ch21_22 modern port theory.pdf

ch21_22 modern port theory.pdf - Chapter 21 Modern...

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Unformatted text preview: Chapter 21 Modern Portfolio Theory & Chapter 22 Equilibrium Asset Pricing 1 21.2 Basic Mean-Variance Portfolio Theory "MODERN PORTFOLIO THEORY" (aka "Mean-Variance Portfolio Theory", or “Markowitz Portfolio Theory” – Either way: “MPT” for short) ¾ DEVELOPED IN 1950s (by MARKOWITZ, SHARPE, LINTNER) (Won Nobel Prize in Economics in 1990.) ¾ WIDELY USED AMONG PROFESSIONAL INVESTORS ¾ FUNDAMENTAL DISCIPLINE OF PORTFOLIO-LEVEL INVESTMENT STRATEGIC DECISION MAKING. 2 See Chapter Appendix I. REVIEW OF STATISTICS ABOUT PERIODIC TOTAL RETURNS: (Note: these are all “time-series” statistics: measured across time, not across assets within a single point in time.) "1st Moment" Across Time (measures “central tendency”): “MEAN”, used to measure: ) Expected Performance ("ex ante", usually arithmetic mean: used in portf ana.) ) Achieved Performance ("ex post", usually geometric mean) "2nd Moments" Across Time (measure characteristics of the deviation around the central tendancy). They include… 1) "STANDARD DEVIATION" (aka "volatility"), which measures: ) Square root of variance of returns across time. ) "Total Risk" (of exposure to asset if investor not diversified) 2) "COVARIANCE", which measures "Co-Movement", aka: ) "Systematic Risk" (component of total risk which cannot be "diversified away") ) Covariance with investor’s portfolio measures asset contribution to portfolio total risk. 3) "CROSS-CORRELATION" (just “correlation” for short). Based on contemporaneous covariance between two assets or asset classes. Measures how two assets "move together": ) important for Portfolio Analysis. 4) "AUTOCORRELATION" (or “serial correlation”: Correlation with itself across time), which reflects the nature of the "Informational Efficiency" in the Asset Market; e.g.: ) Zero Î "Efficient" Market (prices quickly reflect full information; returns lack predictability) Î Like securities markets (approximately). ) Positive Î "Sluggish" (inertia, inefficient) Market (prices only gradually incorporate new info.) Î Like private real estate markets. ) Negative Î "Noisy" Mkt (excessive s.r. volatility, price "overreactions") Î Like securities markets (to some extent). 3 "Picture" of 1st and 2nd Moments . . . 30 Actual Asset Value 1st & 2nd Moment of Return 25 L.R. Trend Asset Value 1st Moment Only of Return Asset Value 20 15 10 5 0 1977 2002 2027 2052 2077 2102 2127 2152 2177 2202 2227 Year Figure by MIT OCW. First Moment is "Trend“. Second Moment is "Deviation" around trend. Food for Thought Question: IF THE TWO LINES ABOVE WERE TWO DIFFERENT ASSETS, WHICH WOULD YOU PREFER TO INVEST IN, OTHER THINGS BEING EQUAL? . . . 