The Efficient Frontier in Modern Portfolio Theory Weaknesses and How to Overcome Them

FEATURE

By Marcus Schulmerich, PhD, CFA®, FRM

n 1952, Harry Markowitz set a mile- give up on their bond values, are associ- to look at all possible combinations of stone in financial theory.1 By intro- ated with credit risk, i.e., the risk of los- weights and calculate for each of these Iducing mean-variance optimization ing money. combinations the portfolio return and (MVO), often referred to as “Markowitz This is, however, not what the volatility. To show this idea graphi- optimization,” he originated modern Markowitz meant when he talked cally, let’s look at figure 1, which shows portfolio theory (MPT). In doing so, about risk. In MPT, risk refers to mar- what is known as the μ-σ (mu-sigma) he provided a comprehensive theoreti- ket risk. Specifically, risk refers to space: Each dot represents a different cal framework for investment analysis the fluctuations in the security price. portfolio as a result of a certain combi- and guidelines for optimal portfolio Markowitz defined risk as the standard nation of weights for stocks A, B, and construction. Even 60 years later his deviation of the percentage returns of C. The portfolio’s expected return, μP model is still widely used among private a security—be they daily, weekly, or is shown on the vertical μ-axis and the and professional investors, despite its monthly returns in practice. This stan- associated volatility, σP, can be found on shortcomings. dard deviation is known as the volatility the horizontal σ-axis. This article presents the theory of of the security. In figure 1, we have shown only Markowitz, taking a closer look at the Therefore, Markowitz characterizes a dozen different combinations of assumptions and the issues associated each investment opportunity by: weights, so the picture is not striking with the model. It closes with answers to • its expected return (μ), and yet. However, if we look at substantially some of the most critical shortcomings • its volatility (σ). more combinations of weights for secu- in the theory of the efficient frontier. rities A, B, and C, such as we do in fig- Let’s go back to our example and ure 2, the picture gets clearer: The dots Modern Portfolio Theory: A Review try to construct a portfolio comprising do not seem to be evenly spread in the Let’s consider the following situation: three stocks: A, B, and C. The question μ-σ space. Even if you look at all possible You have won $100,000 in a lottery and now arises what the weights of each combinations of weights for stocks A, B, want to invest this amount in global stock ought to be. Theoretically we have and C, the resulting dots are not evenly equity securities. What stocks should you pick? What should your decision be FIGURE 1: PORTFOLIOS IN THE μ-σ SPACE based on? Thousands of stocks are available from which to choose, so you need Expected Return (µ) a systematic approach to investing. Naturally, you’d start looking at the µ expected return of each of these securi- P ties. On the other hand, you know that equity investing is associated with risk. These are the two dimensions you need to pay attention to: return versus risk. While return is easy to understand, the term “risk” needs clarification. For a layperson, risk is mainly associated with losing money. This is even more true σP Volatility (σ) after all the recent events in Greece. • Possible Portfolios Events such as the March 2012 Greek Source: SSgA, for illustrative purposes only. debt haircut, where investors had to

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spread in the μ-σ space. As a matter of FIGURE 2: PORTFOLIOS AND THEIR BOUNDARY IN THE SPACE fact, there are areas in the μ-σ space μ-σ where no dots lie. They are concen- Expected Return (µ) trated in a certain area that seems to be bounded. This boundary leads to what is known as the efficient frontier, as indi- µP cated in figure 3. Markowitz looks at an efficient portfolio. A portfolio is efficient if no other portfolio has the same expected return with a lower volatility. Graphically, this is explained in figure 3. On the green horizontal line lie all portfolios with an

expected return of μP. The efficient port- σP Volatility (σ) folio of all of those dots on the green line • Possible Portfolios is the one with the lowest volatility σ. Source: SSgA, for illustrative purposes only. If we undertake this procedure with all expected returns we get the curved line FIGURE 3: EFFICIENT FRONTIER IN THE SPACE in figure 3, comprising a blue part and a μ-σ gray part. While all portfolios located on Expected Return (µ) the blue and gray line are efficient in the sense of the definition above, obviously

