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1

RiskRisk--adjusted Performance Measurementadjusted Performance Measurement

Alpha to OmegaAlpha to OmegaDownside to DrawdownDownside to Drawdown

Appraisal to PainAppraisal to Pain

Carl Bacon

Zurich6th February 2008

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Risk ManagementRisk Management

Risk Managers Front Office

Paid to take risk

Risk is good Risk Controllers

Middle office

Paid to monitor/reduce risk Risk is bad

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Risk MeasuresRisk Measures

Ex-Post

Risk after the event

Historical

Ex-Ante

Risk before the event

Prospective

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Simple Risk MeasuresSimple Risk Measures

Absolute Variability, Sharpe Ratio

Relative Tracking Error, Information Ratio

Regression

,

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VariabilityVariability

mean

r

= [ri-mean]2

n

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6.4

6.6

6.8

7.0

7.2

7.4

7.6

7.8

8.0

A B Benchmark

Portfolio ReturnsPortfolio Returns

7.9%

6.9%

7.5%

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Portfolio RiskPortfolio Risk

0.0

1.0

2.0

3.0

4.0

5.0

6.0

A B Benchmark

5.50%

3.20%

4.50%

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Sharpe RatioSharpe Ratio

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

Risk

Retur

n

B ABenchmark

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MM22

0%1%

2%

3%

4%5%

6%

7%

8%9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

Risk

Return

B

A

Benchmark

Return adjusted

to risk M2 for B

Return adjusted

to risk M2 for A

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Risk Adjusted ReturnsRisk Adjusted Returns

Sharpe

Ratio P

FP rrSR

=

P = P o r t f o l i o R i s k M = M a r k e t R i s k R P = P o r t f o l i o R e t u r nR F = R i s k F r e e R a t e

or

( )pMP SRrM +=

2

( ) FP

MFP rrrM +=

2

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Modigliani Excess ReturnModigliani Excess Return

4.5%

B

Return adjusted for

Risk M2= 8.9%

M2 (Excess) =8.9% - 7.5% = 1.4%

Risk adjusted excess return

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Tracking errorTracking error

Tracking Error

Standard Deviation of excess return

Measures consistency of excess return

Ex-Post or Ex-ante Normally annualised

Information RatioAnnualised Excess Return

Annualised Tracking Error

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Information RatioInformation Ratio

Excess

Return

Tracking Error

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Information RatioInformation Ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.41.6

Q1

97

Q3

97

Q1

98

Q3

98

Q1

99

Q3

99

Information ratio

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Risk Efficiency RatioRisk Efficiency Ratio

Risk Efficiency Ratio

or

Measures the quality of the tracking error forecast

Observed Tracking ErrorForecast Tracking Error

Ex-Post Tracking ErrorEx-Ante Tracking Error

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Regression EquationRegression EquationR

B

=0.1

xx x

xx x

x

x x

xx

x

x x

x

x =1.0

R = + (B-RF) +

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Bull & Bear BetasBull & Bear BetasR

B

xx x

xx

xx

x x

xx

x

xx

xx+ =1.1

=0.9

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Regression EquationRegression Equation

Regression Equation

Beta Gradient of the regression equation

Jensens Alpha Intercept of the Regression Equation

( )

( )

++= )( FMFp rrrr

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Regression StatisticsRegression Statistics

Correlation ( r or ) =

Coefficient of =

Determination (R2)

Systematic RiskPortfolio Risk

Systematic VariancePortfolio Variance

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Treynor RatioTreynor Ratio

0%

1%2%

3%

4%

5%

6%

7%

8%

9%

10%

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Beta

Return

B

A Benchmark

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Risk MeasuresRisk Measures

Treynor Ratio

Systematic Risk

Appraisal Ratio

ms =

FPrr

TR

=

Modified Jensen

=

RiskSpecific

AlphasJensen'

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FamaFama DecompositionDecomposition

Early form of performance attribution Return above risk free rate decomposed into two

components:

Systematic Risk Investors risk

Managers risk

Selectivity ( Jensens alpha) Diversification

Net selectivity

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FamaFama DecompositionDecomposition

Return

Total

Excess

Return

Return

from

Selectivity

Return from

Managers Risk

Return from

Investors risk

Diversification

Net Selectivity

A1

Benchmark

A3

A2

I P F

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FamaFama Decomposition: Systematic riskDecomposition: Systematic risk

Return due to systematic risk

Return due to investors systematic risk

Return due to managers systematic risk

( )FMPp rrr =

( )FMII rrr =

( ) ( )FMIPM rrr =

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FamaFama DecompositionDecomposition

Fama Equivalent F

Diversification (d) (Return required to justified specific risk taken)

Net Selectivity (s)

M

PF

=

ds =

)()(FMP

F rrd =

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GH 1 ( Graham & Harvey)GH 1 ( Graham & Harvey)

