Notes on Real Estate Returns, and the Markowitz Portfolio Model. S. Malpezzi

Transcription

1 Notes on Real Estate Returns, and the Markowitz Portfolio Model S. Malpezzi

2 Know this up front The Markowitz model is to be used with great caution. Many assumptions are embedded. Naïve use can be very misleading. I think of it as a heuristic tool, rather than as a good guide to actual portfolio construction. Also, software exists that does the calculations below automatically. So, then, why this module? Markowitz model still gives insight, if used appropriately. This module teaches a number ofstatistical concepts, named ranges, use of statistical tools (the covariance tool), matrix algebra, Solver, and some charting principles, among other things. It will give us a chance to look at some properties of actual return data (including shortcomings).

3 Some Risks Real Estate Investors Face Values and rents: what goes up (must come down?) Business risk; economic activity: the business cycle, regional variation. Legislative risk (taxes, regulations, etc.) Tenants come and go. Financial risk: rates, availability go up and down. Purchasing power risk what if you lag inflation? Liquidity risk Management risk Natural disasters

4 Two General Types of Risk Idiosyncratic risk: special risks faced by a particular property, location, etc. Can be diversified away Recall intro lecture: real estate is heterogenous. Each property p is unique. Market or systemic risk: we re all in this together (more or less). Cannot be diversified away To manage idiosyncratic risk, hold a portfolio.

5 Risk and Return We all want higher returns. Most of us would also like to lower risk, at least for a given return. We re risk averse. (If we re risk averse, why do people gamble?) (Why do people gamble stupidly, e.g. play state lotteries?) Risk is ubiquitous in the economy. Let s examine a very broad measure: changes in GDP per capita.

6 Annu ual Per rcentag ge Cha ange 15% 10% 5% 0% -5% -10% -15% -20% 2002 S. Malpezzi Change in Real GNP/GDP PC

7 The safest way to double your money, Mr. Bond, is to fold it twice and put it in your pocket. Auric Goldfinger (Famous Investor)

8 The risk characteristics of assets differ Generally, riskier assets command higher average returns. There are exceptions. Can we explain them? Problem: how do we define an asset? EG the stock market as a whole? Which index? Or some other basket of stocks? Real estate: a national average, or a smaller area g, index?

9 2003, S. Malpezzi 9 From Brealey and Myers

10 2003, S. Malpezzi 10 From Brealey and Myers

11 Source: Brealey & Myers

12 18% Risk vs. Return, According to a Morgan Stanley Study Venture Capital 16% Annual Return 14% 12% 10% 8% Farmland Com Hsg RE S&P 500 Art Small Stocks 6% 4% T Bills CPI T Bonds Gold Silver Standard Deviation of Returns 2003, S. Malpezzi 12

13 Housing: Risk and Return by MSA Real Housing Prices, Growth Rate of Ho ousing Pri ices 2.5% 2.0% 1.5% 10% 1.0% 0.5% 0.0% -0.5% -1.0% NSH MEM ORL KCM IND TPA COL DC ABQ CHI MIN LVL KNX BIR BAL SAC SAT ELPDETSYR SLK DES GRR MIL AKR Coef. of Variation, Prices TUL PHL ALB OKC SDI HRT LA HOU BAT NY PRV SF 2003, S. Malpezzi 13

14 A few measures of risk and return Single period total return Expected return Variance Standard deviation Coefficient of variation Covariance Correlation

15 Single period total return r t (Vt Vt 1 ) V t 1 R t r is the return V is the value of the asset R is the rent (or dividend) t is the time period

16 Expected Return Since everything we know is wrong, any forecast of return is subject to error. If we can say something about the errors in particular, assign probabilities to different outcomes ( states of the world ) we can construct a weighted average of these different outcomes. Common approach: proxy expected return using a weighted average of past returns, where weights are (say) based on market capitalization.

17 Variance For a single asset, the variance is the expected squared deviations from the expected return, r Variance 2 n i 1 ( r r ) 2 i P i where is the expected return, and P i are the weights (probability of each outcome). So we calculate the expected return first, then calculate the variance.

