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60-663: In 2014, the Prague Stock Exchange introduced the total return index PX-TR . It shares the same base as the PX Index but, unlike the PX index, the PX-TR takes dividends into account. After an initial boom encouraged by voucher privatization (the top was in February 1994 retroactively calculated on 1245 points) the index started to decline fast, and ended the year 1994 with 557 points. In 1995

120-470: A r [ w T R ] ∑ i w i = 1 {\displaystyle {\begin{cases}E[w^{T}R]=\mu \\\min \sigma ^{2}=Var[w^{T}R]\\\sum _{i}w_{i}=1\end{cases}}} Portfolios are points in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The third equation states that the portfolio should fall on

180-482: A diversified portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk. The mean-variance framework for constructing optimal investment portfolios was first posited by Markowitz and has since been reinforced and improved by other economists and mathematicians who went on to account for the limitations of the framework. If all the asset pairs have correlations of 0—they are perfectly uncorrelated—the portfolio's return variance

240-473: A correctly priced asset in this context. Intuitively (in a perfect market with rational investors ), if a security was expensive relative to others - i.e. too much risk for the price - demand would fall and its price would drop correspondingly; if cheap, demand and price would increase likewise. This would continue until all such adjustments had ceased - a state of " market equilibrium ". In this equilibrium, relative supplies will equal relative demands: given

300-403: A hyperbola in the ( σ , μ ) {\displaystyle (\sigma ,\mu )} plane. The hyperbola has two branches, symmetric with respect to the μ {\displaystyle \mu } axis. However, only the branch with σ > 0 {\displaystyle \sigma >0} is meaningful. By symmetry, the two asymptotes of

360-578: A plane defined by ∑ i w i = 1 {\displaystyle \sum _{i}w_{i}=1} . The first equation states that the portfolio should fall on a plane defined by w T E [ R ] = μ {\displaystyle w^{T}E[R]=\mu } . The second condition states that the portfolio should fall on the contour surface for ∑ i j w i ρ i j w j {\displaystyle \sum _{ij}w_{i}\rho _{ij}w_{j}} that

420-622: A problem in quadratic curves . On the market, we have the assets R 1 , R 2 , … , R n {\displaystyle R_{1},R_{2},\dots ,R_{n}} . We have some funds, and a portfolio is a way to divide our funds into the assets. Each portfolio can be represented as a vector w 1 , w 2 , … , w n {\displaystyle w_{1},w_{2},\dots ,w_{n}} , such that ∑ i w i = 1 {\displaystyle \sum _{i}w_{i}=1} , and we hold

480-406: A risk-free asset, because they pay a fixed rate of interest and have exceptionally low default risk. The risk-free asset has zero variance in returns if held to maturity (hence is risk-free); it is also uncorrelated with any other asset (by definition, since its variance is zero). As a result, when it is combined with any other asset or portfolio of assets, the change in return is linearly related to

540-462: A second bottom was reached on 18 February 2009 at 629 points. The components as of March 2020 are sorted by reduced market capitalization: In March 2014 Prague Stock Exchange introduced new total return index PX-TR. The index shares same base as PX index, but unlike PX index take into account dividends . As the starting exchange day (a benchmark date) was selected 20 March 2006, when PX-50 and PX-D were merged. Main aim of creation of PX-TR index

600-426: A total return index. The TRI is also used to develop a portfolio as a weighted combination of assets, as it is described in modern portfolio theory . Though this theory is working with historical data, the models following this theory are trying to calculate the expected return based on a selected combination of assets. For example, in this way a stock portfolio representing a part of a stock index can be compared with

660-422: Is also called diversifiable, unique, unsystematic, or idiosyncratic risk. Systematic risk (a.k.a. portfolio risk or market risk) refers to the risk common to all securities—except for selling short as noted below, systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with

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720-481: Is as close to the origin as possible. Since the equation is quadratic, each such contour surface is an ellipsoid (assuming that the covariance matrix ρ i j {\displaystyle \rho _{ij}} is invertible). Therefore, we can solve the quadratic optimization graphically by drawing ellipsoidal contours on the plane ∑ i w i = 1 {\displaystyle \sum _{i}w_{i}=1} , then intersect

780-514: Is called the capital allocation line (CAL), and its formula can be shown to be In this formula P is the sub-portfolio of risky assets at the tangency with the Markowitz bullet, F is the risk-free asset, and C is a combination of portfolios P and F . By the diagram, the introduction of the risk-free asset as a possible component of the portfolio has improved the range of risk-expected return combinations available, because everywhere except at

840-454: Is different between capital gains and dividends , so that the total return index only forms a rough approximation of what a long term investor can expect to keep after taxation. Many stock indexes are calculated as a price return index and a total return index as well: The US stock index S&P 500 is an example of a price return index and the German stock market index DAX is an example of

900-416: Is different from a price return index. A price return index only considers price movements (capital gains or losses) of the securities that make up the index, while a total return index includes dividends, interest, rights offerings and other distributions realized over a given period of time. Looking at an index's total return is usually considered a more accurate measure of performance. Typically, taxation

960-416: Is easily solved using a Lagrange multiplier which leads to the following linear system of equations: One key result of the above analysis is the two mutual fund theorem . This theorem states that any portfolio on the efficient frontier can be generated by holding a combination of any two given portfolios on the frontier; the latter two given portfolios are the "mutual funds" in the theorem's name. So in

1020-501: Is free money, breaking the no arbitrage assumption. Suppose ∑ i v i ≠ 0 {\displaystyle \sum _{i}v_{i}\neq 0} , then we can scale the vector to ∑ i v i = 1 {\displaystyle \sum _{i}v_{i}=1} . This means that we have constructed a risk-free asset with return v T R {\displaystyle v^{T}R} . We can remove each such asset from

1080-410: Is hyperbolic, and the upper part of the hyperbolic boundary is the efficient frontier in the absence of a risk-free asset (sometimes called "the Markowitz bullet"). Combinations along this upper edge represent portfolios (including no holdings of the risk-free asset) for which there is lowest risk for a given level of expected return. Equivalently, a portfolio lying on the efficient frontier represents

1140-434: Is less than the return of the global MVP, in order that the tangency portfolio exists. However, even in this case, as μ R F {\displaystyle \mu _{RF}} approaches μ M V P {\displaystyle \mu _{MVP}} from below, the tangency portfolio diverges to a portfolio with infinite return and variance. Since there are only finitely many assets in

1200-402: Is the sum over all assets of the square of the fraction held in the asset times the asset's return variance (and the portfolio standard deviation is the square root of this sum). If all the asset pairs have correlations of 1—they are perfectly positively correlated—then the portfolio return's standard deviation is the sum of the asset returns' standard deviations weighted by the fractions held in

1260-507: The above problem is possible, with potential caveats (poor numerical accuracy, requirement of positive definiteness of the covariance matrix...). An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return R T w . {\displaystyle R^{T}w.} This version of the problem requires that we minimize subject to and for parameter μ {\displaystyle \mu } . This problem

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1320-635: The above problem, called the critical line algorithm , that can handle additional linear constraints, upper and lower bounds on assets, and which is proved to work with a semi-positive definite covariance matrix. Examples of implementation of the critical line algorithm exist in Visual Basic for Applications , in JavaScript and in a few other languages. Also, many software packages, including MATLAB , Microsoft Excel , Mathematica and R , provide generic optimization routines so that using these for solving

1380-410: The absence of a risk-free asset, an investor can achieve any desired efficient portfolio even if all that is accessible is a pair of efficient mutual funds. If the location of the desired portfolio on the frontier is between the locations of the two mutual funds, both mutual funds will be held in positive quantities. If the desired portfolio is outside the range spanned by the two mutual funds, then one of

1440-417: The amount paid for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model that derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole. The CAPM is usually expressed: A derivation

1500-458: The assets according to w T R = ∑ i w i R i {\displaystyle w^{T}R=\sum _{i}w_{i}R_{i}} . Since we wish to maximize expected return while minimizing the standard deviation of the return, we are to solve a quadratic optimization problem: { E [ w T R ] = μ min σ 2 = V

1560-430: The assets can be exactly replicated using the other assets at the same price and the same return. Therefore, there is never a reason to buy that asset, and we can remove it from the market. Suppose ∑ i v i = 0 {\displaystyle \sum _{i}v_{i}=0} and v T R ≠ 0 {\displaystyle v^{T}R\neq 0} , then that means there

1620-462: The change in risk as the proportions in the combination vary. When a risk-free asset is introduced, the half-line shown in the figure is the new efficient frontier. It is tangent to the hyperbola at the pure risky portfolio with the highest Sharpe ratio . Its vertical intercept represents a portfolio with 100% of holdings in the risk-free asset; the tangency with the hyperbola represents a portfolio with no risk-free holdings and 100% of assets held in

1680-403: The combination offering the best possible expected return for given risk level. The tangent to the upper part of the hyperbolic boundary is the capital allocation line (CAL) . Matrices are preferred for calculations of the efficient frontier. In matrix form, for a given "risk tolerance" q ∈ [ 0 , ∞ ) {\displaystyle q\in [0,\infty )} ,

1740-595: The context of proportional reinsurance, under a stronger assumption. The paper was obscure and only became known to economists of the English-speaking world in 2006. MPT assumes that investors are risk averse , meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact trade-off will not be

1800-425: The contours with the plane { w : w T E [ R ] = μ  and  ∑ i w i = 1 } {\displaystyle \{w:w^{T}E[R]=\mu {\text{ and }}\sum _{i}w_{i}=1\}} . As the ellipsoidal contours shrink, eventually one of them would become exactly tangent to the plane, before the contours become completely disjoint from

1860-475: The decline continued (influenced by the Mexican crisis , that discouraged foreign investors from emerging markets) and the index reached its first bottom on 29 June with 387 points. Then the index increased slowly and ended the year 1995 with 426 points. The slow increase continued in 1996 and the index reached its second top on 25 February 1997 with 629 points. In 1998, influenced by the 1998 Russian financial crisis ,

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1920-400: The efficient frontier is found by minimizing the following expression: where The above optimization finds the point on the frontier at which the inverse of the slope of the frontier would be q if portfolio return variance instead of standard deviation were plotted horizontally. The frontier in its entirety is parametric on q . Harry Markowitz developed a specific procedure for solving

1980-480: The historical variance and covariance of returns is used as a proxy for the forward-looking versions of these quantities, but other, more sophisticated methods are available. Economist Harry Markowitz introduced MPT in a 1952 essay, for which he was later awarded a Nobel Memorial Prize in Economic Sciences ; see Markowitz model . In 1940, Bruno de Finetti published the mean-variance analysis method, in

2040-426: The horizontal axis (volatility). Volatility is described by standard deviation and it serves as a measure of risk. The return - standard deviation space is sometimes called the space of 'expected return vs risk'. Every possible combination of risky assets, can be plotted in this risk-expected return space, and the collection of all such possible portfolios defines a region in this space. The left boundary of this region

2100-649: The hyperbola intersect at a point μ M V P {\displaystyle \mu _{MVP}} on the μ {\displaystyle \mu } axis. The point μ m i d {\displaystyle \mu _{mid}} is the height of the leftmost point of the hyperbola, and can be interpreted as the expected return of the global minimum-variance portfolio (global MVP). The tangency portfolio exists if and only if μ R F < μ M V P {\displaystyle \mu _{RF}<\mu _{MVP}} . In particular, if

2160-429: The idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. The variance of return (or its transformation, the standard deviation ) is used as a measure of risk, because it is tractable when assets are combined into portfolios. Often,

