Optimization What do, in this text, I mean when I write optimization? When talking about optimization I refer to the mathematical method of minimize or maximize a certain function The solution to this problem is only optimal in a mathematical sense and of course restricted by the mathematical model it is a solution to This means that the solution to the optimization problem is not necessarily the best or even a good plan of action to the problem for which the, in this case, investor uses the model Still, since we cannot model all the complexities in the world and calculate an optimal solution, I think a tool like a mathematical model handling some aspects of a problem can help an investor in his or hers investment decisions Traditional, mean-variance portfolio optimization When Markowitz published the article Portfolio Selection in 1952, he initiated the academic field of Portfolio Theory Before this little research were done in the mathematical relations of portfolio selection John Burr William s Theory of Investment Value inspired Markowitz Williams claimed that the value of an asset should equal its future dividends But, since future dividends are unknown, Markowitz asserted that the value of an asset should equal its expected future dividends Still if the investor only considers expected future dividends the portfolio only need to contain one asset, the one with the highest expected future dividends Markowitz showed mathematically that investors not only care about the expected future dividends but also risk On the basis of this Markowitz developed the portfolio theory saying that investor chooses between a set of Pareto optimal risk-return combinations He also proved how and why investors diversify and that diversification leads to portfolios with maintained expected return but lower risk Markowitz model, mean-variance portfolio theory Trade-offs between risk and return Diversification The feasible set The Minimum-variance set The efficient frontier The two-fund theorem Inclusion of a risk-free asset The one-fund theorem Markowitz assumptions Rational investment behaviour Theory of rational choice under uncertainty Portfolio optimization and assumptions In portfolio theory you are supposed to provide an matrix with expected covariance between all assets and a vector of expected returns However, in the field there is more discussions on 1 av 9
the vector of expected returns than on the expected covariance matrix There is often a discussion about whether to use historical values or not in the prognosis of the expected returns Practical problems in mean-variance optimization Unconstrained optimization generates portfolios with large negative weights in assets with low level of capitalization Constrained optimization results in corner solutions Require the investor to specify views, in form of expected return, for every asset in investment universe Extremely sensitive to the assumptions made Black-Litterman The Black-Litterman model was first presented in 1992, when the two initiators, Fisher Black and Robert Litterman, published the article Global Portfolio Optimization (1992) The aim was to solve many of the problems faced by traditional mean-variance optimization The well-known problems in mean-variance optimization are considered to be the main reason to the limited use of models in actual investment decisions Unconstrained meanvariance optimization generates recommended portfolios with large negative positions in assets with low level of capitalization (varför är detta ett problem?) Many investors are not allowed to go short When adding this constraint, the optimizer ends up in a corner solution, resulting in recommendations with null weights in most of the assets in the investor universe One other problem in mean-variance optimization is that the user must input a complete set of expected returns Investors, however, often concentrate on a small segment of their potential investment universe, choosing assets they assume is under- or overvalued (Black & Litterman, 1992) This means that in mean-variance optimization, the investor must input expected return on assets where she/he doesn t have an opinion Black and Litterman apply, what they call, an equilibrium approach, meaning that they use the idealised market equilibrium as a point of reference The investor specifies a chosen number of market views in form of expected returns and a level of confidence for each expected return The model then combines the views with the equilibrium returns, creating a portfolio in which bets are taken on the assets where the investor has views and nowhere else, 2 av 9
the size of the bets in relation to the equilibrium portfolio weights depends on the confidence levels specified by the user This approach has shown to generate more realistic portfolios As I wrote above, in mean-variance optimization the investor must input expected returns for every asset in his or her investment universe The model then optimizes these, generating a recommended portfolio In the Black-Litterman model the investor can choose exactly how many views he or she would like to specify, leaving some or many assets in the model with no opinions Still, there is a complete vector of expected returns handed over, by the model, to the optimizer The Black-Litterman model combines the views with the equilibrium-expected returns The equilibrium-expected returns calculated, by the model, for every asset the model handles So the model hands over a complete set of modified expected returns to the optimizer Market Efficiency and the Equilibrium This section aims to provide an understanding of the equilibrium approach