利用Ledoit-Wolf方法测度对冲基金的超额回报比率
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本文档主要探讨了对冲基金同行绩效比率（Peer Performance Ratios）的测量与评估方法，特别关注的是夏普比率（Sharpe Ratio），这是一项在投资行业中广泛使用的绝对风险调整性能指标。夏普比率由William F. Sharpe在1992年提出，它衡量的是每承受一单位总风险所获得的超额回报，即风险调整后的收益。
Ledoit和Wolf在2002年提出的方法被用于该研究中，用来测试两个基金间的夏普比率差异。作者David Ardia、Kris Boudt等人从统计学角度出发，定义了基金的超额表现（outperformance）或不足表现（underperformance）：即与同行基金相比，如果一个基金的真实业绩更好，则视为超额表现；反之则为不足表现。他们强调，通过比较基金间的绩效对等检验的p值，可以得到关于超额表现和不足表现比率的估计，这些比率对于假阳性发现（即错误地识别出显著的alpha差异）有较强的抗干扰性。
当应用到对冲基金领域时，论文发现，根据基金的超额表现排名，可以构建出一个高排名的前五分位组合。这个组合不仅具有较高的绝对收益，而且其风险调整后的业绩也优于仅仅基于估计alpha的排名。这意味着，通过这种方法，投资者能够更有效地筛选和选择那些在控制风险的前提下，实际表现优秀的对冲基金。
作者的研究结果对于投资者和基金经理来说具有重要的实践意义，因为它提供了一种更为稳健和全面的方式来评价基金的表现，有助于做出更明智的投资决策。同时，本文也展示了如何运用现代金融计量方法来衡量和解释投资产品的相对优势，对于理解和优化投资组合管理策略具有深远的影响。
Electronic copy available at: https://ssrn.com/abstract=2000901
pairs for which the null hypothesis is correct) and p–values that are close to zero (when the null hypothesis
is false). This key insight is provided by Storey (2002) and it is used by Barras et al. (2010) to estimate the
proportion of funds that perform equally well in the same manner as a passive investor in style indices.
7
The next step is then to attribute the remaining segment of the peer group to funds that significantly
underperform and those that outperform. Thus, for each fund, we obtain an equal–performance ratio, an
outperformance ratio, and an underperformance ratio. The proposed equal–performance ratio is robust to
false discoveries, and unless the fund performance differs significantly from its peers, it will tend toward
100%. When the return data are sufficiently informative about differences in performance, the outperfor-
mance ratio is suitable for classifying funds into top–performing funds. The estimators rely on pairwise
tests to calculate the p–values, thus they can use the longest common time series span for each pair in an op-
timal manner, and parallel computing can be employed to calculate these p–values in a numerically efficient
way.
2.4. The equal–performance ratio
A crucial feature of the proposed peer performance ratios is that they exploit the difference in the
distribution of the p–values bp
i−j
when ∆
i−j
= 0 versus ∆
i−j
6= 0.
If the test is sufficiently powerful, a threshold value λ
i
exists such that almost surely the p–value of the
two–sided equal–performance test is less than λ
i
if the two funds have different performances:
(A1) : P[bp
i−j
< λ
i
| ∆
i−j
6= 0] = 1 . (7)
The validity of this assumption depends on the magnitude of ∆
i−j
, the test statistic itself, the calculation of
its p–value (e.g., asymptotic versus bootstrap methods), and the sample size.
In the case of equal performance, and provided that the estimated
b
F
i−j
coincides with the true F
i−j
used to calculate the p–value, the p–value is uniformly distributed for a given pair (i, j). This implies that
the probability of bp
i−j
exceeding λ
i
when ∆
i−j
= 0 is 1 − λ
i
:
(A2) : P[bp
i−j
≥ λ
i
| ∆
i−j
= 0] = 1 − λ
i
. (8)
In practice, the cumulative distribution function F
i−j
is not fully known and the calculation of the p–values
requires parameter estimates. Asymptotically, the p–value is uniformly distributed if consistent estimators
are used (Rosenblatt, 1952), whereas in finite samples, assumption (A2) is only approximately satisfied.
A key result related to the definition of the proposed equal–performance ratio is that under (A1) and
(A2), the expected number of p–values exceeding λ
i
is (1 − λ
i
)n
0
i
, with n
0
i
the number of peer funds that
perform equally well as the focal fund:
E
X
j6=i
I{bp
i−j
≥ λ
i
}
= E
X
j6=i
∆
i−j
=0
I{bp
i−j
≥ λ
i
}
| {z }
= (1−λ
i
) n
0
i
+ E
X
j6=i
∆
i−j
6=0
I{bp
i−j
≥ λ
i
}
| {z }
= 0
= (1 − λ
i
) n
0
i
.
7
Instead of estimating the market–wide equal–performance ratio, as in Barras et al. (2010), we estimate the individual equal–
performance ratio for each fund, which broadens the application scope. The aggregate equal–performance ratio allows us to answer
general economic questions such as the usefulness of actively managed funds, but our proposed individual equal–performance ratio
can be used directly by investors to evaluate the performance of a specific fund.
7
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