Evaluation of a Portfolio in Dow Jones Industrial Average Optimized by Mean-Variance Analysis

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INOM

EXAMENSARBETE TEKNIK, GRUNDNIVÅ, 15 HP

STOCKHOLM SVERIGE 2020,

Evaluation of a Portfolio in Dow

Jones Industrial Average

Optimized by Mean-Variance

Analysis

DANIEL LIU

ALEXANDER STRID

KTH

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Evaluation of a Portfolio in

Dow Jones Industrial Average

Optimized by Mean-Variance

Analysis

Daniel Liu

Alexander Strid

ROYAL

Degree Projects in Applied Mathematics and Industrial Economics (15 hp) Degree Programme in Industrial Engineering and Management (300 hp) KTH Royal Institute of Technology year 2020

Supervisor at KTH: Johan Karlsson Examiner at KTH: Sigrid Källblad Nordin

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TRITA-SCI-GRU 2020:103 MAT-K 2020:004

Royal Institute of Technology School of Engineering Sciences KTH SCI

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Abstract

This thesis evaluates the mean-variance analysis framework by comparing the performance of an optimized portfolio consisting of stocks from the Dow Jones Industrial Average to the performance of the Dow Jones Industrial Average index itself. The results show that the optimized portfolio performs better than the corresponding index when evaluated on the period between 2015 and 2019. However, the variance of the returns are high and therefore it is difficult to determine if mean-variance analysis performs better than its corresponding index in the general case. Furthermore, it is shown that individual stocks can still influence the movement of an optimized portfolio significantly, even though the model is supposed to diversify firm-specific risk. Thus, the authors recommend modifying the model by restricting the amount that is allowed to be invested in a single stock, if one wishes to apply mean-variance analysis in reality. To be able to draw further conclusions, more practical research within the subject needs to be done.

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Utv¨ardering av en portf¨olj i Dow Jones Industrial Average

optimerad genom mean-variance analysis

Alexander Strid, Daniel Liu May 22, 2020

Sammanfattning

Denna uppsats utvärderar ramverket ”mean-variance analysis” genom att jämföra prestandan av en optimerad portfölj best˚aende av aktier fr˚an Dow Jones Industrial Ave- rage med prestandan av indexet Dow Jones Industrial Average självt. Resultaten vi- sar att att den optimerade portföljen presterar bättre än motsvarande index när de utvärderas p˚a perioden 2015 till 2019. Dock är variansen av avkastningen hög och det

är därför sv˚art att bedöma om mean-variance analysis generellt sett presterar bättre än sitt motsvarande index. Vidare visas det att individuella aktier fortfarande kan p˚averka den optimerade portföljens rörelser, fastän modellen antas diversifiera företagsspecifik risk. P˚a grund av detta rekommenderar författarna att modifiera modellen genom att begränsa mängden som kan investeras i en individuell aktie, om man önskar att tillämpa mean-variance analysis i verkligheten. För att kunna dra vidare slutsatser s˚a krävs mer praktisk forskning inom omr˚adet.

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1 Introduction

In 2016, 49.3 percent of the households in the United States stated that they kept some part of their savings in the stock market [1]. Saving money for the future is beneficial, whether it is for retirement, to buy a house or to send ones children to college. Investing in the stock market is one way to make the money grow and potentially be worth more in the future. While the market might grow in the long run, there is no guarantee that a given investment will yield a positive return during a fixed time period. Very risk averse investors may choose to save all their money in a practical risk-free rate, but such an investment usually has a low yield. It is reasonable to assume that most investors want a high return on their investment, but at the same time are reluctant to expose themselves to unnecessary risk. One method that could be considered is to use a framework called mean-variance analysis (MV analysis). In theory, mean-variance analysis minimizes the risk the investors expose themselves to, while still maximizing the return they expect to receive. This thesis is an empirical investigation of the performance of the mean-variance analysis framework on the Dow Jones Industrial Average.

1.1 Background

The process of investing among various assets to gain a positive return is a game of uncer- tainty. In 1952, Harry Markowitz marked the beginning of modern portfolio theory with his article ”Portfolio Selection” [2]. For the first time, the problem of portfolio selection is clearly stated and solved [3]. The essence of the theory does not only imply diversification, but the ”right kind” of diversification for the ”right reason” [2]. For instance, the adequacy of diversification should not solely be assessed by the number of securities in the portfolio but rather by low covariance of returns across different types of securities. Furthermore, as with all models and frameworks, assumptions need to be made regarding the nature of the actors in question. In mean-variance analysis it is assumed that the investors are rational and risk averse with concave utility functions [3]. It is also shown that some conventional

”truths” within finance must be rejected.

One way of diversifying a portfolio without extensive research is to invest in mutual funds.

Ever since the bull market began in 2009, index funds have grown substantially in popularity and in 2019, assets invested in U.S. passive equity funds topped those in actively managed funds [4]. This is mainly because of the increasingly substantiated belief that actively managed funds on average do not appear to provide higher returns than passively managed funds [5]. Considering the mean variance analysis and the superiority of index mutual funds, an intriguing question that could be asked is whether it is possible to achieve positive alpha, excess return of a benchmark index at the same risk, by optimizing the diversification of the stocks for a given index.

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One index that is appropriate for this kind of investigation is the Dow Jones Industrial Average (DJIA). DJIA is an index consisting of 30 blue chip stocks in the U.S. stock market. At least historically, the index has been used to reflect the U.S. market as a whole [6], but the S&P 500 might be a better indicator since it encompasses more stocks [7].

Despite not representing the whole market, the DJIA serves as a good ”test” index for portfolio optimization because of primarily three reasons:

1. It is a stable index with few company changes and the changes that have occurred are well documented.

2. All the stocks that are, or have been a part of the index in recent years, have been listed for at least 10 years (at the time of writing).

3. It only contains 30 stocks which limits the required computational power when calculated in practice.

The first two reasons facilitate the process of eliminating the so-called survivorship bias which is the logical error of concentrating, in this case, on the wrong types of companies.

An example of the survivorship bias would be to exclude companies that have ceased to exist but were active during the backtesting period. This is elaborated later on in the thesis. A list of the stocks that are used in the calculations is presented in Appendix A.

1.2 Problem Statement

The aim of this thesis is to investigate whether a portfolio optimized according to the theory of MV analysis can, in practice, perform better than its corresponding index. In other words, this thesis investigates if it is possible for an investor to construct a portfolio consisting of stocks in a defined market index so that the actual return of the portfolio is greater than that of the index, while exposing themselves to the same risk.

