building portfolios in a non-normal world...

Building Portfolios in a Non-Normal WorldBuilding Portfolios in a Non Normal World

× Paul D. Kaplan, Ph.D., CFA× Paul D. Kaplan, Ph.D., CFAQuantitative Research Director, Morningstar Europe, Ltd.

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© 2011 Morningstar, Inc. All rights reserved.

“We seem to have a once-in-a-lifetime crisis every three or f ”four years.”Leslie Rahl, Founder of Capital Market Risk Advisors

Source: Christopher Wright “Tail Tales ” CFA Institute Magazine March/April 2007Source: Christopher Wright, Tail Tales, CFA Institute Magazine, March/April 2007

The Black Turkey

× “An event that is entirely i t t ith t d t b tconsistent with past data but

that no one thought would happen” Larry Siegel

A Flock of Turkeys

Asset Class Time Period Peak to Trough Decline

U.S. stocks (real total return) 1911-1920 51%U.S. stocks (real total return) 1911 1920 51%

U.S. stocks (DJIA, daily) 1929-1932 89%

Long U.S. Treasury bond (realtotal return)

1941-1981 67%

U.S. stocks 1973-1974 49%

U.K. stocks (real total return) 1972-1974 74%

Gold 1980-1985 62%

Oil 1980-1986 71%

Japan stocks 1990-2009 82%

U.S. stocks (S&P) 2000-2002 49%U.S. stocks (S&P) 2000 2002 49%

U.S. stocks (NASDAQ) 2000-2002 78%

U.S. stocks (S&P) 2007-2009 57%

Nominal price return unless otherwise specified.

U.S. Stock Market History, 1871 – April 2011y, p10,000

Dot Com Bubble Burst

Crash of 2007-2009

1,000

ve V

alue

Inflationary BearMarket of 1973-74

Crash of 1987

100

Rea

l Cum

ulat

iv

Great Depression

Postwar Bear Market

10 Panic of 1907 Auto

Bubble Burst

11870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Year

Source: 2011 Ibbotson Stocks, Bonds, Bills, and Inflation (SBBI) Classic Yearbook, Morningstar, Inc. Morningstar EnCorr. Goetzmann, William N., Roger G. Ibbotson, and Liang Peng, “A New Historical Database for the NYSE 1815 to 1925: Performance and Predictability,” Journal of Financial Markets, December 2000. Pierce, Phyllis S., ed., The Dow Jones Averages, 1885—1980, Homewood, IL: Dow Jones Irwin, 1982. www.econ.yale.edu/~shiller/data.htm.

Cracks in the Bell Curve: U.S. Real Monthly Returns,J 1886 A il 2011January 1886 – April 2011

Lognormal Historical

128

Historical Frequency (Months)

256

8

16

3264

1

24

-30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45%

Monthly ReturnSource: 2011 Ibbotson Stocks, Bonds, Bills, and Inflation (SBBI) Classic Yearbook, Morningstar, Inc. Morningstar EnCorr. Goetzmann, William N., Roger G. Ibbotson and Liang Peng “A New Historical Database for the NYSE 1815 to 1925: Performance and Predictability ” Journal of Financial Markets DecemberIbbotson, and Liang Peng, A New Historical Database for the NYSE 1815 to 1925: Performance and Predictability, Journal of Financial Markets, December 2000. Pierce, Phyllis S., ed., The Dow Jones Averages, 1885—1980, Homewood, IL: Dow Jones Irwin, 1982. www.econ.yale.edu/~shiller/data.htm.

Covariation of Returns: Linear or Nonlinear?S&P 500 EAFE M thl T t l R t J 1970 S 2010S&P 500 vs. EAFE, Monthly Total Returns: Jan. 1970 – Sep. 2010

Limitations of Mean-Variance Analysis

× Fat tails in returns not modeled× Covariation of returns assumed linear, cannot handle optionality× Single period investment horizon (arithmetic mean)× Risk measured by volatility× Risk measured by volatility

Building A Better Optimizer

Issue Markowitz 1.0 Markowitz 2.0

Return Distributions Mean-Variance Framework(No fat tails)

Scenarios+Smoothing(Fat tails possible)

Return Covariation Correlation MatrixLi

Scenarios+SmoothingN li ( ti )Linear Nonlinear (e.g. options)

Investment Horizon Single PeriodArithmetic Mean

Can use Multiperiod Kelly CriterionCan use Geometric Mean

Risk Measure Standard Deviation Can use Conditional Value at Risk andRisk Measure Standard Deviation Can use Conditional Value at Risk and other risk measures

