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Yale School of Management
Portfolio Management I
William N. Goetzmann
Yale School of Management,1997
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Overview
Risk and return technologyOptimizationPast performance of asset classesBetas and factor models
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Investment Problem
Choice of securitiesAsset class choiceTiming decisionForecasting
Time
Index ValuesCumulative Wealth
22.83
12.0010.93
6.05
20.53
28.18
10.49
Dec1969
Dec1996
Dec1970
Dec1975
Dec1980
Dec1985
Dec1990
Dec1995
0.8
30
1
10
S&P 500 TR U.S. LT Corp TRU.S. LT Gvt TR U.S. 30 Day TBill TRMSCI World TR BZW Extended Equity TRGold TR
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Technology of Return and Risk
Harry Markowitz , 1959Reduced investment to two dimensionsShowed that portfolio mix matters mostTurned investing into statistics
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Mean and Standard Deviation
Mean measures expected return
Standard deviation measures investor risk
Example: six asset classes 1970 - 1996
Risk (STD)
Return (AM)Risk vs. Return
0% 26% 3% 6% 9% 12% 15% 18% 21% 6%
17%
8%
10%
12%
14%
16%
S&P 500 TR
U.S. Small Stk TR
U.S. LT Gvt TR
MSCI EAFE TR
U.S. 30 Day TBill TR
U.S. LT Corp TR
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Correlation: the Third Statistic
Correlation and co-movement
One asset “hedges” the other
Two assets are better than one
Risk
Return
Correlation = 1
Correlation = -1Asset A
Asset B
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Gold and the Stock Market
Correlation of -.3 since 1970
Hedged 70’s crash
Time
Return ValuesReturns
Dec1970
Dec1996
Dec1975
Dec1980
Dec1985
Dec1990
0%
-32%
127%
0%
20%
40%
60%
80%
100%
0%
-20%
S&P 500 TR Gold TR
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Gold in the Portfolio?
25% risk reduction3/4 stocks, 1/4 goldIs gold dominated?
Standard Deviation (Risk)
Expected ReturnEfficient Frontier
75% Stocks 25% Gold
Gold TR
S&P 500 TR
0 34 4 8 12 16 20 24 2813
13
13
13
13
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The Efficient Frontier
More assets move frontier
Frontier is a continuous set of efficient portfolios
Highest return for each level of risk Risk
ReturnAsset A
Asset E
Asset B
Asset C
Asset D
New Frontier
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The First Frontier
Markowitz took stocks from the NYSE
Mixed them with cashCreated the first
frontier
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Applying Portfolio Theory
Select a Universe of AssetsForecast Risk, Return and CorrelationInput to an OptimizerCalculate Portfolio with Highest Mean For
Each Level of Standard Deviation
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A Universe of Assets
Time
Index ValuesCumulative Wealth
28.1830.11
47.8545.36
8.518.097.35
26.41
17.2718.72
8.05
22.83
Dec1969
Dec1996
Dec1970
Dec1975
Dec1980
Dec1985
Dec1990
Dec1995
0.7
50
1
10
BZW Extended Equity TR BZW Interm Cap TRBZW Micro Cap TR BZW Small Cap TRLB Corp TR LB Gvt TRLB Mortgage TR MSCI EAFE TRWilshire Large Growth TR Wilshire Large Value TRS&P 500 Cap App S&P 500 TR
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Historical Statistical Inputs
N Periods Geometric Arithmetic Standard
Mean (%) Mean (%) Deviation
BZW Extended Equity TR 22.00 16.39 17.36 15.24 BZW Interm Cap TR 22.00 16.74 17.78 15.92 BZW Micro Cap TR 22.00 19.22 21.34 22.47 BZW Small Cap TR 22.00 18.93 20.31 18.28
LB Corp TR 24.00 9.33 9.81 10.60 LB Gvt TR 24.00 9.10 9.31 6.99 LB Mortgage TR 21.00 9.96 10.39 10.32