4 Historical statistics, annual periodic total returns: Stocks, Bonds, Real Estate, 1970-2003… S&P500 LTG Bonds Private Real Estate Mean (arith) 12.7% 9.7% 9.9% Std.Deviation 17.5% 11.8% 9.0% 100% 27.2% 16.6% 100% -21.0% 1st Moments Correlations: S&P500 LTG Bonds Priv. Real Estate 2nd Moments 100% PORTFOLIO THEORY IS A WAY TO CONSIDER BOTH THE 1ST & 2ND MOMENTS (& INTEGRATE THE TWO) IN INVESTMENT ANALYSIS. 5 21.2.1. Investor Preferences & Dominant Portfolios SUPPOSE WE DRAW A 2-DIMENSIONAL SPACE WITH RISK (2ND-MOMENT) ON HORIZONTAL AXIS AND EXPECTED RETURN (1ST MOMENT) ON VERTICAL AXIS. A RISK-AVERSE INVESTOR MIGHT HAVE A UTILITY (PREFERENCE) SURFACE INDICATED BY CONTOUR LINES LIKE THESE (investor is indifferent along a given contour line): RETURN P Q RISK THE CONTOUR LINES ARE STEEPLY RISING AS THE RISK-AVERSE INVESTOR WANTS MUCH MORE RETURN TO COMPENSATE FOR A LITTLE MORE RISK. 6 A MORE AGGRESSIVE INVESTOR MIGHT HAVE A UTILITY (PREFERENCE) SURFACE INDICATED BY CONTOUR LINES LIKE THESE. RETURN P Q RISK THE SHALLOW CONTOUR LINES INDICATE THE INVESTOR DOES NOT NEED MUCH ADDITIONAL RETURN TO COMPENSATE FOR MORE RISK. BUT BOTH INVESTORS WOULD AGREE THEY PREFER POINTS TO THE "NORTH" AND "WEST" IN THE RISK/RETURN SPACE. THEY BOTH PREFER POINT "P" TO POINT "Q". 7 FOR ANY TWO PORTFOLIOS "P" AND "Q" SUCH THAT: EXPECTED RETURN "P" ≥ EXPECTED RETURN "Q" AND (SIMULTANEOUSLY): RISK "P" ≤ RISK "Q" IT IS SAID THAT: “Q” IS DOMINATED BY “P”. THIS IS INDEPENDENT OF RISK PREFERENCES. Î BOTH CONSERVATIVE AND AGGRESSIVE INVESTORS WOULD AGREE ABOUT THIS. IN ESSENCE, PORTFOLIO THEORY IS ABOUT HOW TO AVOID INVESTING IN DOMINATED PORTFOLIOS. DOMINATES "Q" RETURN P DOMINATES "Q" Q DOMINATED BY "Q" RISK PORTFOLIO THEORY TRIES TO MOVE INVESTORS FROM POINTS LIKE "Q" TO POINTS LIKE "P". 8 Section 21.2.2… III. PORTFOLIO THEORY AND DIVERSIFICATION... "PORTFOLIOS" ARE "COMBINATIONS OF ASSETS". PORTFOLIO THEORY FOR (or from) YOUR GRANDMOTHER: “DON’T PUT ALL YOUR EGGS IN ONE BASKET!” WHAT MORE THAN THIS CAN WE SAY? . . . (e.g., How many “eggs” should we put in which “baskets”.) In other words, GIVEN YOUR OVERALL INVESTABLE WEALTH, PORTFOLIO THEORY TELLS YOU HOW MUCH YOU SHOULD INVEST IN DIFFERENT TYPES OF ASSETS. FOR EXAMPLE: WHAT % SHOULD YOU PUT IN REAL ESTATE? WHAT % SHOULD YOU PUT IN STOCKS? TO BEGIN TO RIGOROUSLY ANSWER THIS QUESTION, CONSIDER... 9 AT THE HEART OF PORTFOLIO THEORY ARE TWO BASIC MATHEMATICAL FACTS: 1) PORTFOLIO RETURN IS A LINEAR FUNCTION OF THE ASSET N WEIGHTS: r P = ∑ wn r n n=1 IN PARTICULAR, THE PORTFOLIO EXPECTED RETURN IS A WEIGHTED AVERAGE OF THE EXPECTED RETURNS TO THE INDIVIDUAL ASSETS. E.G., WITH TWO ASSETS ("i" & "j"): rp = ωri + (1-ω)rj WHERE ωi IS THE SHARE OF PORTFOLIO TOTAL VALUE INVESTED IN ASSET i. e.g., If Asset A has E[rA]=5% and Asset B has E[rB]=10%, then a 50/50 Portfolio (50% A + 50% B) will have E[rP]=7.5%. 