2. A FRAMEWORK FOR FIXED-INCOME PORTFOLIO MANAGEMENT 3. MANAGING FUNDS AGAINST A BOND MARKET INDEX

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1 2. A FRAMEWORK FOR FIXED-INCOME PORTFOLIO MANAGEMENT The four activities in the investment management process are as follows: 1. Setting the investment objectives i.e. return, risk and constraints. 2. Developing a portfolio strategy 3. Implementing the strategy 4. Monitoring and adjusting the portfolio Source: Exhibit 1, Volume 4, Reading 23, P. 7 Types of investor based on Investment Objectives: 1. Investor whose objective is to replicate the performance of the chosen index e.g. bond mutual fund. This approach is also known as investing on a benchmark-relative basis. A specific bond market index is usually selected as a benchmark for the portfolio. Portfolio s objective is to either match or exceed the rate of return on that index. Risk of bond holdings is evaluated both in relation to the benchmark index and to the contribution to the risk of the overall (multi-asset-class) portfolio. 2. Investor whose objective is to generate sufficient cash from bond investment to satisfy the liabilities e.g. money borrowed to leverage the portfolio, defined-benefit pension plans, quasi-liabilities (i.e. retirement needs) etc. Since meeting a liability is the investment objective, it also becomes the benchmark for the portfolio. Portfolio s objective is to ensure that the rate of return earned in the portfolio satisfies the return promised to liability holders. 3. MANAGING FUNDS AGAINST A BOND MARKET INDEX A passive management strategy assumes that the market s expectations are essentially correct and manager has no reason to disagree with these expectations because he/she has no superior forecasting skills. Characteristics of Passive Management Strategy: Portfolio s risk profile is matched with that of benchmark s risk profile. Portfolio will have average risk level and average rate of return. Manager is not required to make independent forecasts. Portfolio should very closely track the benchmark index. Active management strategy is based on manager s forecasting ability i.e. when manager s forecasts are more accurate than the market forecast, the portfolio return increases. 3.1 Classification of Strategies 1. Pure Bond Indexing (or Full Replication Approach): In this approach, portfolio will have exactly the same risk exposures as the benchmark. Full replication is done by owning all the bonds in the index in the same percentage as the index. Full replication is inefficient, costly & difficult to implement due to presence of illiquid & infrequently traded issues in the bond index i.e. high transaction costs. Full replication generates return less than that of benchmark e.g. if benchmark earns 10%, the strategy earns 9.5%. This approach fairly closely tracks the index; thus, has zero tracking error. 2. Enhanced Indexing by matching primary risk factors: In this approach, sampling is done to match the primary index risk factors. Primary risk factors include i. Changes in the level of interest rates, ii. Twists in the yield curve, iii. Changes in the spread between Treasuries and non-treasuries. It achieves a higher return as compared to full replication; however, return is still less than that of benchmark. This approach less closely tracks the index relative to full replication i.e. tracking risk increases. It is less costly to implement but involves higher management fees. It has following two approaches: A. Construction & maintenance costs of the portfolio are reduced by investing in a sample of bonds instead of the whole Index. Example: Benchmark earns 10%; this strategy earns 9.9%. B. By matching the primary risk factors, the portfolio manager may try to enhance the portfolio s return by using valuation techniques i.e. investing in undervalued bonds. Example: Benchmark earns 10%; this strategy earns 9.95%. NOTE: Both A & B requires sampling. 1

2 3. Enhanced Indexing by small risk factor Mismatches: 1) Duration must be matched with that of benchmark in this approach. 2) However, portfolio s return can be marginally increased by small mismatches in other risk factors i.e. twists in the yield curve, changes in the spread. It can achieve a higher return as compared to first two strategies aforementioned i.e. return is equal to that of benchmark. Example: Benchmark earns 10%; this strategy earns 10%. 3) It leads to increase in risk & tracking error. 4) This approach is less costly in terms of transaction costs but involves greater management fees. 5) Objective is to earn return that is enough to overcome the difference in administrative costs between the portfolio and the index. 4. Active Management by Larger risk factor Mismatches: In this approach, manager actively pursues opportunities in the market to increase return i.e. by opting for large mismatches on the primary risk factors (including duration). For example, manager may overweight A rated bonds relative to AA rated, overweight corporate relative to treasuries, or adjust portfolio s duration different from the benchmark etc. Objective is to earn return that is enough to overcome the higher transaction and risk involved i.e. it can generate return that is greater than that of benchmark. Example: Benchmark earns 10%; this strategy earns 10.1%. This approach involves higher tracking error, risk and increase in management fees. 5. Full-blown Active Management: Full-blown active management is based on aggressive mismatches on duration, sector weights and other factors. It can generate return that is greater than that of benchmark. Example: Benchmark earns 10%; this strategy earns 10.2%. This approach involves higher tracking error, risk and increase in management fees. NOTE: Pure Bond Indexing (or Full Replication Approach), Enhanced Indexing by matching primary risk factors & Enhanced Indexing by small risk factor Mismatches are categories of Passive styles with varying degree of enhancements. Active Management by Larger risk factor Mismatches & Full-blown Active Management are categories of Active styles. Any difference in return is due to strategies employed by the manager, not structural differences between the portfolio and the benchmark. Each enhancement strategy is intended to increase the portfolio return, but is not guaranteed to do so. 3.2 Indexing (Pure and Enhanced) Advantages of Indexing include: 1. Indexed portfolios have lower fees than actively managed portfolios i.e. both advisory and nonadvisory fees (custodial fees) are lower for indexed portfolios. 2. It is difficult to outperform a broadly based market index on a consistent basis. 