Value-added, also called active return, is the difference between the managed portfolio return and the benchmark portfolio return. It is calculated using the following equation:

$$ R_A=R_P-R_B $$

Where:

Value added is positive if an investor outperforms the passive benchmark portfolio. On the other hand, it is negative if an investor underperforms the benchmark portfolio.

The return of the benchmark portfolio, \(R_B\), is given by:

$$ R_B=\sum_{i=1}^{n}{w_{b,i}\times R_i} ………1 $$

Where:

On the other hand, \(R_P\), the return on the managed portfolio is given by:

$$ R_P=\sum_{i=1}^{n}{w_{p,i}\times R_i} ………2 $$

Where:

Note that the value added is driven by differences in weights between the managed portfolio and the benchmark portfolio.

Combining equation 1 and 2 gives:

$$ R_A=\sum_{i=1}^{n} \Delta w_iR_i $$

Where:

Value-added can emerge from security selection, asset class allocation, and decompositions into economic sector weightings and geographic weights.

The following table shows the weights and the corresponding portfolio and benchmark returns of four securities of ABC, a hypothetical company.

$$ \begin{array}{c|c|c|c} \textbf{Security} & \textbf{Portfolio weight} & \textbf{Benchmark weight} & \textbf{Return} \\ \hline A & 35\% & 30\% & 16\% \\ \hline B & 30\% & 25\% & 18\% \\ \hline C & 20\% & 25\% & 14\% \\ \hline D & 15\% & 20\% & 10\% \end{array} $$

Calculate the active return (value added) of ABC.

$$ R_A=\sum_{i=1}^{n} \Delta w_iR_i $$

Where:

The following table shows the calculations of value added:

$$ \begin{array}{c|c|c|c|c|c} \textbf{Security} & \textbf{Portfolio} & \textbf{Benchmark} & \textbf{Return} & \bf{ {w}_{{p},{i}}} & \bf{({w}_{{p},{i}}-{w}_{{b},{i}})}\\ {} & \textbf{weight} & \textbf{weight} & {} & \bf{ – {w}_{{b},{i}}} & {\bf{\times {R}_{i}}} \\ \hline A & 35\% & 30\% & 16\% & {5\%}& 0.8\%\\ \hline B & 30\% & 25\% & 18\% & {5\%} & 0.9\% \\ \hline C & 20\% & 25\% & 14\% & {-5\%}& -0.7\% \\ \hline D & 15\% & 20\% & 10\% & {-5\%} & -0.5\% \end{array} $$

Therefore,

$$ R_A=0.8+0.9-0.7-0.5=0.5\% $$

XYZ is a hypothetical pension scheme with investments in various asset classes, as shown in the following table. The expected portfolio returns and the passive benchmark return is shown alongside each asset class.

$$ \begin{array}{c|c|c|c|c} \textbf{Asset} & \textbf{Portfolio} & \textbf{Benchmark} & \textbf{Portfolio} & \textbf{Benchmark} \\ \textbf{Class} & \textbf{weight} & {\textbf{weight } \bf{(\%)}} & \textbf{Return} & \textbf{Return} \\ \hline \text{Quoted} & 20\% & 25\% & 20\% & 18\% \\ \text{equities} & & & & \\ \hline \text{Treasury} & 50\% & 40\% & 15\% & 10\% \\ \text{bonds} & & & & \\ \hline \text{Offshore} & 30\% & 35\% & 10\% & 4\% \\ \text{investments} & & & & \\ \end{array} $$

The ex-ante value added is closest to:

The correct answer is B.

Ex-ante value added is the difference between the expected return of an actively managed portfolio and the expected return of its benchmark:

$$ E(R_A)={E(R}_P)-{(R}_B) $$

Return of the benchmark portfolio, \(R_B\), is given by:

$$ \begin{align*} E({R}_B) &=\sum_{i=1}^{n}{w_{b,i}\times E(R_i)} \\ R_B &=\left(0.25\times0.20\right)+\left(0.4\times0.15\right)+\left(0.35\times0.10\right)=0.145 \end{align*} $$

Return on the managed portfolio is given by:

$$ \begin{align*} {E(R_P)} & ={\sum_{i=1}^{n}{w_{p,i}\times E(R_i)} }\\ &=\left(0.20\times0.20\right)+\left(0.50\times0.15\right)+\left(0.30\times0.10\right)=0.145 \\ R_A & =0.145-0.145=0.0\% \end{align*} $$

Reading 44: Analysis of Active Portfolio Management

LOS 44 (a) Describe how value added by active management is measured.