only the blue line is beneficial to the µP investor because it provides increased returns. This blue line is called the MVP ef ficient frontier. The point of the line with the lowest volatility is called mini- mum-variance portfolio (MVP). So far no lending or borrowing of money is included in the model. But in σ reality you can also deposit your money P Volatility (σ) in the bank or borrow some more. The Efficient Frontier • Possible Portfolios • Portfolios with µ • MVP interest rate associated with these activi- P

ties is called the risk-free rate (Rf), i.e., it is Source: SSgA, for illustrative purposes only. the return on investment with no volatility. Please note that this concept already FIGURE 4: EFFICIENT FRONTIER WITH RISK-FREE ASSET assumes that lending and borrowing is possible at the same risk-free rate. Expected Return ( µ) Let’s come back to our original investment problem: How do I invest $100,000 in three stocks A, B, and C? We learned that the optimal combination of these three stocks is an efficient portfolio on the efficient frontier. However, it R is not clear yet which portfolio on the f MVP ef ficient frontier to choose. The higher return I demand the higher my associated risk will be, as figure 4 shows. The key idea is now to include the Volatility (σ) risk-free asset as a possible investment Efficient Frontier • Possible Portfolios • Tangency Portfolio

in your investment problem: Invest a Source: SSgA, for illustrative purposes only. portion of your $100,000 in an efficient

TABLE 1: ASSUMPTIONS REQUIRED FOR THE MARKOWITZ OPTIMIZATION Assets Investors Markets • Asset returns are nor- • Are rational and behave in a manner as to max- • Guarantee free access to fair and correct infor- mally distributed. imize their utility. mation on the returns and risk. • Everyone agrees on • Base decisions on expected returns and stan- • Are efficient and absorb the information quickly their distribution. dard deviations of the returns. and perfectly. • Assets are highly divisi- • Are risk averse and try to minimize the risk and • Have no transaction costs or taxes. ble into small parcels. maximize return. • Allow unrestricted short selling. • There is a risk-free • Perceive risk as the standard deviation of • Every asset is tradable at any point of time. asset. returns. • Maximize the single-horizon utility. • Are price takers, i.e., cannot influence prices.

