CH-25-26 Bond Performance Measurement and Evaluation – Bond Markets, Analysis and Strategies : Book Guide

This Document Contains Chapters 25 to 26 CHAPTER 25 BOND PERFORMANCE MEASUREMENT AND EVALUATION CHAPTER SUMMARY In this chapter we will see how to measure and evaluate the investment performance of a fixed-income portfolio manager. Performance measurement involves the calculation of the return realized by a portfolio manager over some time interval, which we call the evaluation period. Performance evaluation is concerned with two issues. The first is to determine whether the manager added value by outperforming the established benchmark. The second is to determine how the manager achieved the calculated return. REQUIREMENTS FOR A BOND PERFORMANCE AND ATTRIBUTION ANALYSIS PROCESS There are three desired requirements of a bond performance and attribution analysis process. The first is that the process be accurate. The second requirement is that the process be informative. The final requirement is that the process be simple. PERFORMANCE MEASUREMENT The starting point for evaluating the performance of a manager is measuring return. Because different methodologies are available and these methodologies can lead to quite disparate results, it is difficult to compare the performances of managers. Alternative Return Measures Let’s begin with the basic concept. The dollar return realized on a portfolio for any evaluation period (i.e., a year, month, or week) is equal to the sum of (i) the difference between the market value of the portfolio at the end of the evaluation period and the market value at the beginning of the evaluation period, and (ii) any distributions made from the portfolio. In equation form, the portfolio’s return can be expressed as follows: Rp = where Rp = return on the portfolio, MV1 = portfolio market value at the end of the evaluation period; MV0 = portfolio market value at the beginning of the evaluation period; and, D = cash distributions from the portfolio to the client during the evaluation period. There are three assumptions in measuring return as given by the above equation. First, it assumes that a period’s cash inflow into the portfolio from interest is either distributed or reinvested in the portfolio. Second, if there are distributions from the portfolio, they occur at the end of the evaluation period or are held in the form of cash until the end of the evaluation period. Third, no cash is paid into the portfolio by the client. From a practical point of view the three assumptions limit its application. The longer the evaluation period, the more likely the assumptions will be violated. Not only does the violation of the assumptions make it difficult to compare the returns of two managers over some evaluation period, but it is also not useful for evaluating performance over different periods. The way to handle these practical issues is to calculate the return for a short unit of time such as a month or a quarter. We call the return so calculated the subperiod return. To get the return for the evaluation period, the subperiod returns are then averaged. There are three methodologies that have been used in practice to calculate the average of the subperiod returns: (1) the arithmetic average rate of return, (2) the time-weighted rate of return (also called the geometric rate of return), and (3) the dollar-weighted rate of return. Arithmetic Average Rate of Return The arithmetic average rate of return is an unweighted average of the subperiod returns. The general formula is RA = where RA = arithmetic average rate of return; Rpk = portfolio return for subperiod k for k = 1, . . . , N; and, N = number of subperiods in the evaluation period. It is improper to interpret the arithmetic average rate of return as a measure of the average return over an evaluation period. The proper interpretation is that it is the average value of the withdrawals (expressed as a fraction of the initial portfolio market value) that can be made at the end of each subperiod while keeping the initial portfolio market value intact. Time-Weighted Rate of Return The time-weighted rate of return measures the compounded rate of growth of the initial portfolio market value during the evaluation period, assuming that all cash distributions are reinvested in the portfolio. It is also commonly referred to as the geometric rate of return because it is computed by taking the geometric average of the portfolio subperiod returns computed from equation. The general formula is RT = [(1 + RP1)(1 + RP2) . . . (1 + RPN)]1/ N – 1 where RT is the time-weighted rate of return, RPk is the return for subperiod k, and N is the number of subperiods. In general, the arithmetic and time-weighted average returns will give different values for the portfolio return over some evaluation period. This is because in computing the arithmetic average rate of return, the amount invested is assumed to be maintained (through additions or withdrawals) at its initial portfolio market value. The time-weighted return, on the other hand, is the return on a portfolio that varies in size because of the assumption that all proceeds are reinvested. In general, the arithmetic average rate of return will exceed the time-weighted average rate of return. The exception is in the special situation where all the subperiod returns are the same, in which case the averages are identical. The magnitude of the difference between the two averages is smaller the less the variation in the subperiod returns over the evaluation period. Dollar-Weighted Rate of Return The dollar-weighted rate of return is computed by finding the interest rate that will make the present value of the cash flows from all the subperiods in the evaluation period plus the terminal market value of the portfolio equal to the initial market value of the portfolio. Cash flows are defined as follows: • A cash withdrawal is treated as a cash inflow. So, in the absence of any cash contribution made by a client for a given time period, a cash withdrawal (e.g., a distribution to a client) is a positive cash flow for that time period. • A cash contribution is treated as a cash outflow. Consequently, in the absence of any cash withdrawal for a given time period, a cash contribution is treated as a negative cash flow for that period. • If there are both cash contributions and cash withdrawals for a given time period, then the cash flow is as follows for that time period: If cash withdrawals exceed cash contributions, then there is a positive cash flow (which is the cash difference). If cash withdrawals are less than cash contributions, then there is a negative cash flow (which is also the cash difference). The dollar-weighted rate of return is simply an internal rate-of-return calculation and hence it is also called the internal rate of return. The general formula for the dollar-weighted return is V0 = where RD = dollar-weighted rate of return; V0 = initial market value of the portfolio; VN = terminal market value of the portfolio; and, Ck = cash flow for the portfolio (cash inflows minus cash outflows) for subperiod k for k = 1, . . . , N. Notice that it is not necessary to know the market value of the portfolio for each subperiod to determine the dollar-weighted rate of return. The dollar-weighted rate of return and the time-weighted rate of return will produce the same result if no withdrawals or contributions occur over the evaluation period and all investment income is reinvested. The problem with the dollar-weighted rate of return is that it is affected by factors that are beyond the control of the manager. Specifically, any contributions made by the client or withdrawals that the client requires will affect the calculated return. This makes it difficult to compare the performance of two managers. Annualizing Returns The evaluation period may be less than or greater than one year. Typically, return measures are reported as an average annual return. This requires the annualization of the subperiod returns. The subperiod returns are typically calculated for a period of less than one year. The subperiod returns are then annualized using the following formula: annual return = (1 + average period return)number of periods in year – 1. PERFORMANCE ATTRIBUTION ANALYSIS Bond attribution models seek to identify the active management decisions that contributed to the portfolio’s performance and give a quantitative assessment of the contribution of these decisions. The performance of a portfolio can be decomposed in terms of four active strategies in managing a fixed-income portfolio: interest-rate expectation strategies, yield curve expectations strategies, yield spread strategies, and individual security selection strategies. Benchmark Portfolios To evaluate the performance of a manager, a client must specify a benchmark against which the manager will be measured. There are two types of benchmarks that have been used in evaluating fixed-income portfolio managers: (i) market indexes published by dealer firms and vendors, and (ii) normal portfolios. A normal portfolio is a customized benchmark that includes “a set of securities that contains all of the securities from which a manager normally chooses, weighted as the manager would weight them in a portfolio.” Thus a normal portfolio is a specialized index. It is argued that normal portfolios are more appropriate benchmarks than market indexes because they control for investment management style, thereby representing a passive portfolio against which a manager can be evaluated. The construction of a normal portfolio for a manager requires (i) defining the universe of fixed-income securities to be included in the normal portfolio, and (ii) determining how these securities should be weighted (i.e., equally weighted or capitalization weighted). Plan sponsors work with pension consultants to develop normal portfolios for a manager. The consultants use vendor systems that have been developed for performing the needed statistical analysis and the necessary optimization program to create a portfolio displaying similar factor positions to replicate the “normal” position of a manager. A plan sponsor must recognize that there is a cost to developing and updating the normal portfolio. A more appropriate benchmark for institutional investors such as defined benefit pension plans is one that reflects its liability structure. It has been argued that the major reason for the failure of both public and private defined benefit plans is the wrong benchmarks have been used. Instead of using a bond index as is commonly used, the appropriate benchmark should be one that is customized liability index based on a specific pension plan’s actuarially determined liability structure. Performance Attribution Analysis Models Clients of asset management firms need to have more information than merely if a portfolio manager outperformed a benchmark and by how much. They need to know the reasons why a portfolio manager realized the performance relative to the benchmark. It is possible that the manager can outperform a benchmark due to a mismatch in duration and invested in specific securities that did poorly. There is no way that the client can determine that by simply looking at the portfolio’s return relative to the benchmark’s return. There are single metrics that have been commonly used to measure performance. Although useful, single metric do not provide sufficient more information about performance to address the questions that need answers. The model that can be used is performance attribution analysis, a quantitative technique for identifying the sources of portfolio risk and performance so that the contributions of members of the portfolio management team can be measured and the major portfolio decisions can be quantified. There are several performance attribution models that are available from third-party entities. In selecting a third-party model, there are requirements that a good attribution model should possess in order to evaluate the decision-making ability of the members of the portfolio management team: additivity, completeness, and fairness. Additivity means that contribution to performance of two or more decision makers of the portfolio management team should be equal to the sum of the contributions of those decision makers. Completeness means that when the contribution to portfolio performance of all decision makers is added up, the result should be equal to the contribution to portfolio performance relative to the benchmark. Fairness means that the portfolio management team members should view the performance attribution model selected as being fair with respect to representing their contribution. Types of Performance Attribution Models Today, performance attribution models can be classified into three types: sector-based attribution models, factor-based attribution models, and hybrid sector-based/factor-based attribution models. The simplest model is the sector-based attribution, also referred to as the Brinson model. In this model, the portfolio return relative to the benchmark is represented by two decisions: (1) the allocation of funds among the different sectors and the (2) the selection of the specific securities within each sector. The first decision is referred to as the asset allocation decision and the second the security selection decision. Factor-based attribution models actually allow a decomposition of the yield curve risk into level risk and changes in the shape of the yield curve. For example, suppose that the attribution due to yield curve risk is determined to be as follows: Risk Factor Portfolio D Portfolio E Portfolio F Yield curve risk 140 1 –60 Level risk 135 60 3 Shape risk 5 −59 −63 Notice that once yield curve risk is decomposed as shown above, it turns out that the manager of Portfolio E did indeed make interest rate bets. It turns out that the two bets almost offset each other so that net there was only a one basis point return attributable to the interest rate bet. Portfolio D’s manager basically made a major duration bet but virtually no bet on changes in the shape of the yield curve. The interest rate bet by the manager of Portfolio F was on changes in the shape of the yield curve but otherwise was basically duration neutral. As the name suggests, a hybrid sector-based/factor-based attribution model combines the previous two attribution models. This model allows for much more detail regarding not only the bets on the primary systematic risk factors driving returns but the impact of decisions with respect to sector and security selection. KEY POINTS Performance measurement involves calculation of the return realized by a portfolio manager over some evaluation period. Performance evaluation is concerned with determining whether the portfolio manager added value by outperforming the established benchmark and how the portfolio manager achieved the calculated return. The rate of return expresses the dollar return in terms of the amount of the initial investment (i.e., the initial market value of the portfolio). Three methodologies have been used in practice to calculate the average of the sub-period returns: (1) the arithmetic average rate of return, (2) the time-weighted (or geometric) rate of return, and (3) the dollar-weighted return. The arithmetic average rate of return is the average value of the withdrawals (expressed as a fraction of the initial portfolio market value) that can be made at the end of each period while keeping the initial portfolio market value intact. The time-weighted rate of return measures the compounded rate of growth of the initial portfolio over the evaluation period, assuming that all