portfolio P,

µpbe the expected or mean return of the portfolio P, and

(24)

The expected return and variance of a portfolio are expressed as follows: µp= E(Rp) = N

∑

i=1 w_iR_i (4.1.0.5) σ_p2= N

∑

i=1 N

∑

j=1 w_iw_jσi j (4.1.0.6) = N

∑

i=1 (w2_iσ_i2) + N

∑

i=1 N

∑

j=1 i6= j (wiwjσi j) (4.1.0.7)

Expressing the above in matrix notation, the expected or mean return on the portfolio is µp= E(Rp) = w>RRR (4.1.0.8)

and the variance of the portfolio having weights w is

w>ΣΣΣw (4.1.0.9) where w> = (w1, w2, ..., wn) is the vector of portfolio weights such that w1+ w2+ ... + wn=

1>w= 1 , RRR= (R1, R2, ..., Rn) is the vector of returns and ΣΣΣ is the positive definite, symmetric

N_{× N variance-covariance matrix of the returns of the N assets shown as}

Σ Σ Σ=      σ₁2 σ12 σ13 ··· σ1n σ21 σ22 σ23 ··· σ2n .. . ... ... . .. ... σn1 σn2 σn3 ··· σn2      (4.1.0.10)

The weights also satisfy the following constraints:

1. N

∑

i=1 w_i= 1 , 2. 0_{≤ w}i≤ 1

As the weights satisfy the two constraints listed above, diversification tends to reduce risk. The variance expression stated in Equation (4.1.0.7) also shows the expediency of diversific-ation in reducing risk attributed to the correldiversific-ation that exists between the asset returns. An investor will habitually settle for lower expected returns than selecting riskier assets with high mean returns when reducing risk through diversification. Markowitz’s theory of optimal port-folios relates, in this sense, to the optimal trade-off between the portfolio’s mean return and its standard deviation. Why does diversification work? Assuming a portfolio of N assets with equal amounts of vested capital in each asset and with such an assumption in mind, the variance

(25)

σ_p2= N

∑

i=1 1 N 2 σ_i2+ N

∑

i=1 N

∑

j=1 i_{6= j} 1 N 1 Nσi j (4.1.0.11) = 1 N N

∑

i=1 σ_i2 N +N− 1 N N

∑

i=1 N

∑

j=1 i6= j σi j N(N_{− 1)} (4.1.0.12)

From Equation (4.1.0.12), both of the terms in the brackets are averages. For the second term in the bracket, there are N values of i and(N− 1) values of j, as i cannot be equal to j. This implies there are N(N_{− 1) covariance terms. The second term, therefore, yields an average as} it is the summation of the covariances divided by the number of covariances.

Mathematically, σ_p2= 1 Nσ 2 i + N_{− 1} N σi j (4.1.0.13) From the above, as the number of assets N increases, the first term approaches zero with an increment to infinity, while the second term approaches σi j, the covariance term.

σ_p2= lim N→∞ 1 Nσ 2 i+ N_{− 1} N σi j (4.1.0.14) = σi j

This implies the individual risk of securities can be diversified, and what contributes to the total portfolio risk, the covariance, cannot be diversified away and this is inferred to as the system-atic or market risk1.

It is possible, however, to sell an asset not owned. This is known as short selling and it basically involves borrowing the asset and then selling it to a buyer. At a later period, one buys back the asset and returns it to the lender. The constraint 0_{≤ w}i≤ 1 is ignored should short

selling be allowed. The theory becomes much less complex in this scenario. Investors usually short sell to profit from falling stock prices. The restriction on short selling is that

N

∑

1=1

x₁= 1 (4.1.0.15)

and this intuitively implies that the total lending and borrowing of assets must sum to the total capital used.

Another definition is by Lintner(1965). His definition which seems to be more realistic states that when an investor sells stock short, cash is not earned instead, held in collateral. In the process as well, the investor must put up an amount equal to the amount of stock short sold. His constraint is

N

∑

1=1

|xi| = 1. (4.1.0.16) 1_{Diversification cannot do away with systematic risk. Hedging or using the best asset allocation approach} provides a better solution in controlling systematic risk.