10 THE 2ND FACT: 2) PORTFOLIO VOLATILITY IS A NON-LINEAR FUNCTION OF THE N N ASSET WEIGHTS: VAR P = ∑ ∑ wi w j COVij I =1 J =1 SUCH THAT THE PORTFOLIO VOLATILITY IS LESS THAN A WEIGHTED AVERAGE OF THE VOLATILITIES OF THE INDIVIDUAL ASSETS. E.G., WITH TWO ASSETS: sP = √[ ω²(si)² + (1-ω)²(sj)² + 2ω(1-ω)sisjCij ] ≤ ωsi + (1-ω)sj WHERE si IS THE RISK (MEASURED BY STD.DEV.) OF ASSET i. e.g., If Asset A has StdDev[rA]=5% and Asset B has StdDev[rB]=10%, then a 50/50 Portfolio (50% A + 50% B) will have StdDev[rP] < 7.5% (conceivably even < 5%). Î This is the beauty of Diversification. It is at the core of Portfolio Theory. It is perhaps the only place in economics where you get a “free lunch”: In this case, less risk without necessarily reducing your expected return! 11 The diversification effect is greater the less correlated are the assets… Stocks & bonds (+30% correlation): Each dot is one year's returns. Stock & Bond Ann. Returns, 1970-2003: +30% Correlation 50% 40% Bond Returns 30% 20% 10% 0% -30% -20% -10% 0% 10% 20% 30% 40% 50% -10% -20% Stock Returns 12 The diversification effect is greater the less correlated are the assets… Stocks & real estate (+17% correlation): Each dot is one year's returns. Real Est. & Stock Ann. Returns, 1970-2003: +17% Correlation 30% 25% 20% R.E. Returns 15% 10% 5% 0% -30% -20% -10% 0% 10% 20% 30% 40% 50% -5% -10% -15% Stock Returns 13 The diversification effect is greater the less correlated are the assets… Bonds & real estate (-21% correlation): Each dot is one year's returns. Real Est. & Bond Ann. Returns, 1970-2003: -21% Correlation 30% 25% Real Estate Returns 20% 15% 10% 5% 0% -20% -10% 0% 10% 20% 30% 40% 50% -5% -10% -15% Bond Returns 14 For example, a portfolio of 50% bonds & 50% stocks would not have provided much volatility reduction during 1981-98, though over the longer 1970-2003 period it would have reduced the half&half portfolio to just bond volatility: Annual Periodic Total Returns, Long-Term Bonds and Stocks, 1970-2003 50% 40% 30% 20% 10% 0% -10% -20% Stocks Returns: Mean Std.Dev. Bonds 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 -30% Bonds/Stocks Bonds Stocks Half&Half 9.7% 12.7% 11.2% 11.8% 17.5% 11.8% 15 Here the portfolio of 50% bonds & 50% real estate would have provided a more consistent diversification during 1970-2003, with less volatility than either asset class alone even though a very similar return: Annual Periodic Total Returns, Long-Term Bonds and Real Estate, 1970-2003 50% 40% 30% 20% 10% 0% -10% -20% Real Estate Returns: Mean Std.Dev. Bonds 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 -30% Bonds/Real Estate Bonds R.Estate Half&Half 