In the formulation of the base MV problem, which later on is expanded, the following notation is used: x is an n × 1 vector with elements x1, ..., xn that denote the proportion of the investor’s total capital allocated in the i:th asset in a portfolio with n securities.

The elements of vector µ = [µ₁, . . . , µ_n]^T are the expected returns of each of the n assets.

C denotes the n × n covariance matrix of the assets where we make the assumption that C is nonsingular, i.e. that none of the asset returns has a perfect linear relationship with the returns of the remaining securities. Nonsingularity of matrix C also implies that none of the assets has zero risk, which makes sense since the securities in question are stocks.

Note that the risk-free asset is covered and elaborated later in the thesis. Finally, it is also determined that the investor must invest all their funds, resulting inPn

i=1x_i = 1. In summary, the problem is defined as:

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(P ) :











minimize

x x^TCx,

subject to µ^Tx = r, 1^T_nx = 1, x ≥ 0,

(1.1)

where 1n is an n × 1 vector consisting of ones and r is a specified portfolio return.

The reference index to measure the performance of the model is the DJIA, which is back- tested from 2015 to 2019 using five years of historical data for each optimization. In this thesis, an optimal portfolio is calculated for each month. In addition to comparing the risk adjusted return to the DJIA index, the problem is also expanded with additional constraints. Relevant questions such as how much the optimal portfolio and the Sharpe ratio, the ratio between reward and variability, will change when the proportions of each asset are limited to a certain level are answered.

1.3 Earlier Research

Modern portfolio theory is introduced and developed by Markowitz [2][8]. Tobin later introduces the risk-free rate in combination with the tangency portfolio [9]. While the MV framework is a well-researched subject, the results of its applications are not very well documented. Mainik evaluates a non-adjusted MV optimized portfolio on the S&P 500 index, and shows that it yields higher cumulative returns than the index between years 2007 and 2011 [10]. Kresta also evaluates the MV framework on the DJIA between years 1995 and 2015 and concludes that, while there seems to be a clear risk-return trade-off, the model reacts to the market too slowly, even when only basing assumptions on one year of historical data [11]. Righi et al. discuss the results of using variance as a risk measure when optimizing a portfolio, compared to optimizing a portfolio using other risk measures, with the conclusion that there is no clear preference for which risk measure to use [12].

Since it is rather simple to simulate a portfolio according to the MV framework, many of the statements made about the performance of MV analysis are briefly stated as a sidenote to other research subjects. However, the authors of this thesis have not found any earlier research that evaluates a portfolio based on its realized returns in combination with risk adjusting or weighting so that the portfolio can be directly comparable to an index.

1.4 Scope and Delimitations

It should be stated that the main focus of the thesis is the application of mean-variance analysis and how it performs given certain constraints and settings. During the process of determining optimal portfolios, one must predict future returns, which in turn is a science

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in itself. Thus, to prevent an overly broad scope, extensive research and complex algorithms are not conducted within this topic.

Furthermore, the duration of the data is only based on the years 2010 to 2019 and only evaluated on the years 2015 to 2019. This thesis does not try to explain the events in society that happened before 2010 and their impact on the stock market. Likewise, it does not try to predict structural changes in future markets.

The problem is also simplified in the sense that all transaction costs and the capital gains tax are set to zero. Obviously, the presence of proportional transaction costs could distort what otherwise would be calculated as the optimal portfolio, especially if the portfolio is frequently optimized. Furthermore, the presence of taxes could incentivize tax planning, i.e. ensuring maximum tax efficiency, which would also distort the standard mean-variance policy.

Also, since the stocks in the DJIA are defined as the assets in the portfolio, no efforts are made to customize a portfolio outside of the index. Markowitz stated that the process of selecting a portfolio might begin with observations and experience to choose appropriate assets [2]. Logically, this would be to identify companies within industries with low covariances. Despite choosing the DJIA straight off, it can be observed by the nature of the index that the encompassing stocks at least belong to different industries to a fairly high degree.

Lastly, this thesis does not attempt to prove nor disprove the general efficient-market hypothesis, which implies that it should not be possible to beat the market on a risk adjusted basis [13]. The concern about the efficient-market hypothesis is that neither the market nor the risk measure is defined, and it is therefore hard to test the hypothesis. In this thesis, the ”market” is restricted to a particular index and the risk is defined as the standard deviation of a portfolio.

1.5 Purpose

The purpose of this thesis is to provide educational and transparent information regarding the performance of MV analysis. It can be viewed as a rational and mathematical basis which investors may use to make their investment decisions. It may be especially interesting for investors who prefer passively managed funds, since the results are presented as an alternative to investing in the corresponding index. Aside from presenting results, this thesis also discusses the risks and pitfalls that an investor must consider, if they decide to use the framework. The feasibility for individual investors to use MV analysis in practice is also discussed. If used correctly, MV analysis seeks to eliminate unnecessary risk that an investor might expose themselves to when investing in the stock market.

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2 Theoretical Framework

Portfolio theory is a well-developed paradigm [14]. Modern portfolio theory is one of the simpler frameworks within portfolio theory, which might explain the popularity of the framework. The original formulation of modern portfolio theory, as defined by Markowitz [2], is the robust MV analysis. Most theory that is required in the methodology and the discussion is presented in this section. Both mathematical and economical aspects are covered.

2.1 Modern Portfolio Theory

Modern portfolio theory based on Harry Markowitz’ original framework relies on a few assumptions regarding the investor [2]. Firstly, it is assumed that the investor finds expected return as a desirable thing and variance of return, both up and down, as an undesirable thing. Furthermore, it is commonly assumed in finance that the investor wants to maximize discounted expected returns. Unlike the first assumption, this maxim must be rejected for the following analysis. The reason for this is the basic idea of diversification. If arbitrage and market imperfections are ignored, the above presented maxim would never imply that a diversified portfolio is preferred to a non-diversified portfolio. A formal proof is not be presented but the idea is rather simple; no matter how the discount rates are determined and how they vary over time, the maxim implies that the investor should place all their funds in the security with the highest discounted value, which is not coherent with the strategy of risk aversion [2].

2.1.1 E − V Rule

In this thesis, we consider the relationship between expected returns and the variance of the returns, which constitutes the basis of the optimization problem. Markowitz entitles this relationship as the E − V rule [2]. Basic statistical concepts are presented so that the reader can fully follow the implications of the E − V rule.

