Building indices using risk components.
In the expanding index universe, new categories of risk-weighted and fundamental indices are garnering interest alongside well-established market-weighted indices. As their distinguishing feature, risk-weighted indices employ quantitative forecasts of risk to achieve better diversification. In contrast, fundamental indices focus on future expected performance to achieve diversification. Market-weighted indices do not directly employ any forecasts and rely on current valuations as the metric that summarises all forecasting information.
The theoretical foundation for risk-weighted indexing can be traced back to Markowitz's mean-variance efficiency. More recently, in the equity asset class there have been several commercial implementations that rely on modified versions of the mean-variance optimisation1. In fixed income, some incipient efforts have been made to mirror the approach used for equities, but the only full-fledged implementations simply link weights with key characteristics that drive risk, such as duration or/and spread2. Across asset classes, risk-based methods are used mainly for portfolio construction, such as the "risk-parity" approach, and index implementations are missing, even though the same methodologies that are applied to portfolio construction could be applied to index construction as well.
WHY RISK-WEIGHTED INDICES?
Risk-based indices are typically useful for asset owners and managers who incorporate quantitative measures of risk in their portfolio construction. Such asset owners may employ risk-based methods to allocate assets across various managers and to create benchmarks for judging the managers' performance. Portfolio managers, whether they are asset allocators or product specialists, may use these benchmarks as a starting point for portfolio construction.
Risk-based portfolios give more weight to assets that either reduce the overall portfolio risk or increase it by less, aiming to maximise diversification benefits. If asset owners leave risk weightings out of the benchmark's construction methodology, they implicitly allow managers to reap the benefits of diversification and claim them as managerial skill, an undesirable outcome.
An appropriate benchmark should also match the investment philosophy of the associated portfolio. A benchmark must incorporate all relevant public information, while being careful to be agnostic about information that is the purview of the portfolio manager. Therefore, when well-known risk metrics, such as volatility, are part of the investment process, asset owners may prefer to use a risk-based index as a benchmark.
Let's consider the example of a manager who allocates to equity and fixed income. The goal is for each allocation to contribute to portfolio risk proportionally to its risk-adjusted performance. For clarity, let's take the Sharpe ratio as the metric for risk-adjusted performance and let's define risk as volatility. As a benchmark, the asset owner may use a base case when both assets are expected to have the same Sharpe ratio. Therefore, fixed income and equity must contribute equally to the total benchmark risk. These conditions meet the asset owner's requirements for a good benchmark, since it includes public information that is used in the portfolio construction process but which is not part of the manager's performance assessment. Furthermore, it avoids information that the owner considers to be the purview of the portfolio manager, such as the relative performance of equity and fixed income. Any deviation of the portfolio from the benchmark will reflect directly the manager's skill at forecasting that relative performance.
In our example, the risk-based benchmark is composed of 25% equity and 75% fixed income3. This benchmark differs from a typical market-weighted combination of equity and fixed income (60% equity and 40% fixed income, or "60/40") because the equity part drives most of the "60/40" portfolio while the fixed income part has little influence. The last point is illustrated in Figure 1, which shows the properties of the 60/40 portfolio and its two components. The portfolio volatility is very similar to the volatility of the equity allocation alone—and much higher than the volatility of the fixed income part. Moreover, equity has an almost perfect correlation with the portfolio, while fixed income has a correlation of less than 50%. The equity dominance over the 60/40 portfolio showcases its lack of diversification.
Now let's say the manager believed throughout the last decade that equities would perform better than fixed income and as a consequence followed a 35/65 portfolio over the period. His portfolio is more correlated to equity than to fixed income, as he intended. The manager's prediction of equity outperformance turned out to be wrong and that is shown by his underperformance of the "25/75" benchmark over 2001 to 2011 (see Figure 2). However, he would still have outperformed significantly a market-based benchmark such as "60/40". This result shows that the decision about which benchmark to use has significant consequences on how the manager's performance is ultimately viewed. Furthermore, for managers investing in less conventional asset classes such as commodities, which are traded via futures contracts, market-weighting is not well defined, making risk-based weighting a more plausible alternative.
Risk benchmarking need not be limited to managers who allocate across assets. Product specialists may employ it as well, as the basic principle of balancing quantitative risk across portfolio components is general in nature. Consider the example of a manager allocating to investment grade corporate bonds across the rating spectrum (AAA/AA/A/BBB). He constructs his portfolio using spread returns, defined as returns in excess of a portfolio of duration-matched Treasuries. He may want to start his portfolio allocation by being exposed equally to events in each rating bucket4 – i.e., each rating bucket contributes equally to portfolio risk. He would then tilt his portfolio based on views about which part of the ratings spectrum outperforms after accounting for risk.