9.7% 9.9% 9.8% 11.8% 9.0% 6.6% 16 This “Diversification Effect” is greater, the lower is the correlation among the assets in the portfolio. NUMERICAL EXAMPLE . . . SUPPOSE REAL ESTATE HAS: EXPECTED RETURN = 8% RISK (STD.DEV) = 10% SUPPOSE STOCKS HAVE: EXPECTED RETURN = 12% RISK (STD.DEV) = 15% THEN A PORTFOLIO WITH ω SHARE IN REAL ESTATE & (1-ω) SHARE IN STOCKS WILL RESULT IN THESE RISK/RETURN COMBINATIONS, DEPENDING ON THE CORRELATION BETWEEN THE REAL ESTATE AND STOCK RETURNS: C = 100% C = 25% C = 0% C = -50% rP sP rP sP rP sP rP sP ω 0% 12.0% 15.0% 12.0% 15.0% 12.0% 15.0% 12.0% 15.0% 25% 11.0% 13.8% 11.0% 12.1% 11.0% 11.5% 11.0% 10.2% 50% 10.0% 12.5% 10.0% 10.0% 10.0% 9.0% 10.0% 6.6% 75% 9.0% 11.3% 9.0% 9.2% 9.0% 8.4% 9.0% 6.5% 100% 8.0% 10.0% 8.0% 10.0% 8.0% 10.0% 8.0% 10.0% where: C = Correlation Coefficient between Stocks & Real Estate. (This table was simply computed using the formulas noted previously.) 17 This “Diversification Effect” is greater, the lower is the correlation among the assets in the portfolio. Correlation = 100% Portf Exptd Return 12% 11% 1/4 RE 10% 1/2 RE 9% 3/4 RE 8% 9% 10% 11% 12% 13% Portf Risk (STD) 14% 15% 14% 15% Correlation = 25% Portf Exptd Return 12% 11% 1/4 RE 10% 1/2 RE 3/4 RE 9% 8% 9% 10% 11% 12% 13% Portf Risk (STD) 18 IN ESSENCE, PORTFOLIO THEORY ASSUMES: YOUR OBJECTIVE FOR YOUR OVERALL WEALTH PORTFOLIO IS: Î MAXIMIZE EXPECTED FUTURE RETURN Î MINIMIZE RISK IN THE FUTURE RETURN GIVEN THIS BASIC ASSUMPTION, AND THE EFFECT OF DIVERSIFICATION, WE ARRIVE AT THE FIRST MAJOR RESULT OF PORTFOLIO THEORY. . . 19 To the investor, the risk that matters in an investment is that investment's contribution to the risk in the investor's overall portfolio, not the risk in the investment by itself. This means that covariance (correlation and variance) may be as important as (or more important than) variance (or volatility) in the investment alone. (e.g., if the investor's portfolio is primarily in stocks & bonds, and real estate has a low correlation with stocks & bonds, then the volatility in real estate may not matter much to the investor, because it will not contribute much to the volatility in the investor's portfolio. Indeed, it may allow a reduction in the portfolio’s risk.) THIS IS A MAJOR SIGNPOST ON THE WAY TO FIGURING OUT "HOW MANY EGGS" WE SHOULD PUT IN WHICH "BASKETS". 20 21.2.4 STEP 1: FINDING THE EFFICIENT FRONTIER SUPPOSE WE HAVE THE FOLLOWING RISK & RETURN EXPECTATIONS… Mean STD Corr Stocks Bonds RE Stocks 10.00% 15.00% Bonds 6.00% 8.00% RE 7.00% 10.00% 100.00% 30.00% 100.00% 25.00% 15.00% 100.00% INVESTING IN ANY ONE OF THE THREE ASSET CLASSES WITHOUT DIVERSIFICATION ALLOWS THE INVESTOR TO ACHIEVE ONLY ONE OF THREE POSSIBLE RISK/RETURN POINTS… 21 INVESTING IN ANY ONE OF THE THREE ASSET