Fixed income portfolios
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Fixed Income Portfolios. Overview. Setting Investment Objectives Establishing investment policy Selecting a portfolio strategy Selecting assets Measuring and Evaluating performance. Setting Investment Objectives. Varies with type of financial institution

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Fixed Income Portfolios

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Fixed income portfolios

Fixed Income Portfolios



  • Setting Investment Objectives

  • Establishing investment policy

  • Selecting a portfolio strategy

  • Selecting assets

  • Measuring and Evaluating performance

Setting investment objectives

Setting Investment Objectives

  • Varies with type of financial institution

    • pension fund -- generate cash flow sufficient to cover obligations

    • life insurance -- meet obligations in insurance (long term) and generate profit

    • banks earn spread over short term deposits, timing of liabilities

Establishing investment objectives

Establishing Investment Objectives

  • Asset allocation

    • Match assets and liabilities based on goals of the financial institution

  • Client and Regulatory constraints

    • limits on credit ratings

  • Tax and Financial Reporting implications

    • mutual funds are tax exempt so munis are not attractive

Selecting a portfolio strategy

Selecting a Portfolio Strategy

  • Active vs. passive strategies

    • Active - Attempts to forecast and exploit changes in future rates and macro economic variables. Change portfolio composition often in response to expectations.

    • Passive - Closer to buy and hold. Goal is to replicate or benchmark for example to an index.

    • Combinations of both

Selecting a strategy

Selecting a Strategy

  • Structured portfolio strategies

    • goal is to achieve a predetermined benchmark or goal such as matching the timing of future liabilities.

    • Immunization - eliminating the impact of interest rate changes in the cash flows received

    • Cash flow matching or horizon matching

    • Often include low risk active strategies within a passive strategy

What determines strategy choice

What Determines Strategy Choice?

  • Efficiency of market

    • If market is efficient, you cannot beat the market return consistently. This implies indexing as the strategy.

  • Liabilities

    • Must be able to meet future obligations of the firm (think about a bank, pension fund or insurance firm)

Selecting assets

Selecting Assets

  • Identifying individual securities (identifying mispriced securities if not indexing or matching cash flows

  • Identifying cash flow characteristics

Measuring and evaluating performance

Measuring and Evaluating Performance

  • Measuring against a benchmark

  • Meeting liability constraints

Sources of active portfolio returns

Sources of Active Portfolio Returns

  • Changes in the level of Interest Rates

  • Changes in the shape of the Yield Curve

  • Changes in the yield spreads among bond sectors

  • Changes in the option adjusted spread

  • Changes in the yield spread of a particular bond

  • Changes in asset allocation within bond sector

Manager expectations vs market consensus

Manager Expectations vs.Market Consensus

  • The market consensus should be reflected in the current market prices and yields.

  • This may or may not agree with the manager expectations.

Interest rate expectations

Interest rate expectations

  • Expected change in interest rates will often force manager to make a change in strategy.

  • This may not include actually changing the underlying assets for example swaps may be used to shorten or lengthen the duration of a portfolio.

  • Problem – No reason to believe that you can forecast accurately

Yield curve strategies

Yield Curve Strategies

  • Positioning your portfolio to capitalize on expected changes in the shape of the Treasury Yield Curve

Parallel shifts

Parallel Shifts







Flattening Twist



Steepening Twist



Butterfly shifts

Butterfly Shifts

Positive Butterfly






Common shifts

Common Shifts

  • Most common shifts are combinations of the types above

  • Downward shift combined with steepening

    • More likely to also be combined with a negative butterfly

  • Upward shift combined with flattening

    • More likely to be combined with a positive butterfly

Portfolio strategies

Portfolio Strategies

  • Need to consider the timing of the cash flows and therefore the duration of the portfolio and / or maturity.

  • Look at expectations of future yield curve shifts

  • Match liabilities

Ladder or spaced maturity

Ladder or Spaced Maturity

  • Maturity is capped and then the portfolio is spread out evenly across the range of maturities

  • Assume 5 year cap – then 20% of portfolio is in each year.











Ladder or spaced

Ladder or Spaced

  • Once a year matures it is assumed to be reinvested in new 5 year bonds. Therefore the trend continues

  • Advantages

    • Reduces Investment income fluctuations

    • Requires little investment expertise

    • Since it continues to roll over into cash it provides flexibility

Front end load bullet strategy

Front End Load (bullet) Strategy

  • Place all of securities in a short period of time








Front end load

Front End Load

  • Uses the portfolio as a source of liquidity since it is so short term

  • Advantages

    • Avoids large capital losses if rates increase since short run securities are less sensitive to interest rate changes.