3. Broadly based bond Index portfolios provide excellent diversification which results in lower risk for a given level of return than other less diversified portfolios ) Selecting of a Benchmark Bond Index: General Considerations That Benchmark should be selected whose characteristics match closely with the desired characteristics of the portfolio. Benchmark is selected based on the following four factors: 1. Market value risk: The market risk of the portfolio and benchmark index should be comparable. For example, for investors who are risk averse, it is more preferable to use short-term or intermediate-term index as a benchmark than the long index because the market risk is directly related to maturity and duration of a portfolio. 2. Income risk: The portfolio and benchmark should have comparable assured income streams. If investor s objective is to earn stable and dependable income (i.e. retirees), then long-term portfolio is the least risky and short-term portfolio is the most risky. 3. Credit risk: The average credit risk of the benchmark index should satisfy any constraints placed on credit quality in the investor s IPS. The diversification among issuers in the benchmark index should also be satisfactory to the investor. 4. Liability framework risk: In order to reduce risk associated with liabilities (applicable only to in case of liability-matching objective), the benchmark index should be selected that reflects the nature of the liabilities e.g. investors with long-term liabilities should select a long index. For the taxable investor, returns and risk must be evaluated on an after-tax basis. If a taxable investor can hold the bond portfolio within a taxable or a tax-deferred account, investors should select the benchmark index with similar risk-return characteristics in taxable account and tax-deferred account as that of his/her portfolio. Although the bond market index may serve as a realistic benchmark portfolio, it is not a real portfolio and only exists on paper or computer software. Therefore, the bond market index is constructed to mimic the overall market and the manager s portfolio is constructed to mimic the bond market index. Consequently, the manager s portfolio will also mimic the overall market. 2

3 Reading 23 Fixed-Income Portfolio Management Part I FinQuiz.com NOTE: Asset allocation and asset location are two different decisions i.e. asset location is the decision concerning the accounts in which to hold assets. For example, stocks are preferred to be held in tax-deferred accounts because of higher tax-savings since they have potential to earn higher returns. Practice: Example 1 Volume 4, Reading 23, P ) Risk in Detail: Risk Profiles The identification and measurement of risk factors plays a role in benchmark selection and in a major benchmark construction. The Major Source of Risk for most bonds is related to yield curve (the relationship between interest rates and time to maturity). Yield Curve changes include: 1) A parallel shift in the yield curve (this accounts for about 90% of the change in value of a bond). 2) A twist of the yield curve 3) Other curvature changes of the yield curve. The portfolio must have the same exposures to various risks as the benchmark index. Therefore, the manager must examine each index s risk profile in order to construct an effective indexed portfolio. A completely effective indexed portfolio will have the exact same risk profile as the selected benchmark. Two commonly used techniques to construct a portfolio that replicates an index are: 1) A Cell-Matching Technique (or Stratified Sampling): The procedure is explained as follows: The manager first divides the bonds in the index into cells according to risk factors, such as sector, quality rating, duration, callability etc. The manager then measures the total value of the bonds in each of the cells and determines each cell s weight in the index. Finally, the manager selects a sample of bonds from each cell and dollar amount selected from this cell is based on that cell s weight in the index e.g. if A rated corporate bonds represent 4% of the entire index, then A rated bonds will be sampled and added until they represent 4% of the manager s portfolio. 2) A Multifactor Model Technique: This technique is based on the idea of constructing a portfolio with exactly the same risk factor exposures as benchmark but with different securities. Primary Risk Factors needed to be matched are as follows: 1) Duration: Effective duration is used to measure the change in the value of a portfolio given a small parallel shift in interest rates. The indexed portfolio s duration should be matched with the benchmark so that portfolio has same interest rate risk exposure as that of benchmark index. However, duration underestimates the increase and overestimates the decrease in the value of the portfolio; thus for large parallel shift in interest rates, convexity adjustment must also be considered. If portfolio s duration > benchmark, portfolio has greater exposure to parallel changes in interest rates and portfolio tracking risk (it is explained below) increases. 2) Key Rate Duration and Present Value Distribution of Cash flows: Key rate duration is used to measure the effect of nonparallel shifts in the yield curve i.e. change in slope or twist in the curve. In this method, spot rate for particular key maturity is changed (all other spot rates are kept constant) to measure a portfolio s sensitivity to a change in that maturity. This sensitivity is called the rate duration. The portfolio and benchmark must have securities with same maturities, key rate durations and the same overall effective duration. Mismatches in key rate duration increase tracking risk. Present Value Distribution (PVD) of Cash flows describes how the total duration of the benchmark index is distributed across its maturity. In other words, it describes the fraction of the portfolio s duration that is attributable to cash flows falling in that time period. In order to construct a portfolio to have same exposure to nonand twists, portfolio must have parallel yield curve shifts same present value distribution of cash flows as that of benchmark index. Following are the steps to calculate PVD of Cash flows: i. The present value of the cash flows from the benchmark index for specific periods i.e. every 6- month period is computed. ii. Then each present value is divided by the present value of total cash flows from the benchmark to determine the percentage of the index s total market value attributable to cash flows falling in each period e.g. 3% of the index cash flows fall in the first 6-month period), 3.8% in the second period and so on. iii. Since the cash flows in each 6-month period can be considered zero-coupon bonds, the time period is the duration of the cash flow. For example, the very first 6-month time period has a duration of 0.5. The next time period has duration of 1.0; the next 1.5, and so forth. The duration of each period is multiplied by its weight to calculate the contribution of each period s cash flows to portfolio duration. For example, the contribution of the first 6-month period is calculated as 0.03(0.5) = The contribution of the second period is 0.038(1.0) = 0.038, and so forth. iv. The duration contribution for each of the period is divided by the index duration (i.e., the sum of all the periods duration contributions). The resulting distribution is the benchmark s PVD. For example, suppose total index duration is then PVD distribution is: Period 1 = / 3.28 = 0.46% 3

4 Period 2 = / 3.28 = 1.16% and so on. Example: A benchmark has two Zero-coupon bonds i.e. Bond A & Bond B. Bond A has maturity of 3-years, FV = $5000, i = 10% (annual) Bond B has maturity of 5-years, FV = $5000, i = 10% (annual) PVD is calculated as follows: Maturity Present Value Duration PV as % of Total PV Duration Contribution PVD of Cash flows 3 FV= 5000, i=10%, n=3, 3 55% / 3.9 = 42% pmt=0, comp PV= FV= 5000, i=10%, n=5, 5 45% / 3.9 = 58% pmt=0, comp PV=3105 Total 6, % % 3) Sector and Quality Percent: The portfolio must match the percentage weight in the various sectors and qualities of the benchmark index in order to have the same risk exposure. Mismatches in sector & quality percent increase tracking risk i.e. if portfolio contains mortgage-backed securities but portfolio does not, tracking risk increases or if a portfolio under-weights AA securities relative to benchmark, tracking risk will be increased. 4) Sector Duration Contribution: The portfolio must match the proportion of the index duration that is contributed by each sector in the index to ensure that a change in sector spreads has the same impact on both the portfolio and the index. For example, if portfolio s sector percentages are similar to that of benchmark but the industrial sector s contribution duration is larger for the portfolio than for the benchmark, a mismatch occurs and tracking risk increases. 5) Quality spread duration contribution: Spread duration is a measure used to describe how a non-treasury security s price will change as a result of widening or narrowing of the spread (spread risk). The portfolio must, therefore, match the proportion of the index duration that is contributed by each quality in the index, (where quality refers to categories of bonds by rating). Spread duration is same as duration for Non-Treasury securities only i.e. it represents % change in value of a security due to 100 bps change in spread or risk-free rate. Treasury securities have zero spread duration but duration is a non-zero number. Mismatches in Quality spread duration contribution increases tracking risk. Spread Duration is further discussed in Topic # on Page# 8 Example: Suppose that total contribution to spread duration of a Portfolio is 3.43% and of a benchmark is 2.77%. It indicates that the portfolio has greater spread risk and is thus more sensitive to changes in the sector spread than the benchmark. This results in larger tracking risk. Greater spread duration is the result of the underweight in the Treasury sector combined with the longer adjusted duration in spread sectors, specifically Industrials Source: Exhibit 5, Volume 4, Reading 23, P. 19 NOTE: The number 3.43% indicates that if spread changes by 100 bps (1%), the value of a portfolio will change by 3.43%. In exhibit 5, volume 4, reading 23, page 19, it can be observed that Industrial sector % is significantly mismatched from benchmark; also this sector has greater spread duration and thus greater contribution to spread duration as compared to benchmark. The reason could be difference in selection of bonds i.e. bond with high duration have been selected. Thus, spread duration is greater either because of selecting: i. Bonds with high maturity; or ii. Bonds with Cash flows coming late i.e. zero-coupon bonds. Note: That Differences in Quality do not affect differences in duration. 6) Sector/Coupon/Maturity Cell Weights: Convexity is difficult to measure for callable bonds as they exhibit negative convexity. In order to match the convexity of bonds (call exposure) in the index, the sector, coupon, and maturity weights of the callable sectors in the index should be matched instead of matching the convexity because matching convexity involves high transaction costs. 4

5 Reading 23 Fixed-Income Portfolio Management Part I FinQuiz.com 7) Issuer Exposure: The portfolio should consist of a sufficient number of securities so that the event risk attributable to any individual issuer is minimized. NOTE: MBS primary risk exposures include sector, prepayment, and convexity risk. Source: Exhibit 3, Volume 4, Reading 23, P. 15 Practice: Example 3 Volume 4, Reading 23, P ) Tracking Risk Tracking risk (tracking error) is a measure of the variability with which a portfolio s return tracks the return of a benchmark index i.e. the closer the portfolio tracks the benchmark index, the lower is the tracking error. When there are mismatches between a portfolio s risk profile and the benchmark s risk profile, tracking risk arises (as explained above in Topic# 3.2.2). An actively managed portfolio that has a different risk profile from those of the benchmark index would likely have large observed excess returns, both positive and negative, and thus would have a large observed tracking error. Active Return = Portfolio s Return Benchmark Index s Return And Tracking Risk = Standard deviation of the i.e. Active Return Mean Active Return Tracking Risk n 1 Interpretation of Tracking Risk: For examplee a tracking risk of 20 bps indicates that in approximately two-thirds of the time periods, the portfolio return will be within a band of the benchmark index s return + / - 20 bps. Practice: Example 2 Volume 4, Reading 23, P. 18 Active Returns / negative slope between 25 & 30 years. The portfolio s net return can be increased by overweighting the undervalued areas on the curve and underweighting the overvalued areas. Overall duration must be matched at any point in time. Note that in yield curve positioning if, e.g. long-term bonds are over-weighted, then these include all longof sector or quality. term bonds regardless 4. Sector and Quality Positioning: It has two forms: a. Maintaining a yield tilt towards a specific sector and/or quality e.g. maintaining a yield tilt toward short duration Corporate issues because, historically corporate securities with less than 5 years maturity have provided the best yield spread per unit of duration risk. However, note that this strategy involves risks (e.g. default risk is higher for corporate securities); although these are manageable. Example: Consistently overweighting long-term Municipal Bonds (Here municipal bond refers to a specific sector). b. Periodic over-or-unde weighting of sectors (e.g. Treasuries vs. Corporates) or qualities. For example, when spreads are expected to widen (narrow), portfolio s net return can be enhanced by overweighting (underweighting) Treasuries. NOTE: Unlike active management strategy, the objective of this strategy is not to outperform the benchmark by a large margin. 5. Call Exposure Positioning: For example, when interest rates decrease, callable bonds underperform as compared to non-callable issues. Therefore, in case of declining interest rates, portfolio s net return can be enhanced by underweighting these issues. Practice: Example 4 Volume 4, Reading 23, P ) Enhanced Indexing Strategies Perfectly indexed portfolio underperforms the benchmark because of the transaction costs associated with constructing and rebalancing an indexed portfolio. Therefore, index enhancement strategiess are used by managers to enhance the portfolio s return. There are five index enhancement strategies: 1. Lower Cost Enhancements: The portfolio s net return can be increased by maintaining tight controls on trading costs and management fees. 2. Issue Selection Enhancements: The manager can identify and select issues that are undervalued relative to a theoretical value or can select issues which according to his/her credit analysis will soon be upgraded to enhance portfolio returns. 3. Yield Curve Positioning: Some maturities along the yield curve tend to remain consistently undervalued or overvalued e.g. yield curve frequently have a 3.3 Active Strategies In contrast to pure indexing and enhanced indexing, active management involves large tracking risk and a large positive active return. If a manager has superior forecasting or analytical skills, he/she can generate a portfolio return that is considerably higher than the benchmark return. After selecting the type of active strategy to pursue, active managers are required to implement the following four extra activities. 1. Identify which index mismatches are to be exploited: The choice of mismatches to be exploited is based on the expertise of the manager e.g. if manager has superior interest rate forecasting skill, mismatches in duration is appropriate to be exploited. 2. Extrapolate the market s expectations (or inputs) from the market data: Current market prices reflect market s expectations. By analyzing these prices and yields, additional dataa can be obtained e.g. forward 5

6 rates can be estimated which can provide insight into the direction and level of expected future interest rates. 3. Independently forecast the necessary inputs and compare these with the market s expectations: For example, if manager believes that forward rates expected by the market are too high as compared to his/her forecasts, he/she can increase the portfolio s duration to gain profit from resulting drop in yield curve. 4. Estimate the relative values of securities in order to identify areas of under-or-overvaluation: It also depends on the skill of managers e.g. some managers will make duration mismatches while others will focus on undervalued securities ) Total Return Analysis and Scenario Analysis Total return analysis and scenario analysis are two primary tools used by active managers to assess the risk-return characteristics of a trade. Total Return Analysis: It allows managers to assess the expected effect of a trade on the portfolio s total return (i.e. coupon income, reinvestment income and change in price) given an interest rate forecast. Total return analysis is helpful for only one assumed change in interest rates. Given an investment horizon, expected reinvestment rate for coupon income and expected price of the bond at the end of the time horizon, the semi-annual total return on the trade is calculated as follows: Semi-annual Total Return = (Total Future Dollars / Full Price of the Bond) 1/n 1 Where, n is the number of periods in the investment horizon. Example: Consider a 8-year, 8% semiannual, $100 par corporate bond. The bond is priced to yield 7% ($106.05), and it is assumed that coupons can be reinvested at 6% over the 1-year investment horizon. 1. Horizon price (in one year the bond will have a 6- year maturity): N = 7 2 = 14; FV = 100; I/Y = 7/2 = 3.5%; PMT = 4; CPT PV = Semiannual return i.e. Horizon value of reinvested coupons: N = 1 2 = 2; PV = 0; I/Y = 6/2 = 3.0%; PMT = 4; CPT FV = 8.12 Total horizon value = = $ PV = ; FV = ; N = 2; CPT I/Y = 3.489% 3. BEY = 3.489% 2 = 6.978% 4. EAR = ( )2 1 = 7.10% Scenario Analysis: It is an excellent risk assessment and planning tool. It is used to evaluate the impact of trade on expected total return under varying set of assumptions. It is helpful in variety of ways i.e. 1. By conducting a risk analysis on the portfolio s trades, manager is able to assess the distribution of possible outcomes. 2. The analysis can be reversed by calculating the range of interest rate movements (inputs) and their probabilities of occurrence that would result in a desirable outcome. 3. The contribution of the individual components of inputs to total return can be evaluated i.e. when analyzing amortizing bonds, impact of different reinvestment rates is also considered. 4. The relative merits of entire trading strategies can also be evaluated. 6