Let Y be an arbitrary stochastic variable. Suppose that Y can take on a finite number of values y1, . . . , ym and let pi denote the probability of Y = yi. The expected value of Y is then defined as:

E[Y ] =

m

X

i=1

piyi. (2.1)

Furthermore, the variance of Y is defined as:

V ar(Y ) =

m

X

i=1

pi(yi− E[Y ])². (2.2)

Now suppose we have several stochastic variables: R₁, ..., R_n. A linear combination of all the Ri:s is also a stochastic variable. Let us write R =Pn

i=1xiRi, where xi are constant

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weights. The expected value of the weighted sum is written as: E[R] =Pn

i=1xiE[Ri]. To obtain the joint variability of the stochastic variables we define the covariance:

Cov(Ri, Rj) = E[(Ri− E[R_i])(Rj − E[R_j])]. (2.3) This can also be written as the product of the known correlation coefficient ρ_ij, the standard deviation of Ri and the standard deviation of Rj:

Cov(Ri, Rj) = ρijσiσj. (2.4) The total variance of a linear combination of R_i:s is defined as:

V ar(R) =

n

X

i=1 n

X

j=1

xixjCov(Ri, Rj). (2.5) To go full circle, we now let Ri be the return of security i and hence E[Ri] is the expected return of Ri. We denote xi to be the proportion of the investor’s funds allocated to security i. Observe the implication that the returns of the securities are stochastic variables and that the proportions invested are fixed by the investor. Finally we end up with the expected return E of the portfolio as:

E =

n

X

i=1

xiE(Ri) = µ^Tx, (2.6)

where µ^T is a n × 1 vector with the expected return for each individual stock, and x is a vector with the weights in each stock. The preferred expected return, E, is set to a certain value in Equation 1.1 by the investor, and therefore constitutes the first constraint. The variance is written as:

V =

n

X

i=1 n

X

j=1

xixjCov(Ri, Rj) = x^TCx, (2.7)

where C is the covariance matrix of the stocks. The total variance is the objective function that is supposed to be minimized in the same problem.

2.1.2 The Efficient Frontier

The E − V rule states that the investor chooses a portfolio that maximizes the expected return for a given variance or minimizes the variance for a given expected return [2].

Efficient combinations of (E, V ), also called the efficient frontier, can be illustrated in its most general form in Figure 1.

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Figure 1: All possible E − V combinations Observe that the bold part of the edge of

the attainable (E, V ) combinations denotes the efficient frontier. The fact that it is only the lower right quarter circle makes sense for the risk averse investor, since they under no circumstances would want a portfolio with the same amount of expected return but with a higher variance. Also note that the variance is placed on the y-axis and the expected return on the x-axis. In later literature however, the expected return is usually placed on the y-axis. We will use the latter format in this thesis. Therefore, one can expect the following illustration of the efficient frontier as shown in Fig- ure 2.

Figure 2: The Markowitz bullet Due to the characteristic shape of the

graph, the parabola is sometimes referred to as the Markowitz bullet. Worth noting is that the efficient frontier begins with the portfolio that has the smallest variance (risk) and then goes all the way to the end of the feasible region.

2.2 Risk Aversion Reconciled with Mean-Variance Analysis Recall that MV analysis applies to risk averse investors. A risk averse investor is by definition characterized by a concave utility function over wealth. Intuitively, this phenomena within behavioural economics is generally clear; one dislikes un-

certainty in lifetime wealth since a dollar that prevents poverty is more valuable than a dollar that contributes to a large wealth [15].

The connection between MV analysis and the risk averse investor might seem clear at first sight, but proves to be less obvious in a mathematical sense. Feldstein, among other mathematicians, proves that the reconciliation of the two policies is current when: (1) the concave utility function is quadratic, and/or (2) when assuming that all random returns

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are normally distributed variables. In the latter case, the utility function can be arbitrary as long as it is concave [16].

In order to show the first case, we define the quadratic utility function as: U (x) = x −₂^bx², where b > 0. Observe that a positive b implies concavity and that the utility function is only meaningful in the range where it is increasing. The objective for any investor is to maximize the expected utility. Suppose that the expected value of a portfolio is denoted by the stochastic variable Y . Then we have:

E[U (Y )] = E[Y −b

2Y²] (2.8)

= E[Y ] − b

2E[Y²] (2.9)

= E[Y ] − b

2(E[Y ])²− b

2V ar(Y ). (2.10)

Since the previous observations stand, we note that E[U (Y )] increases with E[Y ]. There- fore, minimizing V ar(Y ) for a given E[Y ] or maximizing E[Y ] for a given V ar(Y ), which is the MV analysis, is equivalent to maximizing E[U (Y )].

To deduce the second case, we choose an arbitrary utility function U (Y ) with stochastic variable Y as expected return for a given security. Suppose that Y ∈ N (µ, σ²). This implies that the expected utility is a function of µ and σ. Furthermore if the utility function is concave, then we have:

E[U (Y )] = f (µ, σ), where ∂f

∂µ > 0 and ∂f

∂σ < 0. (2.11)

Suppose that the returns of all securities are normally distributed. A linear combination of these securities therefore has a normal distribution N (M,P). When selecting the optimal portfolio, the objective is to choose the combinations (proportions) of securities that maximizes f (M,P) with respect to the feasible combinations. In turn, this implies minimization of variance for a given return and hence the solution is mean-variance efficient.

2.3 Quadratic Optimization

A quadratic optimization problem has a quadratic objective function and linear constraints.

We will consider the following general problem as an example:

(minimize ¹₂x^THx + c^Tx + c₀,

subject to Ax = b, (2.12)

where A ∈ IR^mxn, H ∈ IR^nxn is symmetric, b ∈ IR^m, c ∈ IRⁿ and c₀ ∈ IR.

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The objective function is convex (strictly convex) if and only if H is positive semi-definite (positive definite). We will assume that H is (semi)-definite for this theory section. There- fore, since the constraints are linear, we can conclude that the whole problem is convex.

There are several ways of solving the above optimization problem, but an efficient and commonly used method is the Lagrange method. If we consider the problem (2.9) where the objective function is convex, ˆx ∈ IRⁿ is an optimal solution if and only if Aˆx = b and there exists a u ∈ IR^m such that H ˆx + c = A^Tλ. A formal proof is not provided but one can easily follow the idea by creating the relaxed Lagrange problem;

(minimize ¹₂x^THx + c^Tx + c₀+ λ^T(b − Ax),

subject to x ∈ IRⁿ (2.13)

where λ ∈ IR^m is called the vector of Lagrange multipliers. Since convexity is present, the optimal solution is obtained by setting the gradient of the new objective function, which is called the Lagrangian L(x, λ), to 0. We have:

(_dL

dx = H ˆx + c − λ^TA = 0 ⇐⇒ H ˆx + c = A^Tλ

dL

du = b − Aˆx = 0 ⇐⇒ Aˆx = b (2.14)

which yields the same result as the condition described above. This can in turn be extended to quadratic problems with linear constraints that both include equality constraints and inequality constraints [17].