18 Standard Deviation Standard deviation is just the square root of the variance: 2 Std. Dev. n i 1 ( r r) 2 i P i

19 Coefficient of Variation The number calculated for both the variance and the standard deviation depends on the units chosen. Sometimes it s useful to have a unit less measure of variance, in which case we use the coefficient of r variation: CV r Dividing by rbar standardizes our measure of risk. Note: CV doesn t work well when rbar is close to zero.

20 Correlation The number obtained for the covariance, like that for the variance, depends on the units chosen. A unit less measure of association is correlation: Correlation AB Cov( A, B) Correlation coefficients vary between 1 (perfectly correlated) and 11 (perfectly negatively correlated). Correlation of zero means no (linear) association. A B

21 Portfolios, and Covariance So far we ve focused on the risk of a single asset in isolation. How can we evaluate the risk of two or more assets combined in a portfolio? We need to consider their individual risks (variances), but also how their individual risks are related.

22 Risk and Return of a Portfolio The return to a portfolio is just a weighted average of the returns to the properties or assets in the portfolio. The weights are the shares of each asset in the portfolio. The risk of a multi asset portfolio is a little trickier to calculate. For a portfolio containing two assets, A and B, with weights w and returns r, it can be shown to be: Port. Var. σ 2 P w 2 A σ 2 A w 2 B σ 2 B 2w A w B Cov(r A, r B )

23 Measuring Portfolio Risk: The General Case With i assets, denoting weights as w i, it can be shown that the i portfolio variance is: σ 2 P 2 i w σ i 2 i i w w where sigma i,j is the covariance between two asset returns. Using matrix notation, this can be expressed more compactly as: 2 σ P wcw where w is the vector of portfolio weights and C is the matrix of covariances among asset returns. j i j σ ij

24 Common Assumptions Made When Implementing the Markowitz Model Expect the future will be like the past. Base our expected future returns, variances and covariances on past data. For expected return, if we assign an equal weight to each past observation, the arithmeticaverageaverage iscommonlyused asa a measure of expected return. Calculate the covariance matrix based on historical data. Note the covariance matrix ti includes the variance terms (they re on the diagonal). Note these assumptions are often questionable, and we will question them. But we ll still calculate the set of efficient portfolios under these restrictive assumptions, asa a starting point.

25 When Do We Use A Given Measure of Second Moments or Risk? ik? Since we re usually comparing one return to another, and returns have more or less the same metric (units of measure), we most often use the standard deviation. But you will run into the other two. Obviously they re all variations on the same theme. There are other measures of risk, e.g. semivariance, root mean squared error, and manyothers others, covered in advanced treatments.

26 18% Risk vs. Return, According to a Morgan Stanley Study Venture Capital 16% Annual Return 14% 12% 10% 8% Farmland Com Hsg RE S&P 500 Art Small Stocks 6% 4% T Bills CPI T Bonds Gold Silver Standard Deviation of Returns 2002 S. Malpezzi

27 Housing: Risk and Return by MSA Real Housing Prices, Growth Rate of Ho ousing Pric ces 2.5% 2.0% 1.5% 10% 1.0% 0.5% 0.0% -0.5% -1.0% SF LA SDI PHL NY KNX BAL PRV BIR SAC HRT NSH DC ALB MEM ORL ABQ CHI SAT KCM IND TPA TUL HOU COL MIN LVL ELPDETSYR SLK DES OKC BAT MIL AKR GRR Coef. of fvariation, Prices 2002 S. Malpezzi

28 A simple portfolio exercise in Excel We start by gathering time series data on the returns to the following assets: Treasury bills Treasury bonds Stocks Precious metals Housing Commercial real estate (NCREIF) Real estate stocks (NAREIT)

29 Common real estate indexes National Council of Real Estate Investors and Fiduciaries (NCREIF) is the most common index of commercial real estate returns. Problem: NCREIF s index is too smooth. National Association of Real Estate Investment Trusts (NAREIT) produces the most common index of REIT returns. Problem: NAREIT s index is a mix of what s happening in real estate and the stock market. Several national housing price indexes are available.

30 We re already making assumptions Why seven assets? Why not 17? Or 57? We re already aggregating. Never forget: an index is not a building, or a company! Each of the indexes is imperfect. Especially the NCREIF real estate return index! Should we run real, or nominal?

