Reading 20 Fixed-Income Portfolio Management—Part I

Question 1

(1) Decomposing Expected Return for Fixed Income?
(2) How do you calculate each component?

Answer 1

(1)

E(R)≈

Yield income

+Rolldown return

+E(Change in price based on investor’s views of yields and yield spreads)

−E(Credit losses)

+E(Currency gains or losses)

(2)

Yield income = annual coupon payment / current bond price

Rolldown return = (Ending bond price - Beg bond price) / Beg bond price

Note: Rolling yield = yield income + rolldown return

E(Change in price based on investor’s views of yields and yield spreads) = [–MD × ∆Yield] + [½ × Convexity × (∆Yield)²]

Note: to use effective duration and effective convexity for bonds with embedded options (Floating-rate notes have modified duration near zero)

E(Credit losses) = Prob of Default * Loss Given Default

Question 2

Four activities in the investment management process

Answer 2

Four activities in the investment management process:

• setting the investment objectives (with related constraints);
• developing and implementing a portfolio strategy;
• monitoring the portfolio; and
• adjusting the portfolio.

Question 3

Two types of investor based on investment objectives

Answer 3

Broadly, there are two types of investor based on investment objectives.

1. The first type of investor does not have liability matching as a specific objective.
2. The second type of investor has a liability (or set of liabilities) that needs to be met.

Question 4

Classification of Bond Management Strategies

Answer 4

Classification of Strategies:

1. Pure bond indexing (or full replication approach). The goal here is to produce a portfolio that is a perfect match to the index. The pure bond indexing approach attempts to duplicate the index by owning all the bonds in the index in the same percentage as the index. Full replication is typically very difficult and expensive to implement in the case of bond indices. Many issues in a typical bond index (particularly the non-Treasuries) are quite illiquid and very infrequently traded. For this reason, full replication of a bond index is rarely attempted because of the difficulty, inefficiency, and high cost of implementation.

2. Enhanced indexing by matching primary risk factors. This management style uses a sampling approach in an attempt to match the primary index risk factors and achieve a higher return than under full replication. Primary risk factors are typically major influences on the pricing of bonds, such as changes in the level of interest rates, twists in the yield curve, and changes in the spread between Treasuries and non-Treasuries.

• By investing in a sample of bonds rather than the whole index, the manager reduces the construction and maintenance costs of the portfolio. Although a sampling approach will usually track the index less closely than full replication, this disadvantage is expected to be more than offset by the lower expenses.
• By matching the primary risk factors, the portfolio is affected by broad market-moving events (e.g., changing interest rate levels, twists in the yield curve, spread changes) to the same degree as the index.

3. Enhanced indexing by small risk factor mismatches. While matching duration (interest rate sensitivity), this style allows the manager to tilt the portfolio in favor of any of the other risk factors. The manager may try to marginally increase the return by pursuing relative value in certain sectors, quality, term structure, and so on. The mismatches are small and are intended to simply enhance the portfolio’s return and/or risk profile enough to overcome the difference in administrative costs between the portfolio and the index.

4. Active management by larger risk factor mismatches. The difference between this style and enhanced indexing is one of degree. This style involves the readiness to make deliberately larger mismatches on the primary risk factors than in Type 3—definitely active management. The portfolio manager is now actively pursuing opportunities in the market to increase the return.

5. Full-blown active management. Full-blown active management involves the possibility of aggressive mismatches on duration, sector weights, and other factors. Often, the fund manager is seeking to construct a portfolio with superior return and risk characteristics, without much day-to-day consideration of the underlying index composition.

Question 5

Reasons exist for bond indexing

Answer 5

There are several reasons exist for bond indexing.

• Indexed portfolios have lower fees than actively managed accounts.
• Outperforming a broadly based market index on a consistent basis is a difficult task, particularly when one has to overcome the higher fees and transactions costs associated with active management.
• Broadly based bond index portfolios provide excellent diversification.

Question 6

Selection of a Benchmark Bond Index

Answer 6

The choice depends heavily on four factors:

1. Market value risk. The desired market value risk of the portfolio and the index should be comparable. Given a normal upward-sloping yield curve, a bond portfolio’s yield to maturity increases as the maturity of the portfolio increases. Does this mean that the total return is greater on a long portfolio than on a short one? Not necessarily. Because a long duration portfolio is more sensitive to changes in interest rates, a long portfolio will likely fall more in price than a short one when interest rates rise. In other words, as the maturity and duration of a portfolio increases, the market risk increases.
2. Income risk. The chosen index should provide an income stream comparable to that desired for the portfolio. If stability and dependability of income are the primary needs of the investor, then the long portfolio is the least risky and the short portfolio is the most risky.
3. Credit risk. The average credit risk of the index should be appropriate for the portfolio’s role in the investor’s overall portfolio and satisfy any constraints placed on credit quality in the investor’s investment policy statement. The diversification among issuers in the index should also be satisfactory to the investor.
4. Liability framework risk. This risk should be minimized. In general, it is prudent to match the investment characteristics (e.g., duration) of assets and liabilities, if liabilities play any role.

Question 7

Investment grade bond means

Answer 7

Investment grade, which means they are rated Baa or higher

Question 8

Bond Index Investability and Use as Benchmarks

Answer 8

Problems with investability of bond indices:

1. The values of many issues constituting bond indices do not represent recent trading but are often estimated (appraised) on the basis of the inferred current market value from their characteristics (an appraisal approach known as “matrix pricing”). Delays in data on spreads used in estimated prices can cause large errors in valuation. Among the factors that explain infrequent trading are the long-term investment horizon of many bond investors, the limited number of distinct investors in many bond issues, and the limited size of many bond issues. Furthermore, corporate bond market trading data, although improving in many markets, have typically been less readily accessible than equity trading data. As a consequence of these facts, many bond indices are not as investable as major equity indices.The impact on price from investing in less frequently traded bonds can be substantial due to their illiquidity. To minimize problems with illiquidity, some index providers create more liquid subsets of their indices.
2. Secondly, owing to the heterogeneity of bonds, bond indices that appear similar can often have very different composition and performance.
3. A third potential challenge is that the index composition tends to change frequently. Although equity indices are often reconstituted or rebalanced quarterly or annually, bond indices are usually recreated monthly. The characteristics of outstanding bonds are continually changing as maturities change, issuers sell new bonds, and issuers call in others.
4. A fourth issue is referred to as the “bums” problem, which arises because capitalization-weighted bond indices give more weight to issuers that borrow the most (the “bums”). The bums in an index may be more likely to be downgraded in the future and experience lower returns. The bums problem is applicable to corporate as well as government issuers. With global bond indices, the countries that go the most into debt have the most weight.
5. A fifth issue is that investors may not be able to find a bond index with risk characteristics that match their portfolio’s exposure.

In sum, because of the small size and heterogeneity of bond issues, their infrequent trading, and other issues, many bond indices will not be easily replicated or investable . If bond indices are not investable, it is unrealistic and unfair to expect a manager to match its performance. As such, bond indices often do not serve as valid benchmarks.

Question 9

“Bums” problem

Answer 9

“Bums” problem arises because capitalization-weighted bond indices give more weight to issuers that borrow the most (the “bums”).

An index heavily weighted by bums will likely have increased risk compared with an equally weighted index. Investors tracking such indices may hold a riskier portfolio than they might otherwise desire, and the index and portfolio are unlikely to be mean–variance efficient.

• A potential solution to this weighting problem is to use bond indices that limit the weights of component securities from particular issuers. However, such an index is more likely to contain smaller-value securities that are difficult to trade without incurring high transaction costs, hindering its investability.
• Another potential solution to the bums problem is to invest in equal-weighted indices, GDP-weighted indices, fundamental-weighted indices, or indices with other weighting systems. However, such weighting schemes may not solve the bums problem entirely, may contain bonds that are less liquid, or may be constructed using subjective inclusion criteria.

Question 10

Six other criteria for valid benchmarks

Answer 10

Six other criteria for valid benchmarks:

According to the authors, valid benchmarks will be:

1. Specified in advance
2. Appropriate
3. Measurable
4. Unambiguous
5. Reflective of current investment opinions
6. Accountable.

Question 11

Risk Profiles of indeces

Answer 11

The identification and measurement of risk factors will play a role both in index selection and in portfolio construction. The major source of risk for most bonds relates to the yield curve (the relationship between interest rates and time to maturity). Yield curve changes include:

1. a parallel shift in the yield curve (an equal shift in the interest rate at all maturities),
2. a twist of the yield curve (movement in contrary directions of interest rates at two maturities), and
3. other curvature changes of the yield curve.

Among the three, the first component (yield curve shift) typically accounts for about 90 percent of the change in value of a bond.

In assessing bond market indices as potential candidates, the manager must examine each index’s risk profile, which is a detailed tabulation of the index’s risk exposures.

The manager needs to know: “How sensitive is the index’s return to changes in the level of interest rates (interest rate risk), changes in the shape of the yield curve (yield curve risk), changes in the spread between Treasuries and non-Treasuries (spread risk), and various other risks?”

The portfolio manager may use various techniques, perhaps in combination, to align the portfolio’s risk exposures with those of the index.

Examples of techniqes:

• cell-matching technique
• multifactor model technique

Question 12

Cell-matching technique

Answer 12

A cell-matching technique (also known as stratified sampling) divides the index into cells that represent qualities designed to reflect the risk factors of the index. The manager then selects bonds (i.e., sample bonds) from those in each cell to represent the entire cell taking account of the cell’s relative importance in the index. The total dollar amount selected from this cell may be based on that cell’s percentage of the total. For example, if the A rated corporates make up 4 percent of the entire index, then A rated bonds will be sampled and added until they represent 4 percent of the manager’s portfolio.

Question 13

Multifactor Model Technique

Answer 13

A multifactor model technique makes use of a set of factors that drive bond returns. Generally, portfolio managers will focus on the most important or primary risk factors. These measures are described below, accompanied by practical comments.

1. Duration. An index’s effective duration measures the sensitivity of the index’s price to a relatively small parallel shift in interest rates (i.e., interest rate risk). (For large parallel changes in interest rates, a convexity adjustment is used to improve the accuracy of the index’s estimated price change. A convexity adjustment is an estimate of the change in price that is not explained by duration.) The manager’s indexed portfolio will attempt to match the duration of the index as a way of ensuring that the exposure is the same in both portfolios. Because parallel shifts in the yield curve are relatively rare, duration by itself is inadequate to capture the full effect of changes in interest rates.

2. Key rate duration and present value distribution of cash flows. Nonparallel shifts in the yield curve (i.e., yield curve risk), such as an increase in slope or a twist in the curve, can be captured by two separate measures. Key rate duration is one established method for measuring the effect of shifts in key points along the yield curve. In this method, we hold the spot rates constant for all points along the yield curve but one. By changing the spot rate for that key maturity, we are able to measure a portfolio’s sensitivity to a change in that maturity. This sensitivity is called the rate duration. We repeat the process for other key points (e.g., 3 years, 7 years, 10 years, 15 years) and measure their sensitivities as well.

Another popular indexing method is to match the portfolio’s present value distribution of cash flows to that of the index. Dividing future time into a set of non-overlapping time periods, the present value distribution of cash flows is a list that associates with each time period the fraction of the portfolio’s duration that is attributable to cash flows falling in that time period.

3. Sector and quality percent. To ensure that the bond market index’s yield is replicated by the portfolio, the manager will match the percentage weight in the various sectors and qualities of the index.

4. Sector duration contribution. A portfolio’s return is obviously affected by the duration of each sector’s bonds in the portfolio. For an indexed portfolio, the portfolio must achieve the same duration exposure to each sector as the index. The goal is to ensure that a change in sector spreads has the same impact on both the portfolio and the index.

5. Quality spread duration contribution. The risk that a bond’s price will change as a result of spread changes (e.g., between corporates and Treasuries) is known as spread risk. A measure that describes how a non-Treasury security’s price will change as a result of the widening or narrowing of the spread is spread duration. Changes in the spread between qualities of bonds will also affect the rate of return. The easiest way to ensure that the portfolio closely tracks the index is to match the amount of the index duration that comes from the various quality categories.

6. Sector/coupon/maturity cell weights. Because duration only captures the effect of small interest rate changes on an index’s value, convexity is often used to improve the accuracy of the estimated price change, particularly where the change in rates is large. However, some bonds (such as mortgage-backed securities) may exhibit negative convexity, making the index’s exposure to call risk difficult to replicate. A manager can attempt to match the convexity of the index, but such matching is rarely attempted because to stay matched can lead to excessively high transactions costs. (Callable securities tend to be very illiquid and expensive to trade.)

7. Issuer exposure. Event risk for a single issuer is the final risk that needs to be controlled. If a manager attempts to replicate the index with too few securities, issuer event risk takes on greater importance.

Question 14

Present value distribution of cash flows method

Answer 14

Another popular indexing method is to match the portfolio’s present value distribution of cash flows to that of the index. Dividing future time into a set of non-overlapping time periods, the present value distribution of cash flows is a list that associates with each time period the fraction of the portfolio’s duration that is attributable to cash flows falling in that time period. The calculation involves the following steps:

1. The portfolio’s creator will project the cash flow for each issue in the index for specific periods (usually six-month intervals). Total cash flow for each period is calculated by adding the cash flows for all the issues. The present value of each period’s cash flow is then computed and a total present value is obtained by adding the individual periods’ present values. (Note that the total present value is the market value of the index.)
2. Each period’s present value is then divided by the total present value to arrive at a percentage for each period.
3. Next, we calculate the contribution of each period’s cash flows to portfolio duration. Because each cash flow is effectively a zero-coupon payment, the time period is the duration of the cash flow. By multiplying the time period times the period’s percentage of the total present value, we obtain the duration contribution of each period’s cash flows. For example, if we show each six-month period as a fractional part of the year (0.5, 1.0, 1.5, 2.0, etc.), the first period’s contribution to duration would be 0.5 × 3.0 percent, or 0.015. The second period’s contribution would be 1.0 × 3.8 percent, or 0.038.
4. Finally, we add each period’s contribution to duration (0.015 + 0.038 + …) and obtain a total (3.28, for example) that represents the bond index’s contribution to duration. We then divide each of the individual period’s contribution to duration by the total. It is this distribution that the indexer will try to duplicate. If this distribution is duplicated, nonparallel yield curve shifts and “twists” in the curve will have the same effect on the portfolio and the index.

Question 15

Tracking Risk

Answer 15

Tracking risk (also known as tracking error) is a measure of the variability with which a portfolio’s return tracks the return of a benchmark index. More specifically, tracking risk is defined as the standard deviation of the portfolio’s active return, where the active return for each period is defined as

Active return = Portfolio’s return – Benchmark index’s return

Therefore,

Tracking risk = Standard deviation of the active returns

Statistically, the area that is one standard deviation either side of the mean captures approximately 2/3 of all the observations if portfolio returns approximately follow a normal distribution.

Tracking risk arises primarily from mismatches between a portfolio’s risk profile and the benchmark’s risk profile.

Question 16

Ways (i.e., index enhancement strategies) to enhance the portfolio’s return

Answer 16

Ways (i.e., index enhancement strategies) to enhance the portfolio’s return:

1. Lower cost enhancements. Managers can increase the portfolio’s net return by simply maintaining tight controls on trading costs and management fees. Although relatively low, expenses do vary considerably among index funds. Where outside managers are hired, the plan sponsor can require that managers re-bid their management fees every two or three years to ensure that these fees are kept as low as possible.

2. Issue selection enhancements. The manager may identify and select securities that are undervalued in the marketplace, relative to a valuation model’s theoretical value. Many managers conduct their own credit analysis rather than depending solely on the ratings provided by the bond rating houses. As a result, the manager may be able to select issues that will soon be upgraded and avoid those issues that are on the verge of being downgraded.

3. Yield curve positioning. Some maturities along the yield curve tend to remain consistently overvalued or undervalued. For example, the yield curve frequently has a negative slope between 25 and 30 years, even though the remainder of the curve may have a positive slope. These long-term bonds tend to be popular investments for many institutions, resulting in an overvalued price relative to bonds of shorter maturities. By overweighting the undervalued areas of the curve and underweighting the overvalued areas, the manager may be able to enhance the portfolio’s return.

4. Sector and quality positioning. This return enhancement technique takes two forms:

• Maintaining a yield tilt toward short duration corporates. Experience has shown that the best yield spread per unit of duration risk is usually available in corporate securities with less than five years to maturity (i.e., short corporates). A manager can increase the return on the portfolio without a commensurate increase in risk by tilting the portfolio toward these securities. The strategy is not without its risks, although these are manageable. Default risk is higher for corporate securities, but this risk can be managed through proper diversification. (Default risk is the risk of loss if an issuer or counterparty does not fulfill contractual obligations.)
• Periodic over- or underweighting of sectors (e.g., Treasuries vs. corporates) or qualities. Conducted on a small scale, the manager may overweight Treasuries when spreads are expected to widen (e.g., before a recession) and underweight them when spreads are expected to narrow. Although this strategy has some similarities to active management, it is implemented on such a small scale that the objective is to earn enough extra return to offset some of the indexing expenses, not to outperform the index by a large margin as is the case in active management.

5. Call exposure positioning. A drop in interest rates will inevitably lead to some callable bonds being retired early. As rates drop, the investor must determine the probability that the bond will be called. Should the bond be valued as trading to maturity or as trading to the call date? Obviously, there is a crossover point at which the average investor is uncertain as to whether the bond is likely to be called. Near this point, the actual performance of a bond may be significantly different than would be expected, given the bond’s effective duration (duration adjusted to account for embedded options). For example, for premium callable bonds (bonds trading to call), the actual price sensitivity tends to be less than that predicted by the bonds’ effective duration. A decline in yields will lead to underperformance relative to the effective duration model’s prediction. This underperformance creates an opportunity for the portfolio manager to underweight these issues under these conditions.

Question 17

Extra Activities Required for the Active Manager

Answer 17

After selecting the type of active strategy to pursue, the active manager will:

1. Identify which index mismatches are to be exploited. The choice of mismatches is generally based on the expertise of the manager. If the manager’s strength is interest rate forecasting, deliberate mismatches in duration will be created between the portfolio and the benchmark. If the manager possesses superior skill in identifying undervalued securities or undervalued sectors, sector mismatches will be pursued.
2. Extrapolate the market’s expectations (or inputs) from the market data. As discussed previously, current market prices are the result of all investors applying their judgment to the individual bonds. By analyzing these prices and yields, additional data can be obtained.
3. Independently forecast the necessary inputs and compare these with the market’s expectations. For example, after calculating the forward rates, the active manager may fervently believe that these rates are too high and that future interest rates will not reach these levels.
4. Estimate the relative values of securities in order to identify areas of under- or overvaluation. Again, the focus depends on the skill set of the manager.

Question 18

Total Return Analysis and Scenario Analysis

Answer 18

What tools does the manager have in his or her tool bag to help assess the risk and return characteristics of a trade? The two primary tools are total return analysis and scenario analysis.

The total return on a bond is the rate of return that equates the future value of the bond’s cash flows with the full price of the bond. As such, the total return takes into account all three sources of potential return: coupon income, reinvestment income, and change in price. Total return analysis involves assessing the expected effect of a trade on the portfolio’s total return given an interest rate forecast.

Even though this total return is the manager’s most likely total return, this computation is for only one assumed change in rates. This total return number does very little to help the manager assess the risk that he faces if his forecast is wrong and rates change by some amount other than that forecast. A prudent manager will never want to rely on just one set of assumptions in analyzing the decision; instead, he or she will repeat the above calculation for different sets of assumptions or scenarios. In other words, the manager will want to conduct a scenario analysis to evaluate the impact of the trade on expected total return under all reasonable sets of assumptions.

Question 19

Advantages of scenario analysis

Answer 19

Scenario analysis is useful in a variety of ways:

1. The obvious benefit is that the manager is able to assess the distribution of possible outcomes, in essence conducting a risk analysis on the portfolio’s trades. The manager may find that, even though the expected total return is quite acceptable, the distribution of outcomes is so wide that it exceeds the risk tolerance of the client.
2. The analysis can be reversed, beginning with a range of acceptable outcomes, then calculating the range of interest rate movements (inputs) that would result in a desirable outcome. The manager can then place probabilities on interest rates falling within this acceptable range and make a more informed decision on whether to proceed with the trade.
3. The contribution of the individual components (inputs) to the total return may be evaluated. The manager’s a priori assumption may be that a twisting of the yield curve will have a small effect relative to other factors. The results of the scenario analysis may show that the effect is much larger than the manager anticipated, alerting him to potential problems if this area is not analyzed closely.
4. The process can be broadened to evaluate the relative merits of entire trading strategies.

The purpose of conducting a scenario analysis is to gain a better understanding of the risk and return characteristics of the portfolio before trades are undertaken that may lead to undesirable consequences. In other words, scenario analysis is an excellent risk assessment and planning tool.

Question 20

Dedication Strategies

Answer 20

Dedication strategies are specialized fixed-income strategies that are designed to accommodate specific funding needs of the investor. They generally are classified as passive in nature, although it is possible to add some active management elements to them.

Immunization aims to construct a portfolio that, over a specified horizon, will earn a predetermined return regardless of interest rate changes. Another widely used dedication strategy is cash flow matching, which provides the future funding of a liability stream from the coupon and matured principal payments of the portfolio.

Obviously, the more uncertain the liabilities, the more difficult it becomes to use a passive dedication strategy to achieve the portfolio’s goals. For this reason, as liabilities become more uncertain, managers often insert elements of active management.

Question 21

Classes of Liabilities

Answer 21

Amount of Liability/Timing of Liability/ Example

1. Known/ Known/ A principal repayment
2. Known/ Unknown/ A life insurance payout
3. Unknown/ Known/ A floating rate annuity payout
4. Unknown/ Unknown/ Post-retirement health care benefits

Question 22

Immunization Strategies, idea

Answer 22

• As interest rates increase, the decrease in the price of a fixed-income security is usually at least partly offset by a higher amount of reinvestment income. As rates decline, a security’s price increase is usually at least partly offset by a lower amount of reinvestment income.
• For an arbitrary time horizon, the price and reinvestment effects generally do not exactly offset each other: The change in price may be either greater than or less than the change in reinvestment income.
• The purpose of immunization is to identify the portfolio for which the change in price is exactly equal to the change in reinvestment income at the time horizon of interest.
• If the manager can construct such a portfolio, an assured rate of return over that horizon is locked in.

Question 23

Classical Single-Period Immunization

Answer 23

In its most basic form, the important characteristics of immunization are:

1. Specified time horizon.
2. Assured rate of return during the holding period to a fixed horizon date.
3. Insulation from the effects of interest rate changes on the portfolio value at the horizon date.

Immunization requires offsetting price risk and reinvestment risk. To accomplish this balancing requires the management of duration. Setting the duration of the portfolio equal to the specified portfolio time horizon assures the offsetting of positive and negative incremental return sources under certain assumptions, including the assumption that the immunizing portfolio has the same present value as the liability being immunized. Duration-matching is a minimum condition for immunization.

• Keep in mind that to immunize a portfolio’s target value or target yield against a change in the market yield, a manager must invest in a bond or a bond portfolio whose*
• 1) duration is equal to the investment horizon and*
• 2) initial present value of all cash flows equals the present value of the future liability.*

Question 24

Rebalancing an Immunized Portfolio

Answer 24

How often should a portfolio be rebalanced to adjust its duration? The answer involves balancing the costs and benefits of rebalancing. On the one hand, more frequent rebalancing increases transactions costs, thereby reducing the likelihood of achieving the target return. On the other hand, less frequent rebalancing causes the duration to wander from the target duration, which also reduces the likelihood of achieving the target return. Thus, the manager faces a trade-off.

Question 25

Determining the Target Return

Answer 25

In general, for an upward-sloping yield curve, the immunization target rate of return will be less than the yield to maturity because of the lower reinvestment return.

Conversely, a negative or downward-sloping yield curve will result in an immunization target rate of return greater than the yield to maturity because of the higher reinvestment return.

Alternative measures of the immunization target rate of return include the yield implied by a zero coupon bond of quality and duration comparable with that of the bond portfolio and an estimate based on results of a simulation that rebalances the initial portfolio, given scenarios of interest rate change.

The most conservative method for discounting liabilities—the method resulting in the largest present value of the liabilities—involves the use of the Treasury spot curve (the term structure of Treasury zero coupon bonds).

A more realistic approach utilizes the yield curve (converted to spot rates) implied by the securities held in the portfolio.

Question 26

Time Horizon

Answer 26

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.

Securities in the portfolio should be limited to high-quality, very liquid instruments, because portfolio rebalancing is required to keep the portfolio duration synchronized with the horizon date.

Question 27

Dollar Duration and Controlling Positions

Answer 27

Dollar duration is a measure of the change in portfolio value for a 100 bps change in market yields. It is defined as

Dollar duration = Duration × Portfolio value × 0.01

In a number of ALM applications, the investor’s goal is to reestablish the dollar duration of a portfolio to a desired level. This rebalancing involves the following steps:

1. Move forward in time and include a shift in the yield curve. Using the new market values and durations, calculate the dollar duration of the portfolio at this point in time.
2. Calculate the rebalancing ratio by dividing the desired dollar duration by the new dollar duration. If we subtract one from this ratio and convert the result to a percent, it tells us the percentage amount that each position needs to be changed in order to rebalance the portfolio.
3. Multiply the new market value of the portfolio by the desired percentage change in Step 2. This number is the amount of cash needed for rebalancing.

Question 28

Spread Duration

Answer 28

Spread duration is a measure of how the market value of a risky bond (portfolio) will change with respect to a parallel 100 bps change in its spread above the comparable benchmark security (portfolio). Spread duration is an important measurement tool for the management of spread risk. Spreads do change and the portfolio manager needs to know the risks associated with such changes.

A characteristic of bonds with credit risk (risk of loss because of credit events such as default or downgrades in credit ratings)—sometimes called “spread product”—is that their yield will be higher than a comparable risk-free security. The large spectrum of bond products available in the marketplace leads to differing types of spread duration. The three major types are:

1. Nominal spread, the spread of a bond or portfolio above the yield of a certain maturity Treasury.
2. Static spread or zero-volatility spread, defined as the constant spread above the Treasury spot curve that equates the calculated price of the security to the market price.
3. Option-adjusted spread (OAS), the current spread over the benchmark yield minus that component of the spread that is attributable to any embedded optionality in the instrument.

Question 29

Assumptions of the Classical immunization theory

Answer 29

Classical immunization theory is based on several assumptions:

1. Any changes in the yield curve are parallel changes, that is, interest rates move either up or down by the same amount for all maturities.
2. The portfolio is valued at a fixed horizon date, and there are no interim cash inflows or outflows before the horizon date.
3. The target value of the investment is defined as the portfolio value at the horizon date if the interest rate structure does not change (i.e., there is no change in forward rates).

Question 30

Extensions of Classical Immunization Theory

Answer 30

1. A natural extension of classical immunization theory is to extend the theory to the case of nonparallel shifts in interest rates. Two approaches have been taken.

• The first approach has been to modify the definition of duration so as to allow for nonparallel yield curve shifts, such as multifunctional duration (also known as functional duration or key rate duration).
• The second approach is a strategy that can handle any arbitrary interest rate change so that it is not necessary to specify an alternative duration measure. This approach establishes a measure of immunization risk against any arbitrary interest rate change. The immunization risk measure can then be minimized subject to the constraint that the duration of the portfolio equals the investment horizon, resulting in a portfolio with minimum exposure to any interest rate movements.