Back end bullet strategy

Back End (bullet) Strategy

  • Places all of portfolio at the upper end of the maturity














Back end strategy

Back End Strategy

  • Stresses Investment income instead of liquidity

  • Advantages

    • Increases gain if interest rates decrease since long term bonds are more sensitive to rate changes (but also larger decline in value if rates increase)

  • Forces institution to depend upon money market for short term returns.

Barbell strategy

Barbell Strategy

  • Combination of front end and back end load. The goal is to balance the desire for liquidity and income.



















  • Combines both goals of liquidity and income

  • Advantages

    • Not as responsive to interest rates (either increase or decrease) as back end load, more responsive than front end load.

Rate expectations

Rate Expectations

  • Aggressive strategy based on expected rates

Shift if rates are expected to decrease

Shift if rates are expected to increase











Rate expectations1

Rate Expectations

  • Very aggressive, attempts to match portfolio to rate expectations.

  • Advantages

    • If successful, capital gains will be increased and capital losses will be decreased.

Analysis of the portfolios

Analysis of the portfolios

  • How the portfolios actually respond will be dependent upon changes in the yield curve (steepening etc.) Not just a static measure of interest rates.

  • Given the duration of portfolio, and estimating the value change is implicitly assuming that the the yield on each of the assets in the portfolio changes by the same amount.

Want to look at total return

Want to look at total return

  • The best way to compare across portfolios is to compare total return if a shift actually occurs.

Compare two portfolios

Compare two portfolios

  • Bullet: 100% in Bond C

    • $ duration = 6.434

    • $ convexity = 55.4506

    • YTM 9.25%

  • Barbell: 50.2% in bond A and 49.8% in bond B

    • $ duration = (0.502)(4.005)+(.498)(8.882)

      = 6.434

    • $ convexity = (0.502)(19.8164)+(.498)(124.17)


    • YTM = .502(.0850)+.498(.0950) = 8.998%

Cost of convexity

Cost of Convexity

  • The barbell has a higher convexity but a lower yield. The bullet has a yield 25.5 basis points higher than the barbell. This is the cost of convexity.

  • Which portfolio does better for a yield change? It depends on the yield shift (parallel or twist etc)

Key point

Key Point

  • Looking at just the duration, convexity, YTM etc. does not provide a good indication of which portfolio is “better.”

Measuring yield curve risk

Measuring Yield Curve Risk

  • Key Rate Duration, Calculating the change in value for a security or portfolio after changing one key interest rate keeping other rates constant.

  • Each point on the spot yield curve has a separate duration associated with it.

  • If you allowed all rates to change by the same amount, you could measure the response to the security or portfolio to a parallel shift in the yield curve.

Key rates and portfolios

Key Rates and Portfolios

  • By focusing on a group of key rates it is possible to investigate the impact of changes in the shape of the yield curve on specific parts of a portfolio, we will cover this in more detail in the portfolio section of the course.

Using key rate durations

Using Key Rate Durations*

  • Assume you have three key rtes 2 years, 16 years and 30 years. Assume that you are investing in zero coupon instruments at each maturity (the duration will be equal to the maturity).

  • Therefore each bond will respond to changes in its portion of the yield curve.

From Fabozzi Fixed Income for the CFA p310 - 312

Consider 3 portfolios

Consider 3 portfolios

  • Portfolio 1 (Barbell)

    • $50 in the 2 year, 0 in the 16 year, and $50 in the 30 year

  • Portfolio 2 (Bullet)

    • 0 in the 2 year, $100 in the 16 year, and 0 in the 30 year

  • Portfolio 3 (Spread)

    • $33.33 in each of the possible bonds.

Portfolio duration

Portfolio Duration

  • The weighted average of the key rate durations similarly the effective duration will be the weighted average of the durations of the securities in the portfolio.

Key rate duration

Key Rate Duration

  • For each maturity (key rate) we need to find the key rate duration.

    • Let D(1) be the duration for the 2 year part of the curve

    • Let D(2) be the duration for the 16 year part of the curve

    • Let D(3) be the duration for the 30 year part of the curve

  • Portfolio 1

    • For portfolio 1 the only portion of the portfolio that is sensitive to a change in the 2 year rate is the two year security, the similar result happens for each of the other maturities.

