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Essential Concept 99: The Full Fundamental Law

According to the full fundamental law, the expected active return is expressed as:

\mathrm{E}\left({\mathrm{R}}_{\mathrm{A}}\right)\mathrm{=\ }\left(\mathrm{TC}\right)\left(\mathrm{IC}\right)\sqrt{\mathrm{BR}}\mathrm{\ }{\mathrm{\sigma}}_{\mathrm{A}}

Information ratio is expressed as:

\mathrm{IR\ =\ }\left(\mathrm{TC}\right)\left(\mathrm{IC}\right)\sqrt{\mathrm{BR}}.

The transfer coefficient is calculated as:

\mathrm{TC\ =\ COR}\left({\mathrm{\mu}}_{\mathrm{i}}/{\mathrm{\sigma }}_{\mathrm{i}}\ ,\ \mathrm{\Delta }{\mathrm{w}}_{\mathrm{i}}{\mathrm{\sigma }}_{\mathrm{i}}\right)

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.

\noindent \mathrm{E}{\left({\mathrm{R}}_{\mathrm{A}}\right)}^{\mathrm{*}}\mathrm{=\ }\left(\mathrm{IC}\right)\sqrt{\mathrm{BR}}\mathrm{\ }{\mathrm{\sigma }}_{\mathrm{A}}$ and IR = $\left(\mathrm{IC}\right)\sqrt{\mathrm{BR}}

The optimal amount of active risk in an actively managed portfolio with constraints is expressed as:

\mathrm{\sigma}}_{\mathrm{A}}\mathrm{=\ TC*}\frac{\mathrm{I}{\mathrm{R}}^{\mathrm{*}}}{\mathrm{S}{\mathrm{R}}_{\mathrm{B}}}\mathrm{*}{\mathrm{\sigma }}_{\mathrm{B}}

The maximum value of the constrained portfolio’s Sharpe ratio is given as:

\mathrm{SR}}^{\mathrm{2}}_{\mathrm{P}}\mathrm{=\ }{\mathrm{SR}}^{\mathrm{2}}_{\mathrm{B}}\mathrm{+}{\left(\mathrm{TC}\right)}^{\mathrm{2}}{\left({\mathrm{IR}}^{\mathrm{*}}\right)}^{\mathrm{2}}


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?


\mathrm {Optimal\ amount\ of\ active\ risk\ }${\mathrm{\sigma}}_{\mathrm{A}}\mathrm{=\ TC*}\frac{\mathrm{I}{\mathrm{R}}^{\mathrm{*}}}{\mathrm{S}{\mathrm{R}}_{\mathrm{B}}}\mathrm{*}{\mathrm{\sigma}}_{\mathrm{B}}
\mathrm{\sigma }}_{\mathrm{A}}=0.5\ *\frac{0.3}{0.4}*16\% = 6\%


If the actively managed portfolio is constructed with this amount of active risk, what is the Sharpe ratio?


\mathrm{SR}}^{\mathrm{2}}_{\mathrm{P}}\mathrm{=\ }{\mathrm{SR}}^{\mathrm{2}}_{\mathrm{B}}\mathrm{+}{\left(\mathrm{TC}\right)}^{\mathrm{2}}{\left({\mathrm{IR}}^{\mathrm{*}}\right)}^{\mathrm{2}}


\mathrm{SR}}^{\mathrm{2}}_{\mathrm{P}}={0.4}^2+{0.5}^2*{0.3}^2 = 0.1825


\mathrm{SR}}_{\mathrm{P}}\mathrm{=\ 0.43}

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 \frac{6.0\%}{8.0\%}=\ 75\%. 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.

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