suggested by Black and Litterman and the implications of this To understand what the equilibrium approach implicates, we need first to understand the debate on market efficiency In this state it is written from a behavioural finance point of view and is therefore not very objective Rationally behaved markets? There is an ongoing debate on weather markets are rationally behaved The efficient markets hypothesis has been the dominating view since Fama (1970) presented the efficient financial theory as one in which securities always are prised in respect to all available information The efficient market hypothesis then claims that real-world financial markets are efficient according to this definition In the last 20 years this view of markets has been challenged The forces that are supposed to attain the efficiency, such as arbitrage trading, are likely to be much weaker than the defenders of the hypothesis stress (Shleifer, 2000) Behavioural finance, both theoretically and empirically, offer an alternative approach Behavioural finance claims that errors, as they are discussed in EMH, are both systematic and significant and also that they are expected to persist for long periods of time The efficient market hypothesis (EMH) rests, according to Shleifer (2000), on three arguments relying on progressively weaker assumptions: 1 Investors are assumed to be rational and hence to value securities rationally 2 If some investors are not rational, their irrational trades are random and therefore cancel each other out 3 If investors should be acting irrational in the similar ways, rational arbitrageurs act on the market and eliminate the influence on prices 3 av 9
A rational investor is here considered as an investor who values securities on basis on their fundamental value, the net present value of its future cash flows, discounted using their risk characteristics If we start looking at the firs argument, it is difficult to sustain that investor acts fully rationally Black (1986) claims that investor often trade on noise rather than on information, fail to diversify, sell winning securities and hold on to the losers etc As Kahneman and Reipe (1998) state, people deviate form the standard decision making model in many essential ways One of the most well known examples of this is what Tversky and Kahneman (1979) calls loss aversion, meaning that the loss function is steeper than the gain function Tversky and Kahneman (1973) show also that individuals violate Bayes rule and other rules of probability theory Kahneman and Tversky have shown that people assess the same probability distribution to the empirical mean value of small and large numbers This bias they refer to as The law of small numbers Kahneman and Tversky also question the second argument in the efficient market hypothesis, saying that irrational investors trades are random and therefore cancel each other out Kahneman and Tversky dispose this entirely Most often people deviate from rationality in the same way Investors are often evaluated according to a benchmark and therefore often act to minimize the risk of falling behind They also often selects the same stock that other investment managers select to again avoid falling behind Lets look at the last of the arguments of the efficient market hypothesis Even if the trades of noisy investors are correlated, the arbitrageurs act to bring prices back to their fundamental values However, behavioural finance claims that arbitrage trades are risky and because of this limited Arbitrage relies heavily on the existence of close substitutes Yet, in many cases securities do not have good substitutes and therefore arbitrage trading cannot work to push prices back to fundamental values For example an investor believing that stocks are overpriced cannot go short in stocks and buy a substitute portfolio But even when there are almost perfect substitutes and the prices of the two securities ultimately converge, the trade may lead to temporary losses Sometimes the arbitrageur cannot maintain the position trough the loss Most arbitrageurs do not manage their own money Instead they are agents for other people These investors evaluate their portfolios regularly and in quite short intervals If the evaluation horizon is shorter than the trade, the investor may not be satisfied whit the performance of the arbitrageur and therefore withdraw money If enough people withdraw money the fund may have to liquidate the position, leading to further performance problems Empirical evidence supporting the efficient market hypothesis in the 1960s and 1970s was however immense Shleifer divides the empirical evidences for the hypothesis into two categories First, when news affecting the value of a security hits the market, it should quickly and correctly affect the price of the security Quickly means that an investor who receives the information late should not be able to profit from this information Correctly, means that the price movement in response the new information should be accurate on average Second, 4 av 9