(26)

4.2 Feasible Portfolio Opportunities

Considering the case where n= 2 risky assets with returns R1and R2and proportionated with

wand 1_{− w weights respectively. Also, ρ}12is the correlation coefficient between the two risky

assets. The mean return on the portfolio is µp = wR1+ (1− w)R2 with the variance of the

portfolio being σ_p2= w2_σ2

1+ (1− w)2σ22+ 2w(1− w)σ12.

When the correlation value is changed, as seen from Figure 4.2.1, the curve tends to change in that respect.

Figure 4.2.1: Varying curves as correlation changes

This outcome brings the notion, in generality, of possible portfolio opportunities for n as-sets corresponding to a set of points in the risk-return space that corresponds to the returns of portfolios for the n assets. That is to say, for cases where n_{≥ 3, with varying weight and} correlation combinations, a reasonable figure is acquired (Figure 4.2.2).

Figure 4.2.2: Varying returns for n assets

(27)

allowed). From the figure above, with having leverage, that is the ability to sell short, investors are open to more possibilities of higher returns (and also higher risks). In both cases, for the same amount of risk, there will always be portfolios yielding higher returns than others. These portfolios are called efficient portfolios, and they lie on the efficient frontier. Intuitively, such portfolios are more desirable by prudent investors and discussed in Section 4.4.

4.3 The Sharpe Ratio

The Sharpe ratio is the measure used in the calculation of risk-adjusted returns. It is defined as the expected excess return per unit of risk Ri−Rf

σi where Rf is the risk-free rate.

R_f is the minimum rate of return, and any prudent investor requires assets to generate returns above the risk-free rate. Risk premium Ri− Rf denotes this difference and usually referred to

as excess return. Decreasing a portfolio risk when an asset is added to a portfolio increases the Sharpe ratio which is a diversification benefit. The higher the value of a Sharpe ratio, the better a portfolio’s risk-adjusted performance. A negative Sharpe ratio shows that the risk-free rate would be of much value than the asset being analyzed. Sharpe ratio is basically identifying or noticing the return an investor makes for the additional risk held for having a risky asset over the risk-free rate Rf.

4.4 The Efficient Frontier

For the same amount of risk, there will always be portfolios yielding higher returns than others. These portfolios, as mentioned using illustrations in Section 4.2, are called efficient portfolios, and they lie on the efficient frontier. The efficient frontier is also the set of all efficient (feasible) portfolios. For a given value of the mean or expected return, the possible point with the smallest risk lies on the left boundary; this is referred to as the Minimum-Variance Portfolio (MVP). For a given value of risk, investors will prefer the portfolio with the largest mean return, which is attained at the upper left boundary point of the feasible portfolio area. The top part of the portfolio possibilities curve or the minimum-variance set, as mentioned, is called the efficient frontier. Figure 4.4.1 gives a vivid outlook of what has been described.

In its generality, the efficient frontier is the set of portfolios with

• Expected return greater than other portfolios having the same or lesser risk or • Lesser risk than other portfolios also having same or greater return.

Generating or finding the efficient frontier will be discussed in detail in the next section as this plays a fundamental aspect in this project.

4.5 Generating the Efficient Frontier

Finding the efficient frontier involves finding weights of the assets that make up each portfolio. This approach is equivalent to the computation of the efficient frontier discussed in the previous section, with specified risk or return conditions coming up with an optimal set of weights.

(28)

Figure 4.4.1: Minimum-variance

Assuming there exist a target value µp for the mean or expected return of the portfolio.

When N = 2, the target expected return is only reached by one portfolio, and the weight value wfinds µpas µp= wµ1+ (1− w)µ2= µ2+ w(µ1− µ2).

For N_{≥ 3, an infinite number of portfolios will achieve the target return µ}p. When the target

return µp is varied, the Quadratic Programming (QP) model will result in a set of efficient

points showing the behaviour of a minimum variance for a given target or expected return and a maximum return for a given risk. These are efficient portfolios as well. The return achieved with the smallest or minimum variance is called the “efficient” portfolio. As the number of assets N increases, the MV optimization problem becomes more computationally complicated to solve.

