A History of Commercially Available Risk Models

Abstract

Multi-factor risk models have been used in portfolio selection since the 1960s and early 1970s. The work of Barr Rosenberg led to the creation of Barra, the first major commercially available portfolio selection software. Rudd and Clasing brought Barra to the academic audience as professional managers were embracing it in industry. It is important to see how quantitative analysis was developed in investment research and analysis. Harry Markowitz, Bill Sharpe, Jan Mossin, and Jack Treynor pioneered capital market equilibrium and the creation and estimation of the Capital Asset Pricing Model. In the 1970s, multi-factor risk models were developed and estimated by Barr Rosenberg, Andrew Rudd, and their colleagues at Barra; John Blin and Steve Bender at APT; and Sebastian Ceria at Axioma. Multi-factor models pushed out the efficient frontier relative to a single factor risk model. McKinley Capital Management developed its Horse Race risk model philosophy to assess the most effective risk models for maximizing the Geometric Means, Sharpe Ratios, and Information Ratios. Several special issues of practitioner-oriented applied investment research featured the McKinley Horse Race results and reported that properly constructed multi-factor risk models created portfolios in which tilt variables that have statistically significant ICs produced statistically significant Active Returns. One must remove the market effect and extra-market covariances to properly estimate the contribution of a variable to the creation of efficient portfolios. APT and Axioma have reigned supreme in those tests. The integration of Axioma and AAF and ITG transactions costs on the FactSet Investment have led MCM to implement an Axioma portfolio.

Appendices

Appendix 1: US-E3 Descriptor Definitions

This appendix gives detailed definitions of the descriptors that underlie the risk indices in US-E3. The method of combining these descriptors into risk indices is proprietary to Barra.

1. Volatility

• (i) BTSG: Beta times sigma

  This is computed as $\sqrt{\beta {\sigma}_{\varepsilon }}$, where β is the historical beta and σ_ε is the historical residual standard deviation. If β is negative, then the descriptor is set equal to zero.

• (ii) DASTD: Daily standard deviation

  This is computed as:

$$ \sqrt{N_{days}\left[{\sum}_{t=1}^T{w}_t{r_t}^2\right]} $$

where r_t is the return over day t, w_t is the weight for day t, T is the number of days of historical returns data used to compute this descriptor (we set this to 65 days), and N_days is the number of trading days in a month (we set this to 23).

• (iii) HILO: Ratio of high price to low price over the last month

  This is calculated as:

$$ \mathit{\log}\left(\frac{P_H}{P_L}\right) $$

where P_H and P_L are the maximum price and minimum price attained over the last 1 month.

• (iv) LPRI: Log of stock price

  This is the log of the stock price at the end of last month.

• (v) CMRA: Cumulative range

  Let Z_t be defined as follows:

$$ {Z}_t={\sum}_{s=1}^t\log \left(1+{r}_{\mathrm{i}.\mathrm{s}}\right)-{\sum}_{s=1}^t\log \left(1+{r}_{\mathrm{f},\mathrm{s}}\right) $$

where r_i,s is the return on stock I in month s, and r_f,s is the risk-free rate for month s. In other words, Z_t is the cumulative return of the stock over the risk-free rate at the end of month t. Define Z_max and Z_min as the maximum and minimum values of Z_t over the last 12 months. CMRA is computed as:

$$ \log \left(\frac{1+{Z}_{\mathrm{max}}}{a+{Z}_{\mathrm{min}}}\right) $$

• (vi) VOLBT: Sensitivity of changes in trading volume to changes in aggregate trading volume

  This may be estimated by the following regression:

$$ \frac{\varDelta {V}_{\mathrm{i},\mathrm{t}}}{N_{\mathrm{i},\mathrm{t}}}=\mathrm{a}+\mathrm{b}\ \frac{\varDelta {V}_{\mathrm{M},\mathrm{t}}}{N_{\mathrm{m}.\mathrm{t}}}+{\xi}_{\mathrm{i},\mathrm{t}} $$

where ΔV_i,t is the change in share volume of stock I from week t-1 to week t, N_i,t is the average number of shares outstanding for stock I at the beginning of week t-1 and week t, ΔV_M,t is the change in volume on the aggregate market from week t-1 to week t, and N_M,t is the average number of shares outstanding for the aggregate market at the beginning of week t-1 and week t.