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

Risk

Retur

C

D

Benchmark

GH1 C

GH1 D

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GH2 ( Graham & Harvey) for MGH2 ( Graham & Harvey) for M22

0%

1%2%

3%

4%

5%

6%

7%

8%9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

Risk

Retu

r

C

D

Benchmark

GH2 C

GH2 D

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Hedge FundsHedge Funds

A hedge fund constitutes an investment programwhereby the managers or partners seek absolute

returns by exploiting investment opportunities

while protecting principal from potential financialloss

Ineichen (2003)

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RiskRisk

--adjusted Performance for Hedge Fundsadjusted Performance for Hedge Funds

Investment returns are not normal

Particularly hedge funds

Skewness

Kurtosis

Excess Kurtosis

n

rrS

P

i 13

=

=

n

rrK

p

i 14

31

4

=

n

rrK

p

iE

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KurtosisKurtosis

Kurtosis >3

Kurtosis

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SkewnessSkewness

Positive Skew Negative Skew

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BeraBera--Jarque TestJarque Test

Test for normality

Reject if exceeds 5.99 (95%)

Reject if exceeds 9.21 (99%)

+=

46

22 EKS

nBJ

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Adjusted Sharpe RatioAdjusted Sharpe Ratio

+=

2

24

3

6

1 SRK

SRS

SRASR

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Downside RiskDownside Risk

D=Downside semi-standard deviation

Fewer data points

Less information Does upside dispersion matter?

down

TP rr

=RatioSortino

Tr =Mimimum acceptable (or Target) return

[ ]

=

=

n

i

Ti

D

n

rr

1

20),(min

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Sortino RatioSortino Ratio

0%

1%

2%

3%

4%5%

6%

7%

8%

9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

Downside Risk

Retur

B

A

Benchmark

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MM

22 (for downside risk)

0%

1%

2%

3%

4%5%

6%

7%

8%

9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

Downside Risk

Retur

B

A

Benchmark

M2 for downside risk

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Upside Potential Ratio (Discrete)Upside Potential Ratio (Discrete)

UPR= Upside Potential

Downside Risk

( )

D

ni

iTi nrr

UPR

=

= =

10,max

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OmegaOmega

Gain-Loss Ratio ( Bernardo & Ledoit )

Omega-Sharpe Ratio

1=

=

=

=

ni

iiT

TP

rrn

rr

1

)0,max(1

0=Tr

PotentialDownside

PotentialUpside

)0,max(

1

)0,max(1

1 =

=

=

=

iT

ni

i

Ti

rrn

rrn

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Prospect RatioProspect Ratio

Penalises loss greater than gain Based on Prospect Theory

( ) ( )( )

D

T

ni

i

ii rrMinrMaxn

+=

=

=10,25.20,

1

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Value at Risk (VaR)Value at Risk (VaR)

Maximum loss in cash terms over a finite period

given a certain level of confidence (say 95%)

Typically ex-ante

Volatility is an input - not an output Similar to ex-ante tracking error

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VaRVaR

Historical Simulation Re-organises actual historical returns

Monte Carlo Simulation

Multiple hypothetical trials Variance- Co-variance

Assumes normality

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Return on VaRReturn on VaR

Return on VaR

(%)VaR

rrR FPVaR

=

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Return on VaRReturn on VaR

0%

1%

2%

3%

4%

5%

6%

7%

8%9%

10%

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0%

VaR

Retur

BA

Benchmark

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Conditional VaR (Conditional VaR (E x p e c t e d S h o r t f a l l E x p e c t e d S h o r t f a l l ))

Conditional

VaR

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Conditional Sharpe RatioConditional Sharpe Ratio

Conditional Sharpe RatioCVaR

FP rr =

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Modified VaModified VaRR

VaR modified for Kurtosis & Skewness

Cornish-Fisher expansion

+

++=

2332

36

52

24

3

6

1MVa S

zzK

zzS

zzrR

ccE

ccccp

=cz -1.96 with 95% confidence=cz -2.33 with 99% confidence

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Modified Sharpe RatioModified Sharpe Ratio

Modified Sharpe RatioMVaR

FP rr =

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DrawdownDrawdown

MaximumDrawdown

Time

D1

D2

D3

Recovery time

Cumulativereturn

St li C l & B k R tiSt li C l & B k R ti

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Sterling, Calmar & Burke RatiosSterling, Calmar & Burke Ratios

Sterling Ratio

Calmar Ratio

Burke ratio

Drawdown = losing period

Max

D

F

r

rr=

=2

Dj

F

r

rr

=D

R

=DjR

=Max

DR

Average Drawdown

Maximum Drawdown

Drawdown j

D

F

r

rr=

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Versions of the Sterling RatioVersions of the Sterling Ratio

Multiple Variations

Original (Deanne Sterling Jones)

Average or Largest Individual Drawdown (say 3

or 5)

