Risk Budgeting in a Multi-Asset Portfolio: A Practical Walkthrough

Setting the Stage: A Hypothetical Portfolio

To make the concepts of risk budgeting more concrete, let's walk through a practical example of how to apply it to a multi-asset portfolio. Let's assume we are managing a $100 million portfolio for a pension fund. The fund has a long-term investment horizon and a moderate risk tolerance. The investment objective is to achieve a return of 7% per year with a target volatility of 10%.

Our investment universe consists of four broad asset classes:

• Global Equities
• Global Government Bonds
• Commodities
• Real Estate (represented by REITs)

Step 1: Estimating Risk and Return Parameters

The first step is to estimate the expected return, volatility, and correlations for each of our asset classes. This is a important step, and the quality of our output will depend on the quality of these inputs. For this example, let's use the following hypothetical parameters, which are broadly in line with long-term historical averages:

And the following correlation matrix:

Step 2: The Traditional 60/40 Portfolio

First, let's see what a traditional 60/40 portfolio (60% equities, 40% bonds) looks like from a risk perspective. Using our assumed parameters, we can calculate the risk contribution of each asset class. The math is complex, but the results are clear:

• Equities Risk Contribution: 95%
• Bonds Risk Contribution: 5%

Even though bonds make up 40% of the capital, they only contribute 5% of the risk. The portfolio is completely dominated by the equity allocation. This is the problem that risk budgeting is designed to solve.

Step 3: Defining the Risk Budget

Now, let's create a risk budget. We want to build a more balanced portfolio, so let's allocate our risk more evenly across the four asset classes. A reasonable starting point might be:

• Equities: 40% of the risk budget
• Bonds: 20% of the risk budget
• Commodities: 20% of the risk budget
• Real Estate: 20% of the risk budget

This is a strategic decision. We are still giving equities the largest allocation, but we are ensuring that the other asset classes also make a meaningful contribution to the portfolio's risk profile.

Step 4: Optimizing for Capital Weights

Now that we have our risk budget, we can use an optimizer to find the capital weights that will achieve this risk allocation. The optimizer will solve for the weights w_i such that the risk contribution of each asset class matches our target budget. The resulting capital allocation might look something like this:

• Global Equities: 35%
• Global Gov't Bonds: 40%
• Commodities: 10%
• Real Estate: 15%

Notice how different this is from the risk allocation. Bonds, with their low volatility, get a large capital allocation (40%) to achieve their 20% risk budget. Commodities, with their high volatility, only need a 10% capital allocation to fill their 20% risk budget. This is the essence of risk budgeting in action.

Step 5: Review and Refine

The final step is to review the resulting portfolio. Does it meet our objectives? Using our assumed parameters, this risk-budgeted portfolio would have an expected return of around 5.5% and a volatility of 8.5%. This is lower than our target return of 7% and our target volatility of 10%. To reach our targets, we would need to apply a modest amount of leverage (around 1.18x) to the portfolio. This would scale up both the expected return and the volatility to our desired levels.

This example is a simplification, but it illustrates the power of the risk budgeting framework. It provides a disciplined and transparent process for building a truly diversified portfolio. By focusing on the allocation of risk, rather than the allocation of capital, it helps to avoid the concentration of risk that can be so damaging in a crisis. It is a more intelligent and robust way to construct portfolios for the long term.