14.1.1 Term Structure Factors

Changes in the term structure—the curve representing the relationship between yields and time to maturity—are a very significant component of the overall risk of a fixed income portfolio. Common ways to represent term structure risk include average change in interest rates, key rates, and “shift-twist-butterfly” factors. We explain the reasoning behind these modeling choices next.

The portfolio return attributed to interest rate changes is approximately the product of effective duration and the average change in rates, that is,

This is the reason for the average change in interest rates () to be sometimes used as a factor in bond return models. The disadvantage of using this single factor is that the resulting factor model does not capture the change and reshaping of the entire term structure.

The yield curve risk component due to changes in the shape of the yield curve can be accounted for by using key rate durations. As explained in Chapter 13, key rates are the interest rates for a prespecified set of maturities. Shifts in the term structure are represented as a discrete vector, the elements of which are the changes in the key zero-coupon rates of various maturities. Interest rate changes at other maturities are derived from the key rates via linear interpolation. Key rate durations are then defined as the sensitivity of the portfolio value to the given key rates at different points along the term structure. Typical key rates are the 6-month, 1-year, 2-year, 5-year, 10-year, 20-year, and 30-year rates but factor models using key rate durations vary in the choice of key rates. Ho (1992) proposed using as many as 11 key rates to hedge interest rate risk effectively. Later in this chapter, we show an example of how key rates are used for portfolio risk decomposition with the Barclays POINT system.

Neighboring key rates are often more than 90% correlated; hence there is an argument for not using so many factors to explain movements in the yield curve.² Some models are based on a compromise approach, which does not treat the term structure in terms of specific key rates but in terms of factors that describe the way the term structure moves.³ The three factors commonly used include shift (when rates move in parallel), twist (when the curve steepens or flattens, that is, when the ends of the curve move in opposite directions), and butterfly (when the curvature changes or flexes). These factors are derived using principal components analysis⁴ applied to the key rate covariance matrix, and have been found to explain as much as 98% of the variability for portfolios of Treasury securities.

In addition to the selection of factors to represent the term structure of interest rates, an important issue is the selection of the benchmark curve itself. Domestic government bonds (e.g., the U.S. Treasury yield curve in the case of the United States) are the choice in most markets but there are important exceptions. For example, in the case of the Eurozone, there is no natural government yield curve but there is a liquid swap market, so the LIBOR/swap curve is typically used for corporate debt.⁵ At the same time, domestic government debt trades relative to its local government yield curve. As long as the term structure factors completely span the interest rate risk and the risk relative to the benchmark, the choice of a benchmark should technically not matter for modeling the complete interest rate risk exposure of a bond portfolio. However, selecting a particular benchmark may have implications for correctly identifying and hedging critical factors.

14.1.2 Credit Spread Factors

All taxable non-U.S. Treasury securities⁶ trade at a spread relative to U.S. Treasury securities due to credit risk. That is, for every maturity, non-U.S. Treasury securities require that the issuer pay a different—typically larger—interest rate to borrow funds than the U.S. government. Hence, non-U.S. Treasury securities are referred to as “spread products.” As we explained in Chapter 13, the spread is due to several sources of credit risk: market-wide credit spread risk, credit event risk, and downgrade risk. Market-wide credit spread risk arises when events in the market affect the general level of credit spreads in a particular industry or rating category. Credit event risk happens when the issuer suffers changes that affect the company's fundamentals or the fundamentals of a sector. Downgrade risk is the risk that the issuer's credit rating will be downgraded.