• (vii) SERDP: Serial dependence

  This measure is designed to capture serial dependence in residuals from the market model regressions. It is computed as follows:

$$ \mathrm{SERDP}=\frac{\frac{1}{T-2}\ {\sum}_{t=3}^T{\left({e}_t+{e}_{t-1}+{e}_{t-2}\right)}^2}{\frac{1}{T-2}\ {\sum}_{t=3}^T\left({e^2}_t+{e^2}_{t-1}+{e^2}_{t-2}\right)} $$

where e_t is the residual from the market model regression in month t, and T is the number of months over which this regression is run (typically, T = 60 months).

• (viii) OPSTD: Option-implied standard deviation

  This descriptor is computed as the implied standard deviation from the Black-Scholes option pricing formula using the price on the closest to at-the-money call option that trades on the underlying stock.

2. Momentum

• (i) RSTR: Relative strength

  This is computed as the cumulative excess return (using continuously compounded monthly returns) over the last 12 months – i.e.,

  $$ \mathrm{RSTR}\stackrel{\longrightarrow}{\leftarrow} ={\sum}_{t=1}^T\log \left(1+{r}_{\mathrm{i},\mathrm{t}}\right)-{\sum}_{t=1}^T\log \left(1+{r}_{\mathrm{f},\mathrm{t}}\right) $$

where r_i,t is the arithmetic return of the stock in month I, and r_f,t is the arithmetic risk-free rate for month i. This measure is usually computed over the last 1 year – i.e., T is set equal to 12 months.

• (ii) HALPHA: Historical alpha

  This descriptor is equal to the alpha term (i.e., the intercept term) from a 60-month regression of the stock’s excess returns on the S&P 500 excess returns.

3. Size

• (i) LNCAP: Log of market capitalization

  This descriptor is computed as the log of the market capitalization of equity (price times number of shares outstanding) for the company.

4. Size nonlinearity

• (i) LCAPCB: Cube of the log of market capitalization

  This risk index is computed as the cube of the normalized log of market capitalization.

5. Trading activity

• i) STOA: Share turnover over the last year

  STOA is the annualized share turnover rate using data from the last 12 months – i.e., it is equal to V_ann / $\overline{N}$_out, where V_ann is the total trading volume (in number of shares) over the last 12 months and $\overline{N}$_out is the average number of shares outstanding over the previous 12 months (i.e., it is equal to the average value of the number of shares outstanding at the beginning of each month over the previous 12 months).

• ii) STOQ: Share turnover over the last quarter

  This is computed as the annualized share turnover rate using data from the most recent quarter. Let V_q be the total trading volume (in number of shares) over the most recent quarter and let $\overline{N}$_out be the average number of shares outstanding over the period (i.e., $\overline{N}$_out is equal to the average value of the number of shares outstanding at the beginning of each month over the previous 3 months). Then, STOQ is computed as 4V_q / $\overline{N}$_out.

• iii) STOM: Share turnover over the last month

  This is computed as the share turnover rate using data from the most recent month – i.e., it is equal to the number of shares traded last month divided by the number of shares outstanding at the beginning of the month.

• iv) STO5: Share turnover over the last 5 years

  This is equal to the annualized share turnover rate using data from the last 60 months. In symbols, STO5 is given by:

$$ \mathrm{STO}5=\frac{12\ \left[\frac{1}{T}{\sum}_{s=1}^T{V}_s\right]}{\sigma_{\varepsilon }} $$

where V_s is equal to the total trading volume in months, and $\overline{N}$_out is the average number of shares outstanding over the last 60 months.

