The post-event alpha ends in 08/2014 and rests on two fund families. The paper's confidence band deserves a place in due diligence; the outperformance claim needs a narrower reading.
Removing the covariance matrix
Beaulieu, Dufour, Khalaf and Melin start with a multivariate regression of n fund returns on a constant and factors. Alpha occupies the intercept row; betas occupy the slope rows. Joint tests must account for an error covariance matrix with 55 free parameters at n = 10 and 325 at n = 25. On short monthly-style samples, asymptotic approximations struggle with that burden.
The authors use the roots of a single determinantal equation constructed from restricted and unrestricted residuals. LR, Wald and related statistics are functions of these roots. They show that the roots are independent of the covariance matrix under any of three conditions: a restriction imposed on every equation, a block-triangular error structure, or normal-mixture errors, which include multivariate Student-t. Simulated critical values are exact in each case.
The paper adds two results. With a factor Z omitted, least squares estimates Θ = B + B_XZ B_Z. This equals the true loading plus the missing factor's projection on the included factors, multiplied by its loading. The authors describe Θ as ex-post because it reflects the sample co-movement of X and Z. Under Gaussian noise, the usual tests are conservative for Θ. Earlier work by the authors on incomplete regressions covered Hotelling-type tests alone; this paper extends the result to statistics formed from symmetric functions of the roots.
Inverting the Hotelling test for one regressor across all funds and then projecting gives intervals in the single-equation t form. For Gaussian errors, the Scheffé-type critical value is (n/τ_n)F_α(n, τ_n), with τ_n = T − k − n + 1.
The application uses seven Cat bond mutual fund (CBMF) share classes, with returns measured in excess of the one-month T-bill. It compares CAPM, quadratic CAPM (QCAPM) with a coskewness mimicking portfolio, and Fama-French three-factor. The 2011 Tohoku earthquake divides the sample. Before 04/2011, all alpha bands include zero, with endpoints spanning roughly -0.0063 to 0.0057. In the later period, CAPM Gaussian bands exclude zero for six of seven funds. Every beta band includes zero in both periods.
How much evidence do seven share classes add?
Four classes belong to Falcon Cat Bond Fund: CHF A, EUR A, USD A, CHF I. The remaining three are LGT CH Cat Bond Fund classes in CHF, EUR and USD. Currency and fee wrappers account for the seven observations on two portfolios.
For the four Falcon classes, post-2011 CAPM Gaussian alpha bands are (0.00011, 0.00480), (0.00092, 0.00528), (0.00103, 0.00545), (0.00056, 0.00523). The bands look almost interchangeable. All four lie between 0.00011 and 0.00545. These seven near-copies incur the full seven-equation critical value because (n/τ_n)F_α(n, τ_n) takes no account of fund correlation. They supply roughly two funds' worth of independent evidence. We did not find a discussion of the relationship between the CHF and EUR classes and a USD T-bill benchmark.
The scope of the 5% guarantee
The 5% coverage holds jointly across funds for one regressor at a time. It does not extend jointly across regressors, models or the two sub-periods.
Fat tails change little. Fund 1's pre-2011 CAPM market beta has a Gaussian band of (-0.06210, 0.13548). Across 2 ≤ κ ≤ 50, the widest Student-t band is (-0.07034, 0.14371), extending about 0.008 further at either end. The authors report qualitatively similar Cauchy results without showing them.
The theorem carries a qualification. Its finite-sample bound for missing factors is proved with Gaussian noise; a later section supplies an asymptotic extension. The Student-t panels therefore remain exact for the complete model if the true κ is between 2 and 50. For the incomplete model, they depend on large-T arguments.
The missing risk remains in alpha
The authors acknowledge that pseudo-true values may only partly capture the economic content of the original parameters. They answer that the literature increasingly incorporates misspecification into models' economic foundations, giving pseudo-true values, including ex-post values, a usable interpretation. Their application concludes that "Conservative tests imply that rejections are compelling", rejecting zero and negative ex-post alpha as identified. The statistical claim holds.
But the authors also acknowledge that their parametrization places the missing risk factor inside the coefficients being interpreted. In trading terms, the intercept of Θ contains true alpha plus the omitted factor's unexplained sample mean multiplied by the funds' loading on it. Suppose equity factors fail to span an insurance-risk premium. Its earnings from 2011 to 2014 enter the alpha column.
The authors take no position on the mechanism and attribute the alpha to the event's unprecedented character. They nevertheless write that their findings "support outperformance, despite the possibility of missing factors." The rejection establishes a positive ex-post intercept. Reading outperformance into it requires setting aside the missing risk the authors acknowledge it contains.
Reading the post-tsunami results closely
Among the six funds whose CAPM Gaussian bands exclude zero, lower bounds range from 0.00011 to 0.00103. That amounts to about 1 to 10 basis points per period. Across all tables, the highest upper bound is 0.00589. Post-2011 CAPM Gaussian alpha bands have widths of 0.003 to 0.0047.
Fund 1's market beta band contracts from about 0.20 wide before 2011 to 0.13 afterward, at (-0.05991, 0.06988). The LGT classes have bands about 0.09 wide. The text places market betas across both periods between -0.10 and 0.20, although fund 4 reaches -0.10200 in the QCAPM Student-t panel. For coskewness, the stated low is -0.23; fund 3 reaches -0.24295. The text places SMB between -0.25 and 0.23. It reports HML's low as -0.016, against -0.162 in the table.
The prose also overstates the count of positive bands. It says all but one fund have positive bands across all models. Fund 1's lower bound in the QCAPM Student-t panel is -0.000001, leaving two of seven bands covering zero. Its Gaussian QCAPM lower bound is 0.00001.
Even the endpoints disagree.
The second sub-period runs from 04/2011-12/2015 in the text and 04/2011-08/2014 in the tables. The event determined the split date. We did not find a break test or a placebo split. Katrina in 2005 and the 2008 crisis fall in the pre-period, which shows no detectable alpha.
After Tohoku, the Muteki bond lost its entire USD 300 million principal. A before-and-after comparison cannot distinguish that shock from market-wide repricing of Cat risk following a total loss. Among four possible channels, the authors name liquidity and sponsor reputation, then acknowledge uncertainty over whether any would generate consistently positive alpha. Their explanation distinguishes unfamiliar shocks from foreseeable ones. The bands cannot test that interpretation.
Why we did not rerun the bands
We lack Cat bond mutual fund NAV histories. Our fund data cover US-listed ETFs, so applying the procedure to them would test a different universe.
For bounding one coefficient jointly across many funds in short, fat-tailed samples, we regard this as a sound finite-sample option. A break date set in advance, tested on more than two fund families with lower bounds clearly above zero, would turn the post-Tohoku alpha into a finding.