Quadratic programmingbecomes applicable here when such a scenario arises. QP is used to find efficient portfolios with an arbitrary number of assets. QP allows one to impose con-straints such as limiting short sales. The goal primarily is to find the efficient portfolio with optimal portfolio weights. Lagrange multipliers can be applied to come up with an explicit solution for the efficient portfolio weights. A constraint can be imposed, and with quadratic programming show how to solve for the efficient portfolios.

We look at the case whereby short selling is allowed. The two constraints to impose to minimize the objective function which is variance on the portfolio having weights w>ΣΣΣw are:

     w>RRR= µp w>1= 1

That is, given a target return µp, the efficient portfolio minimizes

(29)

subject to      w>RRR= µp w>1= 1 (4.5.0.2)

Using Lagrange multipliers to solve the above:

L= w>ΣΣΣw− λ1(w>RRR− µp)− λ2(w>1− 1) (4.5.0.3)

where w>RRR_{− µ}_p= 0 , w>1_{− 1 = 0 and λ}1, λ2, the multipliers, which belongs to the set of real

numbers.

Based on First Order Condition,              ∂ L ∂ w = 2ΣΣΣw− λ1RRR− λ21= 0 ∂ L ∂ λ1 = µp− w >_R_R_R_{= 0}_{⇒ µ} p= w>RRR= RRR>w ∂ L ∂ λ2 = 1− w >₁₁₁_{= 0}_{⇒ 1 = w}>₁₁₁_{= 1}>_w (4.5.0.4)

From the first part of (4.5.0.4) , w= λ1ΣΣΣ−1RRR+λ2ΣΣΣ−1111

2 . Putting this in the last two equations of

(4.5.0.4), the following is derived:      λ1RRR>ΣΣΣ−1R+λRR 2RRR>ΣΣΣ−1111 2 = µp λ1RRR>ΣΣΣ−11+λ11 2111>ΣΣΣ−1111 2 = 1 (4.5.0.5)

From (4.5.0.5) , let a= 111>ΣΣΣ−1111 , b= RRR>ΣΣΣ−1111 and c= RRR>ΣΣΣ−1RRRwhere a, b, c are constants. From this, (4.5.0.5) becomes

     λ1c+λ2b 2 = µp λ1b+λ2a 2 = 1 (4.5.0.6)

Solving for (4.5.0.6), the multipliers λ1, λ2becomes:

     λ1= 2(aµ_ac_−bp−b)2 λ2= 2(c_−bµp) ac−b2 (4.5.0.7)

Plugging λ1, λ2back into w, w becomes

weff=

µp(aΣΣΣ−1RRR− bΣΣΣ−11) + R11 RR(cΣΣΣ−1111− bΣΣΣ−1)

ac_{− b}2 (4.5.0.8)

w and weffin this context are same. weffis used here to demonstrate optimal portfolio weight

vector or the weight vector of the efficient portfolio.

The variance in the MVP or variance of the efficient portfolio is: σ_p2= w>ΣΣΣw=

aµ2_p_{− 2bµ}p+ c

(30)

To find the coordinates(variance, return) with the lowest or minimum risk of the MVP, usu-ally called the Global Minimum-Variance Portfolio(GMVP), we differentiate (4.5.0.9) above in finding the mean or return for the GMVP,

∂ σ2_p ∂ µp =2aµp− 2b ac_{− b}2 = 0 ⇒ µp= b a (4.5.0.10)

With the mean now known, we plug (4.5.0.10) into (4.5.0.9) to find the variance or risk.

σ_p2= a(b_a)2− 2b(b a) + c ac_{− b}2 = 1 a (4.5.0.11)

The corresponding weight vector is given by: weff=

ΣΣΣ−1111

a (4.5.0.12)

It is worth noting that the weight vector of GMVP does not depend on the expected returns of the stocks or assets under consideration.

