CHAPTER 23 BOND PORTFOLIO CONSTRUCTION CHAPTER SUMMARY In this chapter, we will see how to construct (build) portfolios. We begin with a brief review of the basic principles of the theory of portfolio selection and portfolio risk. Then we explain a key metric used in constructing, monitoring, and controlling portfolio risk: tracking error. In the last two sections of the chapter we then explain two approaches to portfolio construction: cell-based approach and multi-factor model approach. BRIEF REVIEW OF PORTFOLIO THEORY AND RISK DECOMPOSITION Portfolio theory as formulated by Harry Markowitz in the early 1950s provides guidance for the construction of portfolios. The Markowitz framework, also referred to as mean-variance analysis, states there are three parameters that are important in making portfolio selection decisions. The first is the mean or expected value of an asset’s return. The second is the variance of an asset’s return and it is this parameter that quantifies the risk of an individual asset. The covariance, the third input needed for the mean-variance model, is the degree to which the returns on two assets co-vary or change together. The correlation between the returns for two assets is equal to the covariance of the two assets divided by the product of their standard deviations. More specifically, for the simple two asset case (assets 1 and 2) the portfolio mean (or expected return) and the portfolio variance are respectively E(Rp) = w1 E(R1) + w2 E(R2) var(Rp) = w12 var(R1) + w22 var(R2) + 2w1 w2cov(R1, R2) where E(R1 ), E(R2), and E(Rp) are the expected return of asset 1, asset 2, and the portfolio, respectively. w1 and w2 are the weight (allocation) of assets 1 and 2, respectively, in the portfolio at the beginning of the period var(R1), var(R2), and var(Rp) are the variance of asset 1, asset 2, and the portfolio, respectively. cov(R1, R2) is the covariance between the return for asset 1 and 2 In contrast to the portfolio expected return, the portfolio variance is not merely a weighted average of the variance of the two assets. Instead, it also depends on the covariance. The relationship between the covariance and the correlation is as follows: where cor(R1, R2) is the correlation between the return of asset 1 and asset 2, and SD(R1) and SD(R2) are the standard deviation of the return of asset 1 and asset 2, respectively. Rearranging the above equation, we have: cov(R1,R2) =cor(R1,R2) SD(R1) SD(R2) Substituting the right-hand side of the latter covariance equation for the previous variance equation we get var(Rp) = w12 var(R1) + w22 var(R2) + 2 w1 w2 cor(R1,R2) SD(R1) SD(R2) Using this latter variance equation, it is easier to appreciate how the relationship between asset returns as measured by the correlation impacts the portfolio variance. A correlation of zero implies that the returns are uncorrelated. Let’s look at the following three cases: Correlation of +1: var(Rp) = w12 var(R1) + w22 var(R2) + 2 w1 w2 SD(R1) SD(R2) Correlation of 0: var(Rp) = w12 var(R1) + w22 var(R2) Correlation of –1: var(Rp) = w12 var(R1) + w22 var(R2) – 2 w1 w2 SD(R1) SD(R2) The maximum portfolio variance occurs when there is perfect correlation (i.e., correlation of +1) between the return of the two assets. The minimum portfolio variance occurs when asset returns have a correlation of –1. Financial theory as well as empirical evidence tells us that the portfolio variance can be decomposed into two general categories: systematic risk and idiosyncratic risk. The systematic risk is the risk that impacts the return on all assets in the portfolio. Idiosyncratic risk is the risk that is unique to the return of the assets in the portfolio. APPLICATION OF PORTFOLIO THEORY TO BOND PORTFOLIO CONSTRUCTION The Markowitz mean-variance framework has been applied to portfolio construction in two ways. The first is at the asset class level. The second application is the use of the mean-variance framework to select securities to construct portfolio. Moving from the implementation for constructing portfolios within an asset class requires the estimation of the inputs (mean, variance, and covariance) for all of the securities that are candidates for inclusion in the portfolio. If there are N securities that can be included in a portfolio, there are N variances and (N2 – N)/2 covariances to estimate. One model Markowitz proposed to explain the correlation structure among security returns assumed that the return on a security depends on an “underlying factor, the general prosperity of the market as expressed by some index.” In 1963, William Sharpe estimated the relationship between the return on the market index (the explanatory variable) and the return on the stock (the dependent variable). The regression model Sharpe estimated is referred to as the single index market model or simply the market model. The regression coefficient of the market model that is estimated is referred to as beta and is a measure of the sensitivity of a stock to general movements in the market index. The use of portfolio variance as a risk measure is an issue for two reasons. The first is that it is assumed that the return distribution for securities is normally distributed. If this holds, the variance is the appropriate measure of risk. However, empirical and theoretical evidence suggests that stock returns and bond returns are not normally distribution. The second attack on portfolio variance is one that follows from a discussion about the objective of portfolio managers: outperforming a benchmark. The measure used with this objective is a portfolio’s tracking error. This measure is the standard deviation or variance of the difference between the portfolio return and the benchmark return. The key point is that in constructing a portfolio where there is a benchmark, the relevant risk measure is not the portfolio variance but the portfolio tracking error. TRACKING ERROR When a portfolio manager’s benchmark is a bond market index, risk is not measured in terms of the variance or standard deviation of the portfolio’s return. Instead, risk is measured by the standard deviation of the return of the portfolio relative to the return of the benchmark index. This risk measure is called tracking error. Tracking error is also called active risk. Calculation of Tracking Error Tracking error is computed as follows: Step 1: Compute the total return for a portfolio for each period. Step 2: Obtain the total return for the benchmark index for each period. Step 3: Obtain the difference between the values found in Step 1 and Step 2. The difference is referred to as the active return. Step 4: Compute the standard deviation of the active returns. The resulting value is the tracking error. Exhibits 23-1 and 23-2 show the calculation of the tracking error for two hypothetical portfolios, A and B, assuming some benchmark index. Portfolio A’s monthly tracking error is 9.30 basis points with its the monthly returns closely tracking the small returns of the benchmark index. In contrast, for Portfolio B, the active returns are large with the monthly tracking error also large at 79.13 basis points. A tracking error is annualized when observations are monthly as follows: annual tracking error 5 monthly tracking error × A tracking error is annualized when observations are weekly as follows: annual tracking error 5 monthly tracking error × Two Faces of Tracking Error We call tracking error calculated from observed active returns for a portfolio backward-looking tracking error. It is also called ex-post tracking error, historical tracking error, and actual tracking error. A problem with using backward-looking tracking error in bond portfolio management is that it does not reflect the effect of current decisions by the portfolio manager on the future active returns and hence the future tracking error that may be realized. The portfolio manager needs a forward-looking estimate of tracking error to reflect the portfolio risk going forward. The way this is done in practice is by using the services of a commercial vendor or dealer firm that has modeled the factors that affect the tracking error associated with the bond market index that is the portfolio manager’s benchmark. These models are called multi-factor risk models. Given a manager’s current portfolio holdings, the portfolio’s current exposure to the various risk factors can be calculated and compared to the benchmark’s exposures to the factors. Using the differential factor exposures and the risks of the factors, a forward-looking tracking error for the portfolio can be computed. This tracking error is also referred to as predicted tracking error and ex ante tracking error. Given a forward-looking tracking error, a range for the future possible portfolio active return can be calculated assuming that the active returns are normally distributed. For example, assume the following: expected return for benchmark = 10% forward-looking tracking error relative to benchmark = 100 basis points It should be noted that there is no guarantee that the forward-looking tracking error at the start of, say, a year would exactly match the backward-looking tracking error calculated at the end of the year. There are two reasons for this. The first is that as the year progresses and changes are made to the portfolio, the forward-looking tracking error estimate would change to reflect the new exposures. The second is that the accuracy of the forward-looking tracking error at the beginning of the year depends on the extent of the stability of the variances and correlations that commercial vendors use in their statistical models to estimate forward-looking tracking error. Tracking Error and Active versus Passive Strategies A passive strategy relative to the benchmark index occurs when a forward-looking tracking error is very small. When the forward-looking tracking error is large, the manager is pursuing an active strategy. CELL-BASED APPROACH TO BOND PORTFOLIO CONSTRUCTION Under the cell-based approach to constructing a bond portfolio, the benchmark is divided into cells, each cell representing a different characteristic of the benchmark. The most common cells used to break down a benchmark are (1) duration, (2) coupon, (3) maturity, (4) market sectors, (5) credit quality, (6) call factors, and (7) sinking fund features. The last two factors are particularly important because the call and sinking fund features of an issue will affect its performance. For a portfolio manager pursuing a passive strategy, the objective is to match the performance of the benchmark. Following the cell-based approach, the manager selects from all the issues in the bond index one or more issues in each cell that can be used to represent the entire cell. The total dollar amount purchased of the issues from each cell will be based on the percentage of the bond index’s total market value that the cell represents. For a portfolio manager pursuing an active strategy, the manager will intentionally mismatch the dollar amount purchased for specific cells where a view is taken. The number of cells that the indexer uses will depend on the dollar amount of the portfolio. In practice, when the cell-based approach for bond portfolio construction is used, once a model portfolio is constructed, the portfolio’s track error can be estimated. The cell-based approached ignores how mismatches impact portfolio risk as a result of cross-correlation associated with the risks of each cell. Complications in Bond Indexing There are three forms of bond indexing: pure bond index matching, enhanced indexing matching primary risk factors, and enhanced indexing allowing for minor risk-factor mismatches. It almost impossible to implement a pure bond indexing strategy and it is not simple to do so for the other two bond indexing strategies. These difficulties apply to both the cell-based and multi-factor model approaches to portfolio construction. In a pure bond indexing strategy, the portfolio manager must purchase all of the issues in the bond index according to their weight in the benchmark index. Instead of purchasing all issues in the bond index, the manager may purchase just a sample of issues using the cell-based approach. This moves the strategy from being a pure bond indexing strategy to an enhanced bond indexing strategy with minor mismatches in the primary risk factors. In terms of the cell-based approach, the primary risk factors are the characteristics or cells. A portfolio manager faces several other logistical problems in seeking to construct an indexed portfolio. First, the prices for each issue used by the organization that publishes the index may not be execution prices available to the indexer. Second, the prices used by organizations reporting the value of indexes are based on bid prices causing a bias between the performance of the bond index and the indexed portfolio that is equal to the bid-ask spread. There are logistical problems unique to certain sectors in the bond market. Because of the illiquidity of this sector of the bond market, not only may the prices used by the organization that publishes the index be unreliable, but many of the issues may not even be available. Next, consider the agency mortgage-backed securities market. There are more than 800,000 agency pass-through issues. The organizations that publish indexes lump all these issues into a few hundred generic issues. Finally, recall that the total return depends on the reinvestment rate available on coupon interest. If the organization publishing the index regularly overestimates the reinvestment rate, the indexed portfolio could underperform the bond index by a significant number of basis points a year. PORTFOLIO CONSTRUCTION WITH MULTI-FACTOR MODELS Multi-factor models are statistical models that are used to estimate a security’s expected return based on the primary drivers affecting the return on securities. The primary drivers of returns are referred to as risk factors or simply factors. These models are also called multi-factor risk models or just factor models. Our focus is on how multi-factor models for bonds are used to identify the sources of a bond portfolio’s risk and the how to employ them to construct bond portfolios. Risk Decomposition Exhibit 23-3 illustrates how a multi-factor model can be used to identify the risk exposure of a portfolio relative to a benchmark. This portfolio was constructed using a multi-factor model combined with an optimization model. The risk exposure for this portfolio is measured in terms of tracking error. The analysis of the portfolio begins with a comparison of the portfolio to that of the benchmark. Although the information contained in Exhibit 23-4 about the allocation based on percentage market value of sector relative to the benchmark provides a good starting point for our analysis, the information has limited value because it is not known how the exposures to the sectors are related to the exposures to the risk factors that drive the portfolio’s return. Thus, the portfolio manager must look beyond a naïve assessment of portfolio risk relative to the benchmark based on percentage allocation to sectors. Exhibit 23-5 provides information about the relative exposure to interest rate risk as measured by duration, spread risk as measured by spread duration, and call/prepayment risk as measured by vega, as well as the convexity. More information about the portfolio’s relative risk exposure to interest rate risk can be obtained by looking at the contribution to duration for the portfolio and the benchmark. This is shown in Exhibit 23-6. As can be seen, the major reason for the slightly longer duration of the portfolio relative to the benchmark is mainly attributable to the duration of the Treasury securities selected for the portfolio. Suppose that a portfolio has more exposure to a risk factor than the benchmark. This would mean if that risk factor moves, the portfolio will have a greater movement than the benchmark. To address this problem, volatility must be taken into consideration. Exhibit 23-7 shows the monthly volatility of risk factor categories. Isolated risk in this exhibit displays the tracking error/volatility of different exposures of the portfolio in isolation. The risk factor “securitized spread” is the exposure to changes in the spreads in the agency MBS market. The risk factor “volatility” is the risk associated with changes in interest rate volatility. How can we determine the monthly tracking error for the portfolio given the monthly tracking error for each risk factor exposure in Exhibit 23-7? Assuming a zero correlation between any pair of risk factors, the portfolio isolated tracking error attributable to systematic risk is found by squaring each isolated tracking error for each risk factor, summing them, and then taking the square root. That is, for the general case where there are K risk factors is Portfolio isolated systematic TE = [(TE1)2 + (TE2)2 + … + (TEK)2]1/2 where TE denotes tracking error and the subscript denotes the risk factor. The assumption that there is zero correlation between every pair of risk factors is unrealistic. To address this, correlations or covariances must be brought into the analysis. In the case of tracking error, let’s consider the case where there are only two risk factors, F1 and F2. Then the portfolio tracking error is equal to Portfolio TE = [(TEF1)2 + (TEF2)2 + 2 Cov(F1,F2)]1/2 where Cov(F1,F2) is the covariance between risk factor exposures 1 and 2. Exhibit 23-8 gives a breakdown of the standard deviation of the returns for the portfolio and the benchmark in terms of systematic risk and idiosyncratic risk. The portfolio has greater systematic and idiosyncratic risk than the benchmark. For the total risk of the portfolio and the benchmark, the standard deviation of the portfolio and benchmark can be calculated as follows: Total risk (volatility of returns) = [(Systematic risk)2 + (Idiosyncratic risk)2]0.5 The portfolio tracking error is Portfolio tracking error = [(Systematic TE)2 + (Idiosyncratic TE)2]0.5 An extremely important point is that the tracking error (and not the idiosyncratic risk) is what the manager must consider in portfolio construction and monitoring. As with equities where a portfolio beta is computed that shows the movement of an equity portfolio in response to a movement in some equity market index (such as the S&P 500), a beta can be computed for a bond portfolio. A beta-type measure can be estimated for each risk factor. For example, consider the risk factor measuring changes in the level of the yield curve which is the portfolio’s duration. A duration beta can be calculated as follows: Exhibit 23-9 provides information about the portfolio risk across the different categories of risk factors. Shown are the systematic risk and the idiosyncratic risk and the seven components of systematic risk. The second column shows the isolated tracking error. The contribution to tracking error for each group of risk factor is shown in the third column. The fourth column gives a new metric, liquidation effect on tracking error. A detailed analysis of the systematic and idiosyncratic risk applied at the asset class level rather than at the individual risk factor level is provided in Exhibit 23-10. The five asset classes are shown in the first column and in the second column the underweighting or overweighting of each asset class (referred to as the “net market weight”) are shown. The last three columns report the contribution to tracking error for systematic risk, idiosyncratic risk, and total risk. Another important observation to take away from the analysis reported in Exhibit 23-10 is that corporate bonds are the major contributor to idiosyncratic risk due to the overweighting of this sector, carrying relatively higher idiosyncratic risk at the individual security level. An analysis similar to the decomposition of risk shown in Exhibit 23-9 by asset class instead of risk factor group is shown in Exhibit 23-11. Notice that the isolated tracking error for both the Treasury and corporate asset classes exceeds that of the portfolio tracking error. This occurs because exposures to certain asset classes in the portfolio are acting as hedges to certain other asset classes (since Treasuries and corporate could also be hedging each other). The hedging effect can be seen in the fourth column where the liquidation