cash distributions are reinvested in the portfolio. The time-weighted return is the return on a portfolio that varies in size because of the assumption that all proceeds are reinvested. In general, the arithmetic average rate of return will exceed the time-weighted average rate of return. The magnitude of the difference between the two averages is smaller the less the variation in the sub-period returns over the evaluation period. The dollar-weighted rate of return is computed by finding the interest rate that will make the present value of the cash flows from all the sub-periods in the evaluation period plus the terminal market value of the portfolio equal to the initial market value of the portfolio. The dollar-weighted rate of return is an internal rate-of-return calculation and will produce the same result as the time-weighted rate of return if (1) no withdrawals or contributions occur over the evaluation period, and (2) all coupon interest payments are reinvested. The problem with using the dollar-weighted rate of return to evaluate the performance of money managers is that it is affected by factors that are beyond the control of the money manager. Specifically, any contributions made by the client or withdrawals that the client requires will affect the calculated return, making it difficult to compare the performance of two portfolio managers. The role of performance evaluation is to determine if a portfolio manager added value beyond what could have been achieved by a passive strategy in a benchmark portfolio. The analysis requires the establishment of a benchmark. One such benchmark is a normal portfolio. This is a customized benchmark that includes a set of securities that contains the universe of securities that a manager normally selects from and weighted as the manager would weight them in a portfolio. Advocates claim that normal portfolios are more appropriate benchmarks than market indexes because they control for investment management style, thereby representing a passive portfolio against which a manager can be evaluated. Bond indexes are commonly used as benchmarks. For defined benefit pension plans, a more appropriate benchmark would be a customized liability index determined by the fund’s actuarially projected future liabilities. Performance attribution models can be used explain why the active return of a portfolio was realized. The three types of performance attribution models available are sector-based attribution models, factor-based attribution models, and hybrid sector-based/factor-based attribution models. ANSWERS TO QUESTIONS FOR CHAPTER 25 (Questions are in bold print followed by answers.) 1. What is the difference between performance measurement and performance evaluation? Performance measurement involves calculation of the return realized by a portfolio manager over some evaluation period. Performance evaluation is concerned with determining whether the portfolio manager added value by outperforming the established benchmark and how the portfolio manager achieved the calculated return. 2. Suppose that the monthly return for two bond managers is as follows: Month Manager I Manager II 1 9% 25% 2 13% 13% 3 22% 22% 4 –18% –24% What is the arithmetic average monthly rate of return for the two managers? The arithmetic average rate of return is an unweighted average of the subperiod returns. The general formula is RA = where RA = arithmetic average rate of return; Rpk = portfolio return for subperiod k for k = 1, . . . , N; and, N = number of subperiods in the evaluation period. In our problem, we have subperiod or monthly portfolio returns for Manager I of RP1 = 9%, RP21 = 13%, RP3 = 22% and RP4 = 18%, for months 1, 2, 3, and 4, respectively. Solving for N = 4, the arithmetic average rate of return is: RManager I = = 0.0650 or 6.50%. Similarly, for Manager II, we get the portfolio return of: RManager II = = 0.0900 or 9.00%. 3. What is the time-weighted average monthly rate of return for the two managers in Question 2? The time-weighted rate of return measures the compounded rate of growth of the initial portfolio market value during the evaluation period, assuming that all cash distributions are reinvested in the portfolio. It is also commonly referred to as the geometric rate of return because it is computed by taking the geometric average of the portfolio subperiod returns. The general formula is RT = [(1 + RP1)(1 + RP2) . . . (1 + RPN)]1/ N – 1 where RT is the time-weighted rate of return, RPk is the return for subperiod k for k = 1, . . . , N, and N is the number of subperiods. In our problem, we have the portfolio returns for Manager I of RP1 = 9%, RP2 = 13%, RP3 = 22% and RP4 = 18%, for months 1, 2, 3, and 4, respectively. Solving for N = 4, the time-weighted rate of return is: RManager I = [(1 + 0.09)(1 + 0.13)(1 + 0.22)(1 + {1 + 0.18}]1/4 – 1 = [(1.09)(1.13)(1.22)(0.82)]1/4 – 1 = [1.23211943]1/4 – 1 = 1.05358519 – 1 = 0.05358519 or about 5.36%. If the time-weighted rate of return is 5.36% per month, one dollar invested in the portfolio at the beginning of month 1 would have grown at a rate of 5.36% per month during the four-month evaluation period. Similarly, for Manager II, we get the portfolio return of: RManager II = [(1 + 0.25)(1 + 0.13)(1 + 0.22)(1 + {1 + 0.24}]1/4 – 1 = [(1.25)(1.13)(1.22)(0.76)]1/4 – 1 = [1.30967000]1/4 – 1 = 1.06977014 – 1 = 0.06977014 or about 6.98%. 4. Why does the arithmetic average monthly rate of return diverge more from the time-weighted monthly rate of return for manager II than for manager I in Question 2? The table below summarizes the managerial performances and differences between the two types of monthly returns. Two Types of Monthly Returns: Arithmetic Average Return Time-Weighted Return Difference in Returns Manager I 6.50% 5.36% 1.14% Manager II 9.00% 6.98% 2.02% As can be seen in the last column of the above table, the arithmetic average monthly rate of return diverges more from the time-weighted monthly rate of return for manager II than for manager I. This is because the arithmetic average rate of return typically is greater than the time-weighted average rate of return with the magnitude of the difference between the two averages greater when the variation (in the subperiod returns over the evaluation period) is greater. Thus, because there is more variation in returns for Manager II, this causes a greater difference between the arithmetic average monthly rate of return and the time-weighted monthly rate of return. More details are given below. In general, the arithmetic and time-weighted average returns will give different values for the portfolio return over some evaluation period. This is because in computing the arithmetic average rate of return, the amount invested is assumed to be maintained (through additions or withdrawals) at its initial portfolio market value. The time-weighted return, on the other hand, is the return on a portfolio that varies in size because of the assumption that all proceeds are reinvested. In general, the arithmetic average rate of return will exceed the time-weighted average rate of return. The exception is in the special situation where all the subperiod returns are the same, in which case the averages are identical. The magnitude of the difference between the two averages is smaller the less the variation in the subperiod returns over the evaluation period. For example, suppose that the evaluation period is four months and that the four monthly returns are as follows: RP1 = 4%; RP1 = 6%; RP1 = 2%; RP1 = –2%. The average arithmetic rate of return is 2.50% and the time-weighted average rate of return is 2.46%. Not much of a difference. However, in the textbook example elsewhere, there was an arithmetic average rate of return of 25% but a time-weighted average rate of return of 0%. The large discrepancy is due to the substantial variation in the two monthly returns. 5. Smith & Jones is a money management firm specializing in fixed-income securities. One of its clients gave the firm $100 million to manage. The market value for the portfolio for the four months after receiving the funds was as follows: End of Month Market Value (in millions) 1 $ 50 2 $150 3 $ 75 4 $100 Answer the below questions based on the above table. (a) Calculate the rate of return for each month. In equation form, the portfolio’s return can be expressed as follows: Rp = where Rp = return on the portfolio, MV1 = portfolio market value at the end of the evaluation period; MV0 = portfolio market value at the beginning of the evaluation period; and, D = cash distributions from the portfolio to the client during the evaluation period. Since there is no cash distribution (i.e., D = 0), we have: Rp = . For period or month 1, we have: Rmonth 1 = = = = 0.5000 or 50.00%. For month 2, we have: Rmonth 2 = = = 2.000 or 200.00%. For month 3, we have: Rmonth 3 = = = 0.5000 or 50.00%. For month 4, we have: Rmonth 4 = = = 0.333333 or about 33.33%. (b) Smith & Jones reported to the client that over the four-month period the average monthly rate of return was 33.33%. How was that value obtained? The value was obtained by using arithmetic average rate of return, which is an unweighted average of the subperiod returns. The general formula is RA = where RA = arithmetic average rate of return; Rpk = portfolio return for subperiod k for k = 1, . . . , N; and, N = number of subperiods in the evaluation period. In our problem, we have subperiod or monthly portfolio returns for a client of RP1 = 50%, RP21 = 200%, RP3 = 50% and RP4 = 33.33%, for months 1, 2, 3, and 4, respectively. Solving for N = 4, the arithmetic average rate of return is: RSmith & Jones = = 0.3333333 or about 33.33%. (c) Is the average monthly rate of return of 33.33% indicative of the performance of Smith & Jones? If not, what would be a more appropriate measure? The 33.33% monthly rate of return is not indicative of the performance of Smith & Jones. A more appropriate measure would be the time-weighted rate of return or the dollar-weighted rate of return. First, let us look at the time-weighted rate of return, which measures the compounded rate of growth of the initial portfolio market value during the evaluation period, assuming that all cash distributions are reinvested in the portfolio. It is also commonly referred to as the geometric rate of return because it is computed by taking the geometric average of the portfolio subperiod returns. The general formula is RT = [(1 + RP1)(1 + RP2) . . . (1 + RPN)]1/ N – 1 where RT is the time-weighted rate of return, RPk is the return for subperiod k for k = 1, . . . , N, and N is the number of subperiods. In our problem, we have the portfolio returns for the client of RP1 = 50%, RP2 = 200%, RP3 = 50% and RP4 = 33.33%, for months 1, 2, 3, and 4, respectively. Solving for N = 4, the time-weighted rate of return is: RManager I = [(1 + {0.500})(1 + 2.00)(1 + {0.500})(1 + {0.3333}]1/4 – 1 = [(0.50)(2.00)(0.55)(1.33333)]1/4 – 1 = [1.00000]1/4 – 1 = 1 – 1 = 0 or 0%. If the time-weighted rate of return is 0% per month, one dollar invested in the portfolio at the beginning of month 1 would have grown at a rate of 0% per month during the four-month evaluation period. This answer is consistent with the fact that Smith and Jones’ client began with $100 million and ended with $100 million. Note that the computation does not take into account the time value of money which is influenced by the fact inflation causes purchasing power to decline. Thus, the client is actually worse off than they began. Now, let us look at the dollar-weighted rate of return, which is computed by finding the interest rate that will make the present value of the cash flows from all the subperiods in the evaluation period plus the terminal market value of the portfolio equal to the initial market value of the portfolio. Cash flows, referred to above, are defined as follows: A cash withdrawal is treated as a cash inflow. So, in the absence of any cash contribution made by a client for a given time period, a cash withdrawal (e.g., a distribution to a client) is a positive cash flow for that time period. A cash contribution is treated as a cash outflow. Consequently, in the absence of any cash withdrawal for a given time period, a cash contribution is treated as a negative cash flow for that period. If there are both cash contributions and cash withdrawals for a given time period, then the cash flow is as follows for that time period: If cash withdrawals exceed cash contributions, then there is a positive cash flow. If cash withdrawals are less than cash contributions, then there is a negative cash flow. The dollar-weighted rate of return is simply an internal rate-of-return calculation and hence it is also called the internal rate of return. The general formula for the dollar-weighted return is: V0 = where RD = dollar-weighted rate of return; V0 = initial market value of the portfolio; VN = terminal market value of the portfolio; and, Ck = cash flow for the portfolio (cash inflows minus cash outflows) for subperiod k for k = 1, . . . , N. Notice that it is not necessary to know the market value of the portfolio for each subperiod to determine the dollar-weighted rate of return. For our problem, we have: V0 = $100 million, N = 4, C1 = $0 million, C2 = $0 million, C3 = $0 million, C4 = $0 million, and V4 = $100 million. Given these values, RD is the interest rate that satisfies the below equation: $100,000,000 =  $100,000,000 =  (1 + RD)4 =  (1 + RD) = – 1  RD = 1 – 1  RD = 0. Thus, RD = 0% satisfies the equation. Another way of looking at this problem is to consider the change in value each period to be like a cash inflow (withdrawal) or cash outflow (contribution). If so, for our problem, we would have: V0 = $100 million, N = 4, C1 = $50 million, C2 = $100 million, C3 = $75 million, C4 = $25 million, and V4 = $100 million. Given these values, RD is the interest rate that satisfies the below equation: $100,000,000 = Inserting RD = 0% gives: $100,000,000 =  $100,000,000 = $50,000,000 + $100,000,000 + $75,000,000 + $75,0000  $100,000,000 = $100,000,000. Thus, RD = 0% satisfies the equation. Because zero percent is the internal rate of return that satisfies our expression above, zero percent is the dollar-weighted return. The dollar-weighted rate of return and the time-weighted rate of return will produce the same result if no withdrawals or contributions occur over the evaluation period and all investment income is reinvested. The dollar-weighted rate of return can be affected by factors that are beyond the control of the manager. Specifically, any contributions made by the client or withdrawals that the client requires will affect the calculated return. This makes it difficult to compare the performance of two managers when using this method. Finally, the evaluation period may be less than or greater than one year. Typically, return measures are reported as an average annual return. This requires the annualization of the subperiod returns. The subperiod returns are typically calculated for a period of less than one year. The subperiod returns are then annualized using the following formula: annual return = (1 + average period return)number of periods in year – 1. For example, suppose that the evaluation period is three years and a monthly period return is calculated. Suppose further that the average monthly return is 2%. Then the annual return is annual return = (1 + average period return)number of periods in year – 1 = (1.02)12 – 1 = 26.8%. In our problem, the evaluation period is four months and the average monthly return is 0%. Then the annual return is annual return = (1 + average period return)number of periods in year – 1 = (1 + 0.0)12 – 1 = 1 – 1 = 0 or zero percent (which we already knew). In conclusion, either a time-weighted or dollar-weighted rate of return is more indicative of the portfolio’s performance and thus a more appropriate measure. 