Analysis of Active Portfolio Management - Intuitive feel for the formulas

[Update: I think I managed to comprehend the formulas better, so I added my explanations for them below, please correct me if I’m wrong]

I have gone through material and questions on chapter 51, “Analysis of Active Portfolio Management”. While I’m able to use the formulas when answering questions, I lack an intuitive grasp of the formulas that I listed below. Formulas like CAPM or WACC makes intuitive sense to me after I see them, so I don’t have to memorize them, but unfortunately I keep having problems remembering these 3 formulas without memorizing them.

I wonder if you would be able help me to understand them?

  1. Sharpe RatioPortfolio2 = Sharpe RatioBenchmark2 + (Transfer Coefficient x Information Ratio)2 => Starting with TC = 1 => [(rP - rF) / σP-F]2 = [(rB - rF) / σB-F]2 + [(rP - rB) / σP-B]2 => The Sharpe Ratio of the portfolio is the return after risk free rate over the portfolio risk. This can be de-constructed into two parts, the excessive return of the benchmark over the risk free (or the Sharpe Ratio of the benchmark) plus the excessive return of the portfolio over the benchmark (or the Information Ratio, which can be seen as a “Sharpe Ratio over the benchmark”). They are combined as squares because of Pythagorean’s Theorem, where two sides of a triangle with a right angle is equal to the third side, squared. The Transfer Coefficient, or the restrictions on applying the portfolio weights that you want, will cause a linear decrease in the IR (portfolio return over the benchmark). This combined result, TC x IR, is then squared together.
  2. Information Ratio = Transfer Coefficient x Information Coefficient x √(Breadth) => The higher the IC (the skill level of the investor), the higher the IR (the return over the benchmark). If you can apply that skill more times (the Breadth), you will get higher IR. However, there tends to be a trade-off between IC and Breadth, so if you make many decisions, you can’t put the same effort into each decision and your skill will be lower. It’s also difficult to get a high breadth, because the decisions will need to be independent. Not being able to apply your skills or your decisions into actual portfolio positions (a TC < 1) will also decrease the IR. If you divide both sides by σA, this formula turns into the “Full fundamental law of active management”: E(RA) = Transfer Coefficient x Information Coefficient x √(Breadth) x σA
  3. σA = (Transfer Coefficient x Information Ratio / Sharpe RatioBenchmark) x σB => Starting with TC = 1 => σA = (Information Ratio / Sharpe RatioBenchmark) x σB => As per 1., the IR can be seen as the “Sharpe Ratio over the benchmark”. If the IR for example is twice that of the Sharpe Ratio (Information Ratio / Sharpe RatioBenchmark = 2), it makes sense to allocate twice as much active risk as benchmark risk, since it gives twice the gains. If IR = Sharpe Ratio (Information Ratio / Sharpe RatioBenchmark = 1), it makes sense to have the same active risk as benchmark risk. The formula can be thought of as: Active risk = How much we should scale up the benchmark risk x Benchmark risk If we are not able to translate all of our potential active gains into actual benchmark weightings (if TC < 1) you shouldn’t allocate as much active risk, so σA will be lower.