Another part worth noting is considering the scenario whereby short selling is not allowed. Similar to the above, all that is required is to set limits to the weights with wi≥ 0. This scenario

does not produce an explicit result. However, QP would still produce results under the linear equality and non-negativity constraints used in this case.

That is, given a target return µpalso, the efficient portfolio minimizes

min w>ΣΣΣw (4.5.0.13) subject to              w>RRR= µp w>1= 1 w_{≥ 0} (4.5.0.14)

R Programming Language would be used to demonstrate this.

4.6 Fundamental Limitations of the MVE

(31)

applicable hardly in principle. In addition to this, investors might show diverse utility func-tions other than quadratic preferences. Furthermore, investors also might have multi-periodic investment horizon, in contrast to the Mean-Variance one-period framework, which assumes will never change their asset allocation once chosen as they focus on a single period horizon.

The more severe problems or limitations, which is of much importance in the practical application of MVE are instability and ambiguity. By instability and ambiguity, small changes in the parameter inputs (estimated) lead to significant modifications in the optimized portfolio. The practical implementation of the mean-variance efficient model requires the determination of the efficient frontier. The expected returns of the assets and the covariance matrix of the returns of the assets are the required inputs. Typically, as made mention earlier in this project work, these input parameters are estimated using historical data. Markowitz’s mean-variance model is a powerful optimization tool. However, the estimation errors in these input parameters can overcome the theoretical benefits of the mean-variance model. These estimation errors may result in much higher (or lower) allocation which in fact should not be. It also leads to investment in non-relevant portfolios. In effect, small changes in the input parameters often lead to very different portfolio weights and, consequently, diverging efficient frontiers.

4.7 Concept Demonstration: The Efficient Frontier

As mentioned, R Programming language will be used to illustrate some parts above in Chapter 4 to demonstrate some aspects of the efficient frontier.

The Risk - Return figure (Figure 4.7.1) for the monthly (log) returns is

−0.02 −0.01 0.00 0.01 0.02 0.00 0.02 0.04 0.06 Risk Monthly Retur ns

Risk−Return space

(32)

The Riand σi2of the assets under consideration are given in Table 4.7.1 with the minimum

return asset being -0.01974329 _{≈ -1.97% of AAPL and the maximum return, 0.02349239 ≈} 2.35% for AIG. In this regard, the minimum variance is 0.001655623 for KMB and the asset with maximum risk is 0.07590609 which happens to be the also belong to AIG. A closer look at the table shows almost all the assets having negative Ri. This goes to confirm the tails of log

return distributions being heavy-tailed as such distributions have the possibility of an enormous negative return.

4.8 Short Selling Scenario

To further demonstrate using R, assuming a target return portfolio µp of 0.05 = 5% is given,

the minimum-variance portfolio is 0.00959093. These values beat that of the assets; higher returns obviously demands greater risk.

To find the efficient frontier involving risk-efficient portfolios, we consider a range of target returns portfolio µp from a minimum of 0.01 = 1% to a maximum of 0.20 = 20%. Negative

returns are not taken into account since that would not be prudent. The easiest way to identify risk-efficient portfolios are that they have returns higher than the returns of the minimum-variance portfolio. The MVP with a given target return of 0.01 = 1%, results in 0.001410531 value for variance. This can be seen in Figure 4.8.1. The MVP has been indicated with ared

dot. 0.05 0.10 0.15 0.20 0.00 0.03 0.06 0.09 Risk Retur ns

Mean−Variance Efficient Frontier

(33)