since rational investor values securities as its fundamental value, price must not be affected by changes in demand and supply of the security According to the first category no one is suppose to be able to make money on stale information This argument is somewhat difficult to challenge To do this, we need to define the meaning of stale information and making money Making money is hard to define In finance making money means owning superior return after an adjustment for risk Showing that a strategy, based on stale information, on average earns a positive return is not enough to show market inefficiency The profit may be just a fair market compensation for risk bearing, but to evaluate this we need a model for fair relationship between risk and return and so forth Still, when researchers find models for making money of stale information critics suggests a model of risk that reduces these profits to a fair compensation for risktaking One empirical result than can work as an argument that information not always is correctly and quickly incorporated in security prices is the so-called January effect The January effect shows that returns are superior in January, especially for small stocks, but there is no evidence that stock or small stocks are riskier in January than the rest of the year According to the second category rational investor only evaluate securities according to their fundamental values, meaning that changes in demand or supply should not affect prices Still research has shown that prices react to inclusion of stocks into the Standard and Poor s 500 Index Inclusion of an asset into the Index is not suppose to convey any information into the market, but still the asset price increases substantially and the increase is shown to be sustainable and sustainable According to Shole s theory, inclusion of a security in an index should not affect its price because of increased demand When the security is included the initial holders should want to sell Black, Litterman and the equilibrium So, what do Black and Litterman mean by equilibrium? In the book Modern Investment Management An Equilibrium Approach, by Robert Litterman and the Goldman Sachs Asset Management Quantitative Resources Group (2003), Litterman thoroughly discusses the concept of the equilibrium approach The equilibrium, according to Litterman, is an idealized state where supply equals demand Although he stresses that this state never really exists, there are a number of attractive characteristics about the idea According to Litterman there are natural forces, in form of arbitrageurs, in the economic system that works to eliminate deviations from equilibrium Even if there are frictions in markets like noise traders, uncertain information and lack of liquidity that results in situations where deviations are large and where the adjustment takes time, mispricing will over time be corrected Hence, the markets aren t always assumed to be in equilibrium The state of equilibrium is instead viewed as a centre of gravity Markets can deviate from this state, but forces will push the system back to equilibrium 5 av 9
Implied views about the market and efficiency etc Litterman claims that the reason of recommending the equilibrium approach is the belief that it is a good and appropriate frame of reference from which identification of deviations can be made and taken advantage of He admits that no financial theory can ever capture the complexity of financial markets Still, Financial theory has the most to say about markets that are behaving in a somewhat rational manner If we start by assuming that markets are simply irrational, then we have little more to say He refers to the extensive amount of literature we access if we are willing to accept the assumption of an arbitrage free market In order to take advantage of portfolio theory we also need to add the assumption that markets, over time, moves toward a rational equilibrium, according to Litterman Portfolio theory makes predictions about how markets will behave, tells investors how to structure their portfolios, how to minimize risk and also how to take maximum advantage of deviations from equilibrium It seems quite legible that the Black and Litterman believe that financial markets work according to the third argument in the efficient market hypothesis What does this mean? Equilibrium in the model When the Black-Litterman model is run without any constraints or market views, the model recommend the investor to hold the market portfolio This seems quite reasonable If we don t have any opinions about the market we shouldn t take any bets in relation to the benchmark However, if we have opinions about assets in our investment universe it is reasonable that the bets are taken in those assets and the rest of the assets have weights close to the market portfolio Most of what is written about the Black-Litterman model assumes a global asset allocation model and because of this Litterman et al (2003) argue that he domestic Capital Asset Pricing Model (CAPM) is a good starting point for a global equilibrium model In the article Universal Hedging (1990) Black discuss an equilibrium model providing a framework from which the Black-Litterman global asset allocation model has emerged Many have, before Black, tried to globalize the domestic CAPM, but Black is said to be the first to point out a universal hedging constant However, Black-Litterman is not only used in global asset management, but also in domestic equity portfolio management and fixed income portfolio management In these cases the equilibrium weights are easier to find using the domestic CAPM There is an obvious problem in using equilibrium weights as a point of reference This is because the weights are not observable Bevan and Winkleman (1998) however, present a way of dealing with this If the market is in equilibrium, then a representative investor will hold a part of the capitalization-weighted portfolio Many investors are evaluated according to 6 av 9