A risk-based benchmark, constructed along the lines described in the previous example, allocates 35%/30%/20%/15% to the four buckets, reflecting differences in riskiness across ratings. This benchmark differs significantly from the market-weighted one, as shown in Figure 3. It has much lower overall risk, and it is less correlated with the BBB bucket. Using a market benchmark, which is very similar to the BBB index, the benchmark tracking error captures both the relative value calls the manager makes (an intended item) and the performance of BBB securities against other ratings (an unintended item).
An asset owner may choose to combine the risk-based asset allocation described in the first example with the single-asset risk-based benchmarks shown in the second example. Consider an asset owner who allocates to two managers, one specialising in fixed income and one in equity. The owner may allocate using the benchmark we obtained in Example 1, which combines equity and fixed income based on their relative risk, while being agnostic about their relative value.
Furthermore, he can define the benchmarks associated with the two managers following a risk-centric approach similar to Example 2. For equity, he may use an industry-based risk combination, whereby industries that are more risky get a lower weight. For fixed income, he may break down exposure by asset type. A schematic of this process is presented in Figure 4. This example showcases the scalability and flexibility of this risk-based approach.
KEY ISSUES IN INDEX DESIGN
The flexibility of risk-based index construction comes at a price: we must make many choices from benchmark structure ("should we break down corporate risk according to ratings or along industry characteristics?") to construction method details ("how should we account for correlations?") to the risk forecasting model ("should we use a 12-month window or should we use three months of daily data?"). To make appropriate choices we must understand what role these choices play in the final benchmark. In this section we will explore some key issues in risk-weighted index design.
Asset Classes Or Risk Factors As Building Blocks
The first choice to be made concerns the breakdown of the investable universe. Let's take the case of an asset owner creating a benchmark for a fixed-income manager. The question is what building blocks should be used, and should they be based on various asset classes (eg, Treasuries, agencies, securitised, or corporates), or should they be based on risk types (eg, rates, credit spreads, securitised spreads)?
The choice has major consequences for a benchmark: for example the returns of investment grade corporates are dominated by interest rate risk, making them highly correlated with Treasuries. To achieve efficient diversification, building blocks should be as unrelated as possible from a risk perspective. Because corporates and Treasuries exhibit similar behaviour, the asset owner may find it undesirable to have them both as building blocks in the benchmark. However, corporates do have a component that is different from Treasuries, the credit spread return, and it may make sense to isolate it as a separate building block and then to combine the common interest rate risk of Treasuries and corporates into one block (see Figure 5).
The investment grade corporate bonds example points to strong technical reasons why a risk-based approach to building a benchmark should be preferred: asset-based breakdowns usually result in more extreme correlations than risk-based ones. Moreover, correlations among risk types are more stable than the ones among various assets. For example, US corporates and Treasuries (or rates) were highly correlated before the 2008 credit crisis (see the second column in Figure 6) and then had a small correlation during the crisis, only for their correlation to increase sharply again afterwards.
Alternatively, we can separate US corporates risk into rates (or Treasury) risk and spread risk. The last column in Figure 6 shows the correlation between the risk factors associated with rates and spread risk as consistently negative. While this correlation becomes larger over the three periods, it does not reach the extremes observed with asset class correlations.
Fundamentally, risk types are related directly to various macroeconomic phenomena, such as growth and inflation, while asset classes are mostly combinations of risk types. For example, credit spread risk is largely a function of the growth prospects of the economy. However, corporate bonds are a combination of rates and spread risk. Risk types have predictable volatility levels, almost by design. To the extent that the mixture of rates and credit risk varies over time, the risk properties of corporate bonds will be more time-varying and less predictable than those of the underlying risk types (see Figure 6 and Figure 7).
A risk-based index should therefore be designed using risk-type building blocks5 as opposed to asset-based building blocks. This method results in more stable and reliable index construction.
What Risk Types To Include: Expected Performance Assumptions
This conclusion leads us to the next issue: what risk types to consider. For example, for equity should we use a classic three-factor model (market, size and value, as described by Fama and French (1993)) or should we use an industry breakdown, or a combination of the two? For rates, should we use a principal-components breakdown (level, slope and convexity) or should we use a duration-based breakdown?
