Absolute and Relative Risk Budgets

Risk budgeting is a means of making optimal use of risk to pursue return. A risk budget is optimal when the ratio of excess return to marginal contribution to total risk (MCTR) is the same for all assets in the portfolio. The following formulas help quantify and analyze portfolio allocation decisions:

Marginal Contribution to Total Risk “MCTR“

$$ MCTR = \text{Beta of asset class} \times \text{Portfolio standard deviation} $$

Absolute Contribution to Total Risk “ACTR“

$$ {ACTR}=\text{Asset weight} \times {MCTR} $$

% of Risk Contributed by a Position

$$ \% \text{ of risk contributed by a position} =\frac {ACTR}{\text{(Total portfolio risk)}} $$

Ratio of Excess Return

$$ \text{Ratio of excess return to MCTR} =\frac {\text{Expected return}-\text{Risk free rate}}{{MCTR}} $$

The optimal allocation to any asset class appears when the excess return to MCTR ratio is equal for all assets and the portfolio Sharpe Ratio.

$$ \small{\begin{array}{c|c|c|c|c|c|c|c} & \textbf{Weight} & {\textbf{Excess}} & \textbf{Beta} & \textbf{MCTR} & \textbf{ACTR} & \bf{\%} & { \textbf{Ratio of}} \\ & &{\textbf{Return}} & & & & \textbf{Contribution} & \textbf{Excess} \\ & & & & & & {\textbf{to Risk}} & \textbf{Return} \\ & & & & & & & \textbf{to} \\ & & & & & & & {\textbf{MCTR} } \\ \hline \textbf{Equity} & 52\% & 6.57\% & 1.5 & 19.50\% & 10.10\% & 77.67\% & \bf{0.337} \\ \hline \textbf{Bond} & 34\% & 2.85\% & 0.65 & 8.45\% & 2.90\% & 22.33\% & \bf{0.337} \\ \hline \textbf{Cash} & 14\% & 0 & 0 & 0.00\% & 0.00\% & 0.00\% & \\ \hline \textbf{Total} & 100\% & 4.38\% & 1 & & 13.00\% & 100\% & \end{array}} $$

*Calculated values are subject to rounding errors.

The portfolio standard deviation is 13%

The calculation for bonds is as follows:

$$ \text{MCTR}_\text{Bonds}=0.65 \times 13= 8.45\% $$

$$ \text{ACTR}_\text{Bonds} =(0.34) \times 8.45\% \approx 2.90\% $$

$$ \% \text{ Contribution to total } {\text{risk}_\text{Bonds}} = \frac {(2.90\%)}{(13\%)} \approx 22.31\% $$

Ratio of excess return to \(\text{ MCTR}_\text{Bonds}=\left(\frac {(6.57\%)}{(8.45\%)} \right)=77.75\% \)

• \(\text{Portfolio Sharpe ratio} =\frac {(4.38\%)}{(13\%)} = 0.337\)
• The Portfolio Sharpe ratio is equal to the ratio of excess return to MCTR for both equities and bonds.

Therefore, the 52/34/14 allocation is optimal from a risk-budgeting perspective.

Question

A risk budget is most likely optimal when which of the following occurs?

1. The ratio of absolute contribution to total risk is the same for all assets in the portfolio.
2. The ratio of marginal contribution to total risk is higher than the portfolio Sharpe ratio.
3. The ratio of excess return to marginal contribution to total risk is the same for all assets in the portfolio.

Solution

The correct answer is C:

A risk budget is optimal when the ratio of excess return to marginal contribution to total risk is the same for all assets in the portfolio.

A is incorrect. It refers to the ratio of absolute contribution to total risk rather than the ratio of excess return to marginal contribution to total risk. Absolute contribution to total risk does not consider an asset’s expected return. It, therefore, does not provide a complete picture of the trade-off between risk and return.

B is incorrect. It refers to the ratio of marginal contribution to total risk being higher than the portfolio Sharpe ratio. The Sharpe ratio measures risk-adjusted performance, calculated as the excess return of an investment over a risk-free rate, divided by the standard deviation of its returns. The relationship between the Sharpe ratio and the ratio of marginal contribution to total risk is irrelevant in determining whether a risk budget is optimal. Instead, what matters is whether the ratio of excess return to marginal contribution to total risk is the same for all assets in the portfolio.

Asset Allocation: Learning Module 4: Principles of Asset Allocation; Los 4(g) Explain absolute and relative risk budgets and their use in determining and implementing an asset allocation