effect on tracking error is shown for each asset class in Exhibit 23-11. In the Barclays Capital model, there is a different yield curve used for government products. With the exception of Treasuries, the other asset classes have exposure to the swap spread factors on top of the Treasury curve. By decomposing the swap curve into the Treasury curve and swap spreads, the Barclays Capital model gives portfolio managers the flexibility to analyze their spread risk over the Treasury or the swap curve depending on their preferences. There are different measures to look at the exposure to changes in the shape of the yield curve. The most common is key rate duration. Six key rate durations with respect to the U.S. Treasury curve, as well as the option-adjusted or effective convexity, are shown in Exhibit 23-12. Key rate duration is the approximate percentage change in the portfolio value or benchmark value for a 100 basis point change in the rate for a particular maturity holding all other rates constant. In terms of mismatch, it is the approximate differential percentage change in the portfolio relative to the benchmark for a 100 basis point in the rate for a particular maturity holding all other rates constant. Assuming that the factor volatility represents a typical movement for the factor (i.e., key interest rate), then the isolated impact of that movement on the return of a portfolio (versus the benchmark) can be found as follows: Return impact of a typical movement = − (Net key rate duration) × Typical rate movement Exhibit 23-13 shows the exposure of the portfolio to the change in the swap spread. The swap spread is the difference between the swap rate and the Treasury rate. All securities in the portfolio except Treasuries expose the portfolio to this risk. Portfolio Construction Using a Multi-Factor Model and an Optimizer As with the cell-based approach to portfolio construction, a manager has views on the various primary factors driving the return on the benchmark and wants to position the portfolio to capitalize on those expectations. A multi-factor model is used in conjunction with an optimizer to construct a portfolio. In using an optimizer, the optimal value for all of the variables that the decision maker seeks is the output for the model. The decision maker specifies the variables (i.e., decision variables), an objective function, and constraints. Given all of that information, the optimizer finds the optimal value for all of the decision variables. The portfolio manager must specify the objective function. This is the measure or quantity that is to be minimized or maximized. In portfolio construction using multi-factor models, the measure to be optimized is the portfolio’s tracking error. The manager wants that measure to be minimized. The optimization of the objective function is typically done subject to constraints. Portfolio Rebalancing While it is common to illustrate portfolio construction starting with a position of cash and building a portfolio of securities, in practice the more common task is to rebalance an existing portfolio. A multi-factor model along with an optimizer can be used to efficiently rebalance the portfolio so as to realign the portfolio that has drifting away from the characteristics of the benchmark over time and/or tilt the portfolio to reflect a manager’s new views. Rebalancing is also required when a portfolio manager receives additional funds from a client or portfolio cash inflows or when a client withdraws funds. Exhibit 23-14 shows the trades that would have been recommended by the optimizer at the time. Before the manager executes the package of trades proposed in Exhibit 23-14, there must be an evaluation of the change in risk exposure. KEY POINTS The Markowitz mean-variance model is used to construct portfolios. The inputs required are the mean (expected return) and variance for each security that is a candidate for inclusion in the portfolio and the covariance or correlation between each pair of securities. While the insights provided by Markowitz mean-variance model are important, there are several issues that limit its application to the construction of bond portfolios. Tracking error, or active risk, is the standard deviation of a portfolio’s return relative to the benchmark index. There are two types of tracking error—backward-looking tracking error, historical tracking error, and forward-looking tracking error. Backward-looking tracking error is calculated based on the actual performance of a portfolio relative to a benchmark index. Forward-looking tracking error is an estimate of how a portfolio will perform relative to a benchmark index in the future. Forward-looking tracking error is used in risk control and portfolio construction. The higher the forward-looking tracking error, the more the manager is pursuing a strategy in which the portfolio has a different risk profile than the benchmark and there is, therefore, greater active management. In the cell-approach to bond portfolio construction, the benchmark is divided into cells which are the characteristics of the benchmark. The portfolio then picks one or more securities to either match or mismatch a cell depending on whether the strategy is a passive or active strategy. In the former, the mismatch is based on the manager’s view. There are complications in creating an indexed bond portfolio that are not faced by equity portfolio managers who want to create an indexed stock portfolio. Multi-factor models (also called multi-factor risk models or factor models) are statistical models that are used to estimate a security’s expected return based on the primary drivers (factors) affecting the return on securities. The systematic risk factors in multi-factor model include yield curve risk, swap spread risk, volatility risk, government-related spread risk, corporate spread risk, and securitized spread risk. The risk exposure in a factor model is measured in terms of tracking error. Multi-factor models combined with an optimization model are used to construct a portfolio that minimizes tracking error subject to the constraints imposed by the manager (client imposed or self-imposed) and embodying the views of the manager. ANSWERS TO QUESTIONS FOR CHAPTER 23 (Questions are in bold print followed by answers.) 1. What is the major insight provided by the Markowitz framework in portfolio theory? The main insight of the Markowitz framework is that when assets are combined to create a portfolio, the portfolio’s risk (as measured by the portfolio variance) is not merely some weighted average of the risks of the individual assets comprising the portfolio. Instead, the portfolio’s risk depends on the covariance or correlation of the returns between each pair of assets comprising the portfolio. The covariance is the degree to which the returns on two assets co-vary or change together. The correlation is analogous to the covariance between the expected returns for two assets. Specifically, the correlation between the returns for two assets is equal to the covariance of the two assets divided by the product of their standard deviations. 2. Answer the below questions. a. Explain whether you agree or disagree with the following statement: “It is the covariance not the correlation that is important in the mean-variance model for portfolio selection.” One would disagree with the statement because the covariance and correlation can be defined in terms of one another. Specifically, the correlation between the returns for two assets is equal to the covariance of the two assets divided by the product of their standard deviations. This is given below: . We see from this equation that an increase in the covariance occurs when there is also an increase in correlation. Thus, it follows that both are important in the mean-variance model for portfolio selection. b. Explain whether you agree or disagree with the following statement: “In the mean-variance framework, the variance is lower the higher the correlation between the assets in the portfolio.” Consider the below equation representing a simple portfolio of two assets (asset 1 and asset 2): var(Rp) = w12 var(R1) + w22 var(R2) + 2 w1 w2 cor(R1,R2) SD(R1) SD(R2). From this equation, a higher positive correlation is associated with a higher portfolio variance. However, suppose the correlation is negative. Then a greater negative correlation is associated with a lower portfolio variance. Thus, the statement is only true if by “higher” we mean greater negative correlation. Since higher implies a greater positive number, one can arguably disagree with the statement. 3. What are the two ways in which the Markowitz mean-variance framework has been used by investors? The Markowitz mean-variance framework has been applied to portfolio construction in two ways. The first is at the asset class level where investors make an asset allocation decision. This is the decision as to how to allocate funds amongst the major asset classes (stocks, bonds, cash, real estate, and alternative assets). This has probably been the major use of the Markowitz framework. The second application is the use of the mean-variance framework to select securities to construct portfolio. Although the mean-variance framework has been used in equity portfolio management for a good number of years, it has seen very limited use in bond portfolio. Moving from the implementation for constructing portfolios within an asset class requires the estimation of the inputs (mean, variance, and covariance) for all of the securities that are candidates for inclusion in the portfolio. These inputs are not easily estimated and there is an entire literature dealing with the issues associated with estimation risk. 4. What are the difficulties of implementing the Markowitz mean-variance framework in constructing portfolios? The implementation for constructing portfolios requires the estimation of the inputs (mean, variance, and covariance) for all of the securities that are candidates for inclusion in the portfolio. These inputs are not easily estimated and this presents a variety of difficulties. For example, the number of inputs that must be calculated is enormous. For example, if there are N securities that can be included in a portfolio, there are N variances and (N2 – N)/2 covariances to estimate. Hence, for a portfolio of just 50 securities that could be included in a portfolio, there are 1,224 covariances that must be calculated. For 100 securities, there are 4,950 covariances. Holding aside estimation risk, the enormity of the estimations that must be made was clear to Markowitz. It was clear to Markowitz that some kind of model of covariance structure was needed for the practical implementation of the theory to large portfolios. He did little more than point out the problem and suggest some possible models of covariance. The use of portfolio variance as a risk measure presents additional difficulties. First, we have to make an assumption about the return distribution. If we assume normal distribution of returns, then the variance is the appropriate measure of risk. However, empirical and theoretical evidence suggests that stock returns and bond returns are not normally distribution. As a result, extensions of the Markowitz optimization framework have been suggested that include other risk measures such skewness and kurtosis. Second, we use of portfolio variance is questioned in regards to the objective of portfolio managers: outperforming a benchmark. The measure used with this objective is a portfolio’s tracking error. This measure is the standard deviation or variance of the difference between the portfolio return and the benchmark return. The key point is that in constructing a portfolio where there is a benchmark, the relevant risk measure is not the portfolio variance but the portfolio tracking error. Finally, consider the notion of the decomposition of portfolio total risk (i.e., portfolio variance) into systematic risk and idiosyncratic risk. Studies of the stock market indicate that it does not take more than 25 or so randomly selected stocks to remove most of the idiosyncratic risk of a portfolio. That is, a randomly selected portfolio of stocks is mostly exposed to systematic risk. However, when risk is measured in terms of tracking error, it takes a considerably larger number of stocks to remove idiosyncratic risk. Typically, this is not the case when dealing with bonds where the benchmark is one of the standard bond market indexes. 5. What was the purpose for William Sharpe’s development of the single index market model? It was clear to Markowitz that some kind of model of covariance structure was needed for the practical implementation of the theory to large portfolios. He did little more than point out the problem and suggest some possible models of covariance. One model Markowitz proposed to explain the correlation structure among security returns assumed that the return on a security depends on an “underlying factor, the general prosperity of the market as expressed by some index.” We might say that the purpose of Sharpe’s development of the single index market model was in response to Markowitz’s proposal. Thus, in response, Sharpe developed a model in 1963 where he tested the suggestion made by Markowitz through an examination of how stock returns tend to go up and down together with a general stock market index. Specifically, Sharpe estimated the relationship between the return on the market index (the explanatory variable) and the return on the stock (the dependent variable). The regression model Sharpe estimated is referred to as the single index market model or simply the market model. The regression coefficient of the market model that is estimated is referred to as beta and is a measure of the sensitivity of a stock to general movements in the market index. 6. Why is the tracking error more important than portfolio variance of returns when a portfolio manager’s performance is measured versus a benchmark? A key point is that in constructing a portfolio where there is a benchmark, the relevant risk measure is not the portfolio variance but the portfolio tracking error. When performance is measured against a benchmark, tracking error is more important than portfolio variance because on tracking error measures how closely a portfolio follows the index to which it is benchmarked. The most common measure is the root-mean-square of the difference between the portfolio and index returns. Many portfolios are managed to a benchmark, normally an index. Some portfolios are expected to replicate, before trading and other costs, the returns of an index exactly (an index fund), while others are expected to 'actively manage' the portfolio by deviating slightly from the index in order to generate active returns or to lower transaction costs. Tracking error (also called active risk) is a measure of the deviation from the benchmark; the aforementioned index fund would have a tracking error close to zero, while an actively managed portfolio would normally have a higher tracking error. Dividing portfolio active return by portfolio tracking error gives the information ratio, which is a risk adjusted performance metric. 7. What is tracking error? When a portfolio manager’s benchmark is a bond market index, risk is not measured in terms of the standard deviation of the portfolio’s return. Instead, risk is measured by the standard deviation of the return of the portfolio relative to the return of the benchmark index. This risk measure is called tracking error. Tracking error is also called active risk. Tracking error is computed as follows. First, compute the total return for a portfolio for each period. Second, obtain the total return for the benchmark index for each period. Third, obtain the difference between the return values for portfolio and index for each period. The difference for each period is referred to as the active return for that period. Finally, compute the standard deviation of the active returns. The resulting value is the tracking error. One should not that the tracking error measurement is in terms of the observation period. If monthly returns are used, the tracking error is a monthly tracking error. If weekly returns are used, the tracking error is a weekly tracking error. Tracking error is annualized as follows. When observations are monthly: annual tracking error = monthly tracking error × . When observations are weekly: annual tracking error = monthly tracking error × . 8. Explain why backward-looking tracking error has limitations for estimating a portfolio’s future tracking error. A portfolio’s backward-looking tracking error is computed based on actual active returns and reflect the portfolio manager’s decisions during the observation period with respect to the factors that affect tracking error. Consequently, one limitation with using backward-looking tracking error in bond portfolio management is that it does not reflect the effect of current decisions by the portfolio manager on the future active returns and hence the future tracking error that may be realized. Another limitation is that the backward-looking tracking error will have little predictive value and can be misleading regarding portfolio risks going forward. 9. Why might one expect that for a manager pursuing an active management strategy that the backward-looking tracking error at the beginning of the year will deviate from the forward-looking tracking error at the beginning of the year? The portfolio manager needs a forward-looking estimate of tracking error to reflect the portfolio risk going forward. The way this is done in practice is by using the services of a commercial vendor or dealer firm that has modeled the factors that affect the tracking error associated with the bond market index that is the portfolio manager’s benchmark. Given a manager’s current portfolio holdings, the portfolio’s current exposure to the various risk factors can be calculated and compared to the benchmark’s exposures to the factors. Using the differential factor exposures and the risks of the factors, a forward-looking tracking error for the portfolio can be computed. Given a forward-looking tracking error, a range for the future possible portfolio active return can be calculated assuming that the active returns are normally distributed. There is no guarantee that the forward-looking tracking error at the start of, say, a year would exactly match the backward-looking tracking error calculated at the end of the year. There are two reasons for this. The first is that as the year progresses and changes are made to the portfolio, the forward-looking tracking error estimate would change to reflect the new exposures. The second is that the accuracy of the forward-looking tracking error at the beginning of the year depends on the extent of the stability in the variances and correlations that commercial vendors use in their statistical models to estimate forward-looking tracking error. These problems notwithstanding, the average of forward-looking tracking error estimates obtained at different times during the year can be reasonably close to the backward looking tracking error estimate obtained at the end of the year. The forward-looking tracking error is useful in risk control and portfolio construction. The manager can immediately see the likely effect on tracking error of any intended change in the portfolio. Thus scenario analysis can be performed by a portfolio manager to assess proposed portfolio strategies and eliminate those that would result in tracking error beyond a specified tolerance for risk. 10. Answer the below questions. (a) Compute the tracking error from the following information: Month 2001 Portfolio A’s Return (%) Lehman Aggregate Bond Index Return (%) January 2.15 1.65 February 0.89 –0.10 March 1.15 0.52 April –0.47 –0.60 May 1.71 0.65 June 0.10 0.33 July 1.04 2.31 August 2.70 1.10 September 0.66 1.23 October 2.15 2.02 November –1.38 –0.61 December –0.59 –1.20 The tracking error is the standard deviation of the active returns where an active return is the portfolio A’s return minus the benchmark’s return for each month. The below table has each active return in the “Active Return” column. [Note that when subtracting a negative index return from a portfolio return, the negative return is actually added to the portfolio return to get the active return. For example, for February, we have 0.89% – 0.10% = 0.89% + 0.10% = 0.99%).] Month 2001 Portfolio A’s Return Lehman Aggregate Bond Index Return Active Return Differences Squared January 2.15% 1.65% 0.50% 0.0707(%2) February 0.89% –0.10% 0.99% 0.5713(%2) March 1.15% 0.52% 0.63% 0.1567(%2) April –0.47% –0.60% 0.13% 0.0109(%2) May 1.71% 0.65% 1.06% 0.6820(%2) June 0.10% 0.33% –0.23% 0.2155(%2) July 1.04% 2.31% –1.27% 2.2625(%2) August 2.70% 1.10% 1.60% 1.8655(%2) September 0.66% 1.23% –0.57% 0.6467(%2) October 2.15% 2.02% 0.13% 0.0109(%2) November –1.38% –0.61% –0.77% 1.0084(%2) December –0.59% –1.20% 0.61% 0.1413(%2) Sum of Portfolio Returns = 2.81% Mean Active Return = 0.2342% Variance (sum of differences squared / 11) = 0.6947(%2) Standard Deviation = Tracking Error = 0.8335% Tracking error in basis points = 83.35 Tracking error in basis points annualized = 288.74 To compute the standard deviation of these active returns, we subtract the average (or mean) active return from each active return, and then square each difference. Each difference squared value is given in the table above in the “Differences Squared” column. We then divided this sum by 12 – 1 = 11. We then multiply by 100 to convert to basis points. One basis point equals 0.0001 or 0.01%. We can then annualize the monthly basis points by multiplying by the square root of 12. At the bottom of the above table we list details including the mean active return, variance, standard deviation or tracking error (in terms of both percentage and basis points), and the annualized tracking error (in terms of basis points). (b) Is the tracking error computed in part (a) a backward-looking or forward-looking tracking error? The tracking error computed in part (a) is backward-looking because it is calculated based on the actual active returns observed for a portfolio is prior periods. Calculations computed for a portfolio based on a portfolio’s actual active returns reflect the portfolio manager’s decisions during the observation period with respect to the factors that affect tracking error. (c) Compare the tracking error found in part (a) to the tracking error found for Portfolios A and B in Exhibits 23-1 and 23-2. What can you say about the investment management strategy pursued by this portfolio manager? The tracking error found for our problem is greater especially compared to Portfolio A. A greater tracking error means greater deviation from the benchmark. This is seen if we compare active return values from our table with the greater active return values found in the exhibits. For our problem, it appears the manager may be employing a high-risk strategy to enhance the indexed portfolio’s return. This strategy is commonly referred to as enhanced indexing or indexing plus. 11. Assume the following: benchmark index = Salomon Smith Barney BIG Bond Index expected return for benchmark index = 7% forward-looking tracking error relative to Lehman Aggregate Bond Index = 200 basis points Assuming that returns are normally distributed, complete the following table: Number of Standard Deviations Range for Portfolio Active Return Corresponding Range for Portfolio Return Probability 1 2 3 With an expected return of 7% and a standard deviation of 200 basis points or 2%, then a normal distribution implies there is about a 67% probability that values will be found between one standard deviation of either side of the mean. Thus, for a standard deviation of 1, the range on either side of the mean for portfolio active return is 1 standard deviation times 2% = 2%. The 2% deviation will be on both the left and right side of the 7% mean value. Thus, with a portfolio mean return of 7%, the corresponding range for portfolio return will be from 5% (7% – 2% = 5%) left of the mean value to 9% (7% + 2% = 9%) right of the mean value Similarly, for a standard deviation of 2, the range on either side of the mean is 2 standard deviation times 2% = 4%. With a portfolio mean active return of 7%, the corresponding range for portfolio return will be from 7% – 4% = 3% to 7% + 4% = 11%. A normal distribution implies there a 96% probability that values will be found between two standard deviation of either side the mean. Likewise, for a standard deviation of three, the range on either side of the mean is 3 standard deviation times 2% = 6%. With a portfolio mean active return of 7%, the corresponding range for portfolio return will be 7% – 6% = 1% and 7% + 6% = 13%. A normal distribution implies there a 99% probability that values will be found between two standard deviation of either side the mean. The above values can all be found in the below table. Number of Standard Deviations Range for Portfolio Active Return Corresponding Range for Portfolio Return Probability 1 2% 5%–9% 67% 2 4% 3%–11% 95% 3 6% 1%–13% 99% 12. At a meeting between a portfolio manager and a prospective client, the portfolio manager stated that her firm’s bond investment strategy is a conservative one. The portfolio manager told the prospective client that she constructs a portfolio with a forward-looking tracking error that is typically between 250 and 300 basis points of a client-specified bond index. Explain why you agree or disagree with the portfolio manager’s statement that the portfolio strategy is a conservative one. If the chosen benchmark is the desired norm, then greater deviation from the norm implies more risk taking, i.e., less conservative than claimed by the portfolio manager. Regardless, it appears the manager is pursuing an active strategy that involves risk taking. More details are given below. First, one would expect a higher tracking error over a longer horizon. Let’s assume the forward-looking tracking error given in our problem (between 250 and 300 basis points of a bond index) is an annual tracking error. Even for this longer horizon, 250 to 300 basis points represent a large tracking error (especially compared to a zero tracking error which would be obtained if one just mimicked the benchmark). However, the tracking error is also unique to the benchmark used. If an improper benchmark is used then the tracking error measure may not be too meaningful. Second, the strategy is not passive. When a portfolio is constructed to have a forward-looking tracking error of zero, the manager has effectively designed the portfolio to replicate the performance of the benchmark. If the forward-looking tracking error is maintained for the entire investment period, the active return should be close to zero. Such a strategy—one with a forward-looking tracking error of zero or very small—indicates that the manager is pursuing a passive strategy relative to the benchmark index. Third, when the forward-looking tracking error is large the manager is pursuing an active strategy. The larger the deviation from the chose benchmark, the larger the tracking error and thus greater risk taking can be inferred. Forward-looking tracking error indicates the degree of active portfolio management being pursued by a manager. Therefore, it is necessary to understand what factors (referred to as risk factors) affect the performance of a manager’s benchmark index. The degree to which the manager constructs a portfolio that has exposure to the risk factors that is different from the risk factors that affect the benchmark determines the forward-looking tracking error. 13. What is meant by tracking error due to systematic risk factors? By tracking error due to systematic risk factors, we mean tracking error caused by factors that affect the return of securities in the benchmark in varying degrees. More details are given below. When a portfolio manager’s benchmark is a bond market index, risk is not measured in terms of the standard deviation of the portfolio’s return. Instead, risk is measured by the standard deviation of the return of the portfolio relative to the return of the benchmark index. This risk measure is called tracking error. Tracking error is also called active risk. Forward-looking tracking error indicates the degree of active portfolio management being pursued by a manager. Therefore, it is necessary to understand what factors (including systematic risk factors) affect the performance of a manager’s benchmark index. The degree to which the manager constructs a portfolio that has exposure to the risk factors that is different from the risk factors that affect the benchmark determines the forward-looking tracking error. The risk factors affecting the Lehman Brothers Aggregate Bond Index have been investigated by various researchers. The risk factors can be classified into two types: systematic risk factors and nonsystematic risk factors. Systematic risk factors are forces that affect all securities in a certain category in the benchmark index. Nonsystematic factor risk is the risk that is not attributable to the systematic risk factors. When we speak of tracking error due to systematic risk factors, we have two factors in mind because systematic risk factors can be divided into two categories: term structure risk factors and non-term structure risk factors. Term structure risk factors are risks associated with changes in the shape of the term structure (level and shape changes). Non-term structure risk factors include the following: sector risk, quality risk, optionality risk, coupon risk, MBS sector risk, MBS volatility risk, and MBS prepayment risk. 14. You are reviewing a report by a portfolio manager that indicates that a fund’s predicted (forward-looking) tracking error is 94.87 basis points. Furthermore, it is reported that the predicted tracking error due to systematic risk is 90 basis points and the predicted tracking error due to non-systematic risk is 30 basis points. Why doesn’t the sum of these two tracking error components total up to 94.87 basis points? The predicted tracking error is 94.87 basis points. The two major risk categories are systematic and non-systematic risks. For our portfolio, they are respectively 90 basis points and 30 basis points. Now this might seem confusing since adding these two risks we do not get to the predicted tracking error of 94.87 basis points for the portfolio. The reason is that these risk measures are standard deviations and therefore they are not additive. However, the variances are additive. The implicit assumption in this calculation is that there is no correlation or covariance between any of the two components of the risk factors. Consequently, the variance of the two major risk components is: Predicted tracking error for systematic risks2 = variance for systematic risks = 902 = 8,100 Predicted tracking error for nonsystematic risks2 = variance for non-systematic risks = 302 = 900 The total variance is 8,100 + 900 = 9,000. The square root of the total variance is 94.86833 basis points, which rounded off to 94.87 basis points is equal to the predicted tracking error for the portfolio. 15. What are the drawbacks of the cell-based approach for bond portfolio construction? Let us first describe the cell-based approach. Under the cell-based approach, the benchmark is divided into cells, each cell representing a different characteristic of the benchmark. The most common cells used to break down a benchmark are (1) duration, (2) coupon, (3) maturity, (4) market sectors, (5) credit quality, (6) call factors, and (7) sinking fund features. The number of cells that the indexer uses will depend on the dollar amount of the portfolio. In a portfolio of less than $100 million, for example, using a large number of cells entails a problem. The “drawback” faced by the manager is it would require purchasing odd lots of issues. This increases the cost of buying the issues to represent a cell and thus would increase the portfolio’s tracking error. Reducing the number of cells to overcome this problem would increase the portfolio’s tracking error in an undesirable way. If the tracking error is unsatisfactory, the manager must alter the portfolio so that the tracking error is within the acceptable range specified by the manager or the client. This is often not a simple process since modifying the portfolio can result in unintended bets (or views) by changing the allocation to cells. This adds another “drawback” to the equation. Moreover, the cell-based approach faces an additional “drawback” as it ignores how mismatches impact portfolio risk as a result of cross-correlation associated with the risks of each cell. For example, it is possible that there may be large mismatches between two of the cells (characteristics) but the correlation of the risks associated with the cells may result in a very small increase in the portfolio’s tracking error. Fortunately, these drawbacks of the cell-based approach can be dealt with using the more quantitative approach, the multi-factor model approach. 16. Why is it difficult to build a portfolio in pursuing a pure bond indexing strategy? While it is not simple to build a portfolio for enhanced indexing strategies, it is even more difficult to implement a pure bond indexing strategy. These grave difficulties apply to both the cell-based and multi-factor model approaches to portfolio construction. Below we attempt to describe why. In a pure bond indexing strategy, the portfolio manager must purchase all of the issues in the bond index according to their weight in the benchmark index. However, substantial tracking error will result from the transaction costs (and other fees) associated with purchasing all the issues and reinvesting cash flow (maturing principal and coupon interest). A broad-based market index could include more than 6,000 issues, so large transaction costs make this strategy impractical. In addition, some issues in the bond index may not be available at the prices used in constructing the index. Instead of purchasing all issues in the bond index, the manager may purchase just a sample of issues using the cell-based approach. This moves the strategy from being a pure bond indexing strategy to an enhanced bond indexing strategy with minor mismatches in the primary risk factors. Although this approach reduces tracking error resulting from high transaction costs, it increases tracking error resulting from the mismatch of the indexed portfolio and the bond index. In practice, managers who state that they pursue a “pure” bond indexing strategy typically are forced to follow an enhanced bond indexing strategy with minor mismatches in the primary risk factors. A portfolio manager faces several other logistical problems in seeking to construct an indexed portfolio. First, the prices for each issue used by the organization that publishes the index may not be execution prices available to the indexer. In fact, they may be materially different from the prices offered by some dealers. In addition, the prices used by organizations reporting the value of indexes are based on bid prices. Dealer ask prices, however, are the ones that the manager would have to transact at when constructing or rebalancing the indexed portfolio. Thus, there will be a bias between the performance of the bond index and the indexed portfolio that is equal to the bid-ask spread. Furthermore, there are logistical problems unique to certain sectors in the bond market. Consider first the corporate bond market. There are typically about 3,500 issues in the corporate bond sector of a broad-based index. Because of the illiquidity of this sector of the bond market, not only may the prices used by the organization that publishes the index be unreliable, but many of the issues may not even be available. Next, consider the agency mortgage-backed securities market. There are more than 800,000 agency pass-through issues. The organizations that publish indexes lump all these issues into a few hundred generic issues. The portfolio manager is then faced with the difficult task of finding pass-through securities with the same risk–return profiles of these hypothetical issues. Finally, recall that the total return depends on the reinvestment rate available on coupon interest. If the organization publishing the index regularly overestimates the reinvestment rate, the indexed portfolio could underperform the bond index by a significant number of basis points a year. 17. How can a multi-factor risk model be used to monitor and control portfolio risk? A multi-factor risk model can be used to monitor and control portfolio risk through a forward-looking estimate of tracking error. The portfolio manager needs this forward-looking estimate of tracking error to reflect the portfolio risk going forward. The way this is done in practice is by using the services of a commercial vendor or dealer firm that has modeled the factors that affect the tracking error associated with the bond market index (i.e., the portfolio manager’s benchmark). These models are called multi-factor risk models. Given a manager’s current portfolio holdings, the portfolio’s current exposure to the various risk factors can be calculated and compared to the benchmark’s exposures to the factors. Using the differential factor exposures and the risks of the factors, a forward-looking tracking error for the portfolio can be computed. This tracking error is also referred to as predicted tracking error and ex ante tracking error. Given a forward-looking tracking error, a range for the future possible portfolio active return can be calculated assuming that the active returns are normally distributed. For example, assume the following: expected return for benchmark – 10% forward-looking tracking error relative to benchmark – 100 basis points From the properties of a normal distribution, we know the following: Number of Standard Deviations Range for Portfolio Active Return Corresponding Range for Portfolio Return Probability 1 –1% 9%–11% 67% 2 –2% 8%–12% 95% 3 –3% 7%–13% 99% It should be noted that there is no guarantee that the forward-looking tracking error at the start of a period would exactly match the backward-looking tracking error calculated at the end of the year. Regardless the average of forward-looking tracking error estimates obtained at different times during the year will be reasonably close to the backward-looking tracking error estimate obtained at the end of the year. The forward-looking tracking error is useful in risk control and portfolio construction. The manager can immediately see the likely effect on tracking error of any intended change in the portfolio. Thus, scenario analysis can be performed by a portfolio manager to assess proposed portfolio strategies and eliminate those that would result in tracking error beyond a specified tolerance for risk. 18. How can a multi-factor risk model be used to rebalance a portfolio? While it is common to illustrate portfolio construction starting with a position of cash and building a portfolio of securities, in practice the more common task is to rebalance an existing portfolio. A multi-factor model along with an optimizer can be used to efficiently rebalance the portfolio. This rebalancing using a multi-factor model involves realigning the portfolio that has drifting away from the characteristics of the benchmark over time. For example, the “drift” may involve a change in the duration of the benchmark requiring a change in the duration of the portfolio or it may entail an upgrade or downgrade of some issues in the portfolio. The rebalancing can also involve tilting the portfolio to reflect a manager’s new views. Rebalancing is also required when a portfolio manager receives additional funds from a client or portfolio cash inflows or when a client withdraws funds. More details are given below. Multi-factor models are statistical models that are used to estimate a security’s expected return based on the primary drivers affecting the return on securities. The primary drivers of returns are referred to as risk factors or simply factors. These models are also called multi-factor risk models or just factor models. Multi-factor models provide managers with information about the sources of risk in their portfolio. Hence, they are indispensable tools for constructing portfolios so as to realize the desired exposure to the risk factors where a manager has a view. Moreover, these models can be used to monitor and control the risk exposure of the portfolio, which is achievable through rebalancing. A multi-factor model may prove helpful to help the rebalancing