2.4 Solution to the Portfolio Optimization Problem As stated, the base optimization problem in question is the following:

(P ) :











minimize

x x^TCx,

subject to µ^Tx = r, 1^T_nx = 1.

(2.15)

See subsection 1.2 for definitions of notations. This can, and is later on in the thesis expanded with additional linear constraints. However, in this section it is shown how to solve the above optimization problem based upon the previous theory. The reason for the derivation is predominantly to illustrate important properties of the problem. Initially we note that the problem is convex since the covariance matrix is positive definite and the constraints are linear.

We form the Lagrangian

L = x^TCx + λ1(r − µ^Tx) + λ2(1 − 1^T_nx) (2.16)

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where λ1 and λ2 are Lagrange multipliers. An optimal solution is obtained by setting the gradient of the Lagrangian to zero:

∂L

∂x = 2Cx − λ₁µ − λ₂1_n= 0, (2.17)

∂L

∂λ₁ = r − µ^Tx = 0, (2.18)

∂L

∂λ₂ = 1 − 1^T_nx = 0. (2.19)

Solve for x in (2.17):

x = 1

2C⁻¹(λ1µ + λ21n) = 1

2C⁻¹µ 1_nλ₁ λ2

. (2.20)

Rewrite (2.18) and (2.19) to matrix form:

µ^T 1^T_n

x =r

1

. (2.21)

Multiply (2.20) with µ^T 1^T_n

and use (2.21) to obtain:

µ^T 1^T_n

x = 1

2

µ^T 1^T_n

C⁻¹µ 1_nλ₁ λ₂

=r 1

. (2.22)

To reduce the length of the equations we denote A ≡ µ^T 1^T_n

C⁻¹µ 1_n. Matrix A is nonsingular since C is positive definite and hence we can solve for the multipliers:

1 2

λ₁ λ₂

= A⁻¹r 1

. (2.23)

Finally we end up with the minimum variance portfolio for a given expected return as:

x = C⁻¹µ 1_n A⁻¹r 1

. (2.24)

Now that we have the derivation of determining the minimum variance portfolio, it can be shown that the maximum expected return portfolio subject to the above minimum variance is in fact the same portfolio. This is because the Lagrangians of the two problems are equivalent. Hence, the key point is that the efficient frontier of minimizing portfolio variance subject to an interval of expected returns is the same as maximizing portfolio expected returns subject to an interval of variances given that they correspond to the same interval of calculation. Note that this holds for any value of x_i, while in this thesis we only

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2.5 Introduction of the Risk-Free Asset

Previously the portfolio problem exclusively revolved around risky assets. In 1958, James Tobin expanded the problem by introducing the risk-free asset [9]. It turns out that it is a small modification of the original problem and that it can easily be solved mathematically.

Suppose there are n + 1 assets with indices i = 0, 1, . . . , n. Asset 0 is defined as the risk- free asset and has a return of r0. Furthermore, the investor’s funds are allocated in the assets with proportions x₀, x₁, . . . , x_n. However, for mathematical convenience, we will still define vector x as x = [x₁, . . . , x_n]^T. This means that the proportion allocated in the risk-free asset is written as: x0 = 1 −Pn

i=1xi. For the portfolio as a whole, the return becomes:

r = µ^Tx + x0r0, (2.25)

where µ = [E[R₁], . . . , E[R_n]]^T.

Furthermore, since the variance of the risk-free asset is zero, the variance of the portfolio is just as in the original problem:

V = x^TCx (2.26)

where C denotes the correlation matrix of the risky assets. All together, the problem can be written as:

(minimize

x x^TCx,

subject to µ^Tx + (1 − 1^T_nx)r₀= r. (2.27) If the optimal solutions of Equation 1.1 and Equation 2.27 are plotted in the same figure for a given range of returns, the following (general) picture is obtained:

Risk

Expected Return

Markowitz bullet Capital allocation line Tangency portfolio

Figure 3: The capital allocation line, the Markowitz bullet and the tangency portfolio

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The up-sloping straight line which is the efficient frontier of Equation 2.27 is usually referred to as the capital allocation line (CAL). It tangents the Markowitz bullet of Equation 1.1 at the so-called tangency portfolio. Observe that all funds are invested in stocks at the tangency portfolio. Tobin showed that, according to the mean-variance framework, the only strategy that should be used, regardless of risk preference, is to invest in a mix of the tangency portfolio and the risk-free rate only [9].

Regarding Equation 2.27, observe that there are no constraints on the weightings of the assets. For example, the constraint Pn

i=1x_i = 1 is not included since otherwise nothing can be added in the risk-free asset. Nor have we added an upper limit of the sum of the proportions. This means from an economical point of view that the investor can borrow money at the risk-free rate and invest these funds in the market.

2.5.1 Equation of Capital Market Line

Since the risk-free asset does not have any correlation with any stock, it should not be included in the covariance matrix of the optimization problem, otherwise the covariance matrix would be singular and the objective function would be impossible to solve by conventional methods. Tobin suggests finding the optimal strategy by separating the problem into two steps [9]. Firstly, one has to find the tangency portfolio. Secondly, one should decide the weight of the assets invested in the tangency portfolio and the risk-free rate.

We want to be able to express the CAL in terms of the expected return and the variance.

Let rp be the expected return of the tangency portfolio and r0 the risk-free rate. Let xp

be the total fraction of the assets invested in the tangency portfolio. The expected return of the strategy can then be expressed as:

E[R] = x_pr_p+ (1 − x_p)r₀. (2.28) Since the variance and the covariance of the risk-free rate is 0, the only risk from the strategy comes from the tangency portfolio. If the variance of the tangency portfolio is denoted σ²_p, the variance of the strategy, σ², is denoted as:

σ² = x²_pσ_p². (2.29)

By rearranging the (2.29), we can express xp in terms of standard deviations.

xp = σ σp

. (2.30)

Inserting (2.30) into (2.28) gives us an expression for the expected return of a portfolio on

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E[R] = r0+rp− r₀

σ_p σ. (2.31)

Note that r0, rp and σp are fixed, so the expression is actually the same linear function as the efficient frontier produced by Equation 2.27. One may, by inserting the preferred risk σ into (2.31) and thereafter (2.28), obtain the expected return and thereafter the weight of funds that should be invested in the tangency portfolio.

