# LO 64.3: Explain the impact of benchmark choice on alpha, and describe

LO 64.3: Explain the impact of benchmark choice on alpha, and describe characteristics of an effective benchmark to measure alpha.
The choice of benchmark has a significant impact on the calculated alpha for an investment. Strictly benchmarking to an identifiable index, like the S&P 500 Index, assumes that an asset has a beta of 1.0. What if the true beta is some value other than 1.0? Consider an investment that has a beta of 0.73 and tracking error of 6.16%. The alpha for this investment could be estimated by regressing the excess return of the fund (Rt Rp) against the excess return of the benchm ark(R ^00 R p ) . In the following regression equation, you see a calculated alpha of 3.44% and a placeholder for error term (e ) because we never know, in advance, how an individual observation will deviate from our model at any point in time.
Rt RF = 0.0344 + 0.73(RtP500 – RF) + et
We can rearrange this formula to isolate only the expected return on our investment. Doing so, we find that our customized benchmark should actually be invested 27% in the risk-free rate and 73% in the S&P 500 Index. Using a benchmark that recognizes the investments beta as 0.73, we calculate an alpha of 3.44%, which translates into an IR of 0.5584 (= 0.0344/ 0.0616).
Rt = 0.0344 + 0.27(RF) + 0.73(Rt P50) + et
If this same investor were to wrongly regress their investment against only the S&P 500 Index, then they would calculate an alpha of 1.50%, which is incorrect because it assumes a beta of 1.0 when the actual beta is 0.73.
Rt = 0.015+ R tP500 + et
Using the wrong benchmark would produce an IR of 0.2435 (= 0.0150 / 0.0616). This suggests that using an incorrect benchmark will understate both the expected alpha and the IR. Inaccurate information may cause an investor to pass on an investment that they otherwise should have accepted.
This illustration leads an investor to wonder: what is the best way to choose a benchmark? An appropriate benchmark can be selected by applying a few different complementary standards. First, the benchmark should be well-defined. It should be hosted by an independent index provider, which makes it both verifiable and free of ambiguity. The S&P 500 Index and the Russell 1000 Index are both examples of well-defined large-cap indices. Second, an index should be tradeable. If the benchmark is not a basket of tradeable securities that could be directly invested in as an alternative, then the benchmark is not a very good comparison. Third, a benchmark must be replicable. This is closely related to the tradability standard. There are some benchmarks, like absolute return benchmarks, that are
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Topic 64 Cross Reference to GARP Assigned Reading – Ang, Chapter 10
not feasible for an investor to replicate. If it cannot be replicated, then the tracking error will be very high. Fourth, the benchmark must be adjusted for risk. In the previous example, you can see that the alpha and the IR will be calculated too low if the risk level of the benchmark is too high for the investment in question.
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