What's new

Nicole Seaman

Director of FRM Operations
Staff member
Learning objectives: Describe Grinold’s fundamental law of active management, including its assumptions and limitations, and calculate the information ratio using this law. Apply a factor regression to construct a benchmark with multiple factors, measure a portfolio’s sensitivity to those factors and measure alpha against that benchmark.


705.1. Below are displayed the actual results for two of Andrew Ang's regressions of Warren Buffett's Berkshire Hathaway (from 1990 to 2012). On the left is his regression of Berkshire's monthly excess returns against the capital asset pricing model (CAPM) benchmark. On the right is his regression of Berkshire's monthly excess returns against the three-factor (MKT, SMB and HML) Fama-French benchmark:

Please note these are excess returns, R(p) - Rf. About these actual regression results, each of the following statements is true EXCEPT which is false?

a. The -0.50 SMB factor loading and +0.380 HML factor loading imply the Berkshire Hathaway is a large-cap, value investor
b. Berkshire generates significant alpha with at least 90.0% confidence, and the addition of the size (SMB) and value (HML) does IMPROVE the fit of the regression model
c. The statistically insignificant alphas, low coefficients of determination (adjusted R^2), and increase in market beta (from 0.510 to 0.670) when SMB and HML factors are added suggest that Berkshire did not generate sustained alpha
d. The benchmark implied by the Fama–French regression estimates is given by: $0.33 in T-bills + $0.67 in the market portfolio + (long $0.50 in large caps - short $0.50 in small caps) + ($0.38 in value stocks - $0.38 in growth stocks)

705.2. Below are Andrew Ang's actual regression results for the annual gross returns of the CalPERS pension fund against a passive portfolio of index funds in stocks and bonds. Please note the returns are gross returns, not excess returns; i.e., they are NOT net of the riskfree rate.

Which of the following statements about these regression results is TRUE?

a. The high adjusted R^2 validates a hypothesis that the CalPERS active fund managers do add value relative to the benchmark
b. Because the factor loadings are not statistically significant, different factors should be repeatedly tested until a set is found that is significant
c. This regression should be re-run with at least one additional factor because robust benchmark portfolios must include at least one risk-free asset
d. It is acceptable to exclude risk-free assets and regress gross returns against the benchmark portfolio conditional on a constraint that the factor loadings sum to one; in this case, we need β(B) + β(S) = 1.0

705.3. Andrew Ang tells us that a portfolio manager creates alpha relative to a benchmark by making bets that deviate from that benchmark. The more successful these bets, the higher the manager's alpha. Grinold’s Fundamental Law of Active Management formalizes this intuition by asserting that the maximum attainable information ratio is given by IR ≈ IC * sqrt(BR) where IR is the information ratio, IC is the information coefficient (the correlation of the manager’s forecast with the actual returns) and BR is the breadth of the strategy. Breadth is the number of securities that can be traded and how frequently they can be traded.

According to Ang, each of the following statements about the Fundamental Law is true EXCEPT which is false?

a. The empirical evidence suggests that, on average, IC tends to fall as BR increases
b. A compelling advantage of the fundamental law is that it successfully incorporates downside risk and higher moment risk; specifically, it adjusts for skew and excess kurtosis
c. If we require an IR of 0.50, this can be achieved either by a highly skilled stock timer with an IC of 0.25 making only four bets a year; or the same IR can be achieved by a manager with only a slight edge, IC of 0.025, who makes fully 400 bets a year
d. A crucial assumption is that the forecasts are independent of each other. But due to realistically correlated factor bets, it is difficult to make truly independent forecasts in BR; e.g., an equity manager with overweight positions on 1,000 value stocks offset by underweight positions in 1,000 growth stocks has not placed 1,000 different bets

Answers here:
Last edited by a moderator:

David Harper CFA FRM

David Harper CFA FRM
Staff member
As I noted in the answer to 705.1, Ang makes an point about the CAPM and Fama-French (factor) benchmark portfolios that I don't think we've seen elsewhere so elegantly in the FRM: "the factor loadings can be translated directly to a benchmark portfolio, only now the portfolio contains (complicated) long–short positions in small/large and value/growth stocks. But it still represents $1.00 of capital allocated between factor portfolios. Every time we run a factor regression, we are assuming that we can create a factor benchmark portfolio." Specifically,
  • In the case of the CAPM benchmark (which just refers to a single-factor benchmark where the benchmark is the equity risk premium) because E[R(p)] = Rf + β[r(m) - Rf] = Rf + β*r(m) - β*Rf = β*r(m) +(1- β)*Rf, the CAPM benchmark portfolio allocates $1.00 to a long position of (β) dollars in the market plus a long position of (1-β) in risk-free Treasury bills. Note that if βM<1, this implies long both the market and the risk-free asset; if β>1, this implies a short position in the risk-free asset (leveraged). In either case, the benchmark is located somewhere on the SML line (not the CML, right? ;))
  • Then the Fama-French three-factor benchmark analogously allocates the MKT factor loading to the market and (1-MKT) to the risk-free asset, which totals $1.00. But it further is long/short the SMB and HML factors, which is net neutral these two factors; for example if SMB = +0.20, then the benchmark is long +$0.20 is small caps and -$0.20 in large caps.

David Harper CFA FRM

David Harper CFA FRM
Staff member
Also, because the new Andrew Ang text includes the fourth (at least) occurrence of alpha in the 2017 FRM syllabus (and the third occurrence, I think, of tracking error), I wanted to mention that the FRM has now achieved a consistent definition of alpha (see here for some history and my feedback https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ ). Ang appears to have been influenced by Grinold, who remains assigned in P2.T8 (the Portfolio Construction chapter which is almost impossible to comprehend in isolation). The bottom line is that, strictly speaking, alpha (α) is the regression intercept and the information ratio is alpha divided by the volatility of alpha, α/σ(α). We can also refer to this as residual return divided by residual risk. The key criteria is ratio-consistency: the denominator, as the risk, ought to be the standard deviation of the numerator, as the return; information ratio is in the class of many different measures of return per unit of risk (risk-adjusted return, or RAPMs).

In this way, I notice an answer to GARP's 2017 Part 2 practice paper (Question 65) includes "Jensen’s alpha measure is calculated as: Alpha = Actual return – Expected return based on systematic risk = Actual return – (risk-free rate + (Market return – risk-free rate)*Beta)" which of course is totally consistent. Jensen's alpha is an alpha (regression intercept or residual return) but it is the special case where we rely on the single-factor CAPM benchmark. Much of Ang is about multi-factor bemchmarks, such that alpha remains the residual return after we subtract the components of return that are explained by the correlation (i.e., beta or factor loadings) to the factor(s). (Over the years, I've noticed understandable confusion about alpha-as-intercept as compared to how we often learn it as the error--i.e., the vertical distance from the SML or regression line. But these are the same. In terms of excess return, the SML goes through the origin, so the intercept is also the vertical distance from the appropriate location on the regression line.)

There remains a terminology ambiguity with respect to tracking error, but not really ;). If I just use CAPM to illustrate the two Grinold terms of "active return [risk]" and "residual return [risk]:"
  • Per CAPM, (Jensen's alpha), α = R(p) - Rf - β*[R(m) - Rf]. Keep in mind this is the simplest single factor version. If we let R_e(p) be the portfolio's excess return (in excess of the riskfree rate), and the equity risk premium is factor, F(1), this is just R_e(p) = β(1)*F(1) ... and Ang extends this into several factors R_e(p) = β(1)*F(1) + β(2)*F(2) + ... + β(n)*F(n). Further, this can generalize into matrix form, such that we have a portfolio of several positions (rows in the matrix) and several factors (columns in the matrix).
  • If we do not want to account for factor correlations (e.g. systematic risk, small cap premium, value premium), we can set beta to one, in which case this alpha generalizes to α = R(p) - Rf - β*[R(m) - Rf] = R(p) - Rf - 1.0*[R(m) - Rf] = R(p) - Rf - R(m) + Rf = R(p) - R(m); i.e., just the return difference between the portfolio and the benchmark (the riskfree rate nets out). This is what Grinold calls the active return (versus the residual return) and its standard deviation is the active risk. We do see some authors define tracking error as σ(active return); the simpler volatility of the excess returns, where we mean "in excess of the bechmark" without factor loadings. In fact, in my opinion, this is the more common definition of tracking error.
  • But, as noted in the link above, any many other places in this forum (with its citations), we can now safely say that GARP understands there can be two versions of the information ratio, as long as we are ratio consistent. But I hope this note helps to clarify how we can view the active/active version as a special case of the ultimately correct residual/residual version.
Finally, @emilioalzamora1 has a deeper dive into the information ratio which is really helpful here https://www.bionicturtle.com/forum/...e-at-risk-tracking-error-var.3634/#post-48127
Last edited: