The Sense and Nonsense of Risk Budgeting

Abstract

A framework is described for the optimal allocation of active risk among broad asset classes or external asset managers. Unlike most risk allocation models used by practitioners, this framework does not assume that cross-correlations are zero. An analytical expression for the optimal allocation of tracking error among investment decision areas (assets and external managers) in the presence of correlations is provided. The key to understanding optimal risk allocation is the correlation-adjusted information ratio, a novel concept introduced in this article. Also discussed are various approaches to setting realistic input assumptions, such as the expected IR, for deriving optimal risk allocation.

Risk budgeting is popular. Everyone talks about it, but when asked to define it, they give a variety of answers. The growing risk-budgeting literature also does not provide a clear definition. We attempt to define risk budgeting clearly and concisely. To appreciate what risk budgeting is all about, analysts need to realize that risk management consists of three stages: risk measurement, risk attribution, and risk allocation.

Risk budgeting starts with an institutional investor deciding how it wants to allocate risk among, for example, asset classes or active managers to achieve the highest risk-adjusted return. This risk allocation should be monitored on a regular basis to ensure that the institution does not over- or underspend the risk budget. The risk allocation may need to be rebalanced to bring risk-spending activities in line with optimal risk allocation. Risk budgeting consequently entails both risk allocation (where to spend the risk) and risk attribution/decomposition (whether risk is being spent accordingly).

Important criteria in developing a risk allocation framework are that risk budgeting do the following:

1. integrate performance attribution and risk attribution to allow evaluation of whether the risk allocation paid off accordingly;

2. tie in directly with the investment decision process of the investor;

3. effectively allocate risk among asset classes and provide clear guidelines to portfolio managers in terms of their risk budget for the purpose of portfolio construction;

4. create clear accountability and delegate measurable investment responsibilities to portfolio managers;

5. guarantee transparency through the use of a formal, disciplined, but simple approach in which portfolio managers are held accountable only for areas that are within their control or responsibility.

We propose and discuss a framework for risk budgeting that meets these criteria. The optimal risk allocation among asset classes we derive depends on three critical inputs: (1) the overall risk budget (or total tracking error) set by an oversight committee, (2) the target information ratios (IRs) provided by portfolio managers in each asset class, and (3) assumptions about correlations between the various investment activities.

We briefly review the literature and discuss the difference between risk allocation and asset allocation. We introduce the concept of correlation-adjusted IRs and discuss the important role they play in deriving the optimal risk allocation. We also introduce the concept of an implied IR—a useful statistic that can help in tracking how much the current risk allocation deviates from the optimal one. We discuss how our framework can be applied to decide on the optimal allocations among sources of systematic (beta) risk and active (alpha) risk. And finally, we discuss various approaches to setting expected IRs.

We would like to thank Krishnan Chandrasekhar, John Gandolfo, Roy Kouwenberg, Veronica Marghescu, and Kelda Simpson for helpful comments. This article reflects the views of the authors, not those of the World Bank.

Notes

¹ Risk allocation is an ex ante process, whereas risk attribution is anex post analysis.

² Winkelmann (2000), Muralidhar (2001), and Sharpe (2002) provide excellent expositions on risk attribution.

³ We define tracking error as the standard deviation of returns in excess of benchmark returns.

⁴ “Information ratio” is defined as the ratio of average excess return to tracking error.

⁵ A risk-budgeting framework based on the M-cube measure would require the introduction of two additional investment options: cash (to deleverage manager returns] and the underlying manager benchmark (to adjust the overall beta of the portfolio).

⁶ Our own empirical studies confirm this finding.

⁷ Some have tried to bridge the gap between asset allocation and risk allocation by using risk-based rebalancing, such as volatility rebalancing, as part of their asset allocation process (see McCalla 1994).

⁸ This optimization problem, in the context of asset allocation, is common in the literature (e.g., Lee and Lam 2001; Waring, Whitney, Pirone, and Castille 2000).

⁹ By “normalizing,” we mean dividing the return series by its volatility.

¹⁰ The assumption of uncorrelated excess returns among asset classes raises an interesting question about “portable-alpha” strategies, which generate excess returns within a certain market or asset class that can be transported to any other benchmark by using futures contracts or total-return swaps. An example is a manager who generates excess returns in the fixed-income markets but transports those excess returns to the S&P 500 Index. This strategy implicitly increases the amount of active risk taken in the fixed-income markets but disguises it as active management in equities. Portable-alpha strategies are a form of hidden active risk. Another example of hidden active risk is active currency risk taken in a portfolio of non-U.S. fixed income. Taking this currency risk increases the overall active risk in currencies while disguising it as active management in fixed income.

¹¹ The conclusions were not affected by the zero IR assumption. A positive IR simply reduced the relative VAR versus the policy allocation.

¹² If portfolio managers lack skill in selecting good submanagers, the alternative for the investor is to invest passively by hiring an index manager or investing in the benchmark, which results in an IR of zero.

¹³ The ratio of alpha to marginal tracking error for each manager or asset class in the optimal solution is constant and equal to the total IR at the fund level.