A well-designed risk-based benchmark contains all risks that the manager considers as part of the portfolio, but no others. The question then becomes what types of risk a manager would, or should, be given access to in order to construct the portfolio. Intuitively, a manager considers only those risk types for which he receives compensation in the form of a positive expected return, or those which reduce portfolio risk more than portfolio performance. This argument underscores a critical point: expected returns are always part of the benchmark construction, implicitly or explicitly.
It is generally believed that risk-based benchmarks do not rely on any performance forecast, but only on a risk forecast. However, creating a portfolio or a benchmark with the sole intention of minimising risk results in an all-cash portfolio or, if cash is not in the investable universe, the most cash-like instrument available to the manager, for example very low-duration bonds or very low-risk, low-beta equities. For example, in Figure 8 we compare three portfolios of equity and fixed income: one that aims to minimise risk, a second that aims to maximise the portfolio Sharpe ratio assuming both equity and rates have the same Sharpe ratio, and a third that is the "60/40". The results show that the first portfolio, which aims to minimise risk, is strongly correlated with fixed income (the lower-risk component of the portfolio) and has little relation to equities, while the "60/40" has the opposite behaviour. The second portfolio is equally correlated to rates and fixed income, a sensible outcome given the similar Sharpe ratio assumption, and arguably much more diversified than the other two portfolios.
Risk cannot be considered as a goal in itself. It must be combined with a performance goal, which requires assumptions about the future performance (i.e., the risk premium) of various risk types, that is the building blocks of the benchmark. This statement about risk premia assumptions being required for any benchmark design may seem strong, especially given that many risk-based portfolio construction methods6 claim not to rely on any performance information. For example, in a "risk-parity" portfolio, all components contribute equally to total portfolio risk. However, this design makes most financial sense when we assume the same premium (same Sharpe ratio) for each component. Thus, even these "performance-agnostic" methods embed some assumptions about the risk premium, albeit trivial ones.
To determine what risk types to consider as a benchmark's building blocks, we should also ask what assumptions we are making about their premia. These assumptions, implicit or explicit in the chosen construction method, drive the risk types considered. Typically the assumptions are trivial, such as one that all risk premia are the same, because we do not want the benchmark to include information that the manager gets paid for. Although we want to attribute to the manager the entire portfolio return due to views on future performance, we need a base case to judge him against, a set of neutral views on performance, but views nonetheless. As a reference, Figure 9 shows the Sharpe ratio of major risk sources7, broken down by sub-periods. Notice the variability both across periods and across asset types, something to be expected given the notorious difficulty of forecasting them even over the long term.
What are the benefits of making these neutral views explicit? First, in doing so we understand better the source of any deviations from the benchmark, that is what the right or wrong calls made by a manager were and how are they reflected in deviations from the benchmark. We can do even more: we can modify those assumptions if we do not agree with them. For example, given the historically low levels of interest rates in 2011, we may not believe rates risk will carry the same premium (Sharpe ratio) as equity going forward, thus we will not allow them to contribute equally to our benchmark risk and tilt the benchmark more towards equities.
This discussion shows how we can select the building blocks of an index: select only those risk types that carry a risk premium. If we cannot distinguish among the magnitude of various risk premia, then we may choose to assume that all risk premia are the same.
Capacity And Market Weight Tilts
Intuitively, a benchmark that contains 50% Asset-Backed Securities (ABS) and 50% US Treasuries may seem inappropriate for a large fixed income manager, because there is not enough market capacity to invest 50% of the portfolio in ABS. Capacity must be considered even for a risk-based index design. However, capacity is investor-specific. The ABS and Treasuries benchmark above may suit a very small investor. Incorporating capacity in the index design tilts the risk weights towards market weights. The more importance capacity has, the more weights will be tilted towards market weights. At the extreme, for an investor the size of the entire market, the risk-weighted benchmark is the market-weighted one because of capacity. Therefore, capacity ties together risk-based and market value-based indices.
Capacity can be incorporated into the index design by choosing the building blocks to be of comparable market size. Consider an investment grade corporate bond benchmark where risk is broken down by credit quality. Naturally, we may choose to create four risk-type buckets, one for each rating (AAA/AA/A/BBB). However, the AAA bucket contains very few corporate bonds, and for any medium or large-size investor, such a benchmark would not reflect the true opportunity set.
Thus we may change the structure to only three risk types, where the AAA and AA risks are combined using market weights. The building blocks are typically created from individual security-level risks using market weights. Therefore, any risk-based index is a combination of risk weighting at a coarse level and market-weighting at the fine level. The definitions of coarse and fine are driven by capacity.