2220-460: The index reached its historical bottom on 8 October with 316 points. In 1999 the index emerged back to 500 points. With the help of the dot-com bubble and starting banks privatization, the index reached its second top on 24 March 2000 with 691 points. With the end of dot-com bubble and the IPB Bank crisis, the index declined back to 500 points. In 2001, the index slightly decreased and with the help of

2280-561: The influence of the September 11, 2001 attacks reached bottom on 17 September with 320 points. In 2002, the index slowly increased back to 500 points. In 2003 the index was rising and ended the year with 659 points. With the help of the Czech Republic European Union accession, the index continued rising in 2004. It reached 1000 points on 19 November and ended the year with 1032 points. The index continued rising in 2005 and ended

2340-613: The line, { μ = ( w ′ T E [ R ] ) t + w T E [ R ] σ 2 = ( w ′ T ρ w ′ ) t 2 + 2 ( w T ρ w ′ ) t + ( w T ρ w ) {\displaystyle {\begin{cases}\mu &=(w'^{T}E[R])t+w^{T}E[R]\\\sigma ^{2}&=(w'^{T}\rho w')t^{2}+2(w^{T}\rho w')t+(w^{T}\rho w)\end{cases}}} giving

2400-406: The market, constructing one risk-free asset for each such asset removed. By the no arbitrage assumption, all their return rates are equal. For the assets that still remain in the market, their covariance matrix is invertible. The above analysis describes optimal behavior of an individual investor. Asset pricing theory builds on this analysis, allowing MPT to derive the required expected return for

2460-421: The market, such a portfolio must be shorting some assets heavily while longing some other assets heavily. In practice, such a tangency portfolio would be impossible to achieve, because one cannot short an asset too much due to short sale constraints , and also because of price impact , that is, longing a large amount of an asset would push up its price, breaking the assumption that the asset prices do not depend on

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2520-412: The model: In general: For a two-asset portfolio: For a three-asset portfolio: The algebra can be much simplified by expressing the quantities involved in matrix notation. Arrange the returns of N risky assets in an N × 1 {\displaystyle N\times 1} vector R {\displaystyle R} , where the first element is the return of the first asset,

2580-410: The mutual funds must be sold short (held in negative quantity) while the size of the investment in the other mutual fund must be greater than the amount available for investment (the excess being funded by the borrowing from the other fund). The risk-free asset is the (hypothetical) asset that pays a risk-free rate . In practice, short-term government securities (such as US treasury bills ) are used as

2640-445: The performance version of the stock index. This article about stock exchanges is a stub . You can help Misplaced Pages by expanding it . Modern portfolio theory Modern portfolio theory ( MPT ), or mean-variance analysis , is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing,

2700-470: The plane. The tangent point is the optimal portfolio at this level of expected return. As we vary μ {\displaystyle \mu } , the tangent point varies as well, but always falling on a single line (this is the two mutual funds theorem ). Let the line be parameterized as { w + w ′ t : t ∈ R } {\displaystyle \{w+w't:t\in \mathbb {R} \}} . We find that along

2760-433: The portfolio occurring at the tangency point; points between those points are portfolios containing positive amounts of both the risky tangency portfolio and the risk-free asset; and points on the half-line beyond the tangency point are portfolios involving negative holdings of the risk-free asset and an amount invested in the tangency portfolio equal to more than 100% of the investor's initial capital. This efficient half-line

2820-513: The portfolio variance is unchanged. An investor can reduce portfolio risk (especially σ p {\displaystyle \sigma _{p}} ) simply by holding combinations of instruments that are not perfectly positively correlated ( correlation coefficient − 1 ≤ ρ i j < 1 {\displaystyle -1\leq \rho _{ij}<1} ). In other words, investors can reduce their exposure to individual asset risk by holding

2880-527: The portfolio. If the covariance matrix is not invertible, then there exists some nonzero vector v {\displaystyle v} , such that v T R {\displaystyle v^{T}R} is a random variable with zero variance—that is, it is not random at all. Suppose ∑ i v i = 0 {\displaystyle \sum _{i}v_{i}=0} and v T R = 0 {\displaystyle v^{T}R=0} , then that means one of