2. A second extension of classical immunization theory applies to overcoming the limitations of a fixed horizon (the second assumption on which immunization depends). Under the assumption of parallel interest rate changes, a lower bound exists on the value of an investment portfolio at any particular time, although this lower bound may be below the value realized if interest rates do not change. Multiple liability immunization involves an investment strategy that guarantees meeting a specified schedule of future liabilities, regardless of the type of shift in interest rate changes.
3. In some situations, the objective of immunization as strict risk minimization may be too restrictive. The third extension of classical immunization theory is to analyze the risk and return trade-off for immunized portfolios.
4. The fourth extension of classical immunization theory is to integrate immunization strategies with elements of active bond portfolio management strategies (contingent immunization).

Question 31

Contingent immunization

Answer 31

The traditional objective of immunization has been risk protection, with little consideration of possible returns.

A technique called contingent immunization provides a degree of flexibility in pursuing active strategies while ensuring a certain minimum return in the case of a parallel rate shift. In contingent immunization, immunization serves as a fall-back strategy if the actively managed portfolio does not grow at a certain rate.

Contingent immunization is possible when the prevailing available immunized rate of return is greater than the required rate of return.

Cushion spread is the difference between the minimum acceptable return and the higher possible immunized rate.

Question 32

Duration and Convexity of Assets and Liabilities

Answer 32

In order for a manager to have a clear picture of the economic surplus of the portfolio—defined as the market value of assets minus the present value of liabilities—the duration and convexity of both the assets and liabilities must be understood. Focusing only on the duration of a company’s assets will not give a true indication of the total interest rate risk for a company.

Convexity also plays a part in changes in economic surplus. If liabilities and assets are duration matched but not convexity matched, economic surplus will be exposed to variation in value from interest rate changes reflecting the convexity mismatch.

Question 33

Three sources of risk of not being able to pay liabilities when they come due

Answer 33

As the market environment changes, the portfolio manager faces the risk of not being able to pay liabilities when they come due. Three sources of this risk are interest rate risk, contingent claim risk, and cap risk.

Interest rate risk.

Because the prices of most fixed-income securities move opposite to interest rates, a rising interest rate environment will adversely affect the value of a portfolio. If assets need to be sold to service liabilities, the manager may find a shortfall. Interest rate risk is the largest risk that a portfolio manager will face.

Contingent claims risk.

When a security has a contingent claim provision, explicit or implicit, there is an associated risk. In a falling rate environment, the manager may have lucrative coupon payments halted and receive principal (as is the case with mortgage-backed securities when the underlying mortgages prepay principal). The loss of the coupons is bad enough but now the principal must be reinvested at a lower rate. In addition, the market value of a callable security will level out at the call price, rather than continuing upwards as a noncallable security would.

Cap risk.

An asset that makes floating rate payments will typically have caps associated with the floating rate. The manager is at risk of the level of market rates rising while the asset returns are capped. This event may severely affect the value of the assets.

Question 34

Barbell and bullet portfolios

Answer 34

Barbell portfolio—a portfolio made up of short and long maturities relative to the horizon date and interim coupon payments.

Bullet portfolio—the bond maturities are very close to the investment horizon.

When interest rates change in an arbitrary nonparallel way, however, the effect on the value of the two portfolios differs — the barbell portfolio is riskier than the bullet portfolio.

The portfolio that has the least reinvestment risk will have the least immunization risk.

Question 35

Risk Minimization for Immunized Portfolios (in the case of non-parallel changes of interest rates)

Answer 35

The target value of an immunized portfolio is a lower bound on the terminal value of the portfolio at the investment horizon if yields on all maturities change by the same amount. If yields of different maturities change by different amounts, the target value is not necessarily the lower bound on the investment value.

If forward rates change by any arbitrary function, the relative change in the portfolio value depends on the product of two terms.

• The first term, denoted M², depends solely on the structure of the investment portfolio, while the second term is a function of interest rate movement only.
• The second term characterizes the nature of the interest rate shock. It is an uncertain quantity and, therefore, outside the control of the manager.

The first term, however, is under the control of the manager, as it depends solely on the composition of the portfolio. The first term can be used as a measure of immunization risk because when it is small, the exposure of the portfolio to any interest rate change is small. The immunization risk measure M² is the variance of time to payment around the horizon date, where the weight for a particular time in the variance calculation is the proportion of the instrument’s total present value that the payment received at that time represents. The immunization risk measure may be called the maturity variance; in effect, it measures how much a given immunized portfolio differs from the ideal immunized portfolio consisting of a single pure discount instrument with maturity equal to the time horizon.

Given the measure of immunization risk that is to be minimized and the constraint that the duration of the portfolio equals the investment horizon, the optimal immunized portfolio can be found using linear programming (optimization in which the objective function and constraints are linear). Linear programming is appropriate because the risk measure is linear in the portfolio payments.

Question 36

Confidence interval as immunization risk measure

Answer 36

The immunization risk measure can be used to construct approximate confidence intervals for the target return over the horizon period and the target end-of-period portfolio value. A confidence interval represents an uncertainty band around the target return within which the realized return can be expected with a given probability. The expression for the confidence interval is:

Confidence interval=Target return±(k)× (Standard deviation of target return)

where k is the number of standard deviations around the expected target return. The desired confidence level determines k. The higher the desired confidence level, the larger k, and the wider the band around the expected target return.

Standard deviation of the expected target return can be approximated by the product of three terms:

1) the immunization risk measure,
2) the standard deviation of the variance of the one-period change in the slope of the yield curve,and
3) an expression that is a function of the horizon length only.