6. The Mercury Company is a fixed-income management firm that manages the funds of pension plan sponsors. For one of its clients it manages $200 million. The cash flow for this particular client’s portfolio for the past three months was $20 million, $8 million, and $4 million. The market value of the portfolio at the end of three months was $208 million. Answer the below questions. (a) What is the dollar-weighted rate of return for this client’s portfolio over the three-month period? The dollar-weighted rate of return is computed by finding the interest rate that will make the present value of the cash flows from all the subperiods in the evaluation period plus the terminal market value of the portfolio equal to the initial market value of the portfolio. Cash flows are defined as follows: • A cash withdrawal is treated as a cash inflow. So, in the absence of any cash contribution made by a client for a given time period, a cash withdrawal (e.g., a distribution to a client) is a positive cash flow for that time period. • A cash contribution is treated as a cash outflow. Consequently, in the absence of any cash withdrawal for a given time period, a cash contribution is treated as a negative cash flow for that period. • If there are both cash contributions and cash withdrawals for a given time period, then the cash flow is as follows for that time period: If cash withdrawals exceed cash contributions, then there is a positive cash flow. If cash withdrawals are less than cash contributions, then there is a negative cash flow. The dollar-weighted rate of return is simply an internal rate-of-return calculation and hence it is also called the internal rate of return. The general formula for the dollar-weighted return is: V0 = where RD = dollar-weighted rate of return; V0 = initial market value of the portfolio; VN = terminal market value of the portfolio; and, Ck = cash flow for the portfolio (cash inflows minus cash outflows) for subperiod k for k = 1, . . . , N. Notice that it is not necessary to know the market value of the portfolio for each subperiod to determine the dollar-weighted rate of return. For our problem, we consider a portfolio with a market value of $1,000,000 at the beginning of month 1. For months 1, 2, 3, and 4, we have: V0 = $200 million, N = 3, C1 = $20 million, C2 = $8 million, C3 = $4 million, and V3 = $208 million. Given these value, RD is the interest rate that satisfies the following equation: $200,000,000 = . Below we verify that 4.0550924080% or about 4.055% is the internal rate of return satisfies the above expression. $200,000,000 =  $200,000,000 = $19,220,587.4188 + {$7,388,619.6145) + $188,168,032.1957  $200,000,000 = $200,000,000. Because about 4.055% is the internal rate of return that satisfies the above expression, 4.055% is the dollar-weighted return. The dollar-weighted rate of return and the time-weighted rate of return will produce the same result if no withdrawals or contributions occur over the evaluation period and all investment income is reinvested. The dollar-weighted rate of return can be affected by factors that are beyond the control of the manager. Specifically, any contributions made by the client or withdrawals that the client requires will affect the calculated return. This makes it difficult to compare the performance of two managers when using this method. (b) Suppose that the $8 million cash outflow in the second month was a result of withdrawals by the plan sponsor and that the cash flow after adjusting for this withdrawal is therefore zero. What would the dollar-weighted rate of return then be for this client’s portfolio? A cash withdrawal is treated as a cash inflow. So, in the absence of any cash contribution made by a client for a given time period, a cash withdrawal (e.g., a distribution to a client) is a positive cash flow for that time period. However, this withdrawal is not by the client but by the plan sponsor so that C2 no longer equals $8 million but zero. Thus, we now have: V0 = $200 million, N = 3, C1 = $20 million, C2 = $0 million, C3 = $4 million, and V3 = $208 million. Given these values, RD is the interest rate that satisfies the following equation: $200,000,000 = . Below we verify that 5.4059618263% or about 5.406% is the internal rate of return satisfies the above expression. $200,000,000 =  $200,000,000 = $18,974,258.812 + $0 + $181,025,741.188  $200,000,000 = $200,000,000. Because about 5.406% is the internal rate of return that satisfies the above expression, 5.406% is the dollar-weighted return. 7. If the average quarterly return for a portfolio is 1.23%, what is the annualized return? The evaluation period may be less than or greater than one year. Typically, return measures are reported as an average annual return. This requires the annualization of the subperiod returns. The subperiod returns are typically calculated for a period of less than one year. The subperiod returns are then annualized using the following formula: annual return = (1 + average period return)number of periods in year – 1. For our problem, the period used to calculate returns is monthly and the average monthly return is 1.23%. Thus, the annual return is: annual return = (1 + average period return)number of periods in year – 1 = (1.0123)12 – 1 = 1.158006 – 1 = 0.158006 or about 15.80%. 8. If the average quarterly return for a portfolio is 1.78%, what is the annualized return? We use the following formula: annual return = (1 + average period return)number of periods in year – 1. For our problem, the period used to calculate returns is quarterly and the average quarterly return is 1.78%. Thus, the annual return is: annual return = (1 + average period return)number of periods in year – 1 = (1.0178)4 – 1 = 1.073124 – 1 = 0.073124 or about 7.31%. 9. What are the difficulties of constructing a normal portfolio? The difficulties of constructing a normal portfolio involve defining the universe of fixed-income securities to be included in the normal portfolio, and determining how these securities should be weighted. More details are given below. To evaluate the performance of a manager, a client must specify a benchmark against which the manager will be measured. There are two types of benchmarks that have been used in evaluating fixed-income portfolio managers: (i) market indexes published by dealer firms and vendors, and (ii) normal portfolios. A normal portfolio is a customized benchmark that includes “a set of securities that contains all of the securities from which a manager normally chooses, weighted as the manager would weight them in a portfolio.” Thus a normal portfolio is a specialized index. It is argued that normal portfolios are more appropriate benchmarks than market indexes because they control for investment management style, thereby representing a passive portfolio against which a manager can be evaluated. The construction of a normal portfolio for a particular manager is no simple task. The principle is to construct a portfolio that, given the historical portfolios held by the manager, will reflect that manager’s style in terms of assets and the weighting of those assets. The construction of a normal portfolio for a manager requires (i) defining the universe of fixed-income securities to be included in the normal portfolio, and (ii) determining how these securities should be weighted (i.e., equally weighted or capitalization weighted). Defining the set of securities to be included in the normal portfolio begins with discussions between the client and the manager to determine the manager’s investment style. Based on these discussions, the universe of all publicly traded securities is reduced to a subset that includes those securities that the manager considers eligible given his or her investment style. Given these securities, the next question is how they should be weighted in the normal portfolio. The two choices are equal weighting or capitalization weighting of each security. Various methodologies can be used to determine the weights. These methodologies typically involve a statistical analysis of the historical holdings of a manager and the risk exposure contained in those holdings. Plan sponsors work with pension consultants to develop normal portfolios for a manager. The consultants use vendor systems that have been developed for performing the needed statistical analysis and the necessary optimization program to create a portfolio displaying similar factor positions to replicate the “normal” position of a manager. A plan sponsor must recognize that there is a cost to developing and updating the normal portfolio. There are some who advocate the responsibility of developing normal portfolios should be left to the manager. However, many clients are reluctant to let their managers control the construction of normal portfolios because they believe that the managers will produce easily beaten, or “slow rabbit,” benchmarks. Bailey and Tierney demonstrate that under reasonable conditions there is no long-term benefit for the manager to construct a “slow rabbit” benchmark and explain the disadvantage of a manager pursuing such a strategy.8 In addition, they recommend that clients let managers control the benchmarks. Clients should, instead, focus their efforts on monitoring the quality of the benchmarks and the effectiveness of the managers’ active management strategies. 10. Suppose that the active return for a portfolio over the past year was 130 basis points after management fees. What questions would you have to before concluding that the manager’s performance was exceptional? Clients of asset management firms need to have more information than merely if a portfolio manager outperformed a benchmark and by how much. For example, you want to know the reasons. Thus, a first question you might ask is: “What are the reasons for why a portfolio manager realized the performance relative to the benchmark?” This question is important because it is possible a pension fund engaged an external manager based on the manager’s claim that return enhancement can be achieved via security selection. Given that the manager has in fact outperformed the benchmark by more than enough to cover management fees, we need to ask: “Did this manager achieve the stated objective?” This question is important because it is not known what specific risks relative to the benchmark that the manager took to generate the return. It is entirely possible that the outperformance was attributable to being mismatched against the benchmark’s duration. In fact, it is possible that the manager could have outperformed the benchmark due to a mismatch in duration and invested in specific securities that did poorly. There is no way that the client can determine that by simply looking at the portfolio’s return relative to the benchmark’s return. A third question we might want to ask is: “How did members of the team perform?” Not only do clients need information about why the portfolio’s return differed from that of the benchmark, but so do the individuals at the asset management firm engaged by the client. At the firm level, bonuses to members of the portfolio management team will be determined based on performance. Breaking down the performance to the team member level is important for this purpose, because it impacts decisions about the advancement and retention of such personnel. A fourth question we might want to ask is: “What performance models were used?” There are several performance attribution models that are available from third-party entities. (Some of the larger asset management firms have developed their own models.) In selecting a third-party model, there are requirements that a good attribution model should possess in order to evaluate the decision-making ability of the members of the portfolio management team: additivity, completeness, and fairness. 11. Not only do clients find performance attribution analysis helpful but so does the chief investment officer of an asset management firm in evaluating the firm’s bond portfolio team. Explain why. The chief investment officer of an asset management firm finds it useful for fairly evaluating the performance of employees and properly allocating bonuses and promotions within the firm. More details are given below. At the firm level, bonuses to members of the portfolio management team will be determined based on performance. Breaking down the performance to the team member level is important for this purpose, because it impacts decisions about the advancement and retention of such personnel. There are single metrics that have been commonly used to measure performance such as the information ratio. Although useful, single metric do not provide sufficient more information about performance to address the questions posed earlier. The model that can be used is performance attribution analysis, a quantitative technique for identifying the sources of portfolio risk and performance so that the contributions of members of the portfolio management team can be measured and the major portfolio decisions can be quantified. There are several performance attribution models that are available from third-party entities. (Some of the larger asset management firms have developed their own models.) In selecting a third-party model, there are requirements that a good attribution model should possess in order to evaluate the decision-making ability of the members of the portfolio management team: additivity, completeness, and fairness. Additivity means that contribution to performance of two or more decision makers of the portfolio management team should be equal to the sum of the contributions of those decision makers. Completeness means that when the contribution to portfolio performance of all decision makers is added up, the result should be equal to the contribution to portfolio performance relative to the benchmark. Finally, the decision-making process is one that involves the interaction of many members of the portfolio management team. Fairness means that the portfolio management team members should view the performance attribution model selected as being fair with respect to representing their contribution. 