2.5.2 Sharpe Ratio

Let us extend the notations and definitions from the previous section. In this setting, the Sharpe ratio [18] for a given portfolio is:

S = r − r₀

σ , (2.32)

where r is the return of the portfolio and σ is the standard deviation of the returns. The ratio indicates the expected excess return per unit of risk. It can be shown that the tangency portfolio, see Figure 3, maximizes the Sharpe ratio, i.e. maximizes the reward to risk compared to portfolios on the Markowitz bullet [19]. It is clear that the Sharpe ratio is the same for all portfolios on the straight line efficient frontier. Therefore, the correct interpretation of a portfolio on the CAL is that a proportion of the funds is invested in the risk-free rate and the rest of the funds are invested in the tangency portfolio [9].

2.6 Firm-Specific Versus Systematic Risk

As stated previously in the thesis, risk is defined as fluctuations in stock prices and potential dividends which cause the final return to be either higher or lower than the expected return.

According to finance theory, these fluctuations has its roots in two types of news [5]:

1. Firm-specific news is news concerning the company itself. For example, it might surface that a company has been evading taxes for an extended period of time.

2. Market-wide news is news concerning the whole market and therefore affects all stocks. For instance, pandemics such as COVID-19 in 2020 have huge impact on the world economy.

Fluctuations of stock returns caused by firm-specific news are independent risks of the market. For example, the death of a high ranking executive of company A in one market is an independent event of firm B’s successful share growth in another market. This type of risk is referred to as firm-specific risk in this thesis. In contrast, when market-wide news causes fluctuations, such as the Federal Reserve lowering the target federal funds rate, all stocks in the market are affected simultaneously. We denote this type of risk as systematic risk. It is typically said that if a portfolio includes many stocks, the firm-specific risk for

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each stock will average out. In other words, the exposure to hardships and cyclicalities of a single business will be reduced in the portfolio. For this reason, firm-specific risk can also be referred to as diversifiable risk. Systematic risk on the other hand, affects all stocks and therefore the whole portfolio, and can not be diversified away [5].

3 Methodology

There are several ways to investigate and evaluate if a portfolio strategy can beat the corresponding index. The method used in this report is to calculate a portfolio based on underlying assumptions of the future and simulate it on historical data. To evaluate the portfolio strategy, it is then compared to the actual performance of the index.

The underlying assumptions about the future, which are used to make investment decisions, are here based on historical data on stock performances. This section describes how the data is collected, how the underlying assumptions are made, how the portfolio is determined and how the portfolio is evaluated.

3.1 Collection and Processing of Data

The data needed to make the underlying assumptions are daily prices of the individual stocks in an index. To be able to compare these stocks with each other, the dividends and splits must be adjusted for, since they are managed differently by various firms. Therefore, the daily stock price is represented by the adjusted closing price.

Since this thesis calculates a portfolio with stocks from the DJIA, the changes in compo- sition of stocks must be accounted for, to prevent survivorship bias. Thus, all stocks that have existed in the index during the backtesting period are included, which means that the data includes 32 stocks (rather than 30). However, the model only activates the appropriate stocks depending on when the stocks existed in the index. This is elaborated further in subsubsection 5.1.3. Furthermore, since the performance of the calculated portfolio is evaluated against the index, the daily net asset value of the index also must be retrieved.

The amount of data needed depends on how much data the assumptions about the future should be based on. In this thesis the assumptions are based on a five year period. Since the calculated portfolio is evaluated on years 2015 to 2019, the data retrieved is from 2010 to 2019. All of the stocks to be evaluated must have existed and been publicly traded throughout this period. The data is in this case retrieved from Yahoo Finance.

The data is then processed so that it displays the change in price from the previous day, in percentage. This is the return of the stock for one day. The processed data is formatted in a table where the daily return for every stock is listed.

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Date ˆDJI MMM AXP · · · 2019-12-30 -0.6% -0.8% -0.7% · · · 2019-12-27 +0.1% +0.1% -0.1% · · · 2019-12-26 +0.3% -0.0% +0.5% · · ·

... ... ... ... . ..

As mentioned in subsection 2.5, the investor may also invest in the risk-free rate. In this thesis, the 3-month U.S. Treasury bill (T-bill) is used as the risk-free rate. Many analysts and investors agree that this is an appropriate approximation for the theoretical risk-free rate since there is virtually no risk of the government defaulting on its obligations [20]. By choosing the 3-month T-bill, any potential maturity premium is also excluded. Historical rates are collected at the website of the U.S. Department of the Treasury. Figure 4 shows the effective 3-month rate, where the red line is the effective interest rate for every single day during the backtesting period and the blue line is the rate at the start of each month, i.e. the rates used in the calculation of the optimal portfolio of each month.

Figure 4: The risk-free rate between 2009 and 2019

3.2 Beliefs about Future Market Performance

The only beliefs about the future that need to be made in this thesis is the expected return of the stocks and the covariance between each of the individual stocks. Although this is perhaps the most important step to obtain good results of one’s investment, the determination of these beliefs are done in simple ways. This is because, although there is a lot of data available, the factors that contribute to market performance are not fully understood, and it is thus difficult to make a good prediction or forecast on the future [21].

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Hence, the sole purpose is to provide a framework for how MV analysis can be used to optimize a portfolio in the market.

It is reasonable to use quantitative prediction or forecasting based on historical data as a method because of two reasons. Firstly, as mentioned before, there exists a lot of historical numerical data that is available to use. Secondly, it is reasonable to assume that some patterns from the past will also continue into the future [21].

Predictions for each month are based on a rolling window of 60 months of historical data.

The reason for this exact time frame is because firstly, an integer number of years takes seasonal variations into consideration, and secondly, according to the National Bureau of Economic Research, the average length of an economic cycle in the United States since 1856 has been around five years [22]. Hence, using this length of time attempts to take economic cycles into consideration.

3.2.1 Calculation of Expected Returns

Making correct assumptions about the expected returns of stocks is important because the results from the MV analysis are based on them. One of the base assumptions in MV analysis is that stocks with higher expected returns are valued higher than those without, given that they have the same risk [2]. Approximating the expected return of each stock correctly allows investors to invest more in the stocks that are expected to do well and less in the stocks that are expected to not do well.