minimize transaction costs by reducing the need to turnover current holdings unnecessarily. The optimizer is able to evaluate the trade-off of replacing one issue held (i.e., a sale) with another issue (i.e., a purchase). The optimizer can identify a package of transactions (i.e., sells and buys) and identify the reduction (or increase) in risk that would result from the execution of those transactions so that the portfolio manager can assess the risk adjustment benefit relative to the cost of executing the transaction. Exhibit 23-14 shows the trades that would have been recommended for rebalancing. The total market value of the trades was roughly $13 million. Almost half of the sales from the portfolio were for banks and they were replaced with various Treasury notes, a corporate bond, a sovereign bond, and an agency MBS. Before the manager executes the package of trades proposed in Exhibit 23-14, there must be an evaluation of the change in risk exposure. The new systematic tracking error (TE) after rebalancing was 4.2 basis points (the original was 4.6 basis points), idiosyncratic TE is 7.8 basis points (same as before rebalancing), and total TE is 8.8 basis points (9.0 basis points before rebalancing). The decline in the total TE is before there are more than 50 securities in the portfolio after the rebalancing. Exhibit 23-14 Trades for Portfolio Rebalancing Buys Identifier Description Position Amount Market Value 912828LK US TREASURY NOTES 3,133,909 3,235,179 912828LS US TREASURY NOTES 2,814,967 2,924,353 489170AB KENNAMETAL INC 1,959,720 2,087,886 94986EAA WELLS FARGO CAPITAL XIII 1,286,097 1,360,888 912810QD US TREASURY BONDS 1,118,189 1,111,380 465138ZR ISRAEL STATE OF 920,297 1,097,735 912810QB US TREASURY BONDS 1,017,169 991,185 GNG03410 GNMA II Single Family 15yr 117,277 119,672 Total 12,928,278 Sells Identifier Description Position Amount Market Value 912828NV US TREASURY NOTES –2,662,260 –2,586,183 16132NAV CHARTER ONE BANK FSB –2,203,358 –2,332,312 05946NAD BANCO BRADESCO SA –1,564,870 –1,828,328 827065AA SILICON VALLEY BANK –1,692,776 –1,770,613 912828NL US TREASURY NOTES –1,603,631 –1,612,239 912810QC US TREASURY BONDS –1,462,336 –1,468,727 912810QE US TREASURY BONDS –1,298,352 –1,329,875 Total –12,928,278 19. In a factor model, what is meant by isolated tracking error? An isolated tracking error refers to the method of calculating the partial tracking error due to a single group of risk factors in isolation; no other forms of risk are considered. Illustrations giving more details are given below. Let us first illustrate an isolated tracking error by considering the risk factor “securitized spread” in Exhibit 23-7. This risk factor is the exposure to changes in the spreads in the agency MBS market. The value of 2.5 means that if the portfolio only differs from the benchmark with respect to its exposure to changes in the spread in the agency MBS sector, then this mismatch relative to the benchmark would result in a monthly isolated tracking error of 2.5 basis points. Second, we consider Exhibit 23-7 where can notice that there is a risk exposure category labeled “volatility”. This risk factor is the risk associated with changes in interest rate volatility and is critical for quantifying the exposure of a portfolio or benchmark to securities with embedded options such as callable bonds and agency MBS because they are impacted by changes in volatility. Hence, the value of 1.3 is the exposure of the portfolio to the risk factor volatility. The value of 1.3 means that if the portfolio only differs from the benchmark with respect to its exposure to changes in volatility, then this mismatch relative to the benchmark would result in a monthly isolated tracking error of 1.3 basis points. We can compute a portfolio isolated systematic tracking error by assuming a zero correlation between any pair of risk factors. Given this assumption, the portfolio isolated tracking error attributable to systematic risk is found by squaring each isolated tracking error for each risk factor, summing them, and then taking the square root. That is, for the general case where there are K risk factors is Portfolio isolated systematic TE = [(TE1)2 + (TE2)2 + … + (TEK)2]1/2 where TE denotes tracking error and the subscript denotes the risk factor. Consider the 50-security portfolio in Exhibit 23-13 where the monthly isolated TE for each risk factor is shown in Exhibit 23-7. Here the portfolio isolated systematic TE is 6.24 basis points per month as shown below: Portfolio isolated systematic TE = [(3.9)2 + (2.6)2 + (1.3)2 + (0.8)2 + (2.8)2 + (2.5)2]1/2 = 6.24 The assumption that there is zero correlation between every pair of risk factors is unrealistic. Obviously, to address this, correlations or covariances must be brought into the analysis. The calculation of the portfolio risk then involves the use of the variance-covariance matrix for the risk factors. Recall that in mean-variance analysis, the portfolio variance (risk) captures the diversification effect by taking into consideration the covariances. In the case of tracking error, let’s consider the case where there are only two risk factors, F1 and F2. Then the portfolio tracking error is equal to Portfolio TE = [(TEF1)2 + (TEF2)2 + 2 Cov(F1,F2)]1/2 where Cov(F1,F2) is the covariance between risk factor exposures 1 and 2. 20. Following is a portfolio consisting of 50 bonds with a market value of $100 million as of April 29, 2011: Identifier Description Position Amount Market Value 003723AA ABN AMRO BANK NV 1,449,636 1,422,596 00104BAC AES EASTERN ENERGY 1,682,044 1,206,446 02051PAC ALON REFINING KROTZ 592,304 630,655 02360XAL AMERENENERGY GENERATING 707,484 737,343 101137AD BOSTON SCIENTIFC 1,551,232 1,656,030 12527GAA CF INDUSTRIES INC 1,328,707 1,499,778 165167BS CHESAPEAKE ENERGY CORP 797,314 880,013 125896BG CMS ENERGY 1,286,476 1,337,697 251591AY DEVELOPERS DIVERS REALTY 646,714 644,344 FGB08000 FHLM Gold Guar Single F. 30yr 2,683,702 3,040,911 FGB07001 FHLM Gold Guar Single F. 30yr 690,235 780,262 FGB06402 FHLM Gold Guar Single F. 30yr 885,600 1,004,579 FGB07002 FHLM Gold Guar Single F. 30yr 3,751,831 4,235,068 FGB05403 FHLM Gold Guar Single F. 30yr 1,411,009 1,531,707 FGB06003 FHLM Gold Guar Single F. 30yr 1,387,727 1,537,027 FGB06004 FHLM Gold Guar Single F. 30yr 633,691 700,545 FGB05011 FHLM Gold Guar Single F. 30yr 651,568 690,585 FNA07098 FNMA Conventional Long T. 30yr 884,357 1,014,899 FNA08000 FNMA Conventional Long T. 30yr 1,643,844 1,883,297 FNA05402 FNMA Conventional Long T. 30yr 1,707,042 1,854,853 FNA06402 FNMA Conventional Long T. 30yr 1,155,221 1,311,433 FNA07002 FNMA Conventional Long T. 30yr 2,241,336 2,563,939 FNA05003 FNMA Conventional Long T. 30yr 641,485 684,085 FNA05403 FNMA Conventional Long T. 30yr 3,194,556 3,469,103 FNA06003 FNMA Conventional Long T. 30yr 1,548,573 1,715,870 FNA05010 FNMA Conventional Long T. 30yr 794,384 843,855 FNA05011 FNMA Conventional Long T. 30yr 1,105,717 1,173,465 GNB04411 GNMA II Single Family 30yr 2,391,899 2,509,580 381427AA GOLDMAN SACHS CAPITAL II 3,123,435 2,761,546 45905CAA INTERNATL BANK RECON DEV-GLOBA 1,151,247 1,200,080 45950KBJ INTL FINANCE CORPORATION 1,227,607 1,198,808 46513E5Y ISRAEL STATE OF-GLOBAL 1,797,220 1,911,761 500769BR KREDIT FUER WIEDERAUFBAU-GLOBA 3,461,061 1,012,672 500769CH KREDIT FUER WIEDERAUFBAU-GLOBA 3,430,115 941,429 582834AM MEAD CORP 727,352 787,191 58551TAA MELLON CAPITAL IV 3,326,734 3,102,915 651715AF NEWPAGE CORP 6,414,006 1,603,501 665772CE NORTHERN STATES PWR MINN 889,932 907,113 723787AG PIONEER NATURAL RESOURCES 945,542 1,045,275 749685AQ RPM INTERNATIONAL INC 591,159 642,823 797440BM SAN DIEGO GAS & ELECTRIC 957,058 856,840 784635AM SPX CORPORATION 708,877 766,621 91311QAD UNITED UTILITES PLC 844,170 848,272 915436AF UPM-KYMMENE CORP 655,540 648,265 912810PW US TREASURY BONDS 797,859 804,588 912810QA US TREASURY BONDS 8,725,929 7,505,533 912810QK US TREASURY BONDS 4,408,259 4,048,097 912828PA US TREASURY NOTES 3,507,446 3,378,751 912828PF US TREASURY NOTES 21,453,185 20,596,365 962166AV WEYERHAEUSER CO 750,667 871,588 The benchmark for the manager who has constructed this portfolio is a composite index consisting one-third each of the Barclays Capital U.S. Treasury index, Barclays Capital U.S. Credit Index, and Barclays Capital U.S. MBS index. Asset Class Portfolio Benchmark Total 100.0 100.0 Treasury 36.3 33.3 Government Related 6.3 6.8 Corporate Industrials 11.0 13.9 Corporate Utilities 5.9 2.9 Corporate Financials 7.9 9.7 MBS Agency 32.6 33.3 Analytics Portfolio Benchmark Difference Duration 6.87 5.37 1.50 Spread Duration 6.77 5.27 1.50 Convexity 0.47 0.00 0.47 Vega ¬0.01 ¬0.03 0.02 Spread 355 55 300.00 Duration Contribution Portfolio Benchmark Difference Total 6.87 5.37 1.50 Treasury 3.62 1.78 1.84 Government Related 0.92 0.41 0.51 Corporate 1.10 1.74 –0.63 Securitized 1.23 1.45 –0.22 Risk Factor Categories Risk Curve 40.8 Swap Spreads 2.5 Volatility 2.8 Spread Government Related 5.3 Spread Corporate 30.6 Spread Securitized 5.8 Volatility Portfolio Benchmark Tracking Error Systematic 141.9 117.4 37.9 Idiosyncratic 19.3 4.8 18.7 Total 143.2 117.5 42.3 Portfolio Beta 1.18 Risk Factor Group Isolated TEV Contribution to TEV Liquidation Effect on TEV TEV Elasticity (%) Total 42.3 42.3 –42.3 1.0 Systematic Risk 37.9 33.2 –22.4 0.8 Curve 40.8 23.4 –4.3 0.5 Swap Spreads 2.5 0.2 –0.1 0.0 Volatility 2.8 0.5 –0.4 0.0 Spread Government Related 5.3 0.0 0.3 0.0 Spread Corporate 30.6 10.0 0.8 0.2 Spread Securitized 5.8 –0.8 1.1 0.0 Idiosyncratic Risk 18.7 9.1 –4.2 0.2 Describe in detail the risk characteristics of this portfolio. Be sure to discuss where it seems like the manager is taking views on the market? The benchmark for the manager who has constructed this portfolio is a composite index consisting of one-third each of the Barclays Capital U.S. Treasury index, Barclays Capital U.S. Credit Index, and Barclays Capital U.S. MBS index. First, in regards to the Barclays Capital U.S. Treasury index, this index measures the performance of U.S. Treasury securities. Second, in regards to the Barclays Capital U.S. Credit Index, this index includes both corporate and non-corporate sectors where the corporate sectors are