12. A financial institution has hired three external portfolio managers: X, Y, and Z. All three managers have the same benchmark. A performance attribution analysis of the portfolios managed by the three managers for the past year was (in basis points): Risk Factor Portfolio X Portfolio Y Portfolio Z Yield curve risk –1 92 –3 Swap spread risk 20 4 20 Volatility risk 40 3 25 Government related spread risk 35 –5 10 Corporate spread risk –2 6 30 Securitized spread risk –2 –4 5 The financial institution’s investment committee is using the above information to assess the performance of the three external managers. Below is a statement from three members of the performance evaluation committee. Respond to each statement. (a) Committee member 1: “Based on overall performance, it is clear that manager Y was the best performing manager given the 96 basis points.” Below we report totals when the basis points (plus and minus) are added for all six risk factors: Portfolio X Portfolio Y Portfolio Z Total of all 6 risk factors: 95 105 90 Based on overall performance, committee member #1 is correct because those in charge of Portfolio Y have an overall better performance. However, as seen above it achieves a 105 basis point active return beyond the benchmark and not an active return of 96 basis points. Its superior return was caused almost exclusively by its performance attributed to yield curve risk (92 basis point active return). Thus, its achievement is based on an interest rate bet, while its non-interest rate bets were slightly positive. In terms of its interest rate bet, we do not know to what extent its performance can be attributed to either level (duration) risk or shape risk. (b) Committee member 2: “All three of the managers were hired because they claimed that they had the ability to capitalize on corporate credit opportunities. Although they have all outperformed the benchmark, I am concerned about the claims that they made when we retained them.” Clients of asset management firms need to have more information than merely if a portfolio manager outperformed a benchmark and by how much. They need to know the reasons why a portfolio manager realized the performance relative to the benchmark. For the three portfolios, suppose a pension fund engaged an external manager based on the manager’s claim that return enhancement can be achieved via “corporate spread risk” selection. As seen below, portfolio managers for Portfolios Y and Z did in fact do well in this area. Risk Factor Portfolio X Portfolio Y Portfolio Z Corporate spread risk –2 6 30 However, only for Portfolio Z can we say there is strong proof that performance in the corporate spread risk was achieved. Thus, the concern of committee member #2 is valid due to differences in performances for the corporate credit products. (c) Committee member 3: “It seems that managers X and Z were able to outperform the benchmark without taking on any interest rate risk at all.” Interest rate risk is captured by the yield curve risk factor, while non-interest rate risk is captured by the other five risk factors. As seen below, it does appear that committee member #3 is correct in believing that managers of Portfolios X and Z were neutral in regards to interest rate risk (as we find small active returns in the interest rate risk category). Risk Factor Portfolio X Portfolio Y Portfolio Z Yield curve risk –1 92 –3 However, what the above breakdown does not include are the individual components of yield curve risk (level and shape risks). This is explained in more detail below. Factor-based attribution models actually allow a decomposition of the yield curve risk into level (duration) risk and changes in the shape of the yield curve. For example, for the three portfolios just discussed, suppose that the attribution due to yield curve risk is determined to be as follows: Risk Factor Portfolio X Portfolio Y Portfolio Z Yield curve risk –1 92 –3 Level risk –51 80 +40 Shape risk +50 12 –43 Notice that once yield curve risk is decomposed as shown above, it turns out those managers of Portfolios X and Z did indeed make interest rate bets on both level risk and shape risk. It turns out that the two bets almost offset each other so there were only small basis point returns attributable to the interest rate bets. It appears that Portfolio Z’s manager made a major duration bet and a minor bet on changes in the shape of the yield curve. Thus, as it turns out, committee member #3 was incorrect as managers of Portfolios X and Z were making greater interest rate bets than that of Portfolio Y. Thus, given the above breakdown of yield curve risk, their performance was not all related to their lack of interest rate bets. We one cannot make definitive conclusions about portfolio managerial performance in terms of interest rate bets without a more detailed analysis. CHAPTER 26 INTEREST-RATE FUTURES CONTRACTS CHAPTER SUMMARY A futures contract is an agreement that requires a party to the agreement either to buy or sell something at a designated future date at a predetermined price. In this chapter we describe interest-rate futures contracts. With the advent of interest-rate futures and other derivatives, managers can achieve new degrees of freedom. It is now possible to alter the interest rate sensitivity of a bond portfolio or an asset/liability position economically and quickly. MECHANICS OF FUTURES TRADING A futures contract is a firm legal agreement between a buyer (seller) and an established exchange or its clearinghouse in which the buyer (seller) agrees to take (make) delivery of something at a specified price at the end of a designated period of time. The price at which the parties agree to transact in the future is called the futures price. The designated date at which the parties must transact is called the settlement or delivery date. The contract with the nearest settlement date is called the nearby futures contract. The next futures contract is the one that settles just after the nearby contract. The contract furthest away in time from settlement is called the most distant futures contract. Opening Position When an investor takes a position in the market by buying a futures contract, the investor is said to be in a long position or to be long futures. If, instead, the investor’s opening position is the sale of a futures contract, the investor is said to be in a short position or short futures. Liquidating a Position A party to a futures contract has two choices on liquidation of the position. First, the position can be liquidated prior to the settlement date. The alternative is to wait until the settlement date. For some futures contracts, settlement is made in cash only. Such contracts are referred to as cash-settlement contracts. Role of the Clearinghouse Associated with every futures exchange is a clearinghouse, which performs several functions. One of these functions is guaranteeing that the two parties to the transaction will perform. When an investor takes a position in the futures market, the clearinghouse takes the opposite position and agrees to satisfy the terms set forth in the contract. A futures contract is an agreement between a party and a clearinghouse associated with an exchange. The clearinghouse makes it simple for parties to a futures contract to unwind their positions prior to the settlement date. Margin Requirements When a position is first taken in a futures contract, the investor must deposit a minimum dollar amount per contract as specified by the exchange. This amount, called the initial margin, is required as deposit for the contract. At the end of each trading day, the exchange determines the settlement price for the futures contract. This price is used to mark to market the investor’s position, so that any gain or loss from the position is reflected in the investor’s equity account. The maintenance margin is the minimum level (specified by the exchange) by which an investor’s equity position may fall as a result of an unfavorable price movement before the investor is required to deposit additional margin. The additional margin deposited, called the variation margin, is the amount necessary to bring the equity in the account back to its initial margin level. FUTURES VERSUS FORWARD CONTRACTS Just like a futures contract, a forward contract is an agreement for the future delivery of the underlying at a specified price at the end of a designated period of time. Futures contracts are traded on organized exchanges and are standardized agreements as to the delivery date (or month) and quality of the deliverable. A forward contract differs in that it has no clearinghouse, usually has non-standardized contracts (i.e., the terms of each contract are negotiated individually between buyer and seller), and typically has nonexistent or extremely thin secondary markets. Because there is no clearinghouse that guarantees the performance of a counterparty in a forward contract, the parties to a forward contract are exposed to counterparty risk. RISK AND RETURN CHARACTERISTICS OF FUTURES CONTRACTS The buyer of a futures contract will realize a profit if the futures price increases; the seller of a futures contract will realize a profit if the futures price decreases. If the futures price decreases, the buyer of a futures contract realizes a loss while the seller of a futures contract realizes a profit. Leveraging Aspect of Futures When a position is taken in a futures contract, the party need not put up the entire amount of the investment. Instead, only initial margin must be put up. Although the degree of leverage available in the futures market varies from contract to contract, the leverage attainable is considerably greater than in the cash market. Futures markets can be used to reduce price risk. Without the leverage possible in futures transactions, the cost of reducing price risk using futures would be too high for many market participants. INTEREST-RATE FUTURES CONTRACTS The two major exchanges where interest rate futures are traded are those operated by the Chicago Mercantile Exchange Group (CME Group) and Euronext Life. The exchanges offer bond futures and short-term interest rate futures. Below we describe two of the major contracts used for risk control by institutional investors: Eurodollar futures and U.S. Treasury futures. Eurodollar Futures Eurodollar futures contracts are traded on both the International Monetary Market of the Chicago Mercantile Exchange and the London International Financial Futures Exchange. The Eurodollar time deposit with a principal value of U.S. $1 million and three months to maturity is the underlying for this contract. The contract is quoted on an index price basis. For example, the index price might be 94.52. From the futures index price, the annualized futures three-month LIBOR is determined as follows: 100 minus the index price. For example, a Eurodollar futures index price of 94.52 means the parties to this contract agree to buy or sell the three-month LIBOR for 5.48%. Since the underlying is an interest rate that obviously cannot be delivered, this contract is a cash settlement contract. An index price change from 94.52 to 94.53 changes the three-month LIBOR from 5.48% to 5.47%. In terms of basis points, a one-tick change in the index price means a 1-basis-point change in the three-month LIBOR. If LIBOR changes by 1 basis point (where 0.01% = 0.0001), then $1,000,000 × (0.0001 × 90/360) = $25. Hence, a one-tick change in the index price or, equivalently, a 1-basis-point change in the three-month LIBOR means a $25 change in the value of the contract. To see how a Eurodollar futures contract is used for hedging, suppose that a market participant is concerned that its borrowing costs six months from now are going to be higher. To protect itself, it takes a short (selling) position in the Eurodollar futures contract such that a rise in short-term interest rates will benefit. To see this, consider our previous illustration in the Eurodollar futures at 94.52 (5.48% rate). Suppose at the settlement date the three-month LIBOR increases to 6.00% and, therefore, the settlement index price is 94.00. This means that the seller sold the contract for 94.52 and purchased it for 94.00, realizing a gain of 0.52 or 52 ticks. The buyer must pay the seller 52 × $25 = $1,300. The gain from the short futures position is then used to offset the higher borrowing cost resulting from a rise in short-term interest rates. For euro-denominated loans, the reference rate used is typically the Euro Interbank Offered Rate (Euribor). Euribor is the rate on deposits denominated in euros and it is the underlying for the Euribor futures contact. Treasury Futures The most active bond derivatives contracts are the Treasury futures contracts. These contracts are classified by maturity. The underlying for the Treasury bond futures contract are certain Treasury coupon securities that were originally issued as Treasury bonds. Treasury note futures contracts include the two-year, five-year, and 10-year Treasury futures. Treasury Bond Futures The underlying instrument for a Treasury bond futures contract is $100,000 par value of a hypothetical 20-year 8% coupon bond. The futures price is quoted in terms of par being 100. Quotes are in 32nds of 1%. The seller of a Treasury bond futures who decides to make delivery rather than liquidate his position by buying back the contract prior to the settlement date must deliver some Treasury bond. The CME Group allows the seller to deliver one of several Treasury bonds that the CME Group declares is acceptable for delivery. The specific bonds that the seller may deliver are published by the CME Group prior to the initial trading of a futures contract with a specific settlement date. Exhibit 26-1 shows the Treasury issues that the seller can select from to deliver to the buyer of four Treasury bond futures contract by settlement month. The price that the buyer must pay the seller when a Treasury bond is delivered is called the invoice price, which is given as: invoice price = (contract size × futures contract settlement price × conversion factor) + accrued interest. In selecting the issue to be delivered, the short will select from all the deliverable issues the one that is cheapest to deliver. This issue is referred to as the cheapest-to-deliver issue; it plays a key role in the pricing of this futures contract. Knowing the price of the Treasury issue, the seller can calculate the return, which is called the implied repo rate. The cheapest-to-deliver issue is then the one issue among all acceptable Treasury issues with the highest implied repo rate because it is the issue that would give the seller of the futures contract the highest return by buying and then delivering the issue. This is depicted in Exhibit 26-2. In addition to the choice of which acceptable Treasury issue to deliver (sometimes referred to as the quality option or swap option) the short position has two more options granted under CBT delivery guidelines. The short position is permitted to decide when in the delivery month delivery actually will take place. This is called the timing option. The other option is the right of the short position to give notice of intent to deliver after the closing of the exchange on the date when the futures settlement price has been fixed. This option is referred to as the wild card option. In sum, these three options are referred to as the delivery options. Treasury Note Futures There are three Treasury note futures contracts: 10-year, five-year, and two-year. All three contracts are modeled after the Treasury bond futures contract and are traded on the CME Group. The underlying instrument for the 10-year Treasury note futures contract is $100,000 par value of a hypothetical 10-year 6% Treasury note. For the five-year Treasury note futures contract, the underlying is $100,000 par value of a U.S. Treasury note. The underlying for the two-year Treasury note futures contract is $200,000 par value of a U.S. Treasury note with a remaining maturity of not more than two years and not less than one year and nine months. PRICING AND ARBITRAGE IN THE INTEREST-RATE FUTURES MARKET There are several different ways to price futures contracts. Each approach relies on the “law of one price.” This law states that a given financial asset (or liability) must have the same price regardless of the means by which it is created. Pricing of Futures Contracts Suppose that a 20-year 100-par-value bond with a coupon rate of 12% is selling at par. Also suppose that this bond is the deliverable for a futures contract that settles in three months. If the current three-month interest rate at which funds can be loaned or borrowed is 8% per year, what should be the price of this futures contract? Suppose that the price of the futures contract is 107. Consider the following strategy: • Sell the futures contract at 107. • Purchase the bond for 100. • Borrow 100 for three months at 8% per year. The borrowed funds are used to purchase the bond, resulting in no initial cash outlay for this strategy. Three months from now, the bond must be delivered to settle the futures contract, and the loan must be repaid. These trades will produce the following cash flows: From Settlement of the Futures Contract: Flat price of bond 107 Accrued interest (12% for 3 months) 3 Total proceeds (107 + 3) 110 From the Loan: Repayment of principal of loan 100 Interest on loan (8% for 3 months) 2 Total outlay (100 + 2) 102 Profit (110 – 102) 8 This strategy will guarantee a profit of 8. Moreover, the profit is generated with no initial outlay because the funds used to purchase the bond are borrowed. The profit will be realized regardless of the futures price at the settlement date. Obviously, in a well-functioning market, arbitrageurs would buy the bond and sell the futures, forcing the futures price down and bidding up the bond price so as to eliminate this profit. This strategy is called a cash-and-carry trade. If the future price was 92 instead of 107, we could reverse the process to make an arbitrage profit. This strategy is called a reverse cash-and-carry trade. There would be no arbitrage if the future price is 99. Hence, the futures price of 99 is the theoretical futures price because any higher or lower futures price will permit arbitrage profits. Theoretical Futures Price Based on Arbitrage Model Considering the arbitrage arguments just presented, the theoretical futures price can be determined on the basis of the following information: 1. The price of the bond in the cash market. 2. The coupon rate on the bond. In our example, the coupon rate is 12% per year. 3. The interest rate for borrowing and lending until the settlement date. The borrowing and lending rate is referred to as the financing rate. In our example, the financing rate is 8% per year. We will let: r = financing rate; c = current yield or coupon rate divided by the cash market price; P = cash market price; F = futures price; t = time, in years, to the futures delivery date. Now consider the following cash-and-carry trade strategy that is initiated on a coupon date: • Sell the futures contract at F. • Purchase the bond for P. • Borrow P until the settlement date at r. The outcome at the settlement date is From Settlement of the Futures Contract: Flat price of bond F Accrued interest ctP Total proceeds F+ctP From the Loan: Repayment of principal of loan P Interest on loan (8% for 3 months) rtP Total outlay P+rtP Profit (total proceeds – total outlay) (F+ctP) – (P+rtP) In equilibrium the theoretical futures price occurs where the profit from this trade is zero. Solving for the theoretical futures price, we have: F = P[1+t(rL– c)]. Alternatively, we could consider a reverse cash-and-carry trade strategy paralleling the above. The theoretical futures price may be at a premium to the cash market price (higher than the cash market price) or at a discount from the cash market price (lower than the cash market price), depending on (r – c). The term r – c is called the net financing cost because it adjusts the financing rate (r) for the coupon interest rate earned (c). The net financing cost is more commonly called the cost of carry, or simply carry. Positive carry means that the current yield earned is greater than the financing cost. Negative carry means that the financing cost exceeds the current yield. Closer Look at the Theoretical Futures Price To derive the theoretical futures price using the arbitrage argument, we made several assumptions, which have certain implications. Interim Cash Flows No interim cash flows due to variation margin or coupon interest payments were assumed in the model. Incorporating interim coupon payments into the pricing model is not difficult. However, the value of the coupon payments at the settlement date will depend on the interest rate at which they can be reinvested. The shorter the maturity of the futures contract and the lower the coupon rate, the less important the reinvestment income is in determining the futures price. Short-Term Interest Rate (Financing Rate) In deriving the theoretical futures price, it is assumed that the borrowing and lending rates are equal. Typically, however, the borrowing rate (rB) is higher than the lending rate (rL). The futures price that would produce no arbitrage profit is F = P[1+t(rB– c)].The futures price that would produce no profit is F = P[1+t(rL– c)]. These latter two equations together provide boundaries for the theoretical futures price. The upper boundary is F (upper boundary) = $100[1 + 0.25(0.08 – 0.12)] = $99.00 The lower boundary using equation (24.3) is F (lower boundary) = $100[1 + 0.25(0.06 – 0.12)] = $98.50 In calculating these boundaries, we assume that no transaction costs are involved in taking the position. In actuality, the transaction costs of entering into and closing the cash position as well as the round-trip transaction costs for the futures contract must be considered and do affect the boundaries for the futures contract. Deliverable Bond Is Not Known The arbitrage arguments used to derive equation F = P[1+t(r– c)] assumed that only one instrument is deliverable. But the futures contracts on Treasury bonds and Treasury notes are designed to allow the short the choice of delivering one of a number of deliverable issues (the quality or swap option). Because the swap option is an option granted by the long to the short, the long will want to pay less for the futures contract than indicated by F = P[1+t(r– c)]. Therefore, as a result of the quality option, the theoretical futures price as given by this equation must be adjusted as follows: F = P[1+t(r– c)] – value of quality option Market participants have employed theoretical models in attempting to estimate the fair value of the quality option. These models are beyond the scope of this chapter. Delivery Date Is Not Known In the pricing model based on arbitrage arguments, a known delivery date is assumed. For Treasury bond and note futures contracts, the short has a timing and wild card option, so the long does not know when the securities will be delivered. The effect of the timing and wild card options on the theoretical futures price is the same as with the quality option. These delivery options should result in a theoretical futures price that is lower than the one suggested: F = P[1+t(r– c)]– value of quality option – value of timing option – value of wildcard option Market participants attempt to value the delivery option in order to apply this latter equation. Deliverable Is Not a Basket of Securities The municipal index futures contract is a cash settlement contract based on a basket of securities. The difficulty in arbitraging this futures contract is that it is too expensive to buy or sell every bond included in the index. Instead, a portfolio including a smaller number of bonds may be constructed to “track” the index. BOND PORTFOLIO MANAGEMENT APPLICATIONS As described next, there are various ways in which a portfolio manager can use interest-rate futures contracts. Controlling the Duration of a Portfolio Interest-rate futures can be used to alter the interest-rate sensitivity of a portfolio. Portfolio managers with strong expectations about the direction of the future course of interest rates will adjust the durations of their portfolios so as to capitalize on their expectations. In addition to adjusting a portfolio based on anticipated interest-rate movements, futures contracts can be used in constructing a portfolio with a longer duration than is available with cash market securities. By buying the appropriate number and kind of interest-rate futures contracts, a manager can increase the portfolio’s duration to the target level. Suppose that the manager wants to restructure the portfolio so that its duration matches that of the benchmark. That is, the portfolio manager seeks to follow a duration-matched strategy and therefore the portfolio’s target duration is 3.68. For a 100 basis change in interest rates, the portfolio’s target dollar duration is then the product of 3.68% times the current market value of the portfolio (which is $48,109,810). Therefore, portfolio target dollar duration = 3.68% × $48,109,810 = $1,770,110 The current portfolio duration is 2.97, so for a 100 basis point change in interest rates, portfolio current dollar duration = 2.97% × $48,109,810 = $1,428,594 The difference between the target and the current dollar duration for the portfolio is $341,516. This means that to get to the target portfolio duration of 3.68, the portfolio manager must increase the dollar duration of the current portfolio by $341,516. Suppose that the portfolio manager wants to use the 5-Treasury note futures contract. The futures price on March 31, 2011 was 116.79. Based on an analysis of this contract, the portfolio manager determines that for a 100 basis point change in interest rates, the 5-year Treasury note futures contract will change by roughly $5,022. If the portfolio manager buys C contracts, then the dollar duration of the futures position for a 100 basis point change in interest rates is the product of the number futures contract; that is, dollar duration of futures contract = $5,022 × C The portfolio manager wants the above equation to be equal $341,516. Thus, $5,022 × C = $341,516. Solving we get C = 68 contracts. Thus, by buying 68 5-year Treasury note futures contracts, the portfolio manager will increase the dollar duration of the portfolio by $341,516 for a 100 basis point change in interest rates. Since the notional amount of the futures contract is $100,000, this means that the total notional amount of the futures position is $6,800,000. The market value of the futures position (given that the future price is 116.79 on March 31, 2011) is equal to (116.79/100) × $100,000 × 68 = $7,941,720 A formula to approximate the number of futures contracts necessary to adjust the portfolio duration to a target level is approximate number of futures contract A negative value indicates the number of contracts that should be sold; a positive value indicates the number of contracts that should be purchased. In our example, it is Suppose instead that the portfolio manager does not want the duration to match the benchmark but instead wants the duration to be 90% of the benchmark. Since the benchmark duration is 3.68, this means that the manager wants the portfolio’s target duration to be 3.31. The portfolio target dollar duration for a 100 basis point change in interest rates is portfolio target dollar duration = 3.31% × $48,109,810 = $1,592,435 Then the approximate number of futures contract to increase the duration is equal to . Hence, 33 futures contracts should be purchased. Hedging Hedging is nothing more than a special case of interest rate risk management where the target duration is zero. In the case of hedging a portfolio, a simple way to determine the number of futures contract to short is to use. The number of futures contract to short is found by dividing the current portfolio dollar duration by the dollar duration of the futures contract that is used as the hedging vehicle. In hedging an individual bond position with futures, the hedger is taking a futures position as a temporary substitute for transactions to be made in the cash market at a later date. If cash and futures prices move together, any loss realized by the hedger from one position (whether cash or futures) will be offset by a profit on the other position. When the net profit or loss from the positions is exactly as anticipated, the hedge is referred to as a perfect hedge. The difference between the cash price and the futures price is the basis. The risk that the basis will change in an unpredictable way is called basis risk. In bond portfolio management, typically, the bond to be hedged is not identical to the bond underlying the futures contract. This type of hedging is referred to as cross hedging. There may be substantial basis risk in cross hedging. A short (or sell) hedge is used to protect against a decline in the cash price of a bond. A long (or buy) hedge is undertaken to protect against an increase in the cash price of a bond. In a long hedge, the hedger buys a futures contract to lock in a purchase price. The Hedge Ratio The key to minimizing risk in a cross hedge is to choose the right hedge ratio. The hedge ratio depends on volatility weighting, or weighting by relative changes in value. The purpose of a hedge is to use gains or losses from a futures position to offset any difference between the target sale price and the actual sale price of the asset. Accordingly, the hedge ratio is chosen with the intention of matching the volatility (i.e., the dollar change) of the futures contract to the volatility of the asset. Consequently, the hedge ratio is given by hedge ratio = . This equation shows that if the bond to be hedged is more volatile than the hedging instrument, then the more of the hedging instrument will be needed. For hedging purposes, we are concerned with volatility in absolute dollar terms. The relevant point in the life of the bond for calculating volatility is the point at which the hedge will be lifted. Volatility at any other point is essentially irrelevant because the goal is to lock in a price or rate only on that particular day. Similarly, the relevant yield at which to calculate volatility initially is the target yield. Consequently, the “volatility of the bond to be hedged” in the above equation is the price value of a basis point for the bond on the date the hedge is expected to be delivered. To calculate the hedge ratio, we need the volatility not of the cheapest-to-deliver issue, but of the hedging instrument (i.e. of the futures contract). Fortunately, knowing the volatility of the bond to be hedged relative to the cheapest-to-deliver issue and the volatility of the cheapest-to-deliver bond relative to the futures contract, we can easily obtain the relative volatilities that define the hedge ratio: hedge ratio = where CTD issue is the cheapest-to-deliver issue. The second ratio can be shown to equal the conversion factor for the CTD issue. Assuming a fixed yield spread between the bond to be hedged and the CTD issue, we have hedge ratio = × conversion factor for CTD. Given the hedge ratio, the number of contracts that must be short is determined as follows: number of contracts = hedge ratio × . Exhibit 26-4 shows that if the simplifying assumptions hold, a futures hedge using the recommended hedge ratio very nearly locks in the target forward amount of $9,678,000 for $10 million par value of the P&G bond. Adjusting the Hedge Ratio for Yield Spread Changes Yield spreads are not constant over time. They vary with the maturity of the instruments in question and the level of rates, as well as with many unpredictable and nonsystematic factors. Because of this, the hedge ratio has to be adjusted. The formula for the revised hedge ratio that incorporates the impact of the yield beta is: hedge ratio = × conversion factor for CTD × adjustment factor Two approaches have been suggested for estimating the adjustment factor that takes into account the relationship between yield levels and yield spreads: (1) regression approach, and (2) pure volatility approach. First, the regression approach involves estimating from historical data the following regression model: yield change on bond to be hedged = a + b × yield change on CTD issue + error The regression procedure provides an estimate of b, called the yield beta, which is the expected relative change in yield between the two bonds. The formula for the revised hedge ratio that incorporates the impact of the yield beta is: hedge ratio = × conversion factor for CTD × yield beta The second approach for capturing the relative movement in yields and estimating the adjustment is the pure volatility adjustment. This is done by first calculating the daily change in yield for the bond to be hedged and the CTD issue. Then for the change in yield, the standard deviation is computed. The pure volatility adjustment is just the ratio of the two standard deviations. That is, In contrast to the regression approach, one can see that the adjustment differs depending on the time period used, and in addition, the number of contracts to be shorted would be greater since the pure volatility adjustment is greater than one. Change in the CTD Issue The effect of a change in the cheapest-to-deliver issue and the yield spread can be assessed a priori. For example, at different yield levels at the date the hedge is to be lifted, a different yield spread may be appropriate and a different acceptable issue will be the CTD. The portfolio manager can determine what this will do to the outcome of the hedge. Creating Synthetic Securities for Yield Enhancement A cash market security can be created synthetically by taking a position in the futures contract together with the deliverable instrument. If the yield on the synthetic security is the same as the yield on the cash market security, there will be no arbitrage opportunity. Any difference between the two yields can be exploited so as to enhance the yield on the portfolio. By being long a bond and short a futures, an investor can synthetically create a three-month Treasury bill. The return the investor should expect to earn from this synthetic position should be the yield on a three-month Treasury bill. If the yield on the synthetic three-month Treasury bill is greater than the yield on the cash market Treasury bill, the investor can realize an enhanced yield by creating the synthetic short-term security. The fundamental relationship for creating synthetic securities is RSP = CBP + FBP where RSP = riskless short-term security position, CBP = cash bond position, and FBP = bond futures position. A negative sign before a position means a short position. For the long bond position, we have: CBP = RSP + FBP. This equation states that a cash bond position equals a short-term riskless security position plus a long bond futures position. Thus, a cash market bond can be created synthetically by buying a futures contract and investing in a Treasury bill. Solving the latter equation for the bond futures position, we have FBP = CBP + RSP. This equation tells us that a long position in the futures contract can be created synthetically by taking a long position in the cash market bond and shorting the short-term riskless security. Allocating Funds between Stocks and Bonds A pension sponsor may wish to alter the composition of its assets by increasing bonds and decreasing stocks. A manager could undertake costly process of buying bonds and selling stocks. However, an alternative course of action is to use interest-rate futures and stock index futures. Buying an appropriate number of interest-rate futures and selling an appropriate number of stock index futures can achieve the desired exposure to stocks and bonds. The advantages of using financial futures contracts are: (1) transactions costs are lower, (2) market impact costs are avoided or reduced by allowing the sponsor time to buy and sell securities in the cash market, and (3) activities of the portfolio managers employed by the pension sponsor are not disrupted. To determine the approximate number of interest-rate futures contracts needed to change the market value of the portfolio allocated to bonds, we use the following expression: approximate number of contracts = . KEY POINTS A futures contract is an agreement between a buyer (seller) and an established exchange or its clearinghouse in which the buyer (seller) agrees to take (make) delivery of something at a the futures price at the settlement or delivery date. Associated with every futures exchange is a clearinghouse, which guarantees that the two parties to the transaction will perform and allows parties to unwind their position without the need to deal with the counterparty to the initial transaction. A party to a futures contract must comply with margin requirements (initial, maintenance, and variation margin). A forward contract differs from a futures contract in that it is usually non-standardized (i.e., the terms of each contract are negotiated individually between buyer and seller), there is no clearinghouse, and secondary markets are often nonexistent or extremely thin. Futures contracts are traded on short-term interest rates, the most active being the Eurodollar futures and Euribor futures. The most active bond futures are Treasury futures (Treasury bond futures and Treasury note futures). The theoretical price of a futures contract is equal to the cash or spot price plus the cost of carry. The cost of carry is equal to the cost of financing the position less the cash yield on the underlying security. The shape of the yield curve will affect the cost of carry. There are several reasons why the actual futures price will depart from the theoretical futures price. In the case of the Treasury bond futures contracts, the delivery options granted to the seller reduce the actual futures price below the theoretical futures price suggested by the standard arbitrage model. Buying futures adds dollar duration to a portfolio; selling futures reduces a portfolio’s dollar duration. Interest-rate futures contracts can be used by portfolio managers to control a portfolio’s duration, to hedge a portfolio or bond position, to enhance returns when futures are mispriced, and to efficient allocate funds between stocks and bonds. ANSWERS TO QUESTIONS FOR CHAPTER 26 (Questions are in bold print followed by answers.) 1. Explain the differences between a futures contract and a forward contract. Just like a futures contract, a forward contract is an agreement for the future delivery of the underlying at a specified price at the end of a designated period of time. Futures contracts are traded on organized exchanges and are standardized agreements as to the delivery date (or month) and quality of the deliverable. However, a forward contract differs in that it has no clearinghouse, usually has non-standardized contracts (i.e., the terms of each contract are negotiated individually between buyer and seller), and typically has nonexistent or extremely thin secondary markets. Also, unlike a futures contract, a forward contract is an over-the-counter instrument. Because there is no clearinghouse that guarantees the performance of a counterparty in a forward contract, the parties to a forward contract are exposed to the risk that the other party to the transaction will fail to perform. Although both futures and forward contracts set forth terms of delivery, futures contracts are not intended to be settled by delivery. In fact, generally less than 2% of outstanding contracts are settled by delivery. Forward contracts, in contrast, are intended for delivery. A futures contract is marked to market at the end of each trading day, whereas a forward contract may or may not be marked to market. Just how much variation margin may be required by one or both parties of a forward contract depends on the terms negotiated. Therefore, although a futures contract is subject to interim cash flows as additional margin may be required in the case of adverse price movements, or as cash is withdrawn in the case of favorable price movements, variation margin may or may not result from a forward contract. 2. Answer the below questions. (a) What is counterparty risk? Counterparty risk is the risk that the other party to the transaction will fail to perform. That is, a party is exposed to credit or default risk. (b) Why do both the buyer and seller of a forward contract face counterparty risk? Because there is no clearinghouse that guarantees the performance of a counterparty in a forward contract, both the buyer and seller of a forward contract are exposed to counterparty risk even though only one side (party) could experience that negative effect per contract. 3. What does it mean if the cost of carry is positive for a Treasury bond futures contract? Positive carry means that the current yield earned is greater than the financing cost and the futures price sells at a discount. Negative carry means that the financing cost exceeds the current yield and the futures price sells at a premium. More details on the variables used in the cost of carry are given below. The theoretical Treasury bond futures price may be at a premium to the cash market price (higher than the cash market price) or at a discount from the cash market price (lower than the cash market price), depending on (r – c) where r = financing rate and c = current yield where c is the coupon rate divided by the cash market price. The term r – c is called the net financing cost because it adjusts the financing rate for the coupon interest earned. The net financing cost is more commonly called the cost of carry, or simply carry. 4. If the Eurodollar CD futures contract is quoted at 91.75, what is the annualized futures three-month LIBOR? The three-month Eurodollar CD is the underlying instrument for the Eurodollar CD futures contract. As with the Treasury bill futures contract, this contract is for $1 million of principal value and is traded on an index price basis. The index price basis in which the contract is quoted is equal to 100 minus the annualized futures LIBOR. In our problem, a Eurodollar CD futures price of 91.75 means a futures three-month LIBOR of 100 – 91.75 = 8.25. This translates into a rate of return of 8.25%. Thus, the annualized futures three-month LIBOR is 8.25%. 5. Suppose that an investor purchased a Eurodollar futures contract at an index price of 95.00. At the settlement date, suppose that the settlement price is 95.40. Explain whether the buyer or the seller of the futures contract receives a payment at the settlement date. The seller of the futures contract receives a payment from the buyer because interest rates have fallen causing the settlement price to rise giving a profit to the investor who purchased the right to buy at the settlement price. More details are given below. The Eurodollar futures contract is a cash settlement contract. That is, the parties settle in cash based on three-month LIBOR at the settlement date. For our problem, a trade occurs at 95.00 and on the settlement date the settlement index price is 95.40. From the perspective of the buyer, the index price increased. Hence, the seller must pay the buyer 95.40 – 95.00 = 0.40. Since one tick is $25 and 0.40 is 40 ticks, the buyer receives from the seller 40 × $25 = $1,000. An alternative way of thinking about this is that the buyer contracted to receive a three-month interest rate of (100.00% – 95.00%) = 5.00%. At the settlement date, the index price is 95.40. This means a three-month LIBOR of 4.60% interest rate is available in the market. The compensation of $1,000 of the seller from the buyer is for the lower prevailing three-month LIBOR of 4.60% rather than the contracted amount of 5.00%. To see how this contract is used for hedging, suppose that a market participant is concerned that her borrowing costs six months from now are going to be higher. To protect itself, it can take a position in the Eurodollar futures contract such that a rise in short-term interest rates will benefit. With this position in the futures contract, if short-term interest rates do in fact increase, future borrowing costs will rise but they will be offset (in whole or in part) from the position in the Eurodollar futures contract. A position in the futures contract that will benefit from a rise in interest rates is a short position (i.e., sell the futures contract). To see this, consider our previous illustration in the Eurodollar futures at 95.00 (5.00% rate). Suppose at the settlement date the three-month LIBOR increases to 6.00% and, therefore, the settlement index price is 94.00. This means that the seller sold the contract for 95.00 and purchased it for 94.00, realizing a gain of 1.00 or 100 ticks. The buyer must pay the seller 100 × $25 = $2,500. The gain from the short futures position is then used to offset the higher borrowing cost resulting from a rise in short-term interest rates. 6. Explain how a market participant concerned with a decline in three-month LIBOR can hedge that risk using the Eurodollar futures contract. A Eurodollar CD is a dollar-denominated CD issued outside of the United States, typically by a European bank. The interest rate paid on Eurodollar CDs is the London Interbank Offered Rate (LIBOR) and is an add-on-interest. Three-month LIBOR is the underlying for the Eurodollar futures contract. That is, the parties are agreeing to buy and sell “three-month LIBOR.” To see how our three-month Eurodollar futures contract can be used for hedging, let us assume that our market participant is an investor who is concerned that its investment possibilities in Eurodollar CDs three months from now are going to be at a rate that is lower than today’s rate. Further assume that our investor would like to buy the Eurodollar CDs now and lock in the current rate but will not have the funds to do this for three months. In order to hedge against the fall in Eurodollar CD rates, our investor can take a position in the Eurodollar futures contract such that a fall in short-term interest rates will bring a benefit. With a position in the futures contract, if short-term interest rates do in fact decrease, future rates of return will fall but these rates will be offset (in whole or in part) from the position in the Eurodollar futures contract. A position in the futures contract that will benefit from a fall in interest rates is a long position (i.e., buy the futures contract). To see this, consider a trade in the Eurodollar futures at 95.00 (5.00% rate). Suppose at the settlement date the three-month LIBOR falls to 4.00% and, therefore, the settlement index price is 96.00. This means that the seller sold the contract for 95.00 and purchased it for 96.00, realizing a loss of 1.00 or 100 ticks. The seller must pay the buyer 100 × $25 = $2,500 per contract bought. The gain from the short futures position is then used to offset the lower investment rates resulting from a fall in short-term interest rates. Thus, by buying Eurodollar futures contracts, our investor can offset any losses from having to wait three months to purchase Eurodollar CDs. 7. Answer the below questions. (a) What is Euribor? LIBOR is the most commonly used reference rate for floating-rate bank loans and derivative instruments denominated in U.S. dollars and British pounds. For euro-denominated loans and derivatives, when a reference rate is used, it is typically the Euro Interbank Offered Rate (Euribor). Euribor is the rate on deposits denominated in euros. The Euribor futures contract, traded on the NYSE Euronext, and the Eurodollar futures contract are the most actively traded futures contracts in the world. (b) What is Euribor futures contract? The Euribor futures contract is similar to the Eurodollar futures contract. The unit of trading is €1,000,000 where € stands for euro. The Eurodollar futures contract is a cash-settled contract. The underlying is 30-day Euribor. More specifically, it is based on the European Banking Federations’ Euribor for three-month deposits. The index price is 100 minus Euribor. The delivery months are March, June, September, December, and four serial months, such that 25 delivery months are available for trading, with the nearest six delivery months being consecutive calendar months. The minimum price movement (tick size) is 0.005, which is equal to €12.50. 