###### Valuing Embedded Options

According to the arbitrage-free framework, the value of a bond with embedded options... **Read More**

**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:

- \(R_A\) is the value-added.
- \(R_P\) is the investorâ€™s return.
- \(R_B\) is the benchmark portfolio return.

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:

- \(w_{b,i}\) is the benchmark weight of security \(i\).
- \(R_i\) is the return on security \(i\).

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:

- \(w_{p,i}\) is the portfolio weight of security \(i\).

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:

- \(\Delta w_i \text{(active weight)} = w_{p,i}-w_{b,i}\).

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:

- \(\Delta w_i\) is the active weight = \(w_{p,i}-w_{b,i}\).

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,

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

## Question

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:

- -0.5%.
- 0.0%.
- 0.75%.
## Solution

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 45: Analysis of Active Portfolio Management

*LOS 45 (a) Describe how value added by active management is measured.*