A common practice to estimate the expected return of a stock is to add the average historical excess return to the current risk-free rate [23]. The procedure used to approximate the expected return of a stock is as follows: Let tstart be the time of the first observation, t_end the time of the last, qi,t the excess return of stock i at time t, r0,t the risk-free rate at time t and n_obs the number of observations. Then the expected return of a stock, E[R_i], is approximated by:

E[Ri] ≈ 1 n_obs(

tend

X

t=tstart

qi,t) + r0,tend. (3.1)

The excess return of stock i at time t is calculated as the difference between the observed return, ri,t, and the risk-free rate at that same time:

q_i,t = r_i,t− r_0,t. (3.2)

The expected return every month between 2015 and 2019 for each stock is calculated with the data obtained in subsection 3.1. Since the data is daily observations, the dimension of

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3.2.2 Calculation of Covariances

Performance of stocks in the same index can often not be considered independent of each other, but have covariances that are significant. As Markowitz describes it, a good diversification is mix of stocks where the covariances of the stocks in the portfolio are low [2]. The expected covariance between two stocks is calculated as the historical covariance.

Cov(Ri, Rj) = 1 n_obs

tend

X

t=tstart

((ri,t− r_i)(rj,t− r_j)) (3.3)

Here ri,t is the observed return of stock i at time t, ri is the mean of the observed returns during the period and the number of observations is denoted n_obs. The combinations of covariances make up the covariance matrix, C. Each element, c_i,j in the matrix is expressed as follows:

c_i,j= Cov(R_i, R_j) (3.4)

Note that when i = j, the element simply is the variance of a stock. This can be quite useful when evaluating individual stocks.

3.3 Determination of Optimal Portfolio and Adjustment to Risk

Given the risk-free rate, the expected returns of the stocks in the market and their covariances, it is now possible to determine an optimal portfolio. In practice, the optimal portfolios are calculated by using the function quadprog in Matlab. As quadprog solves quadratic optimization problems, the portfolio selection problem is modeled so that the total variance is minimized subject to a given return. In fact, the main reason why the quadratic version is chosen instead of the equivalent problem, where the objective function is linear and one of the constraints quadratic, is to utilize the computational efficiency of built-in quadratic optimization functions.

It has been shown that, at least historically, high risk investments yield on average higher returns than low risk investments [24][25]. Thus, it is reasonable to believe that if a portfolio has a greater risk than an index, it is also expected to yield a greater return. To be able to compare and evaluate portfolios and indexes, one must therefore adjust them so that the risk is equal.

The process of adjusting risk to index starts by first determining the index volatility.

Nevertheless, this can be interpreted in several ways. Initially, suppose that an investor has put all their funds in the tangency portfolio. In order to compare the portfolio and the index on the same basis, one should adjust the portfolio to the actual observed index

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risk. However, if an investor already knows that they want to expose themselves to the index risk before investing, they will not know the actual index risk. They must then approximate the risk by calculating the historical variance of returns. Both scenarios are covered in the result section.

Figure 5: The tangency portfolio is adjusted by moving along the CAL

The first step in the process of determining an optimal portfolio is by finding all (or as many as the computational capacity allows) of the portfolios on the Markowitz bullet and the CAL. Since we know that the tangency portfolio, which is the portfolio with the highest Sharpe ratio, lies on the CAL, any adjusted optimal portfolio will also lie on the CAL.

Therefore, the expected return and risk of the strategy that the investor chooses to use can be represented by the point on the CAL that has the same risk as the index. See Figure 5. After the port-

folio is adjusted to risk, the adjusted tangency portfolio is then determined, and can be tested and evaluated on historical data.

3.4 Expansion of Problem with Allocation Limits

Sometimes, when a stock has performed relatively well over a long time, the model will adapt and invest more in that stock. In reality however, an investor might want to be careful about investing a large fraction of its fund in one stock, since they might expose themselves to unpredictable firm-specific risk, even if the model minimizes variance. Consequently, one may then put some additional constraints on the optimization problem, so that there is a limit of how large of a fraction the model will at maximum invest in an individual stock.

The optimization problem can then be expressed as below, where l ∈ [_n¹, 1] is a fixed allocation limit of the fraction the model, at maximum, invests in a single stock.

(P ) :











minimize

x x^TCx,

subject to µ^Tx = r, 1^T_nx = 1, 0 ≤ x ≤ l.

(3.5)

As mentioned before, the tangency portfolio is the portfolio with the highest Sharpe ratio.

Since the optimization problem is even more constrained with the allocation limit, the

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efficient frontier will be lower and therefore have a lower Sharpe ratio than the original tangency portfolio. How the allocation limit changes the Sharpe ratio and the performance of the portfolio is presented in the results.

3.5 Simulation and Evaluation of Portfolio Performance

In order to realistically evaluate the performance for a month and not let the results be influenced by the outcomes of the period on which the portfolio is evaluated, the calculations of expected returns and the covariance matrix is based on data before the month. For example, a strategy which is evaluated on June 2017 has expected values and covariances based on the time period from May 2012 to May 2017. The strategy is then simulated on the actual outcomes of the returns in the index, and compared to how the index performed during the same period. The source code for the simulation is written by the authors.

During a five year period, it is highly unlikely that any portfolio will beat any index every single day. Thus, we need a way to summarize and evaluate the results so that the evaluation would give a correct picture of what would happen if the simulation would be performed on another subset of data points.

The performance of the optimal portfolio is primarily evaluated on three criteria:

1. Cumulative returns during the five year period 2. Standard deviation of the portfolio

3. Fraction of days the portfolio performed better than the index

If the portfolio beats the index in all three of these criteria, it means not just that the portfolio has yielded higher returns at a lower risk, but also that the result is not solely weighted up by a few good days. If the portfolio performs better than the index during the majority of months, it makes it less likely that the cumulative returns are higher because of chance.

4 Results

Results are presented for three different simulations. The simulations are made when investing in the tangency portfolio and then adjusting to the actual index risk, when investing and adjusting to expected risk and when introducing additional constraints on how much that can be invested in an individual stock. The results show that all simulations beat the index in cumulative return for a five year period, and that the average risk during the period is very close to the actual risk of the index, whether it is adjusted monthly or weighted by the expected risk.

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Figure 6: The tangency portfolio for November 2017

Furthermore, all of the simulations show a similar pattern in movement of returns.

This is because the tangency portfolio used in all of the simulations are almost the same. The first two simulations consider exactly the same tangency portfolio, while the third simulation uses a slightly different but still similar portfolio, depending on the chosen allocation limit.

