Risk Budgeting Concepts in Portfolio Construction

Risk budgeting involves distributing the total portfolio risk efficiently among its components. It's a crucial element of a robust risk management process, which consists of these four steps:

1. Define the relevant risk metrics based on the fund's mandate:
   • Should you use absolute or relative measures?
2. Evaluate the risk associated with various aspects of the strategy:
   • Are you exposed to rewarded factors, or is it about allocations to sectors and securities?
3. Set the appropriate risk budget:
   • Establish an overall risk target for the portfolio.
4. Allocate the risk across individual factors or components.

Causes and Sources of Absolute Risk

Absolute risk in a portfolio arises from its variance, which is only relative to the portfolio itself, not compared to a benchmark. To calculate the contribution of a specific asset (i) to the portfolio's variance, we use this equation:

$$ Cv_i = \sum(W_iW_jC_{ij}) = w_iC_{ip} $$

Where:

• \(W_j\) = Asset j's weight in the portfolio.
• \(C_{ij}\) = The covariance of returns between asset j and asset i.
• \(C_{ip}\) = The covariance of returns between asset I and the portfolio.

This formula is multiplicative, meaning that each variable's increase leads to a higher contribution to portfolio variance. So, if an asset has a higher weight in the portfolio or tends to move closely with the portfolio, it contributes more to the portfolio's variance.

Causes and Sources of Relative/Active Risk

The contribution of asset i to portfolio active variance can be determined using the following equation:

$$ CAV_i = (W_{pi} -W_{bi}) rC_{ip} $$

Where:

• \(W_{pi}\) = Weight of asset i in the portfolio.
• \(W_{bi}\) = Weight of asset i in the benchmark.
• \(rC_{ip}\) = Covariance between the active returns of asset i and the active returns of the portfolio, considering the covariances with all n assets in the portfolio.

The sum of individual CAVs provides the portfoli's active return variance.

Determining the Appropriate Level of Risk

When customizing portfolios for clients, financial professionals should consider the following factors to align risk with individual needs:

• Implementation constraints: Some portfolios come with explicit limitations on what can be included, directly impacting a manager's choices.
• Limited diversification opportunities: As risk increases, the benefits to portfolio performance diminish, emphasizing the importance of considering risk-adjusted returns.
• Leverage and its implications for risk: While leverage can enhance returns, it also introduces higher volatility and trading costs, prompting portfolio managers to conduct a cost-benefit analysis before implementing it.

For those curious about the mathematical aspect of the last point, this formula illustrates the connection between geometric and arithmetic returns. When leverage is increased, both \(R_a\) and \(\sigma\) go up. However, the negative relationship between standard deviation and the exponent means that continually escalating leverage will eventually outweigh the augmented returns.

$$ R_g = R_a – \left(\frac {\sigma^2}{2} \right) $$

Where:

\(R_g\) = Geometric compound returns. \(R_a\) = Arithmetic return. \(\sigma^2\) = Portfolio standard deviation.

Question

Based on the table below, what is asset b's relative contribution to portfolio variance?

$$ \begin{array}{c|c}
\textbf{Asset} & {\textbf{Absolute Contribution} \\ \textbf{to Portfolio Variance}} \\ \hline
A & 0.0434 \\ \hline
B & 0.0019 \\ \hline
C & 0.0355
\end{array} $$

1. 0.0808.
2. 0.0019.
3. 0.0235.

Solution

The correct answer is C.

Step 1: Calculate the overall portfolio variance by summing the absolute contributions:

$$ 0.0434 + 0.0019 + 0.0355 = 0.0808 $$

Step 2: Calculate the relative contribution from asset B:

$$ \frac {0.0019}{0.0808} = 0.0235 $$

The relative contribution from asset B is relatively low because its absolute contribution to portfolio variance is low.

A and B are incorrect. From the calculation the correct value is 0.0235

Reading 26: Active Equity Investing: Portfolio Construction

Los 26 (d) Discuss the application of risk budgeting concepts in portfolio construction