Question 37

Multiple Liability Immunization

Answer 37

A portfolio is said to be immunized with respect to a given liability stream if there are enough funds to pay all the liabilities when due, even if interest rates change by a parallel shift.

Matching the duration of the portfolio to the average duration of the liabilities is not a sufficient condition for immunization in the presence of multiple liabilities. Instead, the portfolio payment stream must be decomposable in such a way that each liability is separately immunized by one of the component streams; there may be no actual securities providing payments that individually match those of the component payment streams.

Conditions that must be satisfied to assure multiple liability immunization in the case of parallel rate shifts. The necessary and sufficient conditions are:

1. The present value of the assets equals the present value of the liabilities (!!! liabilities should be discounted by the IRR on the immunized portfolio !!!).
2. The (composite) duration of the portfolio must equal the (composite) duration of the liabilities.
3. The distribution of durations of individual portfolio assets must have a wider range than the distribution of the liabilities.

An implication of the second condition is that to immunize a liability stream that extends 30 years, it is not necessary to have a portfolio with a duration of 30. The condition requires that the manager construct a portfolio so that the portfolio duration matches the weighted average of the liability durations.

The third condition requires portfolio payments to bracket (be more dispersed in time than) the liabilities. That is, the portfolio must have an asset with a duration equal to or less than the duration of the shortest-duration liability in order to have funds to pay the liability when it is due. And the portfolio must have an asset with a duration equal to or greater than the longest-duration liability in order to avoid the reinvestment rate risk that might jeopardize payment of the longest duration. This bracketing of shortest- and longest-duration liabilities with even shorter- and longer-duration assets balances changes in portfolio value with changes in reinvestment return.

• Relative change in the portfolio value if forward rates change by any arbitrary function depends on the product of two terms: a term solely dependent on the structure of the portfolio and a term solely dependent on the interest rate movement (delta P = Immunization risk term x Interest rate term).
• An optimal immunization strategy is to minimize the immunization risk measure subject to the constraints imposed by these two conditions (and any other applicable portfolio constraints). Constructing minimum-risk immunized portfolios can then be accomplished by the use of linear programming.

Question 38

Immunization for General Cash Flows

Answer 38

• Suppose a manager has a given obligation to be paid at the end of a two-year horizon. Only one-half of the necessary funds, however, are now available; the rest are expected at the end of the first year, to be invested at the end of the first year at whatever rates are then in effect. Is there an investment strategy that would guarantee the end-of-horizon value of the investment regardless of the development of interest rates?
• Under certain conditions, such a strategy is indeed possible. The expected cash contributions can be 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.
• We can illustrate this using the two-year investment horizon. The initial investment should be constructed with a duration of 3. Half of the funds are then in an actual portfolio with a duration of 3, and the other half in a hypothetical portfolio with a duration of 1. The total stream of cash inflow payments for the portfolio has a duration of 2, matching the horizon length. This match satisfies a sufficient condition for immunization with respect to a single horizon.

Question 39

Return Maximization for Immunized Portfolios

Answer 39

• The objective of risk minimization for an immunized portfolio may be too restrictive in certain situations. If a substantial increase in the expected return can be accomplished with little effect on immunization risk, the higher-yielding portfolio may be preferred in spite of its higher risk.
• Instead of minimizing the immunization risk against nonparallel rate changes, however, a trade-off between risk and return is considered. The immunization risk measure can be relaxed if the compensation in terms of expected return warrants it. Specifically, the strategy maximizes a lower bound on the portfolio return. The lower bound is defined as the lower confidence interval limit on the realized return at a given confidence level.

Question 40

Cash Flow Matching Strategies

Answer 40

Cash flow matching is an appealing strategy because the portfolio manager need only select securities to match the timing and amount of liabilities. Conceptually, a bond is selected with a maturity that matches the last liability, and an amount of principal equal to the amount of the last liability minus the final coupon payment is invested in this bond. The remaining elements of the liability stream are then reduced by the coupon payments on this bond, and another bond is chosen for the next-to-last liability, adjusted for any coupon payments received on the first bond selected. Going back in time, this sequence is continued until all liabilities have been matched by payments on the securities selected for the portfolio.

Question 41

Cash Flow Matching versus Multiple Liability Immunization

Answer 41

If all the liability flows were perfectly matched by the asset flows of the portfolio, the resulting portfolio would have no reinvestment risk and, therefore, no immunization or cash flow match risk. Given typical liability schedules and bonds available for cash flow matching, however, perfect matching is unlikely. Under such conditions, a minimum immunization risk approach should be as good as cash flow matching and likely will be better, because an immunization strategy would require less money to fund liabilities. Two factors contribute to this superiority.

• First, cash flow matching requires a relatively conservative rate of return assumption for short-term cash and cash balances may be occasionally substantial. By contrast, an immunized portfolio is essentially fully invested at the remaining horizon duration.
• Second, funds from a cash flow–matched portfolio must be available when (and usually before) each liability is due, because of the difficulty in perfect matching. Because the reinvestment assumption for excess cash for cash flow matching extends many years into the future, a conservative interest rate assumption is appropriate. An immunized portfolio needs to meet the target value only on the date of each liability, because funding is achieved by a rebalancing of the portfolio.

Thus, even with the sophisticated linear programming techniques used, in most cases cash flow matching will be technically inferior to immunization. Cash flow matching is easier to understand than multiple liability immunization, however; this ease of use occasionally supports its selection in dedication portfolio strategies.