industrial, utility, and finance that include both U.S. and non-U.S. corporations. The non-corporate sectors are sovereign, supranational, foreign agency, and foreign local government. The index is calculated monthly on price-only and total-return basis. All returns are market value-weighted inclusive of accrued interest. Third, in regards to the Barclays Capital U.S. MBS index, this index measures the performance of investment grade fixed-rate mortgage-backed pass-through securities of GNMA, FNMA, and FHLMC. The analysis of the portfolio begins with a comparison of the portfolio to that of the benchmark. Identification of the mismatches indicates where the manager has taken a view (unintentional or not). The “asset class” table compares the portfolio and the benchmark in terms of the allocation to the major sectors of the benchmark. From the asset allocation (as shown in the table below that puts in the “differences”) that the portfolio manager is taking a positive view on the treasury and corporate utility sectors by overweighting them and this is achieved by underweighting the other sectors. Although the information contained in the “asset” table (about the allocation based on percentage market value of sector relative to the benchmark) provides a good starting point for our analysis, the information has limited value because it is not known how the exposures to the sectors are related to the exposures to the risk factors that drive the portfolio’s return. Here are three examples. First, consider the Treasury sector. It is possible that the specific Treasury securities contained in the portfolio have a lesser contribution to portfolio duration than the contribution to index duration of the Treasuries in the benchmark despite the overweighting of Treasuries in the portfolio. As a second example of why the manager must look beyond the percentage allocation to a sector, consider the corporate bonds in the financial sector. Corporate financials will have a contribution to spread duration in both the portfolio and the benchmark. It is possible to have an underweight of this sector in the portfolio and yet have a contribution to spread duration that is greater than that of the benchmark. Finally, a portfolio’s convexity relative to the benchmark will impact relative performance. It is possible to underweight the portfolio’s exposure to agency MBS so as to create a portfolio with large negative convexity while the benchmark has much lower negative convexity. It is for this reason that the portfolio manager must look beyond a naïve assessment of portfolio risk relative to the benchmark based on percentage allocation to sectors. The “analytics” table provides information about the relative exposure to interest rate risk as measured by duration, spread risk as measured by spread duration, and call/prepayment risk as measured by vega, as well as the convexity. From these analytics we observe the following: The duration of the portfolio exceeds that of the benchmark so that portfolio has more exposure to changes in the level of interest rates. Due to the underweighting in Treasuries, spread duration is higher. The higher portfolio convexity compared to the benchmark means less exposure to call and prepayment risk which can be attributable to the less exposure to agency MBS. Exposure to call/prepayment risk as measure by vega is small and about the same for the portfolio and the benchmark. More information about the portfolio’s relative risk exposure to interest rate risk can be seen by looking at the contribution to duration for the portfolio and the benchmark. As can be seen, the major reason for the longer duration of the portfolio relative to the benchmark is attributable to the duration of the Treasury securities selected for the portfolio. The analysis thus far is missing a vital element. To understand why, suppose that a portfolio has more exposure to a risk factor than the benchmark. This would mean if that risk factor moves, the portfolio will have a greater movement than the benchmark. But the question is: To what extent does that risk factor move? Another way of asking this is: How volatile is the risk factor? For example, from the analysis of the analytics, we know that the portfolio has greater exposure than the benchmark to changes in the level of interest rates (i.e., a higher duration) but less exposure to changes in spreads (i.e., a higher spread duration). But which exposure (i.e., risk factor) has greater volatility? To address this, volatility must be taken into consideration. The “contribution to duration” table shows the (monthly) volatility of risk factor categories. Let’s look at each one of these volatilities and what they mean. The (isolated) risk in this table displays the tracking error/volatility of different exposures of the portfolio in isolation. Consider first the yield curve risk of 40.8 reported in this table. Yield curve risk is the risk exposure to changes in the interest rates. We know from the “analytic” table that the portfolio duration is greater than the benchmark (6.87 versus 5.37), but how does that translate into what it will cost the manager in terms of additional risk. That is where the 40.8 is useful. Suppose that the portfolio only differs from the benchmark with respect to its exposure to changes in the yield curve. Then the 40.8 means that this mismatch relative to the benchmark creates a risk equal to 40.8 basis points per month of volatility. That is, if rates were the portfolio’s only net exposure, this number would be the tracking error volatility (TEV) of that portfolio versus the benchmark. The “volatility” table gives a breakdown of the standard deviation of the returns for the portfolio and the benchmark in terms of systematic risk and idiosyncratic risk. The portfolio has greater systematic and idiosyncratic risk than the benchmark. For the total risk of the portfolio and the benchmark, since the systematic and idiosyncratic risks are constructed so as to be independent, the standard deviation of the portfolio and benchmark can be calculated as follows: Total risk (volatility of returns) = [(Systematic risk)2 + (Idiosyncratic risk)2]0.5 The total risk for the portfolio and the benchmark using the values in the “volatility” table is 143.2 and 117.5, respectively. Notice that for the benchmark, the percentage of the total risk (117.5) that is explained by the systematic risk factors (117.4) is 99.91%. For the portfolio it is 99.09% (141.9/143.2). It would therefore seem that the idiosyncratic risk is not important. This, however, is not true when dealing with the tracking error of the portfolio (volatility of the net position, portfolio vs. the benchmark). The systematic and idiosyncratic tracking error (per month) is 37.9 basis points and 18.7 basis points per month, respectively. The portfolio tracking error is Portfolio tracking error = [(Systematic TE)2 + (Idiosyncratic TE)2]0.5 Therefore, the portfolio tracking error is 42.3 basis points per month. Consequently, although idiosyncratic risk is minimal for the portfolio on a standalone basis, when risk is assessed relative to a benchmark, there is tracking error risk of 42.3 basis points per month. (The systematic risk is responsible for 18.7/42.3 or 44.21% of the total risk.) This is an extremely important point: It is the tracking error not the idiosyncratic risk (as measured by the standard deviation of the idiosyncratic returns) that the manager must consider in portfolio construction and monitoring. In our illustration, the portfolio tracking error is small, only 42.3 basis points. As with equities where a portfolio beta is computed that shows the movement of an equity portfolio in response to a movement in some equity market index (such as the S&P 500), a beta can be computed for a bond portfolio. As shown in “volatility” table, the portfolio beta is 1.18. Since the benchmark is the Composite Index, a beta of 1.18 means that if that index increases by 10%, the portfolio will increase, on average, by 11.8%. A beta-type measure can be estimated for each risk factor. For example, consider the risk factor measuring changes in the level of the yield curve which is the portfolio’s duration. A duration beta can be calculated as follows: For our portfolio and benchmark, since the duration is 6.87 and 5.37, respectively (see “contribution to duration” table), the duration beta is 1.18. While the information contained in the “volatility” table gives us a starting point for understanding the portfolio’s risk relative to the benchmark, further insight can be gained by looking at how the portfolio risk (as measured by tracking error) is allocated across the different (1) categories of risk factors and (2) asset classes (i.e., sectors of the benchmark). This “risk factor group” table provides information about the portfolio risk across the different categories of risk factors. Shown are the systematic risk and the idiosyncratic risk and six components of systematic risk. The “contribution to TEV” column shows the isolated tracking error. The contribution to tracking error for each group of risk factor is shown in the “liquidation effect on TEV” column. As can be seen, the major risk exposures of the 50-bond portfolio are yield curve risk, corporate spread risk, and systematic risk. The “Liquidation Effect on TEV” column gives a new metric, liquidation effect on tracking error. Barclays Capital defines this metric to be the impact to the portfolio’s tracking error by hedging (i.e., eliminating) the exposure to the respective risk group. For example, consider the systematic risk. The liquidation effect on tracking error shown in the exhibit is −22.4 and is interpreted as follows: if the portfolio manager hedges the systematic risk, then the portfolio’s tracking error will decline by 22.4 basis points per month. Since the portfolio’s tracking error is 42.3 basis points per month, this means that hedging the systematic risk reduces the monthly tracking error for the portfolio to 37.9 basis points per month. Solution Manual for Bond Markets, Analysis and Strategies Frank J. Fabozzi 9780132743549, 9780133796773
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