Table 4.7.1: Expected returns of each asset with its Risk

Asset

Return

Risk

XOM

-0.003855844

0.002249753

COP

-0.002583297

0.005847905

VLO

-0.003288484

0.012367214

CVX

-0.00634827

0.003348626

WMT

-0.004266494

0.002270595

COST

-0.011667605

0.002640548

TGT

-0.003980141

0.004727939

DLTR

-0.018776695

0.005104809

AAPL

-0.01974329

0.009528996

HPQ

0.000237871

0.007508226

MSFT

-0.007615181

0.005089515

ORCL

-0.009552044

0.004787313

GE

-0.002613567

0.007284232

HON

-0.010369919

0.004663845

UTX

-0.006054081

0.003217766

MMM

-0.008283325

0.003253121

AIG

0.023492392

0.075906092

ALL

-0.003739725

0.008385931

MET

-0.001370832

0.010812496

HIG

0.003457498

0.036816237

BRKB

-0.006820906

0.002592562

GS

-0.002977512

0.008623238

MS

0.002782823

0.012853445

BLK

-0.009823001

0.008393883

PFE

-0.005448425

0.003017837

MRK

-0.006909509

0.00413956

MCK

-0.011661631

0.004882724

ABT

-0.00872702

0.00230899

CTL

-0.002561992

0.004389702

VZ

-0.008133287

0.002587769

T

-0.006684361

0.002546517

LVLT

0.000287205

0.028941543

JPM

-0.006301332

0.007716237

WFC

-0.006890913

0.008922788

C

0.017219013

0.026101788

BAC

0.00635731

0.022545939

EL

-0.014164267

0.006452895

PG

-0.004823293

0.001971054

CL

-0.009354931

0.001710938

KMB

-0.01011483

0.001655623

(34)

4.9 No Short Selling Scenario

Unlike the scenario where short selling is allowed, not every desired portfolio return is feasible when short selling is disallowed. With the R-codes to demonstrate and using the same inputs when short sales are disallowed, a given target return portfolio µpof 0.05 = 5% is not feasible

or achievable when short sales are disallowed. Further reduction from 0.05 = 5% to 0.03 = 3% results in the same. However at 0.023 = 2.3% it is feasible. Thus, in this case, the target return portfolio cannot outmatch the maximum return of AIG, which happens to be the asset with the highest return. The MVP becomes 0.0682386.

To find the efficient frontier, the same range of target return portfolio µpfrom a minimum

of 0.01 = 1% to a maximum of 0.20 = 20% is maintained. Portfolio returns not feasible will not be considered as the R program runs. The MVP has with a given minimum target return of 0.01 = 1% results in 0.01162432 variance; a value much higher than that of shorts sales allowed. Figure 4.9.1 illustrates this.

0.010 0.015 0.020 0.02 0.04 0.06 variance Retur ns

Mean−Variance Efficient Frontier

Figure 4.9.1: Mean-Variance Efficient Frontier, QP (Short Selling Not Allowed)

4.10 Tangency Portfolio

(35)

and tangent to the efficient frontier is known as the Capital Market Line(CML). It is defined as µ = Rf+ σp Ri− Rf σi (4.10.0.1) = Rf+ σpSR

where SR is Sharpe ratio. Rf is the intercept on the y-axis and SR is the slope of the CAL.

The Sharpe ratio of any efficient portfolio is also the market portfolio. The point of tangency on the efficient frontier is tangency portfolio and it is also the point where the Sharpe Ratio is maximum. The tangency portfolio(maximum Sharpe ratio):

• with short sales allowed gives 0.073969 ≈ 7.4% as the value of the return and 0.0184744 as the risk and

• with short sales disallowed is 0.0233668 ≈ 2.34% for the return and 0.0738901 as the risk and

These values indicate that with short sales allowed, the data will have the better a portfolio’s risk-adjusted performance. Figure 4.10.1 shows the tangency portfolio (maximum Sharpe Ra-tio) 0.05 0.10 0.15 0.20 0.00 0.03 0.06 0.09 variance Retur ns

Mean−Variance Efficient Frontier

(a) Short Sales allowed

0.010 0.015 0.020 0.00 0.02 0.04 0.06 variance Retur ns

Mean−Variance Efficient Frontier

(b) Short Sales Not Allowed

(36)

Chapter 5 Estimation Error In Essence

5.1 Estimation Error

Markowitz assumes that the input parameters of the MV portfolio optimization are 100% known. However, these inputs, as stated earlier, are measured from historical data and fed into the model making it seem they were known perfectly. Such assumed inputs have been re-searched (Jobson and Korkie et al.) to have significant statistical and specification errors. The MV analysis does not take into account the uncertainty integrated into the input parameters.