As an example, the unconstrained tangency portfolio for November 2017 is shown in Figure 6. Of the 60 tangency portfolios determined, all of the portfolios invest more than 20% of the funds in one stock, 41 portfolios invest more than

30%, 20 portfolios invest more than 40% and 3 portfolios invest more than 50%. No portfolio invests more than 60% of the funds in one stock. On average, 99 percent of the funds are allocated in only 6.5 stocks every month. The most invested stocks for every month are shown in Appendix B.

4.1 Results when Investing in the Tangency Portfolio

Suppose that an investor wants to invest all their existing funds in the tangency portfolio.

After all, no weighting in the risk-free asset could potentially yield a higher return and one might not feel comfortable enough to borrow money at the risk-free rate and further invest in the tangency portfolio. However, as explained in subsection 3.3, it might be perceived as problematic to compare the actual returns of this strategy to the actual returns of index since the risks (standard deviations of returns) most likely will differ. Therefore, it is necessary to determine the risk adjusted returns of the tangency portfolio and compare these with the risk adjusted returns of the index. The authors have chosen to use the Sharpe ratio as the adjusted return measure. If, for instance, the Sharpe ratio of the tangency portfolio is higher than the Sharpe ratio of the index for a given month, then the tangency portfolio has yielded a higher return by the same risk unit. The actual Sharpe ratio for the tangency portfolio and the index are summarized in Figure 7.

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Figure 7: The Sharpe ratio of the tangency portfolio versus the Sharpe ratio of the index

Apparently, the actual Sharpe ra- tios are closely correlated and it is difficult to decide which has the best overall performance just by looking at the picture. It turns out that the Sharpe ratio of the tangency portfolio is higher than that of the index for 28 out of the 60 months. Hence, one can conclude in terms of risk adjusted returns that the tangency portfolio beats index little less than half of the months.

While the above result shows the performance for each month, it does not display the cumulative

performance of the whole period. In order to test the aforementioned, the tangency portfolio is re-weighted with the risk-free asset so that the standard deviation of the actual returns of this weighted portfolio is the same as the index.

Figure 8 shows the daily and cumulative returns for the adjusted optimal portfolio and the index simulated on the sample November and December 2017. For November 2017, the daily returns of the strategy seem to correlate with the daily returns of the index. Looking at the cumulative return, one can see that the strategy performed better than the index for that month. However, this is not always the case. Using the same strategy a month after, in December 2017, the strategy neither beats the index nor seems to correlate as much as the month before, even though the data which the predictions are based on only differs in a month.

Figure 8: Daily and cumulative returns for November (left) and December 2017 (right)

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It turns out that in order to match the risk level of index, often fairly small proportions, yet some up to 43%, of the funds must be put in the risk-free rate for 48 out of the 60 months. For the remaining 12 months, small proportions of the funds must be borrowed and put in the tangency portfolio, i.e. increasing the risk of the tangency portfolio. In Figure 9, we see the cumulative returns the tangency portfolio, the risk adjusted tangency portfolio and the index.

0 200 400 600 800 1000 1200 1400

Business Day -20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Cumulative Return

Tangency portfolio Adjusted tangency portfolio Index

Figure 9: Cumulative returns when investing in tangency portfolio compared to index During the five year period, the tangency portfolio yields a return of 55.60% while the standard deviation for the duration is 0.97 percentage points. When adjusting to the observed index risk, the adjusted optimal portfolio yields a return of 46.74% compared to the index return of 40.39%. Furthermore, the adjusted optimal portfolio performs better than the index 53.54% of all days, and they move in the same direction as each other 79.87% of days. The total standard deviation for the period is 0.85 percentage points for the adjusted optimal portfolio and 0.86 percentage points for the index. Lastly, the correlation coefficient between them is 0.88. A fraction of times the strategy beat index is shown in the table below. All the results for each month is presented in Appendix C.

By days By months by years Fraction of times the

strategy beat index 53.54% 46.67% 60.00%

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4.2 Results when Weighting to Expected Risk

In this simulation the investor wants to match the risk level of the index, but obviously does not know how much the index will vary beforehand. Consequently, the investor will have to make predictions about the variability of the index and choose its strategy before observing the actual outcome. In order to maximize the Sharpe ratio, the investor will use a strategy consisting of investing in a mix of the tangency portfolio and the risk-free rate to expose themselves to the same predicted risk of the index. When using this method, one first finds the tangency portfolio and then weights it by moving along the efficient frontier, as described in the methodology. The expected risk of the index is calculated the same way as the expected risk of each individual stock, using the same historical data.

Figure 10: The tangency portfolio and efficient frontier without borrowing

There is another difference between weighting to an expected risk and adjusting to the actual observed risk. When adjusting to the actual observed risk, the return is adjusted so that the Sharpe ratio is constant. This is equivalent to moving the portfolio along the capital market line. When not knowing the actual risk, the practical implication of moving along the capital market line is to put a fraction of ones investments in the risk-free asset. When the risk of the index used, expected or actual, is below the risk of the tangency portfolio, this means moving down on the capital market line in both cases. However, when

the index risk is higher than that of the tangency portfolio, moving up on the CAL would mean borrowing at the risk-free rate and further invest the money in the tangency portfolio.

This is less applicable in the real world, since an investor may not always be able to borrow money, let alone at the risk-free rate. If the investor cannot borrow at the risk-free rate and still wants to match the index risk, they will then have to invest in a portfolio higher up on the Markowitz bullet that has a lower Sharpe ration than the tangency portfolio. In contrast, this is not a restriction when adjusting to the actual observed risk, since moving up the capital market line has no intuitive meaning other than keeping the Sharpe ratio constant. Therefore, a difference between investing in a mix of the tangency portfolio and the risk-free asset and investing all funds in the tangency portfolio is that the segment of the capital market line which is above the tangency portfolio is not used in the case where the strategy is to match the expected risk without borrowing. The efficient frontier and the adjusted optimal portfolio for November 2017 is illustrated in Figure 10.

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200 400 600 800 1000 1200 Business Day

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Cumulative Return

Optimal portfolio weighted by expected risk Index

Figure 11: Cumulative returns for strategy weighted to expected risk

The cumulative returns of investing in the strategy between years 2015 and 2019 is shown in Figure 11. During the five year period, the strategy yielded a return of 52.36% while the index yielded a return of 40.39%. The strategy performed better than the index 54.33% of days, 50.00% of months and 60.00% of years.