Average annual Maximum Drawdown

%10+

D

p

r

r

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SterlingSterling--Calmar RatioCalmar Ratio

Max

D

Fp

r

rr Sterling-Calmar Ratio =

Combines concepts in Calmar & Sterling Ratios

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Pain & Ulcer IndexesPain & Ulcer Indexes

Reflect pain & worry since high water mark

Pain Index

Ulcer Index =

=

!=

ni

i

i

n

DUI

1

2

=

=

!=

ni

i

i

n

DPI

1

iperiodinpeakpervioussincedrawdown=!i

D

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Pain & Martin RatiosPain & Martin Ratios

Sharpe type ratios

Pain Ratio

Martin Ratio(or Ulcer Performance Ratio)

=

=

!

=

ni

i

i

FP

n

D

rrMR

1

2

=

=

!

=mi

i

i

FP

n

D

rrPR

1

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DurationDuration

Duration is a Fixed Income Risk Measure Macaulay Duration

Macaulay-Weil Duration

Modified Duration

Effective Duration

It measures price sensitivity to changes in interest

rates

B

P

DDDuration =

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Hurst IndexHurst Index

Where:

Between 0.5 and 1 - Persistent

Around 0.5 - Totally Random

Between 0 and 0.5 - Anti-Persistent

( )

( )n

mH

log

log=

( ) ( )

P

ii PPm

minmax =

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Risk Adjusted Attribution

( )FMF RRRR +=

Ri k Adj t d Att ib ti

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Risk Adjusted Attribution

Portfolio Benchmark Portfolio Index Weight Weight Return Return

UK 40% 40% 20.0 10.0 1.3

Japan 30% 20% -5.0 -4.0 1.0

US 30% 40% 6.0 8.0 0.8

Total 100% 100% 8.3 6.4 1.1

Total Excess Return =1.79

Risk Adjusted Attribution (Asset Allocation)

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Risk Adjusted Attribution (Asset Allocation)

Portfolio Benchmark Portfolio Index Semi-

Weight Weight Return Return Notional

UK 40% 40% 20.0 10.0 10.0

Japan 30% 20% -5.0 -4.0 -4.0US 30% 40% 6.0 8.0 8.0

Total 100% 100% 8.3 6.4 5.2

Total Excess Return -1= 1.79

Asset (or country) Allocation

1.083

1.064

[ ] 01064.1

10.1%40%40 =

[ ] 97.01064.1

96.0%20%30 =

[ ] 15.01064.1

08.1%40%30 =

13.11064.1

052.1=

UK

JAPAN

US

TOTAL 0 - 0.97 - 0.15 = -1.13 or alternatively

S t ti Ri kS t ti Ri k

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Portfolio Benchmark Portfolio Index Risk-

Weight Weight Return Return adjusted

UK 40% 40% 20.0 10.0 12.7

Japan 30% 20% -5.0 -4.0 -4.0US 30% 40% 6.0 8.0 6.4

Total 100% 100% 8.3 6.4 5.8

Systematic Risk

03.1052.1

10.1110.1

127.1%]40[ =

0.0052.1

96.01

96.0

96.0%]30[ =

44.0052.1

08.11

08.1

0644.1%]30[ =

58.01052.1

0581.1=

UK

JAPAN

US

TOTAL 1.03 +0.0 - 0.44 = 0.58

Systematic RiskSystematic Risk

or alternatively

S l ti itS l ti it

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Portfolio Benchmark Portfolio Index Risk-

Weight Weight Return Return adjusted

UK 40% 40% 20.0 10.0 12.7

Japan 30% 20% -5.0 -4.0 -4.0US 30% 40% 6.0 8.0 6.4

Total 100% 100% 8.3 6.4 5.8

76.20581.1

127.11127.1

20.1%]40[ =

28.00581.1

96.01

96.0

95.0%]30[ =

12.00581.1

0644.11

0644.1

06.1%]30[ =

35.210581.1

083.1=

UK

JAPAN

US

TOTAL 2.76 - 0.28 - 0.12 = 2.35

SelectivitySelectivity (or Adjusted Stock Selection)(or Adjusted Stock Selection)

or alternatively

C l di R kC l di R k

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Concluding RemarksConcluding Remarks

Risk like beauty is in the eye of the

beholder The basic structure of all composite risk measures

is the same:

Get it wrong and youre in trouble

RiskReward

U ilUnile er M illMerrill

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UnileverUnilever -- MerrillMerrill

Financial Times

1st October 2001

AppendixAppendix Excel FunctionsExcel Functions

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AppendixAppendix -- Excel FunctionsExcel Functions

Sample excess Kurtosis

Sample Skewness

( )

( ) ( ) ( )

( )

( ) ( )32

13

321

12

4

+

=

nn

n

nnn

nnrr

p

i

( ) ( )21

3

= nn nrr P

i