CLASSES WITHOUT DIVERSIFICATION ALLOWS THE INVESTOR TO ACHIEVE ONLY ONE OF THE THREE POSSIBLE RISK/RETURN POINTS DEPICTED IN THE GRAPH BELOW… 3 Assets: Stocks, Bonds, RE, No Diversification 12% Stocks E(r) 10% 8% Real Est 6% Bonds 4% 6% 8% 10% 12% 14% 16% Risk (Std.Dev) Stocks Bonds Real Ests IN A RISK/RETURN CHART LIKE THIS, ONE WANTS TO BE ABLE TO GET AS MANY RISK/RETURN COMBINATIONS AS POSSIBLE, AS FAR TO THE “NORTH” AND “WEST” AS POSSIBLE. 22 ALLOWING PAIRWISE COMBINATIONS (AS WITH OUR PREVIOUS STOCKS & REAL ESTATE EXAMPLE), INCREASES THE RISK/RETURN POSSIBILITIES TO THESE… 3 Assets: Stocks, Bonds, RE, with pairwise combinations 12% Stocks E(r) 10% 8% Real Est 6% Bonds 4% 6% 8% 10% 12% 14% 16% Risk (Std.Dev) RE&Stocks Stks&Bonds RE&Bonds 23 FINALLY, IF WE ALLOW UNLIMITED DIVERSIFICATION AMONG ALL THREE ASSET CLASSES, WE ENABLE AN INFINITE NUMBER OF COMBINATIONS, THE “BEST” (I.E., MOST “NORTH” AND “WEST”) OF WHICH ARE SHOWN BY THE OUTSIDE (ENVELOPING) CURVE. 3 Assets with Diversification: The Efficient Frontier 12% E(r) 10% 8% 6% 4% 6% 8% 10% 12% 14% 16% Risk (Std.Dev) Effic.Frontier RE&Stocks Stks&Bonds RE&Bonds THIS IS THE “EFFICIENT FRONTIER” IN THIS CASE (OF THREE ASSET CLASSES). 24 IN PORTFOLIO THEORY THE “EFFICIENT FRONTIER” CONSISTS OF ALL ASSET COMBINATIONS (PORTFOLIOS) WHICH MAXIMIZE RETURN AND MINIMIZE RISK. THE EFFICIENT FRONTIER IS AS FAR “NORTH” AND “WEST” AS YOU CAN POSSIBLY GET IN THE RISK/RETURN GRAPH. A PORTFOLIO IS SAID TO BE “EFFICIENT” (i.e., represents one point on the efficient frontier) IF IT HAS THE MINIMUM POSSIBLE VOLATILITY FOR A GIVEN EXPECTED RETURN, AND/OR THE MAXIMUM EXPECTED RETURN FOR A GIVEN LEVEL OF VOLATILITY. (Terminology note: This is a different definition of "efficiency" than the concept of informational efficiency applied to asset 25 markets and asset prices.) SUMMARY UP TO HERE: DIVERSIFICATION AMONG RISKY ASSETS ALLOWS: ¾ GREATER EXPECTED RETURN TO BE OBTAINED FOR ANY GIVEN RISK EXPOSURE, &/OR; ¾ LESS RISK TO BE INCURRED FOR ANY GIVEN EXPECTED RETURN TARGET. (This is called getting on the "efficient frontier".) PORTFOLIO THEORY ALLOWS US TO: ¾ QUANTIFY THIS EFFECT OF DIVERSIFICATION ¾ IDENTIFY THE "OPTIMAL" (BEST) MIXTURE OF RISKY ASSETS 26 MATHEMATICALLY, THIS IS A "CONSTRAINED OPTIMIZATION" PROBLEM ==> Algebraic solution using calculus ==> Numerical solution using computer and "quadratic programming". Spreadsheets such as Excel include "Solvers" that can find optimal portfolios this way. 27 21.2.5. STEP 2: PICK A RETURN TARGET FOR YOUR OVERALL WEALTH THAT REFLECTS YOUR RISK PREFERENCES... E.G., ARE YOU HERE (7%)?... Optimal portfolio (P) for a conservative investor: Target=7% 12% max risk/return indifference curve E(r) 10% Efficient Frontier 8% P = 16%St, 48%Bd, 36%RE for 