Portfolio key rate durations

Portfolio Key Rate Durations

  • Portfolio 1

    • D(1)=(50/100)2+(0/100)0+(50/100)0=1

    • D(2)=(50/100)0+(0/100)0+(50/100)0=0

    • D(3)=(50/100)0+(0/100)0+(50/100)30=15

  • Portfolio 1

    • D(1)=(0/100)0+(100/100)0+(0/100)0=0

    • D(2)=(0/100)0+(100/100)16+(0/100)0=16

    • D(3)=(0/100)0+(100/100)0+(0/100)30=0

  • Portfolio 1

    • D(1)=(33.3/100)2+(33.3/100)0+(33.3/100)0=.6666

    • D(2)=(33.3/100)0+(33.3/100)16+(33.3/100)0=5.333

    • D(3)=(33.3/100)0+(33.3/100)0+(33.3/100)30=10

Effective duration

Effective Duration

  • The effective duration of each portfolio would be the weighted average of the securities durations

    • Portfolio 1

      (50/100)2+(0/100)16+(50/100)30 = 16

    • Portfolio 2

      (0/100)2+(100/100)16+(0/100)30 = 16

    • Portfolio 3

      (33.3/100)2+(33.3/100)16+(33.3/100)30 = 16

A parallel shift in the yield curve

A parallel shift in the yield curve

  • Assume that all spot decrease by 10%

  • Given the key rate durations for portfolio 1

    • D(1)=1, D(2)=0, D(3)=15

  • For a 100 Bp decrease in the 2 year rate, the portfolio should see a 1% increase in price, for a 10 Bp decrease price should increase by .1%

  • Similarly a 10 Bp decrease in the 30 year rate should increase price by 1.5%

  • The total change in price is then .1%+ 1.5%

Three possible yield curve shifts

Three possible yield curve shifts

  • Now lets consider the impact of three different possible shifts in the yield curve on each of the three portfolios

    • Scenario 1 Parallel Downward Shift

      All maturities decrease by 10 Bp

    • Scenario 2

      2-yr rate shifts up 10 Bp, 30-yr rate shifts down by 10Bp

    • Scenario 3

      2-yr rate shifts down 10 Bp, 30-yr rate shifts up by 10Bp

Comparison of shifts

Comparison of shifts

Yield spread strategies

Yield Spread Strategies

  • Positioning a portfolio to capitalized on expected changes in yield spreads.

  • Intermarket Spread Swaps – exchanging one bond for another between sectors of the bond market based on the yield spread

Credit spreads

Credit Spreads

  • Credit spreads (Spread between treasury and similar maturity non treasury) generally widen in a declining economy and narrow during expansion.

  • Yield Ratios vs. Spreads. As the level of rates change so should the absolute spread.

Yield spread strategies1

Yield Spread Strategies

  • Positioning a portfolio to take advantage of changes in the spread between two classifications of bonds.

  • One example would be an intermarket spread swap.

  • May recognize differences in credit spreads, or embedded options.

Example credit spreads expected to widen

10% BBB rated Corp,

5 yrs to mat, YTM = .10

8% Treasury, 5 yrs to Mat, YTM = .09072458

Example: Credit Spreads Expected to Widen

Yield Spread = .10-.090724 = .009275742 (92.75742Bp)

What strategy should you undertake?

Purchase the Treasury and Short the Corp

1) Treasury yield falls - price of treasury increases

2) Corp. yield increases - price of corp decreases

Example continued


Time 0

Receive $100

Next Day

Pay $100

Total = $100


Time 0

Buy 1.044 of Treas =$100

Next Day

Sell 1.044 @ 97.21

Total = 101.50635

Example continued

Assume that you hold the positions for 1day. At that time the

treasury yield has decreased to 8.70%

Importance of duration

Importance of Duration

  • When comparing spreads it is imperative to look at positions that have the same duration.

  • If the duration of the new and old position are not the same then you are accepting risk associated with a change in the level of rates as well as a change in the spread.

Example credit spreads expected to widen1

10% A rated Corp,

5 yrs to mat, YTM = .10

Mac Duration = 4.0539

Modified Duration = 4.0539/1.10 = 3.68537

$ duration =



8% Treasury

5 yrs to Mat, YTM = .090724

Mac Duration = 4.19

Modified Duration =

4.19/1.090724 = 3.84836

$ duration = 3.84836(95.7646)


Example: Credit Spreads Expected to Widen

Example continued1


Time 0

Receive $100

Next Day

Pay $99.9614

Total = $99.961


Time 0

Buy 1.044 of Treas =$100

Next Day

Sell 1.044 @ 95.726

Total = 99.957

Example continued

Assume that you hold the positions for 1day. At that time both

yields increased by 1 basis point

The price change was basically the same for both!