• v) FSPLIT: Indicator for forward split

  This descriptor is a 0–1 indicator variable to capture the occurrence of forward splits in the company’s stock over the last 2 years.

• vi) VLVR: Volume to variance

  This measure is calculated as follows:

$$ \mathrm{VLVR}=\log \frac{\frac{12}{T}\ \left[{\sum}_{s=1}^T{V}_s{P}_s\right]}{\sigma_{\varepsilon }} $$

where V_s equals the number of shares traded in month s, P_s is the closing price of the stock at the end of month s, and σ_ε is the estimated residual standard deviation. The sum in the numerator is computed over the last 12 months.

6. Growth

• (i) PAYO: Payout ratio over 5 years

  This measure is computed as follows:

$$ \mathrm{PAYO}=\frac{\frac{1}{T}{\sum}_{t=1}^T{D}_t}{\frac{1}{T}{\sum}_{t=1}^T{E}_t} $$

where D_t is the aggregate dividend paid out in year t and E_t is the total earnings available for common shareholders in year t. This descriptor is computed using the last 5 years of data on dividends and earnings.

• (ii) VCAP: Variability in capital structure

  This descriptor is measured as follows:

$$ \mathrm{VCAP}=\frac{\frac{1}{T-1}{\sum}_{t=2}^T\left(|{N}_{t-1}-{N}_t|{P}_{t-1}+|\mathrm{L}{\mathrm{D}}_{t-1}-\mathrm{L}{\mathrm{D}}_t|+|\mathrm{P}{\mathrm{E}}_t-\mathrm{P}{\mathrm{E}}_{t-1}|\right)}{\mathrm{C}{\mathrm{E}}_T+\mathrm{L}{\mathrm{D}}_T+\mathrm{P}{\mathrm{E}}_T} $$

where N_t-1 is the number of shares outstanding at the end of time t-1; P_t-1 is the price per share at the end of time t-1; LD_t-1 is the book value of long-term debt at the end of time period t-1; PE_t-1 is the book value of preferred equity at the end of time period t-1; and CE_T + LD_T + PE_T are the book values of common equity, long-term debt, and preferred equity as of the most recent fiscal year.

• (iii) AGRO: Growth rate in total assets

  To compute this descriptor, the following regression is run:

$$ {\mathrm{TA}}_{\mathrm{it}}=\mathrm{a}+\mathrm{bt}+{\upxi}_{\mathrm{it}} $$

where TA_it is the total assets of the company as of the end of year t, and the regression is run for the period = 1, …, 5. AGRO is computed as follows:

$$ \mathrm{AGRO}=\frac{b}{\frac{1}{T}{\sum}_{t=1}^T\mathrm{T}{\mathrm{A}}_{\mathrm{it}}} $$

where the denominator average is computed over all the data used in the regression.

• (iv) EGRO: Earnings growth rate over last 5 years

  First, the following regression is run:

$$ {\mathrm{EPS}}_{\mathrm{t}}=\mathrm{a}+\mathrm{bt}+{\upxi}_{\mathrm{t}} $$

where EPS_t is the earnings per share for year t. This regression is run for the period t = 1, …, 5. EGRO is computed as follows:

$$ \mathrm{EGRO}=\frac{b}{\frac{1}{T}{\sum}_{t=1}^T\mathrm{EP}{\mathrm{S}}_t} $$

• (v) EGIBS: Analyst-predicted earnings growth

  This is computed as follows:

$$ \mathrm{EGIBS}=\frac{\left(\mathrm{EARN}-\mathrm{EPS}\right)}{\left(\mathrm{EARN}\stackrel{\longrightarrow}{\leftarrow} +\stackrel{\longrightarrow}{\leftarrow} \mathrm{EPS}\right)/2} $$

where EARN is a weighted average of the median earnings predictions by analysts for the current year and next year, and EPS is the sum of the four most recent quarterly earnings per share.

