July 18, 2000

Stephen Britt & Bill Pauling

Asset Classes in DFA Modeling

2000 CAS DFA Seminar, New York

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What is an asset class?

The concept of an asset class does not appear in finance.

Traditional asset pricing models (APT, CAPM etc) describe ways to price securities, not asset classes.

However, practitioners are very aware of the concept: Performance benchmarks (S&P 500) are

generally based on the concept Investment departments are generally split by

class ( bonds, equities, property) ‘Asset Allocation’ generally refers to setting

distribution by class.

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Theoretical definition

Definition one borrows from Arbitrage Pricing Theory.

Ross (1976) suggests that the change in a security’s price can be accounted for by unanticipated changes in a small number of factors (eg): Inflation Economic activity Term spread Credit spread Currency

This is a useful framework to think about classes.

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Securities with similar factor exposures form a natural asset class

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From an operation viewpoint we might use a different approach

Use the concept of a representative index

A number of purveyors of performance indices exist which report on the performance of groups of securities: MSCI S&P Dow Jones Lehman Bros Merrill Lynch

These provide off-the-shelf asset classes that the DFA modeler can chose

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Why use ‘established’ benchmarks ?

Opportunity sets as well as existing assets

Performance characteristics well publicized

Often a long history of data on a consistent basis

Well understood by the investment function in most companies

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How many asset classes are there?

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For DFA work the number of asset classes will depend on the use

For asset allocation work need to be wary of using too many asset classes (multi-collinearity)

Number of asset classes should relate to the investment function If the client manages mortgages, corporates

and government bonds against one index, perhaps one asset class is adequate

For other work (eg rating agency discussions) additional asset classes may be useful.

July 18, 2000

Stephen Britt & Bill Pauling

Modeling asset classes

General considerations

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DFA requires modeling of two aspects of asset class behavior

Generic Total return Income per unit

investment Duration Convexity

Company specific Market value Book value Cost basis FAS 115 definition Book yield Transaction costs

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We allow clients to choose the most appropriate approach to asset modeling

Index Asset Classes

Continuous Asset Classes

Discontinuous Asset Classes

Seriatim Assets/EPA Individual

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Index Asset Model Type

Aim to be fast

No attempt to model cash flows or book accounting

Variables modeled: Market value Total return Investment expense

Requires a ‘feed’ of total return from a scenario generator

Useful in a total return environment, where book value accounting unnecessary

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Continuous Asset Model Type

‘Wasting’ assets are not suited to asset allocation studies Represent assets owned,

not the opportunity set Characteristics decay

Continuous asset classes overcome this limitation Asset class characteristics

relatively stable Reasonably fast Some support for book

value

One data point per asset class

Additional attributes supported (approximations only) Amortized cost Book income Book Yield Realized/unrealized gains Cash Income FAS Classification Tax status

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Discontinuous Asset Model Type

For detailed asset-liability analysis, we require detailed cash flow Capture the decay of

existing assets Additional asset

characteristics needed Model accounting entries

as accurately as possible

However, this asset class must be relatively fast (much faster than seriatim)

Aim to model all assets in 30 data points or less

Additional variables supported Maturities, defaults Sales, Prepayments Effective duration Effective convexity YTM, WAL, etc.

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Seriatim asset classes

Some applications (statutory cash flow testing for life companies) require each asset to be modeled individuality

Useful where the optionality of individual assets is critical

Time constraints on seriatim asset modeling are onerous.

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What Do We Model?

Price return taxes accounting

+ Income return taxes accounting

= Total return market value of assets

If we know any of the two above, we can calculate the third

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Asset Modeling Methodologies

Mean Variance Covariance

Fixed Income Asset

Core Equity Asset

Capital Asset Pricing Model

Arbitrage Pricing Theory

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Asset Modeling MethodologiesMean Variance Covariance (MVC)

Functional form:

Single period

Returns are driven by inputs mean return standard deviation of return correlation matrix

Returns are assumed to be lognormally distributed

1 Ri eR

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Asset Modeling MethodologiesMVC - Example

Example: cash - T-Bills bonds - Lehman

Aggregate Bond Index equity - S&P 500

Input assumptions annual return annual standard

deviation correlationsCash Bonds Equity

Cash 1.00 0.10 -0.10Bonds 0.10 1.00 0.40Equity -0.10 0.40 1.00

Asset Annual AnnualClass Return Risk

Cash 5.50% 2.00%Bonds 6.50% 6.80%Equity 11.00% 17.00%

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Asset Modeling MethodologiesMVC - Results

Cash returns can be negative - 16 occurrences

Cash returns have negative serial correlation - should be positive

How to determine income return?

