RISK BUDGETING

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2562233/risk-budgeting-1

Learning Outcomes

• explain absolute and relative risk budgets and their use in determining and implementing an asset allocation

• describe how client needs and preferences regarding investment risks can be incorporated into asset allocation

[A] risk budget is simply a particular allocation of portfolio risk. An optimal risk budget is simply the allocation of risk such that the first order of conditions for portfolio optimization are satisfied. The risk budgeting process is the process of finding an optimal risk budget.

Kurt Winkelmann (2003, p. 173)

As this quote from Kurt Winkelmann suggests, there are three aspects to risk budgeting:

• The risk budget identifies the total amount of risk and allocates the risk to a portfolio’s constituent parts.

• An optimal risk budget allocates risk efficiently.

• The process of finding the optimal risk budget is risk budgeting.

Although its name suggests that risk budgeting is all about risk, risk budgeting is really using risk in relation to seeking return. The goal of risk budgeting is to maximize return per unit of risk—whether overall market risk in an asset allocation setting or active risk in an asset allocation implementation setting.

The ability to determine a position’s marginal contribution to portfolio risk is a powerful tool that helps one to better understand the sources of risk. The marginal contribution to a type of risk is the partial derivative of the risk in question (total risk, active risk, or residual risk) with respect to the applicable type of portfolio holding (asset allocation holdings, active holdings, or residual holdings). Knowing a position’s marginal contribution to risk allows one to (1) approximate the change in portfolio risk (total risk, active risk, or residual risk) due to a change in an individual holding, (2) determine which positions are optimal, and (3) create a risk budget. Risk-budgeting tools assist in the optimal use of risk in the pursuit of return.

Exhibit 16 contains risk-budgeting information for the Sharpe ratio–maximizing asset allocation from our original UK example. The betas are from Exhibit 12. The marginal contribution to total risk (MCTR) identifies the rate at which risk would change with a small (or marginal) change in the current weights. For asset class i, it is calculated as MCTRi = (Beta of asset class i with respect to portfolio)(Portfolio return volatility). The absolute contribution to total risk (ACTR) for an asset class measures how much it contributes to portfolio return volatility and can be calculated as the weight of the asset class in the portfolio times its marginal contribution to total risk: ACTRi = (Weighti)(MCTRi). Critically, beta takes account not only of the asset’s own volatility but also of the asset’s correlations with other portfolio assets.

The sum of the ACTR in Exhibit 16 is approximately 10.88%, which is equal to the expected standard deviation of this asset allocation mix. Dividing each ACTR by the total risk of 10.88% gives the percentage of total risk that each position contributes. Finally, an asset allocation is optimal from a risk-budgeting perspective when the ratio of excess return (over the risk-free rate) to MCTR is the same for all assets and matches the Sharpe ratio of the tangency portfolio. So in this case, which is based on reverse-optimized returns, we have an optimal risk budget.

Exhibit 16:

Risk-Budgeting Statistics

Asset Class	Weight	MCTR	ACTR	Percent Contribution to Total Standard Deviation	Ratio of Excess Return to MCTR
UK large cap	3.2%	11.19%	0.36%	3.33%	0.368
UK mid cap	0.9	12.02	0.11	0.98	0.368
UK small cap	0.3	12.44	0.03	0.30	0.368
US equities	34.4	14.51	5.00	45.94	0.368
Europe ex UK equities	8.7	16.68	1.45	13.34	0.368
Asia Pacific ex Japan equities	3.1	16.35	0.51	4.69	0.368
Japan equities	6.6	10.69	0.70	6.46	0.368
Emerging market equities	5.9	17.51	1.02	9.42	0.368
Global REITs	1.8	17.79	0.31	2.86	0.368
Global ex UK bonds	31.8	4.21	1.34	12.33	0.368
UK bonds	3.2	1.22	0.04	0.35	0.368
Cash	0.2	0.00	0.00	0.00	0.368
	100.0		10.88	100.00

For additional clarity, the following are the specific calculations used to derive the calculated values for UK large-cap equities (where we show some quantities with an extra decimal place in order to reproduce the values shown in the exhibit):

• Marginal contribution to risk (MCTR):Asset beta relative to portfolio × Portfolio standard deviation1.0289 × 10.876 = 11.19%

• ACTR:Asset weight in portfolio × MCTR3.2% × 11.19% = 0.36%

• Ratio of excess return to MCTR:(Expected return − Risk-free rate)/MCTR(6.62% − 2.5%)/11.19% = 0.368

EXAMPLE 4

Risk Budgeting in Asset Allocation

1. Describe the objective of risk budgeting in asset allocation.

   Solution to 1:

   The objective of risk budgeting in asset allocation is to use risk efficiently in the pursuit of return. A risk budget specifies the total amount of risk and how much of that risk should be budgeted for each allocation.

2. Consider two asset classes, A and B. Asset class A has two times the weight of B in the portfolio. Under what condition would B have a larger ACTR than A?

   Solution to 2:

   Because ACTRi = (Weighti)(Beta with respect to portfolio)i(Portfolio return volatility), the beta of B would have to be more than twice as large as the beta of A for B to contribute more to portfolio risk than A.

3. When is an asset allocation optimal from a risk-budgeting perspective?

   Solution to 3:

   An asset allocation is optimal when the ratio of excess return (over the risk-free rate) to MCTR is the same for all assets.