• (vi) DELE: Recent earnings change

  This is a measure of recent earnings growth and is measured as follows:

$$ \mathrm{DELE}=\frac{\left(\mathrm{EP}{\mathrm{S}}_t-\mathrm{EP}{\mathrm{S}}_{\mathrm{t}-1}\right)}{\left(\mathrm{EP}{\mathrm{S}}_t+\mathrm{EP}{\mathrm{S}}_{t-1}\right)/2} $$

where EPS_t is the earnings per share for the most recent year, and EPS_t-1 is the earnings per share for the previous year. We set this to missing if the denominator is nonpositive.

7. Earnings Yield

• (i) EPIBS: Analyst-predicted earnings-to-price

  This is computed as the weighted average of analysts’ median predicted earnings for the current fiscal year and next fiscal year divided by the most recent price.

• (ii) ETOP: Trailing annual earnings-to-price

  This is computed as the sum of the four most recent quarterly earnings per share divided by the most recent price.

• (iii) ETP5: Historical earnings-to-price

  This is computed as follows:

$$ \mathrm{ETP}5=\frac{\frac{1}{T}{\sum}_{t=1}^T\mathrm{EP}{\mathrm{S}}_t}{\frac{1}{T}{\sum}_{t=1}^T{P}_t} $$

where EPS_t is equal to the earnings per share over year t, and P_t is equal to the closing price per share at the end of year t.

8. Value

• (i) BTOP: Book-to-price ratio

  This is the book value of common equity as of the most recent fiscal year-end divided by the most recent value of the market capitalization of the equity.

9. Earnings Variability

• (i) VERN: Variability in earnings

  This measure is computed as follows:

$$ \mathrm{VERN}=\frac{{\left(\frac{1}{\mathrm{T}-1}{\sum}_{t=1}^T{\left({E}_t-\overline{E}\right)}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\frac{1}{T}{\sum}_{t=1}^T{E}_t} $$

where E_t is the earnings at time t(t = 1,…,5), and Ē is the average earnings over the last 5 years. VERN is the coefficient of variation of earnings.

• (ii) VFLO: Variability in cash flows

  This measure is computed as the coefficient of variation of cash flow using data over the last 5 years – i.e., it is computed in an identical manner to VERN, with cash flow being used in place of earnings. Cash flow is computed as earnings plus depreciation plus deferred taxes.

• (iii) EXTE: Extraordinary items in earnings

  This is computed as follows:

$$ \mathrm{E}\mathrm{XTE}=\frac{\frac{1}{T}{\sum}_{t=1}^T\mid \mathrm{E}{\mathrm{X}}_t+\mathrm{NR}{\mathrm{I}}_t\mid }{\frac{1}{T}{\sum}_{t=1}^T{E}_t} $$

where EX_t is the value of extraordinary items and discontinued operations, NRI_t is the value of nonoperating income, and E_t is the earnings available to common before extraordinary items. The descriptor uses data over the last 5 years.

• (iv) SPIBS: Standard deviation of analysts’ prediction to price

  This is computed as the weighted average of the standard deviation of IBES analysts’ forecasts of the firm’s earnings per share for the current fiscal year and next fiscal year divided by the most recent price.

10. Leverage

• (i) MLEV: Market leverage

  This measure is computed as follows:

$$ \mathrm{MLEV}=\frac{\mathrm{M}{\mathrm{E}}_t+\mathrm{P}{\mathrm{E}}_t+\mathrm{L}{\mathrm{D}}_t}{\mathrm{M}{\mathrm{E}}_t} $$

where ME_t is the market value of common equity, PE_t is the book value of preferred equity, and LD_t is the book value of long-term debt. The value of preferred equity and long-term debt are as of the end of the most recent fiscal year. The market value of equity is computed using the most recent month’s closing price of the stock.