8. For a Treasury futures contract, how do you think the cost of carry will affect the decision of the short as to when in the delivery month the short will elect to deliver? The short will elect to deliver at that time when it is most advantageous. For example, consider a short position such that future price falls to the extent c > r. By delivering as early as possible the future price would be enhanced. 9. Explain the asymmetric effect on the variation margin and cash flow for the short and long in an interest-rate futures contract when interest rates change. If interest rates rise, the short position in futures will receive margin as the futures price decreases; the margin can then be reinvested at a higher interest rate. In contrast, if interest rates fall, there will be variation margin that must be financed by the short position; however, because interest rates have declined, financing will be possible at a lower cost. The opposite occurs for the long position. 10. What are the delivery options granted to the seller of the Treasury bond futures contract? The delivery options are the quality option or swap option, the timing option, and the wild card option. The options imply that the long position can never be sure of which Treasury bond will be delivered or exactly when it will be delivered. 11. How is the theoretical futures price of a Treasury bond futures contract affected by the delivery options granted to the short? In the case of the Treasury bond futures contracts, the delivery options granted to the seller reduce the actual futures price below the theoretical futures price suggested by the standard arbitrage model. More details are given below. In selecting the issue to be delivered, the short will select from all the deliverable issues the one that is cheapest to deliver. This issue is referred to as the cheapest-to-deliver issue; it plays a key role in the pricing of this futures contract. The cheapest-to-deliver issue is determined by participants in the market as follows. For each of the acceptable Treasury issues from which the seller can select, the seller calculates the return that can be earned by buying that issue and delivering it at the settlement date. Note that the seller can calculate the return because she knows the price of the Treasury issue now and the futures price that she agrees to deliver the issue. The return so calculated is called the implied repo rate. The cheapest-to-deliver issue is then the one issue among all acceptable Treasury issues with the highest implied repo rate because it is the issue that would give the seller of the futures contract the highest return by buying and then delivering the issue. In addition to the choice of which acceptable Treasury issue to deliver—sometimes referred to as the quality option or swap option—the short position has two more options granted under CBT delivery guidelines. The short position is permitted to decide when in the delivery month delivery actually will take place. This is called the timing option. The other option is the right of the short position to give notice of intent to deliver up to 8:00 P.M. Chicago time after the closing of the exchange (3:15 P.M. Chicago time) on the date when the futures settlement price has been fixed. This option is referred to as the wild card option. The quality option, the timing option, and the wild card option (in sum referred to as the delivery options) mean that the long position can never be sure of which Treasury bond will be delivered or when it will be delivered. 12. Explain how the shape of the yield curve influences the theoretical price of a Treasury bond futures contract. The theoretical price of a futures contract is equal to the cash or spot price plus the cost of carry. The cost of carry is equal to the cost of financing the position less the cash yield on the underlying security. In the case of interest-rate futures, carry (the relationship between the short-term financing rate and the current yield on the bond) depends on the shape of the yield curve. When the yield curve is upward-sloping, the short-term financing rate will generally be less than the current yield on the bond, resulting in positive carry. The futures price will then sell at a discount to the cash price for the bond. The opposite will hold true when the yield curve is inverted. 13. Suppose that the conversion factor for a particular Treasury bond that is acceptable for delivery in a Treasury bond futures contract is 0.85 and that the futures price settles at 105. Assume also that the accrued interest for this Treasury bond is 4. What is the invoice price if the seller delivers this Treasury bond at the settlement date? The price that the buyer must pay the seller when a Treasury bond is delivered is called the invoice price. The invoice price is the settlement futures price plus accrued interest on the bonds delivered. The seller can deliver one of several acceptable Treasury issues. To make delivery fair to both parties, the invoice price must be adjusted based on the actual Treasury issue delivered. It is a conversion factor that is used to adjust the invoice price. The invoice price is invoice price = (contract size × futures contract settlement price × conversion factor) + accrued interest. In our problem, the Treasury bond futures contract settles at 105 (the futures contract settlement price of 105 means 105% of par value); the short elects to deliver a Treasury bond issue with a conversion factor of 0.85; the contract size is $100,000; and, the accrued interest is 4 (or $4 per $100 which is $4,000 per $100,000). Inserting these values into our invoice price formula and noting 4 is 4% or 0.04, we get: invoice price = ($100,000 × 1.05 × 0.85) + (0.04 × $100,000) = $89,250 + $4,000 = $93,250. Thus, the invoice price the buyer pays the seller is $93,250. 14. Suppose that bond ABC is the underlying asset for a futures contract with settlement six months from now. You know the following about bond ABC and the futures contract: (1) In the cash market ABC is selling for $80 (par value is $100); (2) ABC pays $8 in coupon interest per year in two semiannual payments of $4, and the next semiannual payment is due exactly six months from now; and (3) the current six-month interest rate at which funds can be loaned or borrowed is 6%. Answer the below questions. (a) What is the theoretical futures price? The theoretical futures price (F) is given by: F = P[1 + t(r – c)] where P = cash market price, t = time, in years, to the futures delivery date, r = financing rate, and c = current yield (coupon rate divided by the cash market price). Inserting in our values, we have: F = P[1 + t(r – c)] = $80[1 + 0.5(0.06 – 0.08)] = $80[0.99] = $79.20. (b) What action would you take if the futures price is $83? You would sell the futures contract at 83, purchase the bond at 80, and borrow 80 for six months at 6% per year. (c) What action would you take if the futures price is $76? You would buy the futures contract at 76, sell (short) the bond for 80, and invest (lend) 80 for six months at 6% per year. (d) Suppose that bond ABC pays interest quarterly instead of semiannually. If you know that you can reinvest any funds you receive three months from now at 1% for three months, what would the theoretical futures price for six-month settlement be? The theoretical futures price (F) is given by: F = P[1 + t(r – c)] where P = cash market price, t = time, in years, to the futures delivery date, r = financing rate, and c = current yield (coupon rate divided by the cash market price). The coupon rate was 4% semiannually or 8% annually. Now it is 2% quarterly and reinvested at 1%. This means we now have 2%(1.01) = 2.02% quarterly or 4(2.02%) = 8.08% annually. Inserting in our new value for c gives: F = P[1 + t(r – c)] = $80[1 + 0.5(0.06 – 0.0808)] = $80[0.9898] = $79.168. (e) Suppose that the borrowing rate and lending rate are not equal. Instead, suppose that the current six-month borrowing rate is 8% and the six-month lending rate is 6%. What is the boundary for the theoretical futures price? The boundary for the theoretical futures price is given by upper and lower boundary formulas. The formula that can be used to obtain the upper boundary for the futures price is F(upper boundary) = P[1 + t(rB – c)] where P = cash market price, t = time, in years, to the futures delivery date, rB = borrowing rate, and c = current yield (coupon rate divided by the cash market price). The formula that can be used to obtain the lower boundary for the futures price is F(lower boundary) = P[1 + t(rL – c)] where rL = lending rate. Inserting in our values for the upper boundary gives: F(upper boundary) = F = P[1 + t(rB – c)] = $80[1 + 0.5(0.08 – 0.08)] = $80.00. Inserting in our values for the lower boundary gives: F(lower boundary) = F = P[1 + t(rL – c)] = $80[1 + 0.5(0.06 – 0.08)] = $79.20. Thus, we have bounds ranging from $79.20 to $80.00. 15. What is the implied repo rate? The implied repo rate is the rate of return given by buying an acceptable Treasury issue, shorting the Treasury bond futures, and delivering the issue at the settlement date. 16. Explain why the implied repo rate is important in determining the cheapest-to-deliver issue. In selecting the issue to be delivered, the short will select from all the deliverable issues the one that is cheapest to deliver. This issue is referred to as the cheapest-to-deliver issue; it plays a key role in the pricing of this futures contract. The cheapest-to-deliver issue is determined by participants in the market as follows. For each of the acceptable Treasury issues from which the seller can select, the seller calculates the return that can be earned by buying that issue and delivering it at the settlement date. Note that the seller can calculate the return because she knows the price of the Treasury issue now and the futures price that she agrees to deliver the issue. The return so calculated is called the implied repo rate. The cheapest-to-deliver issue is then the one issue among all acceptable Treasury issues with the highest implied repo rate because it is the issue that would give the seller of the futures contract the highest return by buying and then delivering the issue. 17. A manager wishes to hedge a bond with a par value of $20 million by selling Treasury bond futures. Suppose that (1) the conversion factor for the cheapest-to-deliver issue is 0.91, (2) the price value of a basis point of the cheapest-to-deliver issue at the settlement date is 0.06895, and (3) the price value of a basis point of the bond to be hedged is 0.05954. Answer the below questions. (a) What is the hedge ratio? Assuming a fixed yield spread between the bond to be hedged and the cheapest-to-deliver issue, the hedge ratio is: hedge ratio = × conversion factor for CTD where PVBP or price value of a basis point refers to the change in price for a one-basis-point change in yield. Inserting in our given numbers, we have: hedge ratio = × 0.91 = 0.8635242 × 0.91 = 0.7858071 or about 0.79. (b) How many Treasury bond futures contracts should be sold to hedge the bond? The number of contracts that must be short is determined as follows: number of contracts = hedge ratio × . Given the hedge ratio in part (a), the amount to be hedged of $20 million, and each Treasury bond futures contract of $100,000, the number of futures contracts that must be sold is: number of contracts = hedge ratio × = 0.7858071 × 200 = 157.16142. Thus, about 157 Treasury bond futures contracts should be sold to hedge the bond. 18. Suppose that without an adjustment for the relationship between the yield on a bond to be hedged and the yield on the hedging instrument the hedge ratio is 1.30. Answer the below questions. (a) Suppose that a yield beta of 0.8 is computed. What would the revised hedge ratio be? The revised hedge ratio would be the hedge ratio times the adjustment factor. For the hedge ratio, we have: hedge ratio = = 1.3. For the adjustment factor, two approaches have been suggested for estimating the adjustment factor that takes into account the relationship between yield levels and yield spreads: (1) regression approach, and (2) pure volatility approach. The regression approach gives the yield beta. For our problem, the yield beta is 0.8. Thus, for the revised hedge ratio, we have: revised hedge ratio = hedge ratio × adjustment factor = 1.3 × 0.8 = 1.04. (b) Suppose that the standard deviation for the bond to be hedged and the hedging instrument are 0.09 and 0.10, respectively. What is the pure volatility adjustment, and what would be the revised hedge ratio? The revised hedge ratio would be the hedge ratio times the adjustment factor. For the hedge ratio, we have: hedge ratio = = 1.3. For the adjustment factor, two approaches have been suggested for estimating the adjustment factor that takes into account the relationship between yield levels and yield spreads: (1) regression approach, and (2) pure volatility approach. The pure volatility approach gives the yield pure volatility adjustment. For our problem, the pure volatility adjustment is given by: pure volatility adjustment = = = 0.9. Thus, for the revised hedge ratio, we have: revised hedge ratio = hedge ratio × adjustment factor = 1.3 × 0.9 = 1.17. 19. Suppose that a manager wants to reduce the duration of a portfolio. Explain how this can be done using Treasury bond futures contracts. Interest-rate futures can be used to alter the interest-rate sensitivity of a portfolio. Portfolio managers with strong expectations about the direction of the future course of interest rates will adjust the durations of their portfolios so as to capitalize on their expectations. Specifically, if a manager expects rates to increase, the duration will be shortened (so as to avoid locking in relatively lower rates for the longer haul); if interest rates are expected to decrease, the duration will be lengthened (so as to lock in relative higher rates for the longer haul). Although portfolio managers can alter the durations of their portfolios with cash market instruments, a quick and inexpensive means for doing so (on either a temporary or permanent basis) is to use futures contracts. In addition to adjusting a portfolio based on anticipated interest-rate movements, futures contracts can be used in constructing a portfolio with a longer duration than is available with cash market securities. As an example of the latter, suppose that in a certain interest-rate environment a pension fund manager must structure a portfolio to have a duration of 20 years to accomplish a particular investment objective. Bonds with such a long duration may not be available. By buying the appropriate number and kind of interest-rate futures contracts, a pension fund manager can increase the portfolio’s duration to the target level of 20. To determine the approximate number of interest-rate futures contracts needed to change the market value of the portfolio allocated to bonds, we use the following expression: approximate number of contracts = . 