By days By months by years Fraction of times the

strategy beat index 54.33% 50.00% 60.00%

Once again, the movement of the strategy seems to have certain patterns with the movement of the index. The returns of the strategy and the index have a correlation coefficient of 0.86 and the value changed in the same direction 79.87% of all times. The total standard deviation for the period is 0.86 percentage points for the strategy and 0.86 percentage points for the index. A summary of the results for each separate month is shown in Appendix D.

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4.3 Limitations of Asset Allocation

Another highly realistic constraint that can be added to the problem is to limit the weight that can be invested in a single asset. For instance, when investing according to the tangency portfolio, 20 optimal portfolios suggested to invest 40% in a single asset during the backtesting period. An investor might be reluctant to choose such a portfolio for many reasons, but the most important rationale is probably to reduce the exposure of unpredictable firm-specific risk. When including these constraints, two aspects are relevant to consider - the optimizational implications and the practical implications. The optimizational implications are how much the maximum Sharpe ratio based on the 5 years of historical data will decrease with the new optimized portfolios, and the practical implications are how much the actual returns of the optimized portfolios will differ. In theory, a portfolio based on historical data with such constraints would perform worse, but since expected future returns are based on this historical data, the actual returns might yield a different result.

Figure 12: The expected Sharpe ration when imposing a allocation limit for November 2017 To illustrate how the expected Sharpe

ratio can change when imposing an allocation limit, the ratio can be calculated for a range of asset limits. As an example, Figure 12 shows the relationship for the tangency portfolio of November 2017. Observe that expected Sharpe ratio begins to decrease from a limit of around 20%. Similar patterns can be found for all months.

In Figure 9, the cumulative returns of tangency portfolios with different asset allocation limits are plotted. The green graph that corresponds to the portfolio that has ”no limit” is the same as the green graph in Figure 4, meaning that there is no asset allocation limit and no risk adjustment with the risk-free rate.

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0 200 400 600 800 1000 1200 1400 Business Day

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Cumulatve Return

No limit 20% limit 10% limit 5% limit

Figure 13: Cumulative returns of tangency portfolios with asset allocation limits Observe in Figure 13 that for any given business day, the cumulative returns of the tangency portfolio with no asset allocation limit is higher or approximately equal to the portfolio with 20% limit until around business day 1050, where the 20% limit portfolio pulls away.

Also note that both the 5% and the 10% limit portfolio are consistently lower than the ”no- limit” portfolio, with an exception around business day 1100 where the tangency portfolio drops to about the same level as the 10% limit portfolio. The outcome of the no-limit portfolio during the period is further discussed in subsubsection 5.1.2.

The daily standard deviation of returns for the whole five year period is also calculated for each of the portfolios. Observe that the standard deviation decreases with the asset allocation limit. See the table below:

No limit 20% limit 10% limit 5% limit Standard

deviation (pp) 0.97 0.92 0.85 0.76

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4.4 Validation of Results

Although the simulations show that the optimized portfolios have performed better than the index, assumptions and simplifications have been made that might inflate the results compared to reality. Hence, we have not proven that optimized portfolios, on average, always will beat index, but rather provided empirical results that support the possibility of it. It is useful to view parts of the analysis in a critical way to validate the assumptions and methods.

When making assumptions about the future, two questions might be considered relevant;

(1) how well the assumptions predict the future, and (2) how much one can trust the result of the prediction. For example, the mean does not say anything about the range of which one finds it probable for a new observations to fall within, even if it is correctly predicted.

(a) Expected and observed return (b) Expected and observed sigma of returns Figure 14: Expected return and risk of the tangency portfolio compared to the actual observed return and risk of the tangency portfolio

Figure 14 shows the predicted return and standard deviation, which are calculated adap- tively on five years of historical data, compared to the actual return and standard deviation of each month. Both graphs show that while the expected values are relatively stable, the observed values vary without any clear pattern. The mean of the difference between the return of the observation and the predicted return is -0.05 percentage points, which means that the prediction is on average 0.05 percentage points too high. Put in context, the explanation to this is that returns between years 2010 to 2015 are higher than the returns between years 2015 to 2019. Note also that the standard deviation is generally large compared to the expected return. This means that even if the mean return is predicted correctly, it may not be a good predictor of the return for a given day.

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It is also useful to know if the returns can be attributed to a certain distribution, and specifically if the returns originate from a normal distribution. Figure 15 shows that the returns of the of the tangency portfolio are not normally distributed. Instead of the normal distribution, some analysts suggest modelling the daily returns with the Laplace distribution [26][27]. While the returns do seem to fit the Laplace distribution slightly better, they still have slight deviations from the theoretical distribution at the tails.

Figure 15: Histogram and QQ-plot of the realized returns for the adjusted tangency portfolio

5 Discussion and Analysis

The results do show that portfolios optimized using modern portfolio theory can beat the index during a fixed time period. However, to be able to draw more conclusions, one has to consider both mathematical and non-mathematical factors. In this thesis, the data and the model are simplified for convenience purposes, both for the authors and the reader.

However, this does not mean that the data and the model are unviable for application in real life. In this chapter, a discussion about the viability of modern portfolio theory and the results from this thesis is held.

5.1 Analysis of Results

Despite that the portfolio consists of 30 stocks, the model often suggests to invest most of the funds in a handful of stocks. This means that the model considers some stocks much better than others and puts a large fraction of the funds in these stocks, despite the fact that the fundamental idea of the model is to encourage diversification. Appendix B shows that the stocks for The Home Depot, UnitedHealth Group and McDonald’s have been particularly dominating. However, even though the strategy invests most of its assets in a few stocks, there is still a clear pattern between the movement of the portfolio and the index. This is likely because of the underlying systematic risk that affects both the portfolio and the index. This result is expected due to the theory that systematic risk cannot be diversified. It can be debated whether a portfolio is diversified sufficiently when only investing in a handful of stocks, but undoubtedly the model considers these as optimal based on the input data.

Evaluation of a Portfolio in Dow Jones Industrial Average Optimized by Mean-Variance Analysis

Evaluation of a Portfolio in Dow

Jones Industrial Average

Optimized by Mean-Variance

Analysis

DANIEL LIU

ALEXANDER STRID

Evaluation of a Portfolio in

Dow Jones Industrial Average

Optimized by Mean-Variance

Analysis

Daniel Liu

Alexander Strid

Utv¨ardering av en portf¨olj i Dow Jones Industrial Average

optimerad genom mean-variance analysis

Contents

1 Introduction

2 Theoretical Framework

3 Methodology

4 Results

5 Discussion and Analysis