Individual security selection

Individual Security Selection

  • Basic goal is to identify undervalued securities

    • Its yield is higher than other comparable securities

    • Its yield is expected to decline

  • In either case a substitution swap -- exchanging a bond for another that offers a higher yield.

Allocation within sectors

Allocation Within Sectors

  • Within a broad sector (corporate for example) a portfolio manager needs to decide how to allocate within the sector (Across credit categories).

  • Combination of past history and future expectations.

Rating transition

Rating Transition

Next step

Next Step

  • Given the rating transition, you then forecast what the spreads will be at the end of the holding period and the return based on the spreads

  • Then use use the probabilities from the matrix to find an expected return

Using leverage

Using Leverage

  • Ability to use leverage to take out a larger position will depend upon the guidelines of the fund.

  • Basic goal is to earn a return greater than the cost of the borrowed funds.

  • Allows the benefit of small price changes to be magnified since relative size of position can be increased.

Duration of levered portfolio

Duration of levered portfolio

  • The duration of the levered portfolio should be calculated based on the “equity position” of the portfolio. (the amount of non borrowed funds)

Calculating duration

Calculating Duration

  • Calculate the duration of the levered portfolio

  • Determine the dollar duration of the portfolio for a given change in interest rates

  • Compute the ratio of the dollar value change in 2) to the initial unlevered portfolio

  • the duration is then:

  • (Ratio in 3))(100/rate change in 2 in bps)100

Creating leverage

Creating Leverage

  • Easiest way to create leverage is via the repurchase market.

  • Repurchase agreement -- sale of security with the agreement to repurchase it the following day (overnight repo) or over a given short term (term repo).

Repo interest

Repo Interest

  • The dollar value of interest is calculated using a 360 day convention.

Reverse repos

Reverse Repos

  • You can also cover a short position with a reverse repo (agreeing to buy the security and then sell it back in the future).

Credit risk

Credit Risk

  • Repo market credit risk can be reduced by over collateralizing the repo transaction.

  • Repo margin - the amount by which the market value of the collateral exceeds the dollar value of the loan

Delivery of collateral

Delivery of Collateral

  • Direct delivery to the other party or the parties agent causes transaction costs to be incurred. The costs are figured into the interest.

  • An alternative to delivery is for the repo to be held in custody (HIC repo)

  • Use of the lenders custodial account at the borrowers clearing bank. Reduces transaction costs and collateral risk.

Determining the repo rate

Determining the Repo Rate

  • The more difficult to obtain the collateral the lower the repo rate (the party lending funds will be willing to pay a lower rate to obtain the collateral.

  • The higher the credit quality and the higher the liquidity the lower the repo rate.

Structured portfolios

Structured Portfolios

  • Structured portfolios are intended to satisfy an investment objective, and are not based upon interest rate expectations.

    • Indexing

    • Match Liabilities and Assets



  • Attempting to match the performance of a given bond index.

  • Performance is measured in terms of total return over a investment horizon.

  • Index Performance is determined relative to the target index (An even split of treasuries and high grade corporate bonds for example, or mortgage backs, or global or….)

Popularity of indexing

Popularity of Indexing

  • Active bond management has traditionally produced poor returns.

  • Indexed portfolio advisory fees are usually less than actively managed portfolios.

  • Nonadvisory fees (custodial etc) are also lower.

  • May limit the risk by limiting the portfolio to certain types of bonds (enhanced control by sponsor).

Problems with indexing

Problems with Indexing

  • Matching index performance, does not mean optimal performance is achieved.

  • Indexing does not guarantee that return objectives will be met. Even if index is matched, it may not match other criteria.

  • Indexing may eliminate some profitable types of investments

Institutional perspective

Institutional perspective

  • My matching some form of “market” index the institution can offer a return similar to the index.

  • Decreases the need for active management.

  • Fee income form management (even though the fees are less)

Selecting an index

Selecting an Index

  • Return objectives -- look at both return and variability

  • Risk factors -- match index to acceptable risk levels

Broad market bond indexes

Broad Market Bond Indexes

  • Lehman Brothers Aggregate Index, Salomon Brothers Broad Investment Grade Bond Index, and Merrill Lynch Domestic Market Index

  • All have over 5,000 issues rated BBB or better, The Salomon index is trader priced while the others include some model priced issues.

  • All exclude issues with less than one year to maturity

Broad market indexes

Broad Market Indexes

  • The three broad indexes produce very similar returns with the correlation of the returns over 98% (Reilly, Kao and Wright (1992).

  • While the correlations of long run returns are high, there is some variation on a month to month basis.

Specific indexes

Specific Indexes

  • Each firm and many others also produce indexes of specific markets such as the government market or Mortgage backed securities.