Sources: Markowitz (1952) and Füss (2008) portfolio on the efficient frontier, invest TABLE 2: HISTORICAL MEAN RETURNS IN US$ AND VOLATILITIES OF THE MSCI WORLD INDEX (NET DIVIDENDS REINVESTED) the remaining in the risk-free asset. To give an example, let’s assume the risk- Period ending December 31, 2011 Return Annualized (%) Risk Annualized (%) free rate is 3 percent and the chosen Past 3 years 8.36 20.41 ef ficient portfolio on the efficient frontier has an expected return of 10 per- Past 5 years –4.53 20.90 cent and a volatility of 12 percent. If I Past 10 years 1.64 17.27 Source: MSCI, as of December 31, 2011. For all calculations, monthly data are used. Past performance is not a invest half of my money in the efficient guarantee of future results. portfolio and the other half in the risk- free asset, my overall investment has transaction costs or taxes, and the inves- tion is where shall we get the required an expected return of 0.5 × 10% + 0.5 × tors’ perception of risk isn’t exactly the input data? As shown above, the data 3% = 6.5%. Because the volatility of the same as the volatility of returns. Also, required to calculate a portfolio return risk-free asset is 0, the volatility of the the existence of a true risk-free asset is and volatility are the expected returns overall investment can be calculated as debatable, but this can be circumvented, for all securities and the covariances 0.5 × 12% = 6%.2 which eventually led to the zero-beta between the percentage returns of The efficient portfolio I should choose capital asset pricing model. all assets. The next question would is the portfolio that delivers the best rela- Some problems associated with MVO be, how does the choice of the input tionship of return versus risk, i.e., port- are even less obvious. As hinted above, parameters affect the output, i.e., how folio return (over risk-free rate) divided investors don’t necessarily perceive risk sensitive is the output versus the input by portfolio volatility. This ratio is known as the standard deviation of returns. The to MVO? as the Sharpe ratio and is also graphi- common perception of risk is that risk Below, we explore the problems aris- cally shown in figure 4: It is the slope of means losing money or doing worse than ing from these questions and advise a straight line that begins at the risk-free expected. Therefore many people, espe- how to overcome them. rate on the μ-axis and goes through a cially after the subprime crisis, use (in The Input Data to MVO and portfolio on the efficient frontier. addition) downside risk measures such Their Output Hence, to maximize the Sharpe ratio as drawdown or Value-at-Risk (VaR) we seek the portfolio with the steepest when looking at investments.3 In light of There are three sources for expected straight line. The efficient portfolio to the events in the stock markets in 2008 returns and covariances for MVO: We choose is the tangency portfolio. In MPT and 2011, these asymmetrical risk mea- derive predictions, or we use histori- this portfolio is also called the market sures that look primarily at the downside cal data to calculate historical returns portfolio and in practice it is replicated of returns might seem more intuitive to and covariances that we use as future by using a capitalization-weighted index. us or have a higher practical relevance to expected returns and covariances, or we For this model to be derived, several investment decisions. use a combination of both. assumptions have to be made about Before we continue on the relevance When using historical data, the vol- the asset returns, the investors, and of volatility as an appropriate risk mea- atility of the markets is a crucial aspect the markets. All these assumptions are sure, let’s take a closer look at MVO as to consider. Obviously, the impact of listed in table 1. In fact, most of these it is, and the relationship between the the chosen historical time period is assumptions have to be questioned (Füss input and the output of the portfolio important for the MVO output. In 2008). For instance, there certainly are optimization. Obviously, the first ques- table 2, the historical mean returns

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(in US$) and volatilities of the MSCI FIGURE 5: DEUTSCHE BANK STOCK PRICES World Equity Index (net dividends MARCH 31, 2007–OCTOBER 31, 2009

reinvested) were calculated for different US$ time periods as of December 31, 2011. 80 As table 2 shows, the choice of the his-

torical time period for MVO input 60 has a significant impact on the resulting data. This clearly affects the opti- 40 mal portfolio weights in MVO and the ef ficient frontier calculation itself. Another important factor to con- 20 sider when historical data are used is the

data frequency: Should daily, weekly, or 0 monthly data be used to, e.g., calculate Apr Aug Jan Jun Oct 2008 2008 2009 2009 2009 the covariances? This has direct impli- Source: Bloomberg, as of April 30, 2012 cations on MVO. If, for example, daily data are used, returns and covariance have to be daily as well such that MVO was detrimental. As the market rallied folios are very sensitive to changes in optimizes on a daily basis. upward, starting in March 2009, like- expected returns (Best and Grauer When using forward-looking estima- wise the analysts’ estimations rose, but 1991). Accordingly, a wrong choice can tions from fundamental analysts we face with a time lag. Figure 5 shows, as an lead to serious misallocations. Table 3 other issues. Stocks of financial institu- example, the stock price of Deutsche gives an overview of important research tions, for example, had great years Bank in US$, as quoted at the New York work on how sensitive the MVO out- before the subprime crisis but suffered a Stock Exchange. come is to its input. lot in 2008 and early 2009. In mid-2008, So how can an investor attain the So far we have looked at the issues most of the estimation for forward earn- right expectation for return and cova- of MVO in terms of the determination ings per share or stock prices prepared riance at any point in time? The impor- of input and how this affects the output. by fundamental analysts was still very tance of that objective was reinforced This leads us to the question of how we positive while the market had already by Michael Best4 and Robert Grauer.5 can improve the problems detected. started to head southward. Here, the lag They showed in a detailed analy- Because everything starts with the in analysts adjusting their estimations sis that the weights of efficient port- input parameters, such as the return