The covariance matrix of asset returns and the sample mean used in creating the MVP act below par out-of-sample because of estimation error. Furthermore, it is accepted that estimation error in the sample mean is much greater than that in the sample covariance matrix. It is for this reason that scholars are focused on the minimum-variance portfolio as it relies heavily on estimates of the covariance matrix, and performs better out-of-sample. Nonetheless, the minimum-variance portfolios are also relatively sensitive to estimation error, and its weights change substantially over time. This is to say that the MVE procedure overuses statistically estimated information and magnifies estimation errors Estimation error can be defined as the difference between the estimated distribution parameters and the actual value of parameters when samples are not large enough. The impact of estimation error on portfolio optimization could be far-reaching.

Portfolio optimization suffers from estimation error or error maximization as highlighted by Bernd Scherer. In Scherer’s article (Scherer 2002), he states that “The optimizer tends to pick those assets with very attractive features (high return and low risk and/or correlation) and tends to short or deselect those with the worst features. These are exactly the cases where estimation error is likely to be highest, hence maximizing the impact of estimation error on portfolio weights. The quadratic programming optimization algorithm takes point estimates as inputs and treats them as if they were known with certainty (which they are not), will re-act to tiny differences in returns that are well within measurement error.” This clearly gives a good reason why mean-variance optimized portfolios suffer from instability and ambiguity. As stated earlier, there is nothing wrong with Markowitz’s procedure or mechanism. How-ever, a refinement of inputs is much required to reduce the estimation error effect on portfolio optimization.

(37)

5.2 Finding Estimators To Reduce Estimation Error

In Section 5.3 and Chapter 6, we will look at approaches that seem to alleviate or reduce or control estimation errors arising from input parameters used in the MV optimization model to derive the efficient frontier. As mentioned in Section 1.4, Resampling and shrinkage ap-proaches will also be discussed. Chapter 6 will primarily focus on the shrinkage estimates of the covariance matrix in this regard.

As already mentioned, to control estimation error in MV optimization inputs, a strong stat-istical technique called resampling (bootstrapping) will be used and this will define a new efficient frontier consistent with most applications of MV efficiency. Also, the shrinkage es-timates of the covariance matrix which seeks to strive a compromise between the instability of the sample covariance estimator and the biases introduced by model-based estimators will also be discussed. The whole idea is to find better estimates to reduce estimation errors or “error maximization” as referred to by Michaud (1989).

5.3 Portfolio Resampling

To control estimation error in MV optimization inputs, a commanding statistical procedure called resampling is applied which is an application of Monte Carlo simulation. The resampling process defines a new efficient frontier consistent with most applications of MV efficiency. In such a process, the bootstrap method is applied. “Bootstrap” was coined by Bradley Efron (1979) and is associated with the phrase “pulling oneself up by one’s bootstraps.” One cannot exactly simulate sampling from an unknown population. However, a sample is an excellent illustrative of the population, and as such sampling can be simulated from the population by sampling from the sample, and this is known as resampling.

Resampling involves the same sample size as the original sample. The reason being that the original or initial sampling is simulated so as to make the resampling as similar as possible to the initial or original sampling. The resampling is drawn with replacement from the original sample because it gives independent observations making the resamples to be i.i.d. just like the original sample.

The resampling procedure looks to deal with a limitation of the MVE generating a res-ampled efficient frontier in the process. The limitations are instability and ambiguity (Sec-tion 4.6). A resampled efficient frontier is generated in this regards. This is not a new procedure and has been introduced by Michaud in Efficient Asset Management by R. Michaud (1998).

5.4 Bootstrap Resampling Process - Generating Resampled

Efficient Frontier

The resampling is performed, as stated earlier, as an application of Monte Carlo Simulation. The sample is the “true population” so that the sample mean and the covariance matrix of the returns are the “true parameters”. That is to say; the actual historical daily returns are used as an estimate of the true population. The standard deviation and mean of the monthly returns of each of the 40 stocks are estimated. Sampling is then made with the estimated mean

(38)

and standard deviation from a multivariate normal distribution. The Monte Carlo simulation produces T = 119 monthly log returns.