7 Reading 23 Fixed-Income Portfolio Management Part I FinQuiz.com 4. MANAGING FUNDS AGAINST LIABILITIES Managing funds against a liability or a set of liabilities has two types of Passive style strategies. 1. Dedication Strategies 2. Derivative Enabled Strategies 4.1 Dedication Strategies Dedication strategies are fixed-income strategies used to meet specific funding needs of the investor. They are generally passive in nature but some active management elements can be added as well. They are further classified as follows: 1) Immunization: Immunization is a strategy used to minimize interest rate risk, and it can be used to fund either single or multiple liabilities. It has following three types: i. Single Period Immunization ii. Multiple Period Immunization iii. Immunization for General Cash Flows 2) Cash flow Matching: It is a strategy which provides the future funding of liability stream from the coupon and matured principal payments of the portfolio. Types (or Classes) of Liabilities: Amount of Timing of Liability Liability Example Known Known A principal repayment Known Unknown A life insurance payout Unknown Known A floating rate annuity payout Unknown Unknown Post-retirement health care benefits The more uncertain the liabilities the more difficult it becomes to use a passive dedication strategy to achieve the portfolio s goals. In such cases, managers prefer to use some elements of active management. These aggressive strategies include active/passive combinations, active/immunization combinations, and contingent immunization ) Immunization Strategies & Classical Immunization Single-period In other words, price risk will exactly offset reinvestment rate risk. Classical immunization theory is based on following assumptions: 1. Any changes in yield curve are parallel changes. (It is the most critical assumption because in case of nonmatching duration to the parallel changes, investment horizon no longer assures immunization). 2. The portfolio is valued at a fixed horizon date. 3. The target value of investment is equal to the portfolio value at the horizon date if the interest rate structure does not change. When these assumptions hold, immunization provides a minimum-risk strategy i.e. target value of the investment is the lower limit of the value of the portfolio at the horizon if there are parallel interest rate changes. Immunization is based on two important conditions: 1) Duration of the portfolio must be set equal to the Investment horizon (time horizon of a liability that needs to be funded). Note that Duration-matching is a minimum condition for immunization. 2) The initial present value of the projected cash flows from the bond (or bond portfolio) equals the present value of the future liability. Example: $100 million liability is due in 10 years. PV of liability is $34,360,700. In order to immunize this single period liability, invest in a bond (or portfolio) with 10-year duration and PV of $34,360,700. NOTE: Investing in a coupon bond with YTM equal to the target yield and maturity equal to the investment horizon does not ensure that the target value will be achieved because when: Market yields rise, total return higher than the target value (target yield) will be achieved because coupon payments can be reinvested at a higher rate. Market yields fall, total return lower than the target value (target yield) will be achieved because coupon payments have to be reinvested at a lower rate. Practice: Example 5 Volume 4, Reading 23, P. 27 Immunization is based on the objective of constructing a portfolio that over a fixed horizon will earn a predetermined return to meet a liability regardless of interest rate changes i.e. If interest rates increase, the gain in reinvestment income loss in portfolio value. If interest rates decrease, the gain in portfolio value loss in reinvestment income. 7

8 Reading 23 Fixed-Income Portfolio Management Part I FinQuiz.com Rebalancing an Immunized Portfolio Duration of a portfolio changes when market yield changes and/or with passage of time. For example, when market yield decreases, there is a need to invest more in short-term bonds because duration is higher at lower yields. Also, duration of a portfolio changes at a different rate from time when the term structure is not flat e.g. if yield curve is upward sloping, and if one year of maturity is decreased, then duration of coupon bonds will be decreased by less than 1. This implies that portfolios are no longer immunized for a single liability when: Interest rates fluctuate more than once. Time passes. Thus, immunization is not a buy-and-holdd strategy. To keep a portfolio immunized, it must be rebalanced periodically. Rebalancing an immunized portfolio involves cost-benefit trade-off i.e. more frequent rebalancing increases transaction costs while less frequent rebalancing causes the duration to stray away from target duration. Effect of mismatch in the duration of the portfolio and the duration of the liability: If portfolio duration is less than liability duration, the portfolio is exposed to reinvestment risk. If interest rates are decreasing, the losses from reinvested coupon and principal payments would be > any gains from appreciation in the value of outstanding bonds. Under this scenario, the cash flows generated from assets would be insufficient to meet the targeted obligation. If portfolio duration is greater than liability duration, the portfolio is exposed to price risk. If interest rates are increasing, this would indicate that the losses from the market value of outstanding bonds would be > any gains from the reinvestment income. Under this scenario, the cash flows generated from assets would be insufficient to meet the targeted obligation Determining the Target Return The immunization target rate of return is defined as the total return of the portfolio assuming no change in the term structure. For a flat yield curve, the yield to maturity (YTM) is approximately equal to the assured target return. For an upward sloping yield curve, the yield to maturity (YTM) of a portfolio can be quite different from its immunization target rate of return i.e. the immunization target rate of return < YTM because of the lower reinvestment return. For a negative or downward sloping yield curve, the immunization target rate of return > YTM because of the higher reinvestment return. Approaches of determining the Target Return: 1) Using a yield implied quality and duration bond portfolio. 2) The most conservative method is to use the lowest rate i.e. Treasury spot rates in order to have the largest present value of liabilities. 3) A more realistic approach is to use yield curve (converted to spot rates) implied by the securities held in the portfolio Time Horizon The immunized time horizon is equal to the portfolio duration. Portfolio duration is equal to a weighted average of the individual security durations where the weights are the relative amounts or percentages invested in each. NOTE: A typical immunized time horizon under Guaranteed Investment Contracts (GICs) is five years. Practice: Example 8 Volume 4, Reading 23, P Dollar Duration and Controlling Positions Dollar duration is a measure of the change in portfolio value for a 100bps (1%) change in market interest rates. Dollar Duration = Duration Portfolio Value 0.01 Portfolio s Dollar Duration = Sum of dollar durations of the individual securities in the portfolio Rebalancing Ratio = Original or Old Dollar Duration / New Dollar Duration Cash required for the rebalancing = (Rebalancing ratio 1) (total new market value of portfolio) Controlling Position = Target (Original) Dollar Duration Current (new) Dollar Duration Needed change in exposure can be obtained from changing the proportion of one or more bonds in the portfolio. Practice: Example 6 & 7 Volume 4, Reading 23, P. 30 & Spread Duration The contribution of an individual bond or sector to the duration of the portfolio is the weight of the bond or sector in the portfolio multiplied by its duration i.e. Contribution of bond or sector i to the portfolio duration = wi Di by a zero coupon bond with comparable with that of the 8