7% Target 6% 4% 6% 8% 10% 12% 14% 16% Risk (Std.Dev) 28 OR ARE YOU HERE (9%)?... Optimal portfolio (P) for an aggressive investor: Target=9% 12% max risk/return indifference curve E(r) 10% 8% P= 67%St, 0%Bd, 33%RE for 9% Target Efficient Frontier 6% 4% 6% 8% 10% 12% Risk (Std.Dev) 14% 16% 29 21.2.6 Major Implications of Portfolio Theory for Real Estate Investment 100% ASSET COMPOSITION OF THE EFFICIENT FRONTIER (based on Exhibit 21-1a expectations) RE Share 75% Bond Share Stock Share 50% 25% 0% 6.0% 7.0% 8.0% 9.0% 10.0% INVESTOR RETURN TARGET Core real estate assets typically make up a large share of efficient (non-dominated) portfolios for conservative to moderate return targets. 30 GENERAL QUALITATIVE RESULTS OF PORTFOLIO THEORY 1) THE OPTIMAL REAL ESTATE SHARE DEPENDS ON HOW CONSERVATIVE OR AGGRESSIVE IS THE INVESTOR; 2) FOR MOST OF THE RANGE OF RETURN TARGETS, REAL ESTATE IS A SIGNIFICANT SHARE. (COMPARE THESE SHARES TO THE AVERAGE U.S. PENSION FUND REAL ESTATE ALLOCATION WHICH IS LESS THAN 5%. THIS IS WHY PORTFOLIO THEORY HAS BEEN USED TO TRY TO GET INCREASED PF ALLOCATION TO REAL ESTATE.) 3) THE ROBUSTNESS OF REAL ESTATE'S INVESTMENT APPEAL IS DUE TO ITS LOW CORRELATION WITH BOTH STOCKS & BONDS, THAT IS, WITH ALL OF THE REST OF THE PORTFOLIO. (NOTE IN PARTICULAR THAT OUR INPUT ASSUMPTIONS IN THE ABOVE EXAMPLE NUMBERS DID NOT INCLUDE A PARTICULARLY HIGH RETURN OR PARTICULARLY LOW VOLATILITY FOR THE REAL ESTATE ASSET CLASS. THUS, THE LARGE REAL ESTATE SHARE IN THE OPTIMAL PORTFOLIO MUST NOT BE DUE TO SUCH ASSUMPTIONS.) 31 21.2.7 SUPPOSE WE EXPAND THE PORTFOLIO CHOICE SET BY ADDING ADDITIONAL SUB-CLASSES OF ASSETS… For example, suppose we add the following expectations for an additional sub-class of stocks (small stocks) and an additional sub-class of real estate (REITs)… Exhibit 21-9a: Possible Risk & Return Expectations for 5 Asset Classes Large Stocks Expected Return (E[r]) 10.00% Small Stocks 12.00% Bonds REITs 6.00% 10.00% Private Real Estate 7.00% Volatility 15.00% 20.00% 8.00% 15.00% 10.00% 100.00% 60.00% 30.00% 45.00% 25.00% 100.00% 0.00% 70.00% 25.00% 100.00% 20.00% 15.00% 100.00% 40.00% Correlation with: Large Stocks Small Stocks Bonds REITs Private Real Estate 100.00% 32 100% ASSET COMPOSITION OF THE EFFICIENT FRONTIER (based on Exhibit 21-9a expectations) Private Real Estate 75% REITs Bonds 50% Small Stocks 25% Large Stocks 0% 6.0% 7.5% 9.0% 10.5% 12.0% INVESTOR RETURN TARGET 33 Section 21.3 VII. INTRODUCING A "RISKLESS ASSET"... IN A COMBINATION OF A RISKLESS AND A RISKY ASSET, BOTH RISK AND RETURN ARE WEIGHTED AVERAGES OF RISK AND RETURN OF THE TWO ASSETS: Recall: sP = √[ ω²(si)² + (1-ω)²(sj)² + 2ω(1-ω)sisjCij ] If sj=0, this reduces to: sP = √[ ω²(si)² = ωsi SO THE RISK/RETURN COMBINATIONS OF A MIXTURE OF INVESTMENT IN A RISKLESS ASSET AND A RISKY ASSET LIE ON A STRAIGHT LINE, PASSING THROUGH THE TWO POINTS REPRESENTING THE RISK/RETURN COMBINATIONS OF THE RISKLESS ASSET AND THE RISKY ASSET. 