We do not have to use market weights to create the building blocks; we also can use risk weights, resulting in an iterative application of risk-weighting. As an example, consider a benchmark for fixed income credit that combines the spread return of corporate bonds, ABS and CMBS. We may prefer not to give the same importance to the ABS or CMBS market as the entire corporate bond market and use a layered approach first to combine ABS with CMBS using risk weights, which is in contrast with the AAA/AA market weights combination in the previous example. Then we may use a different breakdown level, whereby the combined ABS and CMBS segment is considered together with industry sectors of the corporate bond market, as shown in Figure 10.
Creating buckets of comparable market weights may not address the market capacity issue if the resulting weights are very different from equal weights. Combining a low-risk and a high-risk bucket using risk weights may result in a large allocation to the low-risk bucket, possibly making capacity an issue even though the two buckets have comparable market values. Both the market value and typical risk level of the resulting bucket should therefore be taken into account. The constraints are greater for larger investors.
Risk-based indices satisfy an important need both for asset owners and portfolio managers. Such indices aim to achieve a high trade-off between performance and risk, using quantitative risk forecasts and trivial assumptions about performance. A typical index is built from various risk types, chosen to satisfy the implicit or explicit assumptions about performance. Using risk types instead of asset classes as building blocks results in a more stable index and benchmark construction exercise. A manager adds value to the portfolio over and above the benchmark by altering the trivial performance assumptions through relative value calls and by incorporating qualitative risk information alongside quantitative metrics.
Footnotes And References
Capacity issues tilt risk-based indices towards market-based ones, ensuring a continuum between the two index methods. Larger investors prioritise capacity thus making their risk-based benchmarks more tilted towards market-value weights. Risk-based indices differentiate among constituents mostly based on risk metrics, while fundamental-based indices differentiate among constituents mostly based on performance metrics. Thus, the two alternative indexing approaches complement each other.
- TOBAM's "Most Diversified Index" and EDHEC's "Risk Efficient Equity Index". See Chow et al (2011) for a survey of alternative equity indices.
- Barclays Capital Targeted Duration Indices, DJ CBOT Treasury Indices, Ryan/Mergent US Treasury Ladder Indices. See Goltz and Campani (2011) for a survey of construction techniques for corporate bond indices. See Dynkin et al (2011) for a risk-based diversification methodology in corporate bond portfolios.
- We construct this allocation using monthly data between 1976-2011. Throughout this paper equity returns are defined as Russell 3000 index returns and Fixed Income returns are defined as Barclays Capital US Aggregate Index returns. Excess Returns are over the 1 month LIBOR rate and portfolios are rebalanced monthly.
- Each rating bucket is defined as the Spread Return of the Barclays Capital US Corporate sub-index pertaining to that rating. The market-weight combination of sub-indices is the Barclays Capital US Corporate Index.
- See Melas and Kang (2011) and Rennison et al (2011) for a discussion on various risk premia sources and how to use them as building blocks for an index or portfolio.
- See Demey et al (2010) for a discussion on risk-based indexation detailing four construction methodologies, some of which are used in this paper's examples.
- The commodities market risk factor is represented by the S&P GSCI Light Energy Index less the one month LIBOR return. The index is modified from March 1979 to December 1982 to add a 10% oil component because the original index does not contain any energy component before January 1983. The rates risk factor is represented by the return of ten year Treasury futures. Equity and spread returns are as defined in the previous examples.
- Chow, T., Hsu, J., Kalesnik V., and Little, B., 2011. A Survey of Alternative Equity Index Strategies, SSRN Working Paper Id 16963331.
- Demey, P., Maillard, S., and Roncalli, T. 2010. Risk-based Indexation, SSRN Working Paper Id 1582998.
- Dynkin, L., Hyman, J., and Konstantinovsky, V. 2011. Sufficient Diversification in Credit Portfolios: Balancing Two Approaches, Barclays Capital Research.
- Fama, E., and French, K., 1993. Common risk factors in the returns of stocks and bonds, Journal of Ffinancial Economics, 33 (1)
- Goltz, F., and Campani, C., 2011. A Review of Corporate Bond Indices: Construction Principles, Return Heterogeneity, and Fluctuations in Risk Exposures, EDHEC-Risk Institution Publication (June).
- Melas, D., and Kang, X., 2010. Applications of Systematic Indexes in the Investment Process, Journal of Indexes (September/October).
- Rennison, G., Staal, A., Ghia, K. and Lazanas, A. 2011. Barclays Capital Risk Premia Family: Sequencing the strategy genome, Barclays Capital Research.