According to the full fundamental law, the expected active return is expressed as:
Information ratio is expressed as:
The transfer coefficient is calculated as:
It can take values from -1 to +1.
If there are no constraints and the actual portfolio optimal weights are equal to the actual weights, then TC will be equal to 1 and we will have the basic fundamental law.
The optimal amount of active risk in an actively managed portfolio with constraints is expressed as:
The maximum value of the constrained portfolio’s Sharpe ratio is given as:
Example: Consider an actively managed portfolio has a transfer coefficient of 0.50 and an unconstrained information ratio of 0.30. The benchmark portfolio has a Sharpe ratio of 0.40 and risk of 16.0%. What is the optimal amount of aggressiveness in the actively managed portfolio?
If the actively managed portfolio is constructed with this amount of active risk, what is the Sharpe ratio?
If the constrained portfolio has an active risk of 8.0%, how can the active risk be lowered to the optimal level of 6.0%?
The benchmark has a risk of 0% and the constrained portfolio has an active risk of 8.0%. To get an optimal level of 6.05, the weight of the actively managed fund must be . The actively managed portfolio will have an optimal risk of 6.0% if the weight in the benchmark is 25% and 75% in the actively managed fund.