A new reading has been added to Level 2 Portfolio Management (2016 curriculum) - “Analysis of Active Portfolio Management”. In the section 3.3 Constructing Optimal Portfolios, the reading talks about 3 formulas / concepts which are not well explained.

Optimal portfolio allocation between actively managed and benchmark portfolio would maximize the sum of squared Sharpe Ratio of Benchmark and Information Ratio

SR§^2 = SR(B)^2 + IR^2

For unconstrained portfolios, the level of active risk that leads to the optimal result is

σ(RA)=IR σ(RB) /SR(B)

Total Risk of the actively managed portfolio is the sum of the benchmark return variance and active return variance

Can someone please through some light on these formulas - their derivation or how did we get here.

You are a portfolio manager and you can invest in an index portfolio. This is called passive management. However you can boost returns by taking more systematic risk. You have two sources where you can get extra returns from: by factor exposure or by specific exposure. The first one is when you increase or reduce the weight in a certain industry (retail, construction, transport, etc); and the second one is when you increase or reduce the weight in a certain stock / bond (Cocacola, AT&T, Facebook, etc).

You must take note that you won’t 100% active manage a portfolio, and since you are actively managing a portfolio, you won’t invest 100% in the benchmark portfolio. Hence, you will build up a combined portfolio.

How do you optimize the weights between the benchmark portfolio and the actively managed portfolio? This formulas will help us in the task.

Answering your questions (not in order):

Since you built a combined portfolio, you have 2 variances that conform total risk: benchmark return variance + active return variance.

The ex-ante sharpe ratio of a combined portfolio will be higher than the sharpe ratio of a benchmark portfolio. There should be a derivation, however it is out of the scope of the exam.

Personally speaking, I don’t see any optimization in this formula, there should be a derivation that leads to it and also is out of the scope of the exam.