performance calculations

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Performance Calculations101

Monday, October 19, 2009

Public Pension Financial Forum

John D. Simpson, CIPMThe Spaulding Group, Inc.


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What well do today Well cover a few basic formulas that are

used to calculate rates of return and risk

Nature is pleased with simplicityIssac Newton, Principia

We will try to make this easy to comprehend But, we have a fair amount to cover and

limited time

Feel free to ask questions


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Rates of Return:

Time-weighting vs. Money-weighting Time-weighted returns measure the

performance of the portfolio manager

Money-weighted returns measure theperformance of the fund or portfolio


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Time-weighting Time-weighting eliminatesor reducesthe

impact of cash flows

Because managers dont control the flows

Two general approaches:

Approximations, which approximatethe exact,

true, time-weighted rate of return Exact, true, time-weighted rate of return


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Approximation methods

well discuss Original Dietz

Modified Dietz

Modified BAI

(a.k.a. Modified IRR and Linked IRR)


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What Are Cash Flows? Two types:

External: impact the portfolio

Internal: impact securities, sectors Specifics:

External: contributions/withdrawals of cash and/orsecurities

Internal: buys/sells, interest/dividends, corporate actions


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Visualizing Flows

Our Portfolio

TechStocks

Bank

Stocks

CorporateBonds

Munis

Transfer100 sharesDell, Inc.

Withdraw

$1,000

Buy 100 shares BofA

ExternalFlows

GM Bond pays interest

InternalFlows


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The scenario we will use to

demonstrate the various formulas:

5/30 BMV 100,000

6/10 Cash Flow 20,000

6/30 EMV 123,000


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Assumes constant rate of return on the portfolioduring the period

Very easy method to calculate

Provides approximation to the true rate of return

Returns can be distorted when large flows occur

Also, return doesnt take into account market

volatility, which further affects the accuracy

Weights each cash flow as if it occurred at themiddle of the time period

Original Dietz


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Original Dietz

R EMV BMV C

BMV C

OriginalDietz

0 5.5/30 BMV 100,000

6/10 Cash Flow 20,000

6/30 EMV 123,000

ROriginalDietz

123 000 100 000 20 000

100 000 0 5 20 0002 73%

, , ,

, . ,.


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Modified Dietz Method

Assumes constant rate of return on theportfolio during the period

Provides an improvement in theapproximation of true time-weighted rate ofreturn, versus the Original Dietz formula

Disadvantage greatest when: (a) 1 or morelarge external cash flows; (b) cash flowsoccur during periods of high market volatility

Weights each external cash flow by the

amount of time it is held in the portfolio


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Modified Dietz MethodR

EMV BMV C

BMV W C

W CD DCD

WCD D

CD

ModifiedDietz

EOD

SOD

1

WSOD

30 10 1

300 70.


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Modified Dietz Method

5/30 BMV 100,000

6/10 Cash Flow 20,000

6/30 EMV 123,000

R EMV BMV C

BMV W CModifiedDietz

RModifiedDietz

123 000 100 000 20 000

100 000 0 70 20 000

2 63%, , ,

, . ,

.


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Determines internal rate of return for the period

Takes into account the exact timing of each

external cash flow Market value at beginning of period is treated as

cash flow

Disadvantage: Requires iterative process solutiondifficult to calculate manually

Modified BAI

(Modified IRR, Linked IRR)


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Modified BAI Method

5/30 BMV 100,000

6/10 Cash Flow 20,000

6/30 EMV 123,000

0

1 1 1 1

1 2

1 2

InitialValue

CashFlow

r

CashFlow

r

CashFlow

r

Outflow

rt t

n

t tn end ...

0 100 000

20 000

1

123 000

10 30

, , ,

.r r


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Modified BAI Method

0

1 1 1 1

1 2

1 2

InitialValue

CashFlow

r

CashFlow

r

CashFlow

r

Outflow

rt t

n

t tn end ...

2.63% (1.40)

2.62% (14.25)2.64% 7.94

Solving for r through

iteration ( tr ial & error)

RModifiedBAI 2 63%.


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Value portfolio every time external flows occur

Advantage: calculates true time-weighted rate

of return Disadvantage: requires precise valuation of the

portfolio on each day of external cash flow

True, exact TWRR


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True, exact TWRR

5/30 BMV 100,000

6/9 EMV 101,000

6/10 Cash Flow 20,000

6/30 EMV 123,000

ROREMV

BMV

EMV

BMV

EMV

BMV

EMV

BMVTruei

ii

nn

n

1

1

1

2

2

1 1...

RExact 101 000

100 000

123 000

121 0001 2 67%

,

,

,

,.


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Money-weighted returnsInternal Rate of Return (IRR)

Takes cash flows into consideration

Cash flows will impact the return

Only uses cash flows and the closing marketvalue in calculation (dont revalue duringperiod)

Produces the return that equates the presentvalue of all invested capital


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Its an iterative process

We solve for r, by trial-and error

The general rule is to use the Modified Dietz return

as the first order approximation to the IRR

0

1 1 1 1

1 2

1 2

InitialValueCashFlow

r

CashFlow

r

CashFlow

r

Outflow

rt t

n

t tn end ...

Solving for the IRR


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5/30 BMV 100,000

6/10 Cash Flow 20,000

6/30 EMV 123,000

0

1 1 1 1

1 2

1 2

InitialValue

CashFlow

r

CashFlow

r

CashFlow

r

Outflow

rt t

n

t tn end ...

0 100 000

20 000

1

123 000

10 30

, , ,

.r r

IRR Method


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0

1 1 1 1

1 2

1 2

InitialValue

CashFlow

r

CashFlow

r

CashFlow

r

Outflow

rt t

n

t tn end ...

2.63% (1.40)

2.62% (14.25)

2.64% 7.94

Solving for r through

iteration (trial & error)

IRR 263%.

IRR Method


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Why did the Modified BAI and IRR yield the samereturns (2.63%)?

Calculation Question


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Contrasting IRRwith time-weighting

Exact

weight

weight

weight

Revalue

Internal Rate of Return

InceptionMostt

RecentPeriod

CashFlow

CashFlow

CashFlow

Value

Revalu

e

Value

Revalue

Revalue

Revalue

M/E M/E M/E M/E M/E M/E M/E M/E

Valu

e

Modified Dietz, Modified BAI

Revalu

e

Revalue

Revalue

Revalue

Revalue

Revalue

Revalue

Revalue

Revalue

DayW

eight

DayW

eight

DayW

eight

IRR values portfolio at the beginning and end of the period

TWRR values at various times throughout the period


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Our investment is a mutual fund

Where two investors begin with 100 shares

And both make two additional purchases duringthe year of 100 shares each

But at different times

And at different prices

Well use an example tocompare TWRR and MWRR


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10

10.5

11

8

14

9

11

15

9

10

9

11

12

7

8

9

10

11

12

13

14

15

16

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Our funds end-of-month NAVs


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Investor #2Purchases

Investor #1Purchases

BMV for both

Investors

10

10.5

11

8

14

9

11

15

9

10

9

11

12

7

8

9

10

11

12

13

14

15

16

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

BelievesBuy low/Sell high

BelievesBuy high/Sell low

Our investors purchases


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Month NAV Investor #1 Investor #2

Dec 10 1,000 1,000

Jan 10.5

Feb 11

Mar 12

Apr 8 800May 14 1,400

Jun 9

Jul 11

Aug 15 1,500

Sep 9 900

Oct 10

Nov 9

Dec 11

3,900 2,700

3,300 3,300

(600) 600

Total Investment

EMV =

Gain/Loss

Paper gainof

$600!

Paper lossof

$600!

The investments unrealizedgains/losses


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The funds return (using an exact TWRR method):

ROR EMVBMVFund

1 1110

1 10%

Whats our return?


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Investor #1s IRR = -24.86%

Investor #2s IRR = +35.16%

How about money-weighting?


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As a Plan Sponsor

Which returns make more sense to you?

Which are more meaningful?

Investor #1 Investor #2

P&L -$600 +$600

TWRR 10% 10%

MWRR -24.86% +35.16%

TWRR judgesportfolio manager

MWRRjudges the

portfolio


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Multi-period rates of return

We dont just want to report returns for amonth

We want to linkour returns to formquarterly, annual, since inception, etc.returns

How do we do this?


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The process used to linksub-period returns tocreate returns for extended periods:

e.g., We want to take January, February, and Marchreturns to create a return for 1Q

We geometrically link in order to compound ourreturns

Geometric linking


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Step-by-step process:

1. Convert the returns to a decimal

2. Add 13. Multiply these numbers

4. Subtract 1

5. Convert the number to a percent

Geometric linking


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Jan 1.10%

Feb 0.89%

Mar 0.60%

Step 1 Convert to a decimal 0.0110

0.0089

0.0060

Step 2 Add 1 1.0110

1.0089

1.0060

Step 3 Multiply 1.0261

Step 4 Subtract 1 0.0261

Step 5 Convert to a percent 2.61%

Geometric linking


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Before we move to risk, arethere any questions?


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Risk measures

Two categories

Formulas that measure risk

Well look at standard deviation and tracking error

Formulas that adjust the return per unit of risk

Well look at Sharpe Ratio and Information Ratio


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Standard Deviation

Measures volatility of returns over time

The most common and most criticized

measure to describe the risk of a security orportfolio.

Used not only in finance, but also statistics,

sciences, and social sciences. Provides a precise measure of the amount of

variation in any group of numbers.


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Standard Deviation; based on theBell-shaped (normal) curve


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Standard Deviation Formulas

R R

n

i

2

Note: This is represented in Excel as the STDEVPFunction

R R

ni

2

1

Note: This is represented in Excel as the STDEVFunction


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An example ofstandard deviation

A B C D E F G

1 Portfolio

2 Month Return

3 1 6.02%

4 2 4.43%

5 3 -3.34%6 4 4.22%

7 5 3.69%

8 6 -2.58%

9 7 6.47%

10 8 0.18%

11 9 1.42%

12 10 -2.45%

13 11 2.53%

14 12 2.82%

15 13 5.78%

16

17 2.25% =AVERAGE(C3..C15)

18 3.25% =STDEVP(C3..C15)

Average ROR =

Standard Deviation =


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Tracking Error

The difference between the performance ofthe benchmark and the replicating portfolio

Measures active risk; the risk the managertook relative to the benchmark

Measured as annualized standard deviation

Standard deviation of excess returns Standard deviation of the difference in

historical returns of a portfolio and itsbenchmark


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Tracking Formula:Volatility ofPast Returns vs. Benchmark

Tracking error measures how closely theportfolio follows the index and is measuredas the standard deviation of the differencebetween the portfolio and index returns.

TrackingError StdDev Rp Rb

A i i


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An example ofTracking Error

A B C D E F

1

2

3 Month Portfolio Index

4 1 6.02% 5.81% 0.21%

5 2 4.43% 4.23% 0.20%6 3 -3.34% -3.24% -0.10%

7 4 4.22% 4.15% 0.07%

8 5 3.69% 3.65% 0.04%

9 6 -2.58% -2.55% -0.03%

10 7 6.47% 6.35% 0.12%

11 8 0.18% 0.13% 0.05%12 9 1.42% 1.20% 0.22%

13 10 -2.45% -2.55% 0.10%

14 11 2.53% 2.50% 0.03%

15 12 2.82% 2.78% 0.04%

16 13 5.78% 5.74% 0.04%

17 0.09% =STDEVP(D4..D16)

18 0.31% =D17*SQRT(12)

Excess

ROR

Tracking Error

Annualized Tracking Error

To annualize,multiply by

square root of12

The Sharpe Ratio


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The Sharpe RatioAlso known as

Reward-to-Variability Ratio Developed by Bill Sharpe

Nobel Prize Winner

Equity Risk Premium (Return) / StandardDeviation (Risk)


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Sharpe Ratio FormulaEquity Risk Premium divided bystandard deviation of portfolio returns

SharpeRatioR R

p f

p


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An example ofSharpe Ratio

Month Rp RFree

1 6.02% 0.32%

2 4.43% 0.31%

3 -3.34% 0.33%

4 4.22% 0.35%

5 3.69% 0.40%

6 -2.58% 0.39%

7 6.47% 0.37%

8 0.18% 0.29%

9 1.42% 0.34%

10 -2.45% 0.35%

11 2.53% 0.41%

12 2.82% 0.38%13 5.78% 0.39%

Ave 2.25% 0.36%

3.25%

0.36%

1.89%

0.58

2.01Annualized Sharpe =

Standard Deviation =

Ave Risk Free ROR

Av ROR - Av Risk Free

Sharpe Ratio =

To annualize,multiply by

square root of12


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Information Ratio

The Information Ratio measures the excessreturn of an investment manager divided by

the amount of risk the manager takesrelative to the benchmark

Its the Excess Return (Active Return) dividedby the Tracking Error (Active Risk)

IR is a variationof the Sharpe Ratio, wherethe Returnis the Excess Returnand the Riskis the Excess or Active Risk


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Information Ratio

IR serves as a measure of the specialinformation an active portfolio manager has

Value Added (excess return) / Tracking Error

Typically annualize

IRExcess turn

TrackingError

Re

IRAvg R Avg R

R R

( ) ( )


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Information Ratio

Active Returnon the account

Accounts

Active Risk

IR

Avg R Avg R

R R

( ) ( )

l f


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An example ofInformation Ratio

A B C D E F

1

2

3 Month Portfolio Index

4 1 6.02% 5.81% 0.21%

5 2 4.43% 4.23% 0.20%

6 3 -3.34% -3.24% -0.10%7 4 4.22% 4.15% 0.07%

8 5 3.69% 3.65% 0.04%

9 6 -2.58% -2.55% -0.03%

10 7 6.47% 6.35% 0.12%

11 8 0.18% 0.13% 0.05%

12 9 1.42% 1.20% 0.22%

13 10 -2.45% -2.55% 0.10%

14 11 2.53% 2.50% 0.03%

15 12 2.82% 2.78% 0.04%

16 13 5.78% 5.74% 0.04%

17 0.09% =STDEVP(D4..D16)

18 0.31% =D17*SQRT(12)

19 0.99%

20 0.08% =D19/13

21 0.85 =D20/D1722 2.93 =D21*SQRT(12)Annualized IR =

=SUM(D4..D16)Sum of Excess Returns =

Average Excess Return =

Excess

ROR

Tracking Error =

Annualized Tracking Error =

Information Ratio =


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What have we covered today

Hopefully youll agree a lot in a short time

Return measures

TWRR approximation measues

Original Dietz

Modified Dietz

Modified BAI TWRR exact measure

True daily

Geometric Linking


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What have we covered today

Risk measures

Measurements of risk

Standard deviation Tracking error

Measurements of risk-adjusted returns

Sharpe ratio

Information ratio


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Questions?

John D. [email protected]

1.310.500.9640

www spauldinggrp com
mailto:[email protected]://www.spauldinggrp.com/http://www.spauldinggrp.com/mailto:[email protected]

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