MSCI EAFE TR 27.00 12.89 15.00 22.52 Wilshire Large Growth TR 19.00 16.18 17.25 16.39 Wilshire Large Value TR 19.00 16.67 17.27 12.02 S&P 500 Cap App 27.00 8.03 9.20 15.66
S&P 500 TR 27.00 12.28 13.47 16.14
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Covariance Matrix
BZW Extended Equity TRBZW Interm Cap TRBZW Micro Cap TRBZW Small Cap TRLB Corp TR LB Gvt TR LB Mortgage TR MSCI EAFE TR Wilshire Large Growth TRWilshire Large Value TRS&P 500 TRBZW Extended Equity TR1 0.98 0.93 0.95 0.36 0.34 0.28 0.26 0.89 0.81 0.88BZW Interm Cap TR 0.98 1 0.84 0.88 0.38 0.39 0.3 0.28 0.94 0.83 0.92BZW Micro Cap TR 0.93 0.84 1 0.99 0.28 0.2 0.2 0.22 0.72 0.67 0.69BZW Small Cap TR 0.95 0.88 0.99 1 0.31 0.25 0.23 0.22 0.76 0.73 0.74LB Corp TR 0.36 0.38 0.28 0.31 1 0.91 0.97 0.25 0.24 0.48 0.58LB Gvt TR 0.34 0.39 0.2 0.25 0.91 1 0.91 0.16 0.28 0.54 0.47LB Mortgage TR 0.28 0.3 0.2 0.23 0.97 0.91 1 0.03 0.23 0.36 0.45MSCI EAFE TR 0.26 0.28 0.22 0.22 0.25 0.16 0.03 1 0.21 0.36 0.51Wilshire Large Growth TR0.89 0.94 0.72 0.76 0.24 0.28 0.23 0.21 1 0.71 0.93Wilshire Large Value TR0.81 0.83 0.67 0.73 0.48 0.54 0.36 0.36 0.71 1 0.85S&P 500 TR 0.88 0.92 0.69 0.74 0.58 0.47 0.45 0.51 0.93 0.85 1
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Standard Deviation (Risk)
Expected ReturnEfficient Frontier
BZW Extended Equity TRBZW Interm Cap TR
BZW Micro Cap TRBZW Small Cap TR
LB Corp TR LB Gvt TR LB Mortgage TR
MSCI EAFE TR S&P 500 TR
Wilshire Large Growth TRWilshire Large Value TR
0.00 22.523.00 6.00 9.00 12.00 15.00 18.00
8.11
21.34
12.00
15.00
18.00
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Shifts in the FrontierNew assets shift frontier leftHigh correlations make frontier shallowDifferent time periods change the inputsSmall changes have big effectsMore assets than time periods create a
false “riskless” portfolioConstraining weights to be positive can
flatten the frontier
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How Well Does it Work?
Short time periods give poor inputsThe risky end of the frontier is poorly
estimatedThe minimum variance portfolio is well
estimated
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Optimizer Fixes
Extreme correlations are adjusted to the average, “shrinkage”
Extreme weights are decreasedMaximum holdings in any asset class
specified.Means are estimated by equilibrium modelsApplications focus on broad asset classes
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Capital Market History 1926 - 1996
SP Small LTG EAFE TB LTCN Periods 852.00 852.00 852.00 324.00 852.00 852.00Geometric Mean (%)10.71 12.58 5.08 12.89 3.74 5.64Arithmetic Mean (%) 12.82 17.50 5.38 14.49 3.74 5.88Standard Deviation (%)22.13 35.65 8.08 19.30 0.95 7.20Highest Return (%) 42.56 73.46 15.23 17.87 1.35 13.76Lowest Return (%) -29.73 -36.74 -8.41 -14.38 -0.06 -8.90N Positive Periods 528 520 533 197 838 574N Negative Periods 322 331 318 127 13 276
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Long-Term Performance
Time
Index ValuesCumulative Wealth
1370.95
4495.99
33.7326.4113.54
49.03
Dec1925
Dec1996
Dec1930
Dec1940
Dec1950
Dec1960
Dec1970
Dec1980
Dec1990
0.1
5000
0.1
1
10
100
1000
S&P 500 TR U.S. Small Stk TRU.S. LT Gvt TR MSCI EAFE TRU.S. 30 Day TBill TR U.S. LT Corp TR
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Choosing the Optimal Portfolio
Specifying investor risk aversionSpecifying investor floor return
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Investor Choice With Floor
Choose a desired target “floor” of 4% return.