20. What risks are associated with hedging? Hedging with futures calls for taking a futures position as a temporary substitute for transactions to be made in the cash market at a later date. If cash and futures prices move together, any loss realized by the hedger from one position (whether cash or futures) will be offset by a profit on the other position. When the net profit or loss from the positions is exactly as anticipated, the hedge is referred to as a perfect hedge. In practice, hedging is not that simple and embodies risk. For example, the amount of net profit will not necessarily be as anticipated. The outcome of a hedge will depend on the relationship between the cash price and the futures price both when a hedge is placed and when it is lifted. The difference between the cash price and the futures price is the basis. The risk that the basis will change in an unpredictable way is called basis risk. In bond portfolio management, typically, the bond to be hedged is not identical to the bond underlying the futures contract. This type of hedging is referred to as cross hedging. There may be substantial basis risk in cross hedging. An unhedged position is exposed to price risk, the risk that the cash market price will move adversely. A hedged position substitutes basis risk for price risk. Another aspect of risk involved with a cross hedge is choosing the right hedge ratio. The hedge ratio depends on volatility weighting, or weighting by relative changes in value. The purpose of a hedge is to use gains or losses from a futures position to offset any difference between the target sale price and the actual sale price of the asset. Accordingly, the hedge ratio is chosen with the intention of matching the volatility (i.e., the dollar change) of the futures contract to the volatility of the asset. Another adjustment in the hedging strategy is usually necessary for dealing with the risk of hedging non-deliverable securities. This adjustment concerns the assumption about the relative yield spread between the cheapest-to-deliver bond and the bond to be hedged. In the prior discussion, we assumed that the yield spread was constant over time. Yield spreads, however, are not constant over time. They vary with the maturity of the instruments in question and the level of rates, as well as with many unpredictable and nonsystematic factors. 21. How could a portfolio manager use a Treasury bond futures contract to hedge against increased interest rates over the next quarter? A portfolio manager would short (or sell) Treasury bond futures to protect against a decline in the cash price of a bond caused by an increase in interest rates. By establishing a short hedge, the hedger has fixed the future cash price and transferred the price risk of ownership to the buyer of the futures contract. To understand why a short hedge might be executed, suppose that a pension fund manager knows that bonds must be liquidated in three months to make a $5 million payment to the beneficiaries of the pension fund. If interest rates rise during the three-month period, more bonds will have to be liquidated to realize $5 million. To guard against this possibility, the manager can sell bonds in the futures market to lock in a selling price. 22. Consider the portfolio in Exhibit 26-3. Suppose that the dollar duration of the 5-year Treasury note futures contract is $5,022. a. What position would a portfolio manager have to take in the contract to hedge the portfolio? The portfolio manager would take a long position by buying futures contracts. More details are given below. The market value of the portfolio in Exhibit 26-3 is $48,109,810 on March 31, 2011 and its effective duration is 2.97. It is assumed that the manager is managing a portfolio whose benchmark is the Barclays Capital Intermediate Aggregate Index and has a duration of 3.68. Because the portfolio duration of 2.97 is less than that of the benchmark duration, the portfolio has less interest rate exposure (for a parallel shift in the yield curve) than the benchmark. The manager wants to restructure the portfolio so that its duration matches that of the benchmark. That is, the portfolio manager seeks to follow a duration-matched strategy and therefore the portfolio’s target duration is 3.68. For a 100 basis change in interest rates, the portfolio’s target dollar duration is then the product of 3.68% times the current market value of the portfolio. Therefore, portfolio target dollar duration = 3.68% × $48,109,810 = $1,770,110 The current portfolio duration is 2.97, so for a 100 basis point change in interest rates, portfolio current dollar duration = 2.97% × $48,109,810 = $1,428,594 The difference between the target and the current dollar duration for the portfolio is $341,516. This means that to get to the target portfolio duration of 3.68, the portfolio manager must increase the dollar duration of the current portfolio by $341,516. One way to do this is by taking a position in a futures contract. Buying futures contracts increases the dollar duration. The question is what is the dollar duration of the futures contract? For our problem, the portfolio manager will use the 5-Treasury note futures contract. The futures price on March 31, 2011 was 116.79. Based on an analysis of this contract, the portfolio manager determines that for a 100 basis point change in interest rates, the 5-year Treasury note futures contract will change by roughly $5,022. If the portfolio manager buys C contracts, then the dollar duration of the futures position for a 100 basis point change in interest rates is the product of the number futures contract; that is, dollar duration of futures contract = $5,022 × C The portfolio manager wants the above equation to be equal $341,516. Thus, $5,022 × C = $341,516 Solving we get C = 68 contracts Thus, by buying 68 5-year Treasury note futures contracts, the portfolio manager will increase the dollar duration of the portfolio by $341,516 for a 100 basis point change in interest rates. b. What is the market value of the position that the portfolio manager must take? The market value of the position that the portfolio manager must take is $7,941,720. More details are given below. Since the notional amount of the futures contract is $100,000, this means that the total notional amount of the futures position is $100,000 × 68 = $6,800,000. The market value of the futures position given that the future price is 116.79 is equal to (116.79/100) × $100,000 × 68 = $7,941,720 A formula to approximate the number of futures contracts necessary to adjust the portfolio duration to a target level is A negative value indicates the number of contracts that should be sold; a positive value indicates the number of contracts that should be purchased. In our example, it is Suppose instead that the portfolio manager does not want to duration match versus the benchmark but instead wants the duration to be 90% of the benchmark. Since the benchmark duration is 3.68, this means that the manager wants the portfolio’s target duration to be 3.31. The portfolio target dollar duration for a 100 basis point change in interest rates is portfolio target dollar duration = 3.31% × $48,109,810 = $1,592,435 Then the approximate number of futures contract to increase the duration is equal to Hence, 33 futures contracts should be purchased. It is important to remember that although one can match the duration of a benchmark, this does not mean that other interest rate risk attributes match the benchmark. More specifically, the duration can be matched but the convexity and key rate durations of the benchmark and portfolio can be mismatched. This is why portfolio managers will actually use other hedging instruments to neutralize differences in key rate duration and convexity. 23. Consider the portfolio in Exhibit 26-3. Suppose that the dollar duration of the 5-year Treasury note futures contract is $5,022. a. What position would a portfolio manager have to take in the contract to obtain a portfolio of 4? The manager would take a long position by buying about 99 future contracts. More details are given below. Using the information found from the previous problem and Exhibit 23-6, we now want to know what position will be taken if the portfolio manager does not want the portfolio duration to match the benchmark. With a portfolio target duration of 4, the manager wants the duration to be greater than the benchmark by about (4 – 3.68) / 3.68 = 0.086957 or about 8.7%. Given a portfolio market value of $48,109,810, the portfolio target dollar duration for a 100 basis point change in interest rates is portfolio target dollar duration = 4.00% × $48,109,810 = $1,924,392.40. The number of futures contract to increase the duration is given as: = Thus, about 99 futures contracts should be purchased. b. What is the market value of the position that the portfolio manager must take? The market value of the position that the portfolio manager must take is about $11,530,117. The details are given below. Since the notional amount of the futures contract is $100,000, this means that the total notional amount of the futures position is $100,000 × 98.7252 = $9,872,520.91. The market value of the futures position given that the future price is 116.79 is equal to (116.79/100) × $100,000 × 98.7252 = $11,530,117.17. It is important to remember that although one can match the duration of a benchmark, this does not mean that other interest rate risk attributes match the benchmark. More specifically, the duration can be matched but the convexity and key rate durations of the benchmark and portfolio can be mismatched. This is why portfolio managers will actually use other hedging instruments to neutralize differences in key rate duration and convexity. 24. Suppose that an institutional investor wants to hedge a portfolio of mortgage pass-through securities using Treasury bond futures contracts. What are the risks associated with such a hedge? First, we recall that a mortgage pass-through security is created when one or more mortgage holders form a collection (pool) of mortgages and sell shares or participation certificates in the pool. There are three major types of pass-throughs, guaranteed by three organizations: Government National Mortgage Association (Ginnie Mae), Federal Home Loan Mortgage Corporation (Freddie Mae), and Federal National Mortgage Association (Fannie Mae). These are called agency pass-throughs. Institutional investor who hedges a portfolio of mortgage pass-through securities using Treasury bond futures contracts face uncertainty because the two instruments are not perfectly correlated. The amount of net profit from the hedge cannot be perfectly anticipated. The outcome of a hedge will depend on the relationship between the cash price and the futures price both when a hedge is placed and when it is lifted. The difference between the cash price and the futures price is the basis. The risk that the basis will change in an unpredictable way is called basis risk. The hedge between mortgage pass-through securities and Treasury bond futures contracts is referred to as cross hedging. There is risk involved with a cross hedge. The key to minimizing this is to choose the right hedge ratio. The hedge ratio depends on volatility weighting, or weighting by relative changes in value. The purpose of a hedge is to use gains or losses from a futures position to offset any difference between the target sale price and the actual sale price of the asset. Accordingly, the hedge ratio is chosen with the intention of matching the volatility (i.e., the dollar change) of the futures contract to the volatility of the asset. 25. The following excerpt appeared in the following article, “Duration,” in the November 16, 1992, issue of Derivatives Week, p. 9: “TSA Capital Management in Los Angeles must determine duration of the futures contract it uses in order to match it with the dollar duration of the underlying, explains David Depew, principal and head of trading at the firm. Futures duration will be based on the duration of the underlying bond most likely to be delivered against the contract …” Answer the below questions. (a) Explain why it is necessary to know the dollar duration of the underlying in order to hedge. Knowing the dollar duration of the underlying is necessary to hedging if the hedging instrument is to offset any loss through ownership of the asset. For example, consider hedging with futures where the bond to be hedged is not identical to the bond underlying the futures contract. This type of hedge is a cross hedge. The key to minimizing risk in a cross hedge is to choose the right hedge ratio. The hedge ratio depends on volatility weighting, or weighting by relative changes in value. The purpose of a hedge is to use gains or losses from a futures position to offset any difference between the target sale price and the actual sale price of the asset. Accordingly, the hedge ratio is chosen with the intention of matching the volatility (i.e., the dollar change) of the futures contract to the volatility of the asset. If the two bonds have the same dollar duration then their percentage change in price is the same. This implies they will have the same dollar price volatility. By having the same dollar duration, the bonds will have the same price change for a given change in yield and thus achieving the hedging purpose of offsetting any loss or gain. (b) Why can the price value of basis point be used instead of the dollar duration? The relevant point in the life of the bond for calculating volatility is the point at which the hedge will be lifted. Volatility at any other point is essentially irrelevant because the goal is to lock in a price or rate only on that particular day. Similarly, the relevant yield at which to calculate volatility initially is the target yield. Consequently, the “volatility of the bond to be hedged” is the price value of a basis point for the bond on the date the hedge is expected to be delivered. 26. You work for a conservative investment management firm. You recently asked one of the senior partners for permission to open up a futures account so that you could trade interest-rate futures as well as cash instruments. He replied, “Are you crazy? I might as well write you a check, wish you good luck, and put you on a bus to Las Vegas. The futures markets are nothing more than a respectable game of craps. Don’t you think you’re taking enough risk trading bonds?” How would you try to persuade the senior partner to allow you to use futures? You would try to persuade your senior partner by explaining that a primary purpose of using futures is to reduce price risk and not necessarily create more risk. Also, as a form of insurance, it is relatively cheap. More details are given below. When a position is taken in a futures contract to hedge an asset, the hedger is attempting to offset any loss in the asset through changing interest rates. This is a risk reducing strategy. Furthermore the cost for most parties is relatively cheap compared to other alternatives. For example, the party need not put up the entire amount of the investment. Instead, only initial margin must be put up. If Bob has $100 and wants to invest in bond XYZ because he believes its price will appreciate as a result of a decline in interest rates, he can buy one bond if bond XYZ is selling for $100. If the exchange where the futures contract for bond XYZ is traded requires an initial margin of $5, however, Bob can purchase 20 contracts with his $100 investment. (This example ignores the fact that Bob may need funds for variation margin.) His payoff will then depend on the price action of 20 XYZ bonds, and not on the one bond he could buy with $100. Thus he can leverage the use of his funds. Although the degree of leverage available in the futures market varies from contract to contract, the leverage attainable is considerably greater than in the cash market. Without the leverage possible in futures transactions, the cost of reducing price risk using futures would be too high for many market participants. Solution Manual for Bond Markets, Analysis and Strategies Frank J. Fabozzi 9780132743549, 9780133796773