34 If either i or j is riskless . . . E[rj] E[ri] si sj Volatility 35 Î IN PORTFOLIO ANALYSIS, THE "RISKLESS ASSET" REPRESENTS BORROWING OR LENDING BY THE INVESTOR… BORROWING IS LIKE "SELLING SHORT" OR HOLDING A NEGATIVE WEIGHT IN THE RISKLESS ASSET. BORROWING IS "RISKLESS" BECAUSE YOU MUST PAY THE MONEY BACK “NO MATTER WHAT”. LENDING IS LIKE BUYING A BOND OR HOLDING A POSITIVE WEIGHT IN THE RISKLESS ASSET. LENDING IS "RISKLESS" BECAUSE YOU CAN INVEST IN GOVT BONDS AND HOLD TO MATURITY. 36 SUPPOSE YOU COMBINE RISKLESS BORROWING OR LENDING WITH YOUR INVESTMENT IN THE RISKY PORTFOLIO OF STOCKS & REAL ESTATE. YOUR OVERALL EXPECTED RETURN WILL BE: rW = vrP + (1-v)rf AND YOUR OVERALL RISK WILL BE: sW = vsP + (1-v)0 = vsP Where: v = Weight in risky portfolio rW, sW = Return, Std.Dev., in overall wealth rP, sP = Return, Std.Dev., in risky portfolio rf = Riskfree Interest Rate v NEED NOT BE CONSTRAINED TO BE LESS THAN UNITY. v CAN BE GREATER THAN 1 ("leverage" , "borrowing"), OR v CAN BE LESS THAN 1 BUT POSITIVE ("lending", investing in bonds, in addition to investing in the risky portfolio). THUS, USING BORROWING OR LENDING, IT IS POSSIBLE TO OBTAIN ANY RETURN TARGET OR ANY RISK TARGET. THE RISK/RETURN COMBINATIONS WILL LIE ON THE STRAIGHT LINE PASSING THROUGH POINTS rf AND rP. 37 NUMERICAL EXAMPLE SUPPOSE: RISKFREE INTEREST RATE = 5% STOCK EXPECTED RETURN = 15% STOCK STD.DEV. = 15% ________________________________________________________ IF RETURN TARGET = 20%, BORROW $0.5 INVEST $1.5 IN STOCKS (v = 1.5). EXPECTED RETURN WOULD BE: (1.5)15% + (-0.5)5% = 20% RISK WOULD BE (1.5)15% + (-0.5)0% = 22.5% ________________________________________________________ IF RETURN TARGET = 10%, LEND (INVEST IN BONDS) $0.5 INVEST $0.5 IN STOCKS (v = 0.5). EXPECTED RETURN WOULD BE: (0.5)15% + (0.5)5% = 10% RISK WOULD BE (0.5)15% + (0.5)0% = 7.5% ___________________________________________________________ 38 NOTICE THESE POSSIBILITIES LIE ON A STRAIGHT LINE IN RISK/RETURN SPACE . . . RISK & RETURN COMBINATIONS USING STOCKS & RISKLESS BORROWING OR LENDI 35% 30% EX 25% PC TE D 20% RE TU 15% R N 10% 5% BORROW V=150% LEND V=100% V=50% V = WEIGHT IN STOCKS V=0 0% 0% 7.5% 15% RISK (STD.DEV.) 22.5% 39 BUT NO MATTER WHAT YOUR RETURN TARGET, YOU CAN DO BETTER BY PUTTING YOUR RISKY MONEY IN A DIVERSIFIED PORTFOLIO OF REAL ESTATE & STOCKS . . . SUPPOSE: REAL ESTATE EXPECTED RETURN = 10% REAL ESTATE STD.DEV. = 10% CORRELATION BETWEEN STOCKS & REAL ESTATE = 25% THEN 50% R.E. / STOCKS MIXTURE WOULD PROVIDE: EXPECTED RETURN ...
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