9 Where, wi= the weight of bond or sector in the portfolio = market value of bond or sector in the portfolio / total portfolio value Di = the effective duration of bond or sector i Spread duration of a Portfolio = Market weighted average of the sector spread durations of the individual securities For a portfolio of Non-Treasury securities, spread duration = portfolio duration. For a portfolio of both Non-treasury and Treasury securities, spread duration < portfolio duration since spread duration of Treasury securities is zero. Duration v/s Spread Duration: In duration, the parallel shift in the yield curve could be caused by a change in inflation expectations, which causes the yields on all bonds, including Treasuries, to increase/decrease the same amount. Duration is an Absolute measure of portfolio interest rate sensitivity. In the spread duration, the shift is in the spread only, which indicates an overall increase in risk aversion (risk premium) for all bonds in a given class. Spread Duration is a Relative measure of portfolio interest rate sensitivity. There are three spread duration measures used for fixedrate bonds: 1) Nominal spread is the spread between the nominal yield on a non-treasury bond and a Treasury of the same maturity. When spread duration is based on the nominal spread, it represents the approximate percentage price change for a 100 basis point change in the nominal spread. 2) Zero-volatility spread (or static spread) is the spread that, when added to the Treasury spot rate curve makes the present value of the cash flows (when discounted at the spot rates plus the spread) equal to the price of the bond plus accrued interest. In this case, Spread duration is the approximate percentage change in price for a 100 basis point change in the zero-volatility spread, holding the Treasury spot rate curve constant. 3) The option-adjusted spread (OAS) is another spread measure that can be interpreted as the approximate percentage change in price of a spread product for a 100 basis point change in the OAS, holding Treasury rates constant. So, for example, if a corporate bond has a spread duration of 5, this means that, if the OAS changes by 30 basis points, then the price of this corporate bond will change by approximately 1.5% ( ) ) Extensions of Classical Immunization Theory 1. A natural extension of classical immunization theory is to deal with non-parallel shifts in interest rates. It involves two approaches: a) Modifying the definition of duration so as to incorporate effects of nonparallel yield curve shifts i.e. use of multifunctional duration (functional or key rate duration). b) Establishing a measure of immunization risk against any arbitrary interest rate change i.e. the immunization risk can be minimized subject to the constraint that the duration of the portfolio equals the investment horizon. 2. Multiple-liability Immunization: A second extension of classical immunization theory is used to deal with the limitations of a fixed horizon assumption (e.g. multiple-period liability like defined benefit plan s promised payouts). Unlike single period liability, there can be numerous certain or even uncertain liabilities with accompanying numerous horizon dates. In such cases, the liability at a single horizon date cannot be considered as the minimum target value of the portfolio. 3. Return Maximization Approach: It is a third extension of classical immunization theory. As long as the liability can be easily satisfied, this strategy allows managers to pursue increased risk strategies that could lead to excess portfolio value (i.e., a terminal portfolio value greater than the liability). 4. Contingent Immunization: It is a fourth extension of classical immunization theory. It combines immunization strategies with elements of active management. Contingent immunization ensures a certain minimum return in case of parallel shift and also provides a degree of flexibility to use active strategies. In contingent immunization, immunization serves as a fall-back strategy if the actively managed portfolio does not grow at a certain rate. Contingent immunization only possible when the prevailing available immunized rate of return > required rate of return. Cushion Spread = Immunized Rate Minimum acceptable return (Safety net rate of return) As long as the rate of return on the portfolio exceeds a pre-specified safety net return, the portfolio is managed actively. If the portfolio s return declines to the safety net return, the immunization mode is activated. The safety net return is the minimum acceptable return as designated by the client. Example: A portfolio manager has decided to pursue a contingent immunization strategy over a three-year time horizon. He just purchased at par $93 million worth of 10.0% semiannual coupon, 12-year bonds. Current rates of return for immunized strategies are 10.0% and the portfolio manager is willing to accept a return of 8.5%. If interest rates rise to 11% immediately, the dollar safety margin is calculated as follows: 9

10 We must first compute the required terminal value: PV=$93,000,000, N=6, I/Y=8.5/2=4.25%, PMT=0, compute FV=$119,382,132. Next, we calculate the current value of the bond portfolio: PMT= ($93,000,000) (.05) =$4,650,000, N=24, I/Y=11/2=5.5%, and FV=$93,000,000, CPT PV=$86,884,460. Next, compute the present value of the required terminal value at the new interest rate: FV=$119,382,132, PMT=0, N=6, I/Y=11/2=5.5%, CPT PV=$86,581,394. The dollar safety margin is positive ($86,884,460 $86,581,394 = $303,066) and the manager can continue to use contingent immunization. When dollar safety margin is negative, the portfolio must be immunized immediately. The yield level at which the immunization mode becomes necessary is called the trigger point. Factors which affect Rebalancing in Contingent Immunization: The frequency of rebalancing the portfolio is determined by the relationship between the safety net return and current market interest rates (and immunized rates). If safety net rate of return is close to market returns: Less flexibility for active management Probably funds are not sufficient If safety