###### Beta Estimation

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The **information ratio** (IR) is the proportion of the active return to the volatility of the active returns, also known as the active risk. It measures a portfolio’s risk-adjusted rate of return.

The information ratio (IR) of an actively managed portfolio is given by:

$$ =\frac{{(R}_P{-R}_B)}{\sigma(R_A)}=\frac{\text{Active return}}{\text{Active Risk}} $$

Where:

- \(R_P\) is the portfolio return.
- \(R_B\) is the benchmark return.
- \(\sigma(R_A)\) is the standard deviation of the active return.

The **forecasted** **active return** is used when calculating the **ex-ante** **information ratio**. That is:

$$ IR=\frac{{E(R}_A)}{\sigma(R_A)} $$

Where

- \(E\left(R_A\right)= {E(R}_P{)-E(R}_B)\) is the expected active return.

On the other hand, the **realized average active returns** and the **realized sample standard deviation** of the active return would be employed to calculate the **ex-post** **information ratio**.

One of the crucial properties of this ratio is that for an unconstrained portfolio, it is unaffected by the aggressiveness of the active weights. This is because both the active return and the active risk increase proportionally.

The Sharpe ratio is the return earned above the risk-free rate per volatility of a portfolio. It aids an investor in understanding the return of a portfolio relative to its risk (volatility):

$$ SR_p=\frac{R_P-R_F}{\sigma(R_P)} $$

Where:

- \(R_P\) is the portfolio return.
- \(R_F\) is the riskless rate of interest.
- \(\sigma(R_P)\) is the volatility of the portfolio return.

$$ SR_p=\frac{{E(R}_P-R_F)}{\sigma(R_P)} $$

Where we use the expected return and the forecasted volatility.

$$ SR_p=\frac{R_P-R_F}{\sigma(R_P)} $$

Where:

- \(R_P\) is the average realized portfolio return.
- \(R_F\) is the average riskless rate of interest.
- \(\sigma(R_P)\) is the sample standard of the portfolio return.

The Sharpe ratio is not affected by the addition of leverage in a portfolio. This means that the leverage created by borrowing risk-free cash and investing in risky assets does not affect a portfolio’s Sharpe ratio.

Assume that \(w_p\) is the weight of an actively managed portfolio and \(1-w_p\) is the weight of the risk-free cash. Changing \(w_p\) does not change the Sharpe ratio as seen in the following equation:

$$ SR_c=\frac{R_c-R_f}{\sigma\left(R_c\right)}=\frac{w_p\left(R_p-R_f\right)}{w_p\sigma(R_p)}=SR_p $$

An optimal portfolio is the one with the **maximum Sharpe ratio**. To determine the maximum Sharpe ratio, we need to find the optimal amount of risk. The optimal amount of risk is the level of volatility that maximizes the overall Sharpe ratio. Optimal risk is calculated as follows:

$$ \sigma_P^\ast=\frac{IR}{SR_B}\times\sigma_B $$

Where:

- \(\sigma_P^\ast\) is the optimal risk.
- \(IR\) is the information ratio of the active portfolio.
- \(SR_B\) is the Sharpe ratio of the benchmark.
- \(\sigma_B\) is the volatility of the benchmark return.

Moreover, we need to calculate the amount we should invest in an actively managed portfolio and a benchmark portfolio. We use the following formula:

$$ w_P=\frac{\sigma_P^\ast}{\sigma_P} $$

Where:

- \(\sigma_P^\ast\) is the optimal amount of active risk.
- \(\sigma_P\) is the total active risk of the active portfolio, and it is calculated as follows:

$$ \sigma_p^2=\sigma_B^2+\sigma_A^2 $$

Where:

- \(\sigma_B^2\) is the benchmark return variance.
- \(\sigma_A^2\) is the active return variance.

The Sharpe ratio of the combined portfolio can be calculated using the following formula:

$$ SR_P^2=SR_B^2+IR^2 $$

Given the following information:

$$ \begin{array}{c|c} & \textbf{Value} \\ \hline \text{Information ratio} & 14\% \\ \hline \text{Active risk} & 12\% \\ \hline \text{Sharpe ratio benchmark} & 30\% \\ \hline \text{Total risk benchmark} & 20\% \end{array} $$

Calculate the following:

- The optimal active risk.
- The amount that should be invested in the actively managed portfolio.
- The Sharpe ratio of the actively managed portfolio.
- The expected active return.

The optimal risk is calculated as follows:

$$ \begin{align*} \sigma_P^\ast & =\frac{IR}{SR_B}\times\sigma_B \\ & =\frac{0.14}{0.30}\times0.20=9.33\% \end{align*} $$

We employ the following formula to calculate the appropriate weight to be invested in the managed portfolio:

$$ \begin{align*} w_P &=\frac{\sigma_P^\ast}{\sigma_P} \\ & =\frac{0.933}{0.12}=77.78\% \end{align*} $$

Using the formula:

$$ SR_P^2=SR_B^2+IR^2 $$

We obtain:

$$ \begin{align*} SR_P & =\sqrt{SR_B^2+IR^2} \\ & =\sqrt{{0.30}^2+{0.14}^2} \\ & =33.11\% \end{align*} $$

Using the formula:

$$ \begin{align*} \text{Expected active return} & = IR\times \text{Optimal active risk} \\ &=0.14\times0.0933=1.31\% \\ \end{align*} $$

The following table shows the calculations above:

$$ \begin{array}{c|c} & \textbf{Value} \\ \hline \text{Information ratio} & 14\% \\ \hline \text{Active risk} & 12\% \\ \hline \text{Sharpe ratio benchmark} & 30\% \\ \hline \text{Total risk benchmark} & 20\% \\ \\ \text{Optimal active risk} & 9.33\% \\ \hline \text{Weight active strategy} & 77.78\% \\ \hline \text{Sharpe ratio combined portfolio} & 33.11\% \\ \hline \text{Expected active return} & 1.31\% \end{array} $$

## Question

Given the following information:

$$ \begin{array}{c|c} \textbf{Active Portfolio} & \\ \hline \text{Annual return} & 0.3323 \\ \hline \text{Volatility of return} & 0.0079 \\ \hline \text{Sharpe ratio} & 0.0095 \\ \hline \text{Information ratio} & 0.0047 \\ \hline \text{Active return} & 0.0380 \\ \hline \text{Active risk} & 0.0025 \\ \hline \textbf{Benchmark Portfolio} & \\ \hline \text{Annual return} & 0.2849 \\ \hline \text{Volatility of return} & 0.0057 \\ \hline \text{Sharpe ratio} & 0.0105 \end{array} $$

The Sharpe ratio of the optimal portfolio obtained by combining the active and the benchmark portfolios is

closestto:

- 0.0095.
- 0.0105.
- 0.0115.
## Solution

The correct answer is C.Recall that an optimal portfolio is the one with maximum Sharpe ratio.

Using the formula:

$$ \begin{align*} SR_P & =\sqrt{SR_B^2+IR^2} \\ SR_P & =\sqrt{{0.0105}^2+{0.0047}^2} \\ & =0.0115 \end{align*} $$

Reading 45: Analysis of Active Portfolio Management

*LOS 45 (b) Calculate and interpret the information ratio (ex-post and ex-ante) and contrast it to the Sharpe ratio.*