TABLE 3: LITERATURE OVERVIEW: EFFECT OF ESTIMATION ERRORS ON OPTIMAL PORTFOLIOS Authors (Year) Title Published Content R. Klein and The Effect of Risk on Journal of Financial The effect of taking estimation risk into account for portfolio V. Bawa (1976) Optimal Portfolio Choice Economics 3: optimization is examined and possible differences in portfo- 215–231 lio choice are illustrated. J. Jobson Estimation for Markowitz Journal of The performance of a portfolio with estimated expectations and B. Korkie Efficient Portfolios the American returns is compared to an equally weighted and the true (1980) Statistical optimal portfolio. The results reinforce the hypothesis that Association 75, estimation errors are maximized throughout the optimiza- no. 371: 544–554 tion process. M. Best and R. On the Sensitivity of Mean- The Review of Properties of efficient portfolios are analyzed under changes Grauer (1991) Variance-Efficient Portfolios Financial Studies 4, in expected asset returns. A surprisingly small increase in to Changes in Asset no. 2: 315–342 assets’ means substantially restructures portfolio composi- Means: Some Analytical tion. The effect is shown to intensify by increasing the num- and Computational Results ber of assets in the universe. V. Chopra and The Effect of Errors in Journal of Portfolio Empirical evidence indicates that errors in means are W. Ziemba Means, Variances, and Management 19, over ten times as damaging as errors in variances, and (1993) Covariances on Optimal no. 2: 6–11 over twenty times as damaging as errors in covariances. Portfolio Choice Moreover, the relative impact of errors in means is even greater at higher risk tolerances. Source: SSgA, as of April 30, 2012