To create the Resampled Efficient Frontier (Resampled Efficiency introduced by Michaud), the algorithm below generates the efficient frontier. The algorithm is similar to the one de-scribed in Scherer’s paper (Scherer 2002). Portfolios on the resampled frontier are made of assets weight vectors. These asset weight vectors are the average of the risk-effficient portfo-lios weight vectors, with a given level of portfolio return. This method guarantees that after averaging, the weight vector w still sums up to one.

The algorithm is as follows:

First, a standard mean-variance optimization is run. Step 1.

The mean vector and variance-covariance matrix of historical inputs are estimated (the inputs can be prespecified as an alternative).

Step 2.

Resample using inputs (created in Step 1) by taking T draws from input distribution. One thing to take note is that the number of draws reflects the degree of uncertainty in the inputs. Calculate a new variance-covariance matrix and mean from the sampled series. Estimation error will result in different variance-covariance matrices and mean vector from those in Step 1.

Step 3.

Calculate efficient frontier for inputs derived in Step 2. Save optimal portfolio weights for m equally distributed return points along the frontier.

Step 4.

After repeating Steps 2 and 3 many (n) times, calculate average portfolio weights for each return point. This helps to obtain the final portfolio weights that lie on the resampled efficient frontier. The average portfolio weights for each return point obtained by

w= 1 B B

∑

i=1 ˆ wib (5.4.0.1)

where ˆwib represents is the estimated optimal portfolio weight vector which is the weight

vec-tor of the bth portfolio for the ith resampling. Evaluate frontier of averaged portfolios with variance-covariance matrix from Step 1 to plot the resampled frontier.

The bootstrap sample, bth, has the mean vector ˆµµµ_iband covariance matrix ˆΣΣΣibwith the averaged

weight vector w achieved in Equation 5.4.0.1. These estimates can be made to replace µp, ΣΣΣ and w in Equations (4.5.0.1), (4.5.0.2), (4.5.0.13) and (4.5.0.14). A plot of these newly estimated parameters generates Figure 5.4.1.

A closer look at Figure 5.4.1 goes to show that MV Efficient Frontier and the Resampled Efficient Frontier are statistically equivalent at a certain level of risk. It is worth noting that, two statistically equivalent portfolios are not obviously the same in relation to mean return nor risk as defined by Michaud. One might be moved to prefer the MV Efficient Frontier but if an investor is really cautious about what estimation errors brings, as earlier discussed, then the

(39)

Exercises 89 0.0178−5 0.018 0.0182 0.0184 0.0186 0.0188 0.019 0 5 10 15 20 x 10−4

Monthly log return

Monthly standard deviation

Estimated efficient frontier Resampled efficient frontier

Fig. 3.11. The estimated eﬃcient frontier (solid curve) and the resampled eﬃcient frontier (dotted curve) of six U.S. stocks.

to do so. The bootstrap samples _{r∗_b1, . . . , r∗_bn; r∗_b_{}, 1 ≤ b ≤ B, can be used}

to estimate the means E(wT_Pr) and variances Var(w_PTr) of various portfolios

P whose weight vectors wP may depend on the observed data (for which

E(wT_Pr) can no longer be written as wT_PE(r) since wP is random). Details

and illustrative examples are given in Lai, Xing, and Chen (2007).

Exercises

3.1. Prove (3.19).

3.2. Prove (3.27) and (3.28).

3.3. Let r0t be the return of a stock index at time t. Sharpe’s single-index

model assumes that the log returns of the n stocks in the index are

gen-erated by rit = αi+βir0t+εit, 1≤ i ≤ p, where itis uncorrelated with r0t

and Cov(it, jt) = σ21{i=j}. The model also assumes that (r0t, . . . , rpt),

1≤ t ≤ n, are i.i.d. vectors.