Market-wide credit spread factors can include the average spread between swap rates and the Treasury curve. Credit event risk factors can, for example, be based on a historical rating migration matrix and average yield spreads estimated for bonds bucketed by rating,⁷ or by using the Merton (1974) framework for linking issuer credit quality to prices observed in the equity market.⁸

14.1.3 Currency Factors

A bond portfolio consisting of issues whose cash flow is not denominated in the domestic currency is subject to currency risk. However, the extent to which the variability in currencies affects the portfolio risk depends on the portfolio exposure relative to the benchmark's exposure to currency risk. The variability in currencies is often modeled using GARCH models.⁹ Factors that can represent currency risk typically include a volatility model such as GARCH, and may include a correlation model.¹⁰

14.1.4 Emerging Market Factors

Emerging market debt can be issued either in the local currency or in an external currency. The two types of debt are typically considered separately. Debt issued in the local currency is subject to local interest rate and spread factors. Debt issued in an external currency is considered riskier because, in contrast to debt issued in the local currency, the government cannot service it by raising taxes or printing money.

Emerging market debt is subject to risk due to the creditworthiness of the issuer, and the credit risk associated with an emerging market sovereign may be larger than the interest rate and all the other sources of risk combined. This risk can be gleaned from relative spreads of other debt issued by the particular issuer.

Emerging market spreads are typically measured relative to the swap curve. An emerging market bond denominated in U.S. dollars would be exposed to interest rate and spread factors in its local market as well as its emerging market credit spread.

14.1.5 Volatility Factors

Fixed income securities with embedded options are subject not only to term structure and credit risk but also to interest rate volatility risk because their returns depend on the market expectations about future interest rate volatility. The main idea in modeling volatility risk factors is to calibrate a stochastic interest rate model to match market prices observed for traded interest rate options. The variation over time of the model parameters can be used to determine the implied factors.¹¹ In the MSCI Barra (2003) model implementation, for example, the factor is the logarithm of the 10-year yield, which captures the volatility in the portion of the yield curve most relevant for mortgage-backed securities (MBSs) and bonds with embedded options. The exposure of a security to the volatility risk factor is the percentage change in price per percentage increase in the 10-year yield volatility.

14.1.6 Prepayment Factors

Certain fixed income securities such as securitized mortgages (MBS) are subject to prepayment risk. Because borrowers have the option to prepay the underlying loans, MBSs have characteristics similar to callable bonds. Prepayment models are typically used to project the prepayment rate on an MBS as a function of the security's characteristics and the current and past state of the market. To incorporate prepayment risk into the valuation model, one would calculate an implied prepayment model that uses market valuations to adjust modeled prepayment rates to match market expectations. Incorporating prepayment risk into a factor model reduces to introducing one or more factors to capture the changes in expectations of the market price of prepayment risk.

14.2.1 Stratification Approach

In Chapter 12, we gave an example of how an equity portfolio manager could use a stratification strategy with two factors: industry composition and size. In the case of fixed income portfolios, commonly used factors are (1) duration, (2) coupon, (3) maturity, (4) market sectors, (5) credit quality, (6) call features, and (7) sinking fund features.¹²

Consider, for example, the case of a portfolio manager who would like to match the credit quality of an investment-grade benchmark. The portfolio manager could consider four “buckets” or “cells” for credit quality: (1) AAA, (2) AA, (3) A, and (4) BBB. Remember that such benchmark could include as many as 6,000 bonds issues. From this large universe of bonds, the portfolio manager selects a few bonds from each of the four categories for credit quality in such a way that the relative weight of all the bonds in a particular bucket in the portfolio is the same as the relative weight of bonds with the corresponding credit quality in the benchmark.

Suppose the portfolio manager would like to include a second characteristic: duration. The portfolio manager could consider two buckets for effective duration: (1) less than or equal to five years, and (2) greater than five years. There will be a total of buckets or cells. (See Exhibit 14.1.) The portfolio manager can then select a few bonds from these 8 categories in such a way that the relative weight of all the stocks in a particular industry in his portfolio is the same as the relative weight of the industry in the benchmark.

The portfolio manager could use this framework to place bets on important factors. For example, the portfolio manager may overweight AAA bonds with duration greater than five years by increasing the percentage of his portfolio allocated to issues in this bucket (upper-right cell in Exhibit 14.1) compared to the percentage of AAA bonds with duration greater than five years in the benchmark.