Select portfolio that minimizes chance of falling below that floor
Point given by tangent line Standard Deviation (Risk)
Expected ReturnEfficient Frontier
BZW Extended Equity TRBZW Interm Cap TR
BZW Micro Cap TRBZW Small Cap TR
LB Corp TR LB Gvt TR LB Mortgage TR
MSCI EAFE TR
S&P 500 TR
Wilshire Large Growth TRWilshire Large Value TR
0.00 30.004.00 8.00 12.00 16.00 20.00 24.00
0.00
20.00
3.00
6.00
9.00
12.00
15.00
18.00
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The Optimal Portfolio
Position 20 BZW Extended Equity TR 0.00 BZW Interm Cap TR 0.00 BZW Micro Cap T 0.00 BZW Small Cap TR 11.47 LB Corp TR 0.00 LB Gvt TR 10.13 LB Mortgage TR 18.18 MSCI EAFE TR 5.70 Wilshire Large Growth 0.00 Wilshire Large Value 54.51 S&P 500 TR 0.00 Exp Return 15.43 Std Dev 9.98 Yield 0.00 Threshold 3.92 Prob 87.57 Sharpe Risk 17.30
Position 20
BZW Small Cap TR (11.5%)
LB Gvt TR (10.1%)LB Mortgage TR (18.2%)
MSCI EAFE TR (5.7%)
Wilshire Large Value TR (54.5%)
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Quest for the Tangency Portfolio
All investor will choose a mix between T-bills and tangency portfolio
CAPM argues that tangency portfolio is the market portfolio Risk
Return
New Frontier
Tangency Portfolio
Riskless Asset
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CAPM and Expected Returns
Only market exposure matters
Higher means higher expected return
Beta
Expected Return
MarketPortfolio
1
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Betas and Factor Models
Assume price-setters are diversifiedIgnore diversifiable riskExpected return must compensate
remaining risk“Factors” are risk sources
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Risk Reduction by Adding Assets
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Systematic Risk
Non-diversifiable riskMarket RiskBeta Risk
Diversifiable Risk
Market Risk
Number of Assets
Risk
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Measuring Beta
Linear “Response” to Factor Returns
Example: MSCI is about a 50% “hedge” of the S&P 500.
Better Fit = Better Hedge
S&P 500 TR
MSCI EAFE TR Linear Regression
-0.23 0.18-0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15
-0.16
0.19
-0.12
-0.09
-0.06
-0.03
0.00
0.03
0.06
0.09
0.12
0.15
0.18
Jan 1970Feb 1970
Mar 1970
Apr 1970
May 1970
Jun 1970
Jul 1970
Aug 1970Sep 1970Oct 1970
Nov 1970
Dec 1970
Jan 1971
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Mar 1971Apr 1971
May 1971 Jun 1971
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Y = 0.0056854 + 0.530719 * X
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Three Multi-Factor Models
APT = Macro-economic risk factorsBARRA = Security-specific risk factorsFama-French = Size and B/M as risk
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Multi-Factor Model Applications
Tailor-made institutional portfoliosRisk-arbitrageAnalysis of sensitivity to macro-shocksCost-of-capital estimates
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Customizing a Portfolio
Assess sensitivity of client to: inflation shocks interest rate shifts GDP shocks
Tilt portfolio away from stocks matching firm sensitivity
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Risk Arbitrage
Measure ’s on risk sources for securities Estimate expected returnsFind under-valued (or overvalued)
securitiesUse ’s to create “market neutral” position
long in undervalued security short in portfolio matching risk profile
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Sensitivity Analysis
Estimate of investment portfolio weighted average ’s of components
Estimate NAV change for 1% shocks in underlying risk factors
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Cost of Capital Estimates
Cost of capital is expected returnLinear factor model
is the factor “loading” Each factor has a risk premium expected return sums up loadings times premia
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Liability Management
Optimization with liabilities and assetsLiabilities are forecast negative flowsLiabilities have risk, return and correlationThey will fit into the mean-variance
framework
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Liabilities in Mean - Variance
Forecast future outflowsEstimate statistical characteristicsinclude as a negative return asset in model
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Optimizing With Liabilities
Result gives the frontier with liabilities as well as assets
Investor chooses portfolio with risk of not meeting liability obligations
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Conclusion
Modern Portfolio Theory Statistical basis for choice Optimizer reduces problem to 2 dimensions models based on portfolio investors
Applications Asset/liability management Custom portfolios Active investing Cost of capital