31 Markowitz Model (In Process) S. Malpezzi Total Returns Data (Nominal) Enter 0 if you want original, smoothed NCREIF returns; enter 1 to use Geltner's desmoothing algorithm. 0 Raw Nominal Geltner's alpha: 0.8 Nominal Nominal Bridge Commercial Nominal Nominal Geltner's beta: Commercial Nominal Nominal NYSE Precious Real Housing Returns, Real Treasury Treasury S&P Metals Estate Appreciation All REITs Date Estate Bills Bonds Index NCREIF OFHEO NAREIT DATE NCREIF BILLS BONDS STOCKS METALS NCREIF2 HOUSING REITS : : : : : : : : : : : :

32 Nominal Investment Returns :1 1980:1 1982:1 1984:1 1986:1 1988:1 1990:1 1992:1 1994:1 1996:1 1998:1 2000:1 2002:1 2004:1 2006:1 STOCKS COMRE REITS

33 Steps in calculating the portfolio variance Usethe covariance tool in Excel to compute the covariance matrix among assets, C. Let a 7x1 range of cells contain the weights, w. Use matrix algebra (what Excel calls array functions ) to make a transposed copy of w, denoted w. Use array multiplication to compute the portfolio variance: 2 σ P wcw

34 Now, let s compute the risk and return of alternative portfolios UseSolver. Start with the endpoints. Solve for the weights that maximize return. Then solve for the weights that minimize portfolio variance. Include two constraints: Weights must sum to 1. All weights 0 (no short sales). Equation on previous slide yields risk. Matrix multiply weights and expected returns of each asset to obtain return.

35 1. Construct covariance matrix "C" (named range "cov") using Tools: BILLS BONDS STOCKS METALS COMRE HOUSING REITS BILLS BONDS STOCKS METALS COMRE HOUSING REITS Enter Portfolio Weights "W" (named range "weights": Sum W=1: Calculate Intermediate Result WC using MMULT: Transpose W to W': Compute Portfolio Variance P = WCW': Variance: Risk: Return: Check:

36 Compute the risk and return of alternative portfolios Once you ve got the risk and return of the endpoints, pick some intermediate levels of risk. Use Solver to maximize return subject to the level of risk chosen. Keep track of each set of weights for each level of risk, as well as the risk and return calculated.

37 D a ta for Cha rting Markowitz Frontier I: Nominal returns, 1978Q1 to 2000 Q4, original smoothed NCREIF data Individual Portfolio Asset Share Share Share Share Share Share Share Risk Return Return BILLS BONDS STOCKS METALS COMRE HOUSING REITS Min Risk Max Return

38 Apparent Optimal Asset Allocation By Risk (Nominal, Smoothed NCREIF) 100% 90% 80% 70% REITS HOUSING 60% COMRE 50% METALS 40% STOCKS BONDS 30% BILLS 20% 10% 0% Risk

39 Markowitz Portfolio Model (Nominal Data, Smoothed NCREIF) No ominal Retur rn Frontier Individual Asset These results are to be used, but not believed Nominal Risk

40 A Little Knowledge is a Dangerous Thing The benefits of portfolio diversification are beyond doubt. But mechanical application of popular portfolio models can yield misleading results. For example, in the mid 1980s, portfolio models implied lowrisk portfolios should contain 20 25% real estate. In late 1980s, anyone who did that lost money. Same thing in the recent crisis. Why? Mainly (1) bad data (smoothed NCREIF returns), (2) temporal instability, (3) non normal data (skewed, and thick in the tails).

41 One reason it s hard to calculate an optimal portfolio 4.0 Covariance Stability Test (Nominal Data) Covariance RE Returns 4 per.mov. Avg. (RE Returns) 6.0 Covariance, St tocks & Real Estate Quarte er's RE Return End Quarter of 30 Quarters To implement the model, we need to know future returns, and covariances. It s hard to forecast these based on past data. Covariances are particularly unstable.