estimations for each security, we take FIGURE 6: THE BLACK-LITTERMAN MODEL OF PORTFOLIO CONSTRUCTION a closer look at how we can get better estimations that do not suffer from the Equilibrium Step 1 problems described above. An impor- expectations Step 3 tant step in improving the quality of the Revised MVO input parameters is the Black-Litterman expectations Subjective Estimation of model. Step 2 expectations certainty The Black-Litterman Model Source: SSgA The Black-Litterman model is a portfolio-selection approach with a focus on meaningfully estimating asset returns. ments, it’s common to use third-party retail investors and, increasingly in It was developed and published by data. A wide range of analysts’ estimates recent years, also institutional investors. Fischer Black6 and Robert Litterman7 in is publicly available from, e.g., I/B/E/S. VaR, e.g., is nowadays a standard the early 1990s and has attracted a lot This allows the level of certainty to be risk measure in the financial industry of attention ever since. In their model, derived from a large sample of analysts’ and is a foundation of key concepts in subjective prognoses are combined with estimates. the Basel legal framework. VaR mea- formally derived expected asset returns Step 3: Equilibrium expectations sures the risk of losing money in a cer- (Black and Litterman 1992). The deci- are put together with the subjective tain period with a certain probability. sion-making process basically splits into prognoses and their certainty levels. For example, a 10-day 5-percent VaR of three steps (see figure 6). Complex mathematical calculations $1,000 means that there is a 5-percent Step 1: Equilibrium return expec- deliver revised expected returns, which probability that the investment will fall tations are derived. A way to do so is then can be used for classical MVO. in value by more than $1,000 within the to use the capital asset pricing model Because of the influence of equilib- next 10 days. Using VaR as the risk mea- (CAPM), which states that rium returns, it turns out that portfolios sure, Markowitz’s efficient frontier con- are much more broadly diversified and cept can be modified to a VaR efficient R = Rf + ß(RM – Rf) stable, too (He and Litterman 1999). frontier concept in which the optimal That is, the asset’s return (R) equals Thus, even though the mathematics portfolio is derived in μ-VaR space. the risk-free interest rate (Rf) plus the of the Black-Litterman framework The main problem associated with market excess return (RM – Rf) multi- are complex, its benefits for MVO are MVO, however, is that MVO is a con- plied by the coefficient ß (which mea- substantial. cept for absolute return portfolios. sures the correlation of the asset with But in institutional asset management, The Risk Measure Choice the market). Because not all compo- portfolios are almost always managed nents are known, we need to estimate Let’s come back to a key assumption against an index as benchmark. This the expected market return (RM) and that Markowitz starts with when devel- means that the portfolio manager is also ß. For that, statistical tools such as oping his MVO: Risk is the standard not interested in absolute return and the Fama-MacBeth regression provide deviation of the security’s percentage risk any more but has to focus on rela- good results. The asset is first regressed returns. Although standard deviation tive return versus the benchmark (the against risk factors (in CAPM: market is mathematically easy to handle, it is alpha) and relative risk known as track- risk) to obtain ß. Then the asset return highly questionable—especially in light ing error. Therefore, the task for an is regressed against the estimated ß to of the recent crisis with high annual active asset manager is to perform bet- obtain risk premia (in CAPM: mar- losses (like in 2008) or high uncertainty ter than the benchmark (no matter ket excess return). Finally, equilibrium (like in 2011)—that this symmetrical how the benchmark performs) and stay expected returns can be calculated risk measure is really appropriate. within the given tracking error budget. using the equation above. It would seem to be inappropriate Traditional MVO cannot help in Step 2: Subjective return progno- for two reasons. First, as explained this situation. However, it is possible ses are made and their degree of cer- above, symmetrical risk measures fail to transfer the analytical procedure of tainty is determined. A portfolio man- to capture risk in times of crisis. Other Harry Markowitz’s efficient frontier ager could directly determine both. But risk measures such as semi-deviation, such that a benchmark is included in since it is not easy to classify the reli- drawdown, VaR, or shortfall probability the framework, using alpha as return ability of one’s own prognoses, and to would be far more suited to capture and tracking error as the risk measure. avoid exposure to individual misjudg- investors’ perception of risk—both for Then, the active portfolio manager’s