Markowitz 1.0 Inputs: Summary Statistics

Correlation

Asset ClassExpected

ReturnStandard Deviation 1 2 3 4

A 5 00% 10 00% 1 00 0 34 0 32 0 32

Correlation

A 5.00% 10.00% 1.00 0.34 0.32 0.32B 10.00% 20.00% 0.34 1.00 0.82 0.82C 15.00% 30.00% 0.32 0.82 1.00 0.71D 13.00% 30.00% 0.32 0.82 0.71 1.00

Markowitz 2.0 Inputs: Scenarios

1

1.5

2

2.5

3

3.5

4

4.5

0

0.5

‐60% ‐40% ‐20% 0% 20% 40% 60% 80% 100%

1.5

2

2.5

100%

150%

200%

250%

0

0.5

1

‐100% ‐50% 0% 50% 100% 150% 200% 250%

1.2

1.4

‐100%

‐50%

0%

50%

‐60% ‐40% ‐20% 0% 20% 40% 60% 80%

300%

350%

250%

300%

350%

0

0.2

0.4

0.6

0.8

1

‐200% ‐100% 0% 100% 200% 300% 400% 500%‐100%

‐50%

0%

50%

100%

150%

200%

250%

‐60% ‐40% ‐20% 0% 20% 40% 60% 80%

‐100%

‐50%

0%

50%

100%

150%

200%

250%

‐100% ‐50% 0% 50% 100% 150% 200% 250%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

50%

0%

50%

100%

150%

200%

250%

300%

350%

‐60% ‐40% ‐20% 0% 20% 40% 60% 80%50%

0%

50%

100%

150%

200%

250%

300%

350%

‐100% ‐50% 0% 50% 100% 150% 200% 250% 50%

0%

50%

100%

150%

200%

250%

300%

350%

‐100% ‐50% 0% 50% 100% 150% 200% 250% 300% 350%0

‐200% ‐100% 0% 100% 200% 300% 400% 500% ‐100%

‐50%

‐100%

‐50%100% 50% 0% 50% 100% 150% 200% 250%

‐100%

‐50%100% 50% 0% 50% 100% 150% 200% 250% 300% 350%

Markowitz 1.0

Reward = 1-Period Return, Risk = Volatility14%

10%

12% Non-US Stocks

8%

d A

rithm

etic

Mea

n US Stocks

Mix 2

Mix 1

4%

6%

Expe

cted Non-US Bonds

0%

2%US Bonds

0% 5% 10% 15% 20% 25% 30%

Standard Deviation

Value-at-Risk (VaR)

VaR identifies the return at a specific point (e.g. 1st or 5th percentile)

Worst 5th Percentile95% of all returns are better5% of all returns are worse

Worst 1st Percentile99% of all returns are better1% of all returns are worse

Conditional Value-at-Risk (CVaR)

CVaR identifies the probability weighted return of the entire tail

Worst 5th Percentile95% of all returns are better5% of all returns are worse

CVaR vs. VaR

Notice that different return distributions can have the same VaRs, b diff CV Rbut different CVaRs

Worst 5th Percentile95% of all returns are better5% of all returns are worse

Markowitz 2.0: More Relevant Risk & Reward Measures

Reward = Long-Term Return, Risk = How Bad is Bad12%

8%

10%

an

Non-US Stocks

6%

ted

Geo

met

ric M

ea

US StocksMix 2

Mix 1

4%Expe

ct Non-US Bonds

0%

2%

0% 5% 10% 15% 20% 25% 30% 35%

US Bonds

0% 5% 10% 15% 20% 25% 30% 35%

Conditional Value at Risk

Mixes 1 & 2

US StocksNon-US Bonds

37%

33%34%

46%

14%3%

21%12%

Mix 1

N US St k21%

Mix 2

Non-US Stocks

US Bonds

Modeling Fat Tails: Mix 2

3.0

3.5

2 0

2.5

1.5

2.0

Fat-Tailed

0 5

1.0

Lognormal

0.0

0.5

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 110% 120%

Return

Markowitz 2.0: Fat Tails Modeled

10%

12%

Non-US Stocks

8%

c M

ean

Mixes

Lognormal

Fat Tailed

6%

pect

ed G

eom

etric

US Stocks

N US B d

2%

4%Exp Non-US Bonds

0%

%

0% 5% 10% 15% 20% 25% 30% 35% 40% 45%

US Bonds

Conditional Value at Risk

Read More About These and Other Ideas in December

“The breadth and depth of the articles in this book suggest that Paul Kaplan has been thinking about markets for about as long as markets have existed.”

F th f dFrom the foreword