Question 42

Extensions of Basic Cash Flow Matching

Answer 42

• In basic cash flow matching, only asset cash flows occurring prior to a liability date can be used to satisfy the liability. The basic technique can be extended to allow cash flows occurring both before and after the liability date to be used to meet a liability. This technique, called symmetric cash flow matching, allows for the short-term borrowing of funds to satisfy a liability prior to the liability due date. The opportunity to borrow short-term so that symmetric cash matching can be employed results in a reduction in the cost of funding a liability.
• A popular variation of multiple liability immunization and cash flow matching to fund liabilities is one that combines the two strategies. This strategy, referred to as combination matching or horizon matching, creates a portfolio that is duration-matched with the added constraint that it be cash flow–matched in the first few years, usually the first five years. The advantage of combination matching over multiple liability immunization is that liquidity needs are provided for in the initial cash flow-matched period. Also, most of the curvature of yield curves is often at the short end (the first few years). Cash flow matching the initial portion of the liability stream reduces the risk associated with nonparallel shifts of the yield curve. The disadvantage of combination matching over multiple liability immunization is that the cost to fund liabilities is greater.

Question 43

Application Considerations in applying dedication strategies

Answer 43

• Universe Considerations

The lower the quality of the securities considered, the higher the potential risk and return. Dedication assumes that there will be no defaults, and immunization theory further assumes that securities are responsive only to overall changes in interest rates. The lower the quality of securities, the greater the probability that these assumptions will not be met. Further, securities with embedded options such as call options or prepayments options (e.g., mortgage-backed securities) complicate and may even prevent the accurate measurement of cash flow, and hence duration, frustrating the basic requirements of immunization and cash flow matching. Finally, liquidity is a consideration for immunized portfolios, because they must be rebalanced periodically.

• Optimization

Optimization procedures can be used for the construction of immunized and cash flow–matched portfolios. For an immunized portfolio, optimization typically takes the form of minimizing maturity variance subject to the constraints of matching weighted average duration and having the necessary duration dispersion (in multiple-liability immunization). For cash flow matching, optimization takes the form of minimizing the initial portfolio cost subject to the constraint of having sufficient cash at the time a liability arises.

• Monitoring
• Transactions Costs

Question 44

Property of swap spreads in the European market

Answer 44

In the European market, swap spreads serve as a good proxy for credit spreads. It is because of the relative homogeneity of the European bond market. European issues are generally high in quality and intermedieate in maturity.

Swap spreads involve the on-the-run Treasury rate, not seasoned issues.

Question 45

It is possible to increase the yield of the portfolio without a proportionate increase in risk by?

Answer 45

Underweighting 1-5 year Tresuaries and overwieighting 1-5 year corporates.

Short-term (less than 5 years) corporate bonds have the most favorable yield spread per unit of duration risk. Overweithing these issues and underweighting short duration Tresuaries is known as enchanced indexing by small risk factor mismathces.

Question 46

Does the total return payer (in total credit swap) own the underlying securities?

Answer 46

The total return payer may or may not own the underlying securities. Entering into a credit swap as the total return payer without owning the underlying assets is a way to short a bond.

Question 47

Safety margin calculation (example)

Answer 47

A portfolio manager had decided to pursue a contingent immunization strategy over a three-year time horizon. He just purchased at par 93 $M worth of 10% semiannual coupon, 12y bonds. Current rates of return for immunized strategies are 10% and the portfolio manager is willing to accept a return of 8.5%. If interest rates rise to 11% immediately, what is the safety margin? Shoulf the manager continue with contingent immunization?

1. Compute required terminal value: PV=93$M, N=6, 1/Y=8.5/2=4.25%, PMT=0 => FV = 119,382,132$
2. Calculate the current value of the bond: PMT = 93$M*0,05=4,650,000$, N=24, 1/Y=11/2=5.5%, FV = 93 $M => PV= 86,884,460$
3. Compute the present value of the required terminal value at the new interest rate: FV=119,382,132$, PMT=0, N=6, 1/Y=5.5% = > PV= 86,581,394$

The dollar safety margin: (86,884,460$ - 86,581,394$ = 303,066$) and the manager can continue to employ contingent immunization.

Question 48

The purpose of break-even analysis (forward rates) to make relative value decisions?

Answer 48

Break-even analysis is used to determine whether or not to hedge.

The strategic outlook is what you “expect” to happen to the currency. The market price can be determined from the forward rate. Comparing the two dictates whether you should or not to hedge.

Question 49

Difference between enchanced indexing by matching primary risk factors and enchanced indexing that allows minor risk factor mismatches?

Answer 49

Matching primary risk factors means that portfolio will be exposed to the same broad market moving movements as the index.

Enchanced indexing with minor risk factor mismatches means that the portfolio will have the same duration as the index, but will have differential movements from the index due to sector, quality, term structure, etc…. mismatches.

Hence, the matching by primary risk factors will tend to yield smaller differences between index and portfolio returns.

Question 50

Efficiency of Cash flow matching vs. Classical immunization

Answer 50

A cash flow matching strategy is unlikely to be more effective than a classical immunization strategy because:

• the more uncertain, indeterminate, or highly variable the liability stream, the less effective cash flow matching will be compared to classical immunization.
• transaction costs from forced, unanticipated trading necessry to adjust assets cash flows to match the frequently changing liability schedule would make cash flow matching less effective than classical immunization
• classical immunization requires less capital to fund liabilities. This is because
  
  • a) a cash flow matching strategy usually requires return assumption for short-term cash balances (and such balances may at times be significant) while an immunized portfolio is essentially fully invested at the remaining horizon duration;
  • b) funds from a cash flow-matched portfoio must be available on or before each liability due date, which tends to reduce the assumed return. A classically immunizes portfolio needs to meet the target value only on the date of each liability, because funding is achieved by a rebalancing of the portfolio.