Note that when the allocation to a bucket is the same percentage as the percentage of the benchmark this bucket represents, the allocation is said to be neutral. When all of the buckets match the benchmark, this is an indexing strategy. Unfortunately, indexing when it comes to fixed income securities is not as simple as equity indexing because of the cost of purchasing a large enough number of issues in a bucket to increase the likelihood that the performance of a portfolio's bucket matches the performance of the corresponding bucket in the benchmark. When the portfolio manager tolerates minor mismatches in the primary risk factors between the portfolio and the benchmark buckets with the exception of duration (i.e., duration must be neutral), the strategy is referred to as an enhanced indexing strategy.

While stratification helps fixed income portfolio managers structure their decisions, it has a number of shortcomings.

First, the number of buckets increases quickly with the addition of factors. For example, suppose the portfolio manager adds two more factors: Maturity with 3 buckets ([1] less than 5 years, [2] between 5 and 15 years, and [3] greater than 15 years) and Market Sectors with 4 buckets ([1] Treasury, [2] agencies, [3] corporate, and [4] agency MBS). The total number of buckets to consider becomes . In a portfolio of less than $100 million, including representative issues in each cell would require buying odd lots of issues and would increase transaction costs considerably.

Second, a portfolio allocation using the stratification approach does not make it easy to understand how the risk profile of the portfolio changes with changes in the allocation of issues to cells. It is difficult to assess immediately if a particular selection of a bond issue adds to or decreases overall portfolio risk relative to the benchmark, and how that compares to the selection of a different bond issue. Correlations between bond issues in different cells are also not taken into account—but it is possible that the selection of two issues in different buckets reduces overall risk even if by themselves the issues do not represent as attractive investments as other issues.

The optimization approach, described in the next section, addresses some of these issues.

14.2.2 Optimization Approach

An optimization formulation of the portfolio allocation problem requires a specification of¹³

1. Decision variables
2. Objective function
3. Constraints

A typical objective is to minimize the tracking error between the portfolio and a benchmark, but one can also, for example, maximize expected total return, and instead limit the tracking error to be within a prespecified risk budget by stating the limit as a constraint.

The decision variables are the amounts that should be allocated to different bond issues. These bond issues are a part of a tradable universe specified by the portfolio manager. The tradable universe may or may not include only issues from the benchmark; if it does not, then portfolio managers often specify as a constraint the maximum percentage of the portfolio that can be allocated to securities not in the benchmark (i.e., nonbenchmark issues).

By specifying a set of constraints, the portfolio manager can express views on the various factors that drive the return on the benchmark, or limit the risk by restricting the portfolio tracking error to be within a particular risk budget. As in the case of equities (see Chapter 11.1), a fixed income portfolio manager may specify upper and lower bounds on the amounts to be invested in individual securities or sectors, or on the maximum number of securities to be held in the portfolio.

Let us consider the following illustration.¹⁴ Suppose we are trying to allocate $100 million among 50 securities so as to track a composite index (specified by a client) made up of the Barclays Capital U.S. Treasury Index, the Barclays Capital U.S. Credit Index, and the Barclays Capital U.S. MBS Index on an equally weighted basis.¹⁵ The tradable universe is the securities in these three indexes.

To allocate the $100 million among 50 securities, we solve the following optimization problem:

Minimize the tracking error between the portfolio and the benchmark subject to the following constraints:

1. Total invested amount (sum of all allocations) = $100 million.
2. No more than 50 securities in the portfolio.¹⁶
3. No short sales.
4. The duration of the portfolio must not exceed the duration of the benchmark by more than 0.30 and must not be more than 0.15 below the benchmark duration.
5. Spreads should be between 50 and 80 basis points higher than the benchmark.
6. The monthly tracking error should not exceed 15 basis points.
7. The maximum active (under- or over-) weight relative to the benchmark should be no more than 3% per issuer.