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task becomes a solvable problem in the Diego. In his seminal article, “Portfolio He, G., and R. Litterman. 1999. The Intuition alpha-tracking error space: To a given Selection” (Markowitz 1952), he did some behind Black-Litterman Model Portfolios. target alpha, find portfolio weights that pioneering work for modern portfolio Goldman Sachs Investment Management lead to the smallest tracking error possi- theory and formally showed that asset Research. http://papers.ssrn.com/sol3/ ble. This is sometimes referred to as the diversification lowers the risk of a portfolio. papers.cfm?abstract_id=334304. tracking error efficient frontier. More For his achievements in financial theory he Jorion, P. 2003. Portfolio Optimization with on that concept can be found in Jorion was awarded the John von Neumann Theory Constraints on Tracking Error. Financial (2003) or Roll (1992). Prize in 1989 and the Nobel Memorial Prize Analysts Journal 59, no. 5: 70–82. WE’RE A FIRM THAT in Economic Sciences in 1990. Markowitz, H. 1952. Portfolio Selection. Summary 2 The universal formula for portfolio variance Journal of Finance 7, no. 1: 77–91. 2 2 2 The Nobel prize-winning innovation is σP = wi σi + wiwjσiσj ρij , where Roll, R. 1992. A Mean/Variance Analysis Σi ΣΣi i‡j wi is the weight of asset i and ρij the corre- of Tracking Error. Journal of Portfolio KNOWS ITS PLACE: of Harry Markowitz definitely changed investment thinking and established a lation between assets i and j. Their square Management 18, no. 4: 13–22. more analytic and quantitative approach root yields the volatility. Schulmerich, M. 2011. How the Recent to portfolio construction. This article 3 See also Schulmerich (2011). Crisis Has Changed EMEA Investors’ introduced this concept and examined 4 Michael Best is a professor at the University View of Risk. State Street Global Advisors. RIGHT BESIDE YOU. its practical suitability. of Waterloo and focuses on portfolio opti- www.ssga.com. It was shown that the aggregation mization and finance. of required data is a tough task to han- 5 Robert Grauer is a finance researcher and Disclaimer: The views expressed in dle. Historical mean returns often are professor at Simon Fraser University in this material are the views of SSgA For us, it’s all about you. After all, you’re the one who forged a solid career, who not very useful in that context, but the British Columbia. Advanced Research through the period built a growing practice, and who earned the trust and loyalty of a grateful clientele. sophisticated Black-Litterman model 6 Fischer Sheffey Black (1938–1995) was an ended April 19, 2012, and are subject might be a solution. Therein, equilibrium economist and is best-known for the famous to change based on market and other That’s why when it comes to our place in the order of things, we easily rank third – returns are combined with subjective Black-Scholes model and formula. After years conditions. The information provided right behind you and your clients, but never in between the relationships you’ve views, leading to good results in practice. of academic work at The University of Chicago does not constitute investment advice dedicated your time and talent to nurturing. In fact, you might say we’re more business Many asset managers focus on a and MIT Sloan School of Management, he and it should not be relied on as such. benchmark when actively managing joined Goldman Sachs in 1984. All material has been obtained from partner than business, because our job is helping advisors do theirs – more simply, a portfolio. In this situation, the 7 Robert Litterman is an American econo- sources believed to be reliable, but more capably and even more successfully. Learn more at RJOnYourSide.com. Markowitz concept can be transferred mist. He was assistant vice president in the its accuracy is not guaranteed. This to the tracking error efficient frontier research department of the Federal Reserve document contains certain statements concept. Additionally, if VaR is required Bank of Minneapolis and an assistant profes- that may be deemed forward-looking as the risk measure in MVO, this is also sor in the economics department at the MIT statements. Please note that any such possible, leading to the VaR efficient Sloan School of Management. In 1986 he statements are not guarantees of any frontier concept. joined Goldman Sachs and became chairman future performance and actual results of the Quantitative Investment Strategies or developments may differ materially Marcus Schulmerich, PhD, CFA®, group of Goldman Sachs Asset Management. from those projected. Past performance FRM, is a vice president with State is not a guarantee of future results. References Street Global Advisors GmbH (SSgA) Investing involves risk including the in Munich, and is a global portfolio Best, M., and R. Grauer. 1991. On the Sensitivity risk of loss of principal. Risk associated We’d go pretty much anywhere strategist responsible for Europe, of Mean-Variance-Efficient Portfolios to with equity investing include stock for our advisors – except in between Middle East, and Africa. He earned Changes in Asset Means: Some Analytical values which may ﬂuctuate in response them and their clients. BS and MS degrees in mathematics and Computational Results. Review of to the activities of individual companies and an MBA from MIT Sloan School Financial Studies 4, no. 2: 315–342. and general market and economic and a PhD from European Business Black, F., and R. Litterman. 1992. Global conditions. Standard & Poor’s (S&P) School EBS in Germany in Portfolio Optimization. Financial Analysts S&P Indices are a registered trademark quant finance. Contact him at Journal 48, no. 5: 28–43. of Standard & Poor’s Financial [email protected]. Füss, R. 2008. Ansätze zur Optimierung eines Services LLC. Portfolios—Warum (nicht) Markowitz. Endnotes Financial Planning Praxis F. http://www. 1 Harry M. Markowitz (1927–) is a world- treasuryworld.de/sites/default/files/H-1_ renowned economist and professor of Portfoliooptimierung_Adams_Fuess_ RJONYOURSIDE.COM finance at the University of California, San Glueck_Lenz_Tilmes.pdf (in German). 866.693.2813

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