• (ii) BLEV: Book leverage

  This measure is computed as follows:

$$ \mathrm{BLEV}=\frac{\mathrm{CE}{\mathrm{Q}}_t+\mathrm{P}{\mathrm{E}}_t+\mathrm{L}{\mathrm{D}}_t}{\mathrm{CE}{\mathrm{Q}}_t} $$

where CEQ_t is the book value of common equity, PE_t is the book value of preferred equity, and LD_t is the book value of the long-term debt. All values are as of the end of the most recent fiscal year.

• (iii) DTOA: Debt-to-assets ratio

  This ratio is computed as follows:

$$ \mathrm{DTOA}=\frac{\mathrm{L}{\mathrm{D}}_t+\mathrm{DC}{\mathrm{L}}_t}{\mathrm{T}{\mathrm{A}}_t} $$

where LD_t is the book value of long-term debt, DCL_t is the value of debt in current liabilities, and TA_t is the book value of total assets. All values are as of the end of the most recent fiscal year.

• (iv) SNRRT: Senior debt rating

  This descriptor is constructed as a multi-level indicator variable of the debt rating of a company.

11. Currency Sensitivity

• (i) CURSENS: Exposure to foreign currencies

  To construct this descriptor, the following regression is run:

$$ {\mathrm{r}}_{\mathrm{i}\mathrm{t}}={\upalpha}_{\mathrm{I}}+{\upbeta}_{\mathrm{i}}{\mathrm{r}}_{\mathrm{mt}}+{\upvarepsilon}_{\mathrm{i}\mathrm{t}} $$

where r_it is the excess return on the stock and r_mt is the excess return on the S&P 500 Index. Let ε_it denote the residual returns from this regression. These residual returns are in turn regressed against the contemporaneous and lagged returns on a basket of foreign currencies, as follows:

$$ {\upvarepsilon}_{\mathrm{i}\mathrm{t}}={\mathrm{c}}_{\mathrm{i}}+{\upgamma}_{\mathrm{i}1}{\left(\mathrm{FX}\right)}_{\mathrm{t}}+{\upgamma}_{\mathrm{i}2}{\left(\mathrm{FX}\right)}_{\mathrm{t}\hbox{-} 1}+{\upgamma}_{13}{\left(\mathrm{FX}\right)}_{\mathrm{t}\hbox{-} 2}+{\upmu}_{\mathrm{i}\mathrm{t}} $$

where ε_it is the residual return on stock I, (FX)_t is the return on an index of foreign currencies over month t, (FX)_t-1 is the return on the same index of foreign currencies over month t-1, and (FX)_t-2 is the return on the same index over month t-2. The risk index is computed as the sum of the slope coefficients γ_i1, γ_i2, and γ_i3 – i.e., CURSENS = γ_i1 + γ_i2 + γ_i3.

12. Dividend Yield

• (i) P_DYLD: Predicted dividend yield

  This descriptor uses the last four quarterly dividends paid out by the company along with the returns on the company’s stock and future dividend announcements made by the company to come up with a Barra-predicted dividend yield.

13. Non-estimation Universe Indicator

• (i) NONESTU: Indicator for firms outside US-E3 estimation universe

  This is a 0–1 indicator variable: It is equal to 0 if the company is in the BARRA estimation universe and equal to 1 if the company is outside the BARRA estimation universe.

Appendix 2 Barra US-E4, GEM4 Earnings Yields and Momentum Strategies

See Figs. 1, 2, 3, and 4

Fig. 1

USA pure earnings yield factor portfolios. Cumulative performance is reported for (a) market-capitalization regression weights, (b) root-cap regression weights, and (c) equal regression weights

Fig. 2

International pure earnings yield factor portfolios. Cumulative performance is reported for (a) market-capitalization regression weights, (b) root-cap regression weights, and (c) equal regression weights

Fig. 3

USA pure momentum factor portfolios. Cumulative performance is reported for (a) market-capitalization regression weights, (b) root-cap regression weights, and (c) equal regression weights

Fig. 4

International pure momentum factor portfolios. Cumulative performance is reported for (a) market-capitalization regression weights, (b) root-cap regression weights, and (c) equal regression weights