a benchmark Often the benchmark is a capitalization-weighted index (Black & Litterman, 2003) In this case the equilibrium portfolio is easy to find It is simply the benchmark weights From the equilibrium portfolio the equilibrium-expected return vector Π is calculated Π = δ Σw Σ = Covariance matrix of returns w = The weights in the equilibrium portfolio δ = Risk aversion, (According to Black & Litterman, 1991, a proportional constant based on the formulas in Black (1989)??? This is the expected return if the world would continue to be in the same state For each asset where the investor doesn t have a view, this is what will be handed over to the optimizer For the assets to which the investor has views, this will be combined with other stuff?? Views The idea of the Black-Litterman model is to combine a neutral equilibrium with investor specific market views In the model a view consists of more than just expected return for an asset The model provides a way for the investor to give relative views and the possibility to express that he or she is more confident in one view than another The model then combines the equilibrium with the views and takes bets in relation to the equilibrium on assets to which the investor has expressed views The more confident the investor is in each view, the larger the bets Often investor has views in the form of: I believe that equities in country A will outperform equities in country B or I believe domestic bonds will outperform domestic equities This view is in a relative form and in traditional mean-variance portfolio optimization these kind of vies cannot be easily expressed In Black-Litterman, however, this is possible Of course it is still possible to express absolute views on the form: Asset A has 5 % expected excess return For each view the investor also specifies a level of confidence If he or she has a very strong feeling for one view the level of confidence should be large and if the investor feels unsure of a market view the level of confidence for that view should be quite low The confidence level is then used when combining the view with the equilibrium-risk-premiums, for determining how much weight to be assigned to the view relative to the equilibrium The degree of confidence is a standard deviation around the expected excess returns (Litterman et al, 2003) We refer to this set of expected excess returns and levels of confidence, as the view portfolio We use the model to determine the conditional probability of the projected excess returns on each asset an currency (Bevan & Winkelman, 1998) Ω is a matrix consisting of the level of confidence for each view It is a diagonal matrix meaning that every level of confidence is independent of the others or that a view represent independent drawn from the future 7 av 9
distribution of returns or that the deviations of expected returns from the means of the distribution representing each view are independent The level of confidence dampens the influence of a particular view This because views are often not correct, but they tell us where we want the model to take bets Combining views with the equilibrium expected returns The Black-Litterman optimal portfolio is a weighted combination of the equilibrium portfolio and the view portfolio 8 av 9
Why do I write about this? My interest in financial modelling started some time ago After taken some classes in financial mathematics, portfolio theory etc during my graduate studies I got in contact with the Black-Litterman model during my final degree project Commissioned by a large Swedish bank, my task was to implement the model for Swedish government bonds and mortgagebacked bonds Not surprisingly, being a newcomer in the practical work of finance, I believed that this would be quite easy done The commissioner seemed very interested in the model, but acknowledged that he didn t quite understand the model I thought he would willingly participate in the development of their new portfolio allocation programme After a couple of months struggling with the articles written about the model and in constant search for a top down definition of all variables in the model I realised that there weren t and aren t many articles and books discussing the model Most of the existing articles on the Black-Litterman model discuss the general idea and not how to set each variable and late in the project I realised that, the model still is more of a framework than a detailed model Still I have and had the impression that the model is quite popular, accepted and used in portfolio management As the commissioner of the project introduced the model to me and also expressed, what I understood as a genuine interest of the model, I believed that he would participate in the development of the programme However I got almost no help during the development and when the project was finished he still didn t understand the model Since many of the assumptions in the model became obvious, question where raised on the use of models like this I had the feeling that users of these kinds of models should be very knowledgeable about portfolio models and both the explicit and implicit assumptions made, but maybe they aren t The project raised many questions about the application of models in the area of finance and especially in portfolio management When I started my doctoral studies I had a broad interest in portfolio management and started to read scientific articles in order to get to know the field At this time I thought I had left the Black-Litterman model behind Early on in my literature study I got in contact with the field of behavioural finance, which to me was very stimulating, because much of what was discussed in the field I could relate to my final degree project As I continued to read about behavioural finance I started to see connections between the progress??? in behavioural finance and the Black-Litterman model 9 av 9