(a) Suppose Var(r0t) = σ02. Show that the covariance matrix F = (fij) of

the log return of the n stocks under the single-index model is given

by F = σ2

0ββT + σ2I, where β = (β1, . . . , βp)T.

(b) Let σij = Cov(rit, rjt) and Σ = (σij)1≤i,j≤p. Let S = (sij) be the

sample covariance matrix based on (rit, . . . , rpt)T, 1 ≤ t ≤ n. Let

R(α) = αF + (1_{− α)S. Consider the quadratic loss function L(α) =}

||·||2_{, where}_{||A|| is the Frobenius norm of a square matrix A deﬁned}

by _||A||2 _{= tr(A}T_{A). Show that the minimizer α}∗ _{of E[L(α)] is}

given by

Figure 5.4.1: Resampled Efficient Frontier with MV Efficient Frontier

The resampled frontier uses the data to produce much spontaneous portfolio allocations which are less sensitive to input instability caused by estimation error. This is made possible as the resampled efficient frontier is more diversified and less risky, intuitively, than one on a corres-ponding MV efficient frontier. Investment information is used in a more forceful way than MV efficiency. As resampled efficiency is an averaging process, it is very stable.

(40)

Chapter 6 Covariance Estimation

Creating an optimal portfolio in a mean-variance framework requires a measure of covariance between all assets that are available in the investment space. This variance-covariance matrix is at the center of optimizing the risk-adjusted return, but cannot be observed in the market. It is necessary therefore to estimate using statistical techniques on historical data, which, as mentioned in Section 1.3, creates two distinct problems.

Firstly, variances of assets are time dependent making old observation less reliable estimators than current. Secondly, the estimation might contain estimation errors that will subsequently distort the optimization. When selecting historical data, the assumption that market volatility and correlations are time-dependent will make it feasible to focus on shorter horizons with higher frequencies in estimating the risks of the assets. Incorporating too old volatilities will contaminate the estimates with irrelevant data according to Litterman (2003).

Markowitz portfolio theory’s assumption of stationarity of asset returns is an issue well known. This is to imply that, the joint distribution of asset returns does not change over time. The covariance matrix of asset returns is used to determine how much an investor should choose to hold in the context of diversification when Markowitz model is involved. This creates the mean-variance portfolio, which determines how much risk one incurs for an expected return. Accurately estimating a covariance matrix is of much importance in portfolio optimization and where risk management is involved. Markowitz initial proposal was in using the sample covari-ance matrix in estimation. However, research and further studies have shown that this is not the best technique or approach as they perform poorly out-of-sample (Broadie (1993), Jobson and Korkie (1981), Britten-Jones (1999) and DeMiguel et al.(2011)). Since the stationarity of asset returns is assumed, a sample covariance matrix does not provide any meaningful information as to how to invest, given a variety of possible market changes.

A shrinkage method suggested by Ledoit and Wolf (2003) is implemented to minimize the er-rors in the estimation. This leads to a discussion about Ledoit and Wolf shrinkage approach in the next section.

Risk-Efficient Portfolios; Estimation Error In Essence

MASTER THESIS IN MATHEMATICS / APPLIED

MATHEMATICS

RISK-EFFICIENT PORTFOLIOS;

ESTIMATION ERROR IN ESSENCE

DANIEL KOFI ADOBAH-OTCHEY

Masterarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICS

School of Education, Culture and Communication

Division of Applied Mathematics

Acknowledgements

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Background

1.2

Current Discussions and Findings

1.3

Practical hindrances and its implication/effect

1.4

Thesis Objective

Chapter 2

Reviewing Fundamental Concepts

2.1

Returns

2.2

Net Returns

2.3

Multi-period returns

∏

2.4

Log Returns

2.5

Adjusting for Dividends

2.6

Random Walk Synopsis

2.7

Box-plots

Chapter 3

Data

0

40

80

120

100

150

200

Scatter Plot of GS

Index

Pr

ice

3.1

Descriptive Statistics

Chapter 4

The Language of Portfolio

4.1

Risk - Return

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

4.2

Feasible Portfolio Opportunities