net rate of return << market returns High flexibility for active management Sufficient funds are being invested (may be too much) NOTE: Rapid adverse large movement in market yields and the uncertainty that the immunization rate will be achieved once the immunization mode is activated leads to failure to attain the minimum target return in spite of effective monitoring procedures Duration and Convexity of Assets and Liabilities Economic Surplus = Market value of assets present value of liabilities To determine the true impact on the value of the economic surplus, both the duration & convexity of the assets and the liabilities must be considered i.e. both must be matched. If liabilities and assets are duration matched but not convexity matched, economic surplus will be exposed to variation in value from interest rate changes. Source: Exhibit 15 & 16, Volume 4, Reading 23, P. 37 & Types of Risk Risks in managing Against Liability Structures: 1. Interest rate risk: Since the values of most fixed income securities move opposite to changes in interest rates, changing interest rates are a major source of risk. To help avoid interest rate risk, the manager will match the duration and convexity of the liability and the portfolio. 2. Contingent claim risk (call risk or prepayment risk): Callable bonds are typically called when interest rates decrease. This means that the portfolio not only loses the higher stream of coupons that were originally incorporated into the immunization strategy, but it is also faced with reinvesting the principal at a reduced rate of return. 3. Cap risk: Funded investors might invest in a floatingrate bond that has a cap (i.e., a maximum interest rate). To fund the investment in the floater, the investor might borrow funds on a short-term basis. There is no cap on the borrowing cost. Thus, if rates rise above the cap specified for the floater and there is no cap on the liabilities, then, at some point the funding cost will exceed the rate earned on the floater. This risk is called cap risk Risk Minimization for Immunized Portfolios Reinvestment risk determines the immunization risk because reinvestment rate only has impact on asset returns not liability i.e. when interest rates decline, only assets face lower reinvestment income. Price risk is not relevant because when e.g. interest rates decline, values of both assets and liabilities increases. The portfolio that has the least reinvestment risk will have the least immunization risk. When both Barbell Portfolio and Bullet Portfolio have durations equal to the horizon length, both portfolios are immune to parallel rate changes. In case of non-parallel rate changes, barbell portfolio is riskier than the bullet portfolio because it has greater exposure to changes in interest rate structure than the bullet portfolio. Barbell portfolio has high dispersion of cash flows around the horizon date; therefore, it is exposed to higher reinvestment risk. For example, when short rates decline while long rates rise: o Both the barbell and bullet portfolios would experience a capital loss and lower reinvestment rates at the end of investment horizon. o However, barbell portfolio would experience a higher decline in value relative to bullet portfolio due to the following two reasons: i. The barbell portfolio experiences the lower reinvestment rates longer than the bullet portfolio. ii. More of the barbell portfolio is still outstanding at the end of investment horizon which results in higher capital loss. When there are no interim cash flows, reinvestment risk is absent and thus, immunization risk is zero e.g. a pure discount instrument maturing at the investment horizon. 10

11 For non-parallel interest rate changes, the target value is not necessarily the lower bound on the investment value. Relative change in the portfolio value depends on two terms. 1) M 2 known as maturity variance: Maturity variance is the variance of the differences in the maturities of the bonds used in the immunization strategy and the maturity date of the liability. For example, if all the bonds have the same maturity date as the liability, M 2 is zero. As the dispersion of the maturity dates increases, M 2 increases. It depends only on the composition of portfolio and therefore, it is under the control of manager. It can be used as a measure of immunization risk i.e. when it is small, the exposure of the portfolio to any interest rate change is small. 2) Interest rate movements i.e. y: It is an uncertain quantity and therefore, outside the control of the manager. NOTE: Linear programming can be used to find the optimal immunized portfolio because the risk measure is linear in portfolio payments. Confidence interval represents an uncertainty band around the target return within which the realized return can be expected with a given probability. Confidence Interval = Target Return +/- (k) (Standard Deviation of Target Return) Where, k = number of S.D around the expected target return The higher the desired confidence level, the larger k and the wider the band around the expected target return. S.D of expected target return is approximately the product of three terms: i. Immunization risk measure ii. S.D of variance of 1-period change in the slope of the yield curve iii. An expression that is a function of the horizon length only 4.1.3) Multiple-Liability Immunization The basic idea behind multiple liabilities immunization is to decompose the portfolio payment streams in such a way that the component streams separately immunize each of the multiple liabilities. There are three conditions that must be satisfied to assure multiple-liability immunization in case of parallel rate shifts: i. The PV of assets must equal the PV of liabilities. ii. The composite (multiple) duration of the portfolio must equal the composite (multiple) duration of the liabilities. It implies that if there are liabilities beyond 30 years, it is not necessary to have a duration for the portfolio that is 30 years in order to immunize the entire liability stream i.e. the second condition requires that portfolio duration must equal the weighted average duration of liabilities. iii. The distribution of durations of individual portfolio assets must have a wider range than the distribution of the liabilities i.e. The portfolio must have an asset with a duration duration of the Shortest-duration liability in order to have funds to pay the liability when it is due (i.e. to avoid price risk). The portfolio must have an asset with a duration longest-duration liability in order to avoid reinvestment rate risk. It is important to note that satisfying these three