Question:

The trustees for a pension fund need to allocate $100 million among two active managers and the benchmark. They want to maximize the information ratio subject to an overall tracking error volatility (TEV) of 3%. Manager #1 has a TEV of 5% with an information ratio (IR) of 0.5; Manager #2 has a TEV of 4% with an IR of 0.4. Optimization shows that the portfolio IR should be 0.64.

(i) With 99% confidence, what is the risk budget?
(ii) What is the TEV and IR of the benchmark index?
(iii) What are the managers’ implied expected returns?
(iv) What is the optimal allocation?

Answer:

Please see solution spreadsheet here.

(i)
Risk budget is $6.98 million as
$6.98 million = ($100 million)*(3% TEV)*(2.326 @ 99% confidence)

(ii)
The benchmark, by definition, has no tracking error (TEV = 0). As it has no excess return, it has an information ratio of zero (or N/A if you lik, but allocation to the index is contributes zero to the portfolio TEV and portfolio IR)

(iii)
IR = alpha/TEV, therefore
alpha = IR*TEV

For Manager #1, alpha = (5% TEV)*(0.50 IR) = 2.5% alpha
For Manager #2, alpha = (4% TEV)*(0.40 IR) = 1.6% alpha

But on a weighted basis (see spreadsheet),
For Manager #1, alpha = (5% TEV)*(0.50 IR) * 46.9% weight = 1.17% alpha
For Manager #2, alpha = (4% TEV)*(0.40 IR) * 46.9% weight = 0.75% alpha

(iv)
The key idea is, as Jorion says, “the relative risk budgets should be proportional to the information ratios”

Therefore, the allocation to manager #1 =
(Manager #1 IR/portfolio IR)*(Portfolio TEV/Manager #1 TEV) = (0.5/0.64)*(3%/5%) = 46.9%

Similarly, the allocation to manager #2 =
(Manager #2 IR/portfolio IR)*(Portfolio TEV/Manager #2 TEV) = (0.4/0.64)*(3%/4%) = 46.9%

Note:
To get from “relative risk budgets should be proportional IRs,” we start with proportionality:
(allocation)*(Manager TEV/Portfolio TEV) = (Manager IR/Portfolio IR),

and solve for the allocation:
allocation = (Manager IR/Portfolio IR) * (Portfolio TEV/Manager TEV)