July 18, 2000 Stephen Britt & Bill Pauling Asset Classes in DFA Modeling 2000 CAS DFA Seminar, New York

2 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.

3 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.

4 Securities with similar factor exposures form a natural asset class

5 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

6 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

7 How many asset classes are there?

8 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

10 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

11 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

12 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

13 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

14 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.

15 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.

16 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

17 Asset Modeling Methodologies Mean Variance Covariance Fixed Income Asset Core Equity Asset Capital Asset Pricing Model Arbitrage Pricing Theory

18 Asset Modeling Methodologies Mean 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

19 Asset Modeling Methodologies MVC - Example Example: cash - T-Bills bonds - Lehman Aggregate Bond Index equity - S&P 500 Input assumptions annual return annual standard deviation correlations

20 Asset Modeling Methodologies MVC - 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

21 Asset Modeling Methodologies MVC - 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%)

22 Asset Modeling Methodologies Fixed 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

23 Asset Modeling Methodologies Fixed 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

24 Asset Modeling Methodologies Fixed Income Asset - Results Results have same mean, standard deviation and serial correlation as MVC model

25 Asset Modeling Methodologies Fixed Income Asset - Analysis Bond yields are realistic Bond total returns are realistic Bond income returns are linked to bond yields

26 Asset Modeling Methodologies Core 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

27 Asset Modeling Methodologies Core 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 = $ Price return = $ / $100 = 18.98% Income return = ( ) / 2 = 1.23% Total return = 18.98% % = 20.21%

28 Asset Modeling Methodologies Capital 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

29 Asset Modeling Methodologies CAPM - 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

30 Asset Modeling Methodologies CAPM - Results Simulated Small/Mid Cap returns are internally consistent with S&P 500 returns Higher annual return and risk consistent with Beta > 1

31 Asset Modeling Methodologies Arbitrage 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