conditions will assure immunization only against parallel rate shifts. In the case of nonparallel rate changes, linear programming models can be used to construct minimum-risk immunized portfolios for multiple liabilities. NOTE: In case of multiple-liability immunization, matching duration of portfolio to the average duration of liabilities is not a sufficient condition for immunization. Relative change in portfolio depends on structure of portfolio and interest rate movement (as explained above). Linear programming can be used to find the optimal immunized portfolio ) Immunization for General Cash Flows In both single investment horizon and multiple liability cases, it is assumed that the investment funds are initially available in full. However, when investment funds are not available in full at the time the portfolio is constructed, immunization for General Cash Flows strategy is used. In this case, the expected cash contributions are considered the payments on hypothetical securities that are part of the initial holdings. The actual initial investment can then be invested in such a way that the real and hypothetical holdings taken together represent an immunized portfolio. The actual initial investment must have a duration > investment horizon. Example: The investment horizon is 2-years. If only half of funds are available initially then, actual initial investment must be constructed with a duration of > 2 e.g. 3. Then, the remaining funds must have a duration of 1 in this case so that weighted average duration of total investment becomes 2 (matching the horizon length). The rate of return on future contributions is not the current spot rate; rather it will be the forward rate for the date of contribution i.e. in the example above, if current spot rate is 10%, actual initial investment will be invested at 10% but remaining half funds will be invested at 1f1 rate. NOTE: Linear programming can be used to find the optimal immunized portfolio. Immunization for general cash flows can also be extended to multiple contributions and liabilities. 11

12 4.1.5) Return Maximization for Immunized Portfolios According to return maximization for immunized portfolios strategy, if expected return can be increased substantially with little effect on immunization risk, then high-yielding portfolio will be preferred in spite of its high risk. In this strategy, duration of portfolio at all times equal to the horizon length. Thus, portfolio remains fully immunized in the classical sense. This strategy focuses on risk-return trade-off instead of dealing with non-parallel rate shifts. This strategy maximizes the lower bound on the portfolio return. Example: Optimally immunized portfolio A has expected return = 8% & S.D = 20bps (with 95% confidence interval). Optimally immunized portfolio B has expected return = 8.3% & S.D = 30bps (with 95% confidence interval). o Out of these two, portfolio B is more risky due to its higher S.D o However, if target return is 7.8%, then portfolio B will be preferred to portfolio A because, portfolio B has 1 out of 40 chances of realized return > 8% compared to 7.8% on minimum-risk portfolio. NOTE: The greater the cushion spread, the more scope the manager has for an active management policy. Linear programming can be used to find the optimal immunized portfolio with the objective of return maximization. The amount of assets required for the immunization will vary inversely with the expected return on the portfolio e.g. manager could commit fewer assets by constructing an actively-managed, higher expected return portfolio. 4.2 Cash Flow Matching Strategies Cash flow matching is an alternative to multiple liability immunization in asset/liability management. In this strategy, there is a need to select securities to match the timing and amount of liabilities i.e. a i. Bond is selected with a maturity equal to last liability maturity; and ii. Amount of principal = Amount of last liability final coupon payment Or Amount of Principal + Final Coupon payment = Amount of Liability Similarly, remaining liability streams are then reduced by the coupon payment on this bond and another bond is selected for the next-to-last (second last) liability i.e. with an Amount of principal = amount of next-to-last liability coupon on first bond selected (for last liability) coupon on bond selected (for next-to-last liability) Or Amount of principal - coupon on first bond selected (for last liability) coupon on bond selected (for next-to-last liability) = Amount of next-to-last liability This process continues until all liabilities have been matched by payments on the securities selected for the portfolio. NOTE: The more excess cash is available at each period, the greater the risk of the strategy because of the reinvestment risk. Therefore, it is more appropriate to use conservative interest rate assumption i.e. 0%. Source: Exhibit 18 & 19, Volume 4, Reading 23, P. 45 & ) Cash Flow Matching versus Multiple Liability Immunization When liability flows are perfectly matched by the asset flows of the portfolio, there is no reinvestment risk and thus, no immunization or cash flow match risk. When perfect matching is not possible, an immunization strategy is preferred because it requires less money to fund liabilities. Immunization is superior to cash flow matching due to two reasons i.e. 1) Cash flow matching requires a relatively conservative rate of return assumption for shortterm cash & cash balances; in contrast, an immunized portfolio is fully invested at the remaining horizon duration. 2) Funds from cash flow matching must be available when ( usually before) each liability is due; in contrast, an immunized portfolio is required to meet the target value only on the date of each liability because funding is achieved by a rebalancing of the portfolio ) Extensions of Basic Cash Flow Matching: In basic cash flow matching, only asset cash flows available before a liability date can be used to satisfy the liability. There are two extensions of basic cash flow matching. 1) Symmetric Cash Flow Matching: It is an extension of basic cash flow matching that allows cash flows occurring both before and after the liability date to be used to meet a liability. In this technique, shortterm borrowing of funds is allowed to meet a liability prior to the liability due date. This results in reduction in the cost of funding a liability. However, it also introduces price risk. 2) Combination Matching or Horizon Matching: In this strategy, a portfolio is created that is duration matched but with an additional constraint that it is cash-flow-matched in the first few years (usually 5 years). 12