42 Some readings and a spreadsheet Wurtzebach and Miles, Real Estate and Modern Portfolio Theory Byrne and Lee, Computing Markowitz Efficient Frontiers using a Spreadsheet toptimizer i Viezer, Application of MPT to Real Estate: A Brief Survey Young, Revisiting Non normal normal Real Estate Return Distributions Brounen, Prado and Stevenson: Kurtosis and Consequences Markowitz Model US Data Expanded.xls

43 One more thing Forgot your matrix algebra???

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46 Introduction to Matrices (Arrays) in Excel Ecel Stephen Malpezzi 5257 Grainger edu

47 Examples of applications of matrices/arrays Few real estate analysts use arrays, y, but for many problems they can save time and effort. For example, What s the proper weighting of comparables in an appraisal? What s the best mix of assets in our portfolio? What s the likely regional economicimpactimpact of this new project? Any problem in which the underlying calculations involve solving several equations at once, or require manipulating columns or blocks of data, can be solved faster and more easilyusing arrays.

48 Objectives What is matrix algebra? What, in particular, is matrix multiplication? How can we implement matrix multiplication li li i (and other matrix algebra) in Excel?

49 Matrices (arrays) and vectors The numbers we use day to day y are single values, sometimes called scalars, e.g. 1, 2, 3, 5.321,, 100, A matrix or an array is a collection of numbers, arranged in a rectangle with a given number of rows and columns, e.g.: M M is a 2 2 matrix, i.e. 2 rows and 2 columns.

50 Matrices (arrays) and vectors A matrix with only one column, or only one row, is sometimes called a vector. Examples: 2 A 3.25 B A is a 3 1 vector, and B is a 1 3 vector.

51 Why use matrix algebra? It s an investment. It takes a little study to understand, and to set up. But once you understand it, matrix algebra greatly simplifies many calculations, including (forexample): Statistical analyses Portfolio construction Input output models Weighting comparables for valuation In general, solving systems of linear equations

52 Some general points about matrix algebra Our discussion today is extremely incomplete and focused on a particular problem. Highly recommended: a course in matrix/linear algebra, and/or read a short text like Jacob Schwartz, Introduction to Matrices and Vectors

53 More general points about matrix algebra Matrices, like scalars, have rules for arithmetic (addition, subtraction and multiplication). But the rules are a little different. Mti Matrices must conform in a certain ti way for an operation. For example, to conform for addition, two matrices must have identical numbers of rows and columns: A , B, then C A B

54 More rules In our usual (scalar) algebra, addition is commutative, i.e. a+b = b+a. The same is true for matrix algebra. In scalar algebra, multiplication is also commutative, i.e. a b = b a. This is not true for matrix algebra, i.e. in general A B B A. Matrices must conform for multiplication, in the following way. The number of columns in the first matrix must equal the number mberof rows in the second matrix. The size of the resultant matrix is the number of rows in the first matrix and the number of columns in the second. Example: If A is 4 3 and B is 3 2, then A and B conform for multiplication, and the resultant matrix C = A B is 4 2. Note that under these assumptions we cannot compute D = B A, because in this ordertheydonotconform do conform.

55 Schematic Setup for Multiplying A B Source: Schwartz

56 Schematic Setup for Multiplying A B Source: Schwartz

57 Setup for Multiplying A B: Example Source: Schwartz

58 Setup for Multiplying A B: Example Source: Schwartz

59 Setup for Multiplying A B: Final Answer Source: Schwartz

60 Some points for future discussion Matrix division, per se, is not possible. Note that in scalar algebra, a 1 a b b This suggests a possible analog to division: (1) Compute the inverse of the denominator matrix (if it exists). (2) Multiply the result times the numerator matrix (if they conform for multiplication). Computing inverses is a little more difficult than the arithmetic we ve done so far. But given another day we could develop the idea; and Excel will invert matrices ti of small size fairly easily.

61 Matrix Algebra in Excel Lotus, Quattro etc. refer to matrix operations. Excel calls them array functions. Entering array functions (matrix operations) in Excel is weird. To multiply named ranges A and B, type =MMULT(A,B) and then hit <Ctrl><Shift><Enter> simultaneously. The cell will contain {=MMULT(A,B)} Excel creates the {}. You CANNOT enter the brackets by hand (well, you can, but then it s just text representing the brackets; it doesn t work).

62 More About Arrays in Excel The results of many array functions cover a range of cells. For example, if A is 2x3 and B is 3x2, AB is 2x2. You must first activate an appropriate range (i.e. 2x2). Then enter the array function. This implies you know the size i of the answer before you enter the function. To delete an array function you must first highlight the entire relevant range. The most useful array functions include: MMULT MINVERSE TRANSPOSE