2940-424: The portfolio. For given portfolio weights and given standard deviations of asset returns, the case of all correlations being 1 gives the highest possible standard deviation of portfolio return. The MPT is a mean-variance theory, and it compares the expected (mean) return of a portfolio with the standard deviation of the same portfolio. The image shows expected return on the vertical axis, and the standard deviation on

3000-427: The relationship of price with supply and demand, since the risk-to-reward ratio is "identical" across all securities, proportions of each security in any fully-diversified portfolio would correspondingly be the same as in the overall market. More formally, then, since everyone holds the risky assets in identical proportions to each other — namely in the proportions given by the tangency portfolio — in market equilibrium

3060-533: The risk (standard deviation) of the market portfolio. Since a security will be purchased only if it improves the risk-expected return characteristics of the market portfolio, the relevant measure of the risk of a security is the risk it adds to the market portfolio, and not its risk in isolation. In this context, the volatility of the asset, and its correlation with the market portfolio, are historically observed and are therefore given. (There are several approaches to asset pricing that attempt to price assets by modelling

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3120-458: The risk-free return is greater or equal to μ M V P {\displaystyle \mu _{MVP}} , then the tangent portfolio does not exist . The capital market line (CML) becomes parallel to the upper asymptote line of the hyperbola. Points on the CML become impossible to achieve, though they can be approached from below. It is usually assumed that the risk-free return

3180-437: The risky assets' prices, and therefore their expected returns, will adjust so that the ratios in the tangency portfolio are the same as the ratios in which the risky assets are supplied to the market. The result for expected return then follows, as below. Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Specific risk

3240-409: The same for all investors. Different investors will evaluate the trade-off differently based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk vs expected return profile — i.e., if for that level of risk an alternative portfolio exists that has better expected returns. Under

3300-402: The second element of the second asset, and so on. Arrange their expected returns in a column vector μ {\displaystyle \mu } , and their variances and covariances in a covariance matrix Σ {\displaystyle \Sigma } . Consider a portfolio of risky assets whose weights in each of the N risky assets is given by the corresponding element of

3360-418: The stochastic properties of the moments of assets' returns - these are broadly referred to as conditional asset pricing models.) Systematic risks within one market can be managed through a strategy of using both long and short positions within one portfolio, creating a "market neutral" portfolio. Market neutral portfolios, therefore, will be uncorrelated with broader market indices. The asset return depends on

3420-421: The tangency portfolio the half-line gives a higher expected return than the hyperbola does at every possible risk level. The fact that all points on the linear efficient locus can be achieved by a combination of holdings of the risk-free asset and the tangency portfolio is known as the one mutual fund theorem , where the mutual fund referred to is the tangency portfolio. The efficient frontier can be pictured as

3480-495: The weight vector w {\displaystyle w} . Then: and For the case where there is investment in a riskfree asset with return R f {\displaystyle R_{f}} , the weights of the weight vector do not sum to 1, and the portfolio expected return becomes w ′ μ + ( 1 − w ′ 1 ) R f {\displaystyle w'\mu +(1-w'1)R_{f}} . The expression for

3540-450: The year with 1473 points. On 20 March 2006 the PX 50 index was merged with PX-D into the PX index. The index reached bottom on 13 June with 1167 points after legislative elections , but ended the year with 1589 points. On 29 October 2007 it reached its top with 1936 points and ended the year with 1815 points. As result of the 2007–2008 financial crisis , the index reached 700 points on 27 October 2008, losing 50% of its value in two months;

3600-460: Was to promote above average dividend yield of Prague Stock Exchange. Total return index A total return index is an index that measures the performance of a group of components by assuming that all cash distributions are reinvested, in addition to tracking the components' price movements. While it is common to refer to equity based indices , there are also total return indices for bonds and commodities . A total return index (TRI)

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