Constraints (1)–(3) are typical constraints that are specified in a manner similar to the specifications in Chapter 11.1. Constraint (5) expresses the portfolio manager's view: it tilts the portfolio in the direction of the portfolio manager's view so that there is a mismatch between the benchmark and the portfolio when it comes to risk factors representing credit risk. Constraint (6) represents the risk budget, and Constraint (7) is imposed for diversification purposes.

The tracking error is calculated based on the covariances between the primary risk factors used in the model.¹⁷ Because of this, correlations between the securities in the portfolio and their impact on portfolio risk are taken into consideration during portfolio construction. This is an advantage to the optimization method compared to the stratification method described in the previous section.

The optimal portfolio with position amounts obtained from the optimal values of the decision variables and with a market value of $100 million is displayed in Exhibit 14.2. The portfolio has 50 securities, as was requested in the portfolio optimization formulation.

The portfolio allocations to different sectors are summarized in Exhibit 14.3. It is clear that the portfolio manager has taken a positive view on the government-related and corporate (specifically, industrials and financials) sectors. Their active weights in the portfolio are positive: 4.48%, 5.38%, and 2.32%, respectively. The other sectors are underweighted relative to the benchmark.

This portfolio allocation has a tracking error (optimal objective function value) of 7.48 basis points (bps) per month, which is within the risk budget of 15 bps set in the constraints, and is the minimum possible given the set of constraints imposed on the portfolio composition.

14.2.3 Portfolio Rebalancing

While portfolio construction provides a setting within which the stratification and the optimization approaches can be illustrated, as we have mentioned earlier in the book, in practice it happens more often that one needs to solve a different problem: how to rebalance an existing portfolio so that the same level of tracking error is maintained while the transaction costs of the rebalancing are as low as possible. This is because even portfolios constructed to have an initial tracking error within a prespecified range may drift away from the required characteristics to achieve that range over time as different events take place. Four common types of such events include:

1. The portfolio manager changes views.
2. Funds are added to or withdrawn from the portfolio.
3. Some of the attributes of the issues in the portfolio change (e.g., there is an upgrade or a downgrade for some of the securities, or there is a change in the duration of some of the issues in the portfolio).
4. Some of the attributes of the benchmark change.

A portfolio manager could apply the stratification approach and identify “buckets” within which the portfolio weights are beginning to deviate substantially from the target weights desired. However, a portfolio manager using this approach would not necessarily have a clear idea of the best way to prioritize correcting these deviations so as to reduce tracking error at a minimal cost to implement the rebalancing strategy. It is in situations like these—in which there are multiple possible transactions that should be considered to achieve a particular goal—that the optimization approach has clear advantages over the stratification approach because it is designed to weigh multiple alternatives, assess the trade-offs, and produce the optimal strategy.

When using an optimizer to rebalance a portfolio, one typically sets as an objective the minimization of transaction costs. A new set of constraints may be imposed, including a risk budget constraint on the amount of tracking error, a constraint on the maximum number of trades allowed, or a new set of requirements on the total portfolio allocation to particular sectors. The optimizer is able to identify optimal packages of transactions (sales and purchases) and calculate the resulting change in portfolio risk so that the portfolio manager can understand the risk adjustment benefit relative to the cost of executing the transactions.

Let us consider a portfolio rebalancing example based on the portfolio example from Section 14.2.2. The sector allocation summary in Exhibit 14.3 shows that corporate industrials were overweighted in the portfolio. Suppose the portfolio manager's view has changed, and he or she would like to rebalance the portfolio so that corporate industrials are not overweighted by more than 3%. He or she would also like to achieve this with 15 or fewer trades. These two constraints (corporate industrials active weight of less than or equal to 3% and number of trades less than or equal to 15) must be added to the optimization. Exhibit 14.4 shows the trades that accomplish the portfolio manager's goal. For example, the optimizer suggests selling some corporate industrial bonds and replacing them primarily with bonds in the government-related and corporate financial institutions categories. The total market value of the trades is $9,146,100. The tracking error declined from 7.48 basis points to 6.90 basis points. The decline in tracking error was aided by an increase in the total number of securities in the portfolio (seven securities were sold while eight were bought).