Returns are not connected to economic factors

CorrelationsCash Bonds Equity

Cash 1.00 0.08 -0.08Bonds 0.08 1.00 0.35Equity -0.08 0.35 1.00

Asset Annual Compound Standard SerialClass Return Return Deviation Correl

Cash 5.50% 5.48% 2.00% -0.11Bonds 6.50% 6.31% 6.80% -0.12Equity 11.00% 9.85% 17.00% -0.11

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Asset Modeling MethodologiesMVC - Analysis of Bond Returns

Given initial yield, duration and convexity, implied bond yields can be calculated

Implied bond yields can be negative

Implied bond yields can be extremely high (e.g. 40%)

MVC Bond - Scenario: 129

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-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

0 1 2 3 4 5 6 7 8 9 10

Year

Implied Yield Bond Rtn

MVC Bond - Scenario: 109

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

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Implied Yield Bond Rtn

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Asset Modeling MethodologiesFixed Income Asset

Functional form:

Income return is computed

Price returns are driven by changes in bond yields

Both income and price returns are linked to a common economic factor

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Asset Modeling MethodologiesFixed Income Asset - Example

Example: Lehman Aggregate Bond Index

Input assumptions yield spread over 10-

year T-Bond duration convexity residual error

10 year T-Bond yields are simulated by Global CAP:Link’s yield curve model

Yield Spread 0.30%

Duration 4.92

Convexity 18

Residual 2.00%

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Asset Modeling MethodologiesFixed Income Asset - Results

Results have same mean, standard deviation and serial correlation as MVC modelAnnual Compound Standard Serial

Return Return Deviation Correl6.50% 6.29% 6.80% -0.12

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Asset Modeling MethodologiesFixed Income Asset - Analysis

Bond yields are realistic

Bond total returns are realistic

Bond income returns are linked to bond yields

Fixed Income - Scenario: 53

-10.00%

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5.00%

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Yield Price Growth Income Rtn Total Return

Fixed Income - Scenario: 53

-10.0%

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0.0%

5.0%

10.0%

15.0%

1 2 3 4 5 6 7 8 9 10

Year

Chg Yield Total Return Income Rtn Price Growth

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Asset Modeling MethodologiesCore Equity Asset

Functional form:

Returns are driven by a valuation factor and a growth factor

Income return is computed

Price return is driven by changes in the valuation factor and growth factor

Both income and price returns are linked to economic factors

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Asset Modeling MethodologiesCore Equity Asset - Example

1999 S&P Total Return = 21%

Given: 1/1/99 Dividend Yield =

1.32% 12/31/99 Dividend Yield

= 1.14% 1/1/99 Earnings Yield =

3.13% 12/31/99 Earnings Yield

= 3.07% 1999 Earnings Growth =

16.7%

Key formula: Earnings Yield = Earnings / Price

Solution: Initial price = $100 Initial earnings = $3.13 Ending earnings =

$3.13*(1.167) = $3.65 Ending price =

$3.65 / .0307 = $118.98 Price return = $118.98 /

$100 = 18.98% Income return =

(.0132+.0114) / 2 = 1.23%

Total return = 18.98% + 1.23% = 20.21%

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Asset Modeling MethodologiesCapital Asset Pricing Model (CAPM)

Functional form:

Single period; can be used in multi-period context

Returns can be driven by other asset class returns that are connected to economic factors

Income return can be derived from income return of market index

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Asset Modeling MethodologiesCAPM - Example

Example: Small/Mid Cap Equity

Input assumptions beta residual error

Market index (e.g. S&P 500) returns are simulated by Global CAP:Link

Risk-free (e.g. cash) returns are simulated by Global CAP:Link

Beta 1.12

Residual 9.00%

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Asset Modeling MethodologiesCAPM - Results

Simulated Small/Mid Cap returns are internally consistent with S&P 500 returns

Higher annual return and risk consistent with Beta > 1

Annual Compound Standard Serial CorrelReturn Return Deviation Correl (Sm/Mid)

Sm/Mid Eq 11.33% 9.28% 21.56% -0.05S&P 500 10.62% 9.28% 17.56% -0.05 0.90

Sm/Mid Cap vs. S&P 500 - Scenario: 121

-30%

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Sm/Mid S&P 500

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Asset Modeling MethodologiesArbitrage Pricing Theory (APT)

Functional form:

Single period; can be used in multi-period context

Returns are driven by multiple risk factors

Risk factors are unspecified

Risk factors can be economic factors or asset class returns

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