Yale School of Management Portfolio Management I William N. Goetzmann Yale School of...

Yale School of Management

Portfolio Management I

William N. Goetzmann

Yale School of Management,1997


Overview

Risk and return technologyOptimizationPast performance of asset classesBetas and factor models


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


Technology of Return and Risk

Harry Markowitz , 1959Reduced investment to two dimensionsShowed that portfolio mix matters mostTurned investing into statistics


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


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


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


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


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


The First Frontier

Markowitz took stocks from the NYSE

Mixed them with cashCreated the first

frontier


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


A Universe of Assets

Time


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


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


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


Standard Deviation (Risk)


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


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


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


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


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


Long-Term Performance

Time


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


Choosing the Optimal Portfolio

Specifying investor risk aversionSpecifying investor floor return


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)


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


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%)


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


CAPM and Expected Returns

Only market exposure matters

Higher means higher expected return

Beta

Expected Return

MarketPortfolio

1


Betas and Factor Models

Assume price-setters are diversifiedIgnore diversifiable riskExpected return must compensate

remaining risk“Factors” are risk sources


Risk Reduction by Adding Assets


Systematic Risk

Non-diversifiable riskMarket RiskBeta Risk

Diversifiable Risk

Market Risk

Number of Assets

Risk


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

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Y = 0.0056854 + 0.530719 * X


Three Multi-Factor Models

APT = Macro-economic risk factorsBARRA = Security-specific risk factorsFama-French = Size and B/M as risk


Multi-Factor Model Applications

Tailor-made institutional portfoliosRisk-arbitrageAnalysis of sensitivity to macro-shocksCost-of-capital estimates


Customizing a Portfolio

Assess sensitivity of client to: inflation shocks interest rate shifts GDP shocks

Tilt portfolio away from stocks matching firm sensitivity


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


Sensitivity Analysis

Estimate of investment portfolio weighted average ’s of components

Estimate NAV change for 1% shocks in underlying risk factors


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


Liability Management

Optimization with liabilities and assetsLiabilities are forecast negative flowsLiabilities have risk, return and correlationThey will fit into the mean-variance

framework


Liabilities in Mean - Variance

Forecast future outflowsEstimate statistical characteristicsinclude as a negative return asset in model


Optimizing With Liabilities

Result gives the frontier with liabilities as well as assets

Investor chooses portfolio with risk of not meeting liability obligations


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

Yale School of Management Portfolio Management I William N. Goetzmann Yale School of...

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