  • Some also offer customized indexes such as the Salomon Large Pension Fund Baseline Bond Index which is designed to match the long duration of pension fund liabilities.

Size of portfolio

Size of Portfolio

  • Given an index that the manager is going to attempt to match, decisions need to be made concerning the construction of the portfolio.

  • Included in this decision is the number of issues to use to attempt to match the index.

    • As issues are added variance decreases

    • The impact of portfolio size is very similar to equity

    • The number of securities needed to eliminate unsystematic risk may differ by sector.

Tracking error

Tracking Error

  • Tracking error is the difference between the indexed portfolio and the benchmark index.

  • Three sources of tracking error

    • Transaction Costs

    • Differences in index composition.

    • Difference in price used to construct the index and those paid by the portfolio

Tracking error tradeoff

Tracking Error Tradeoff

  • The larger the number of issues in the portfolio the greater the transaction cost and the greater the associated tracking error.

  • The smaller the number of issues in the portfolio the greater the differences in return based upon composition and the greater the tracking error.

Indexing methodologies

Indexing Methodologies

  • Stratified Sampling (or Cell)

  • Optimization Approach

  • Variance Minimization Approach

  • In all three the goal is to minimize or eliminate diversifiable risk leaving only the systematic risk common to the sector.

Stratified sampling cell

Stratified Sampling (Cell)

  • The index is split into cells representing different characteristics of the index such as duration, coupon, maturity, market sector, credit rating, call features, and sinking fund features.

  • The total number of cells is then dependent upon the partitions in each sector.

Stratified sampling example

Stratified Sampling Example

  • Characteristic 1 Duration: < 5 years & > 5 years

  • Characteristic 2 Maturity: < 7 years & > 7 years

  • Characteristic 3 Sector: Treasuries and Corporate

  • Then make cells out of each possible combination of characteristics:

    Cell 1: Duration < 5, Maturity < 7, Treasury

    Cell 2: Duration < 5, Maturity < 7, Corporate

    Cell 3: Duration < 5, Maturity > 7, Treasury etc…

  • Total cells = 2 x 2 x 2 = 8

Stratified sampling

Stratified Sampling

  • For each cell select one or more issues from the index that can represent the entire cell.

  • The total dollar amount form each cell is the proportioned by the total dollar amount form each cell in the index.

  • The number of cells will increase with the size of the portfolio, since more cells require a larger number of issues purchased, a small portfolio should keep the number of cells relatively small (but tracking error increases)



  • First the goal will be to match the cells as in stratified sampling, but then add the goal of optimizing an outcome subject to extra constraints.

  • Outcome Examples: Maximize portfolio yield, maximize convexity, Maximize total returns

  • Constraints: Limit the number of issues purchased from a given issuer, overweighting a cell



  • Given the objective and constraints mathematical programming can then be used to determine which issues to include in the portfolio.

Variance minimization

Variance Minimization

  • Requires historical data for each issue.

  • Based on the historical data a price function is estimated for each issue.

  • The price function is then used to establish the variance of the tacking error.

  • The goal is then to use mathematical programming to minimize the variance of the tracking error of the portfolio.

Problem in implementation

Problem in Implementation

  • Published prices may not be executable. They are often based on bid prices not ask prices.

  • Illiquidity of the market -- some of the issues may not actually be available

  • Aggregation -- often generic issues are established to look like a group of issues (mortgage backs for example)

Enhanced indexing

Enhanced Indexing

  • The goal of enhanced indexing is to consistently outperform the total return of a given index. (this justifies higher advisory fees). It also comes at the cost of a higher risk of under performing the index.

  • The goal is accomplished by being more active in management and accepting greater interest rate risks and duration related risks.

Asset liability management

Asset / Liability Management

  • The goal of asset / liability management is to match the timing and size of assets to the expected cash flows associated with the liabilities.

  • Nature of institution will determine the liabilities and the associated management strategies.

Liability classification

Liability Classification

Liquidity concerns

Liquidity Concerns

  • Will depend upon the type of institution.

    Banking -- depository withdraws

    Life Insurance -- surrender and loan values

  • May also change the nature of expected cash inflows.

Surplus management

Surplus Management

  • Goals -- earn an adequate return and maintain a surplus of assets beyond liabilities.

  • Three types of surpluses

    • Economic -- based on market value

    • Accounting -- based upon GAAP

    • Regulatory -- based upon regulatory accounting principles

Economic surplus

Economic Surplus

Market Value of Assets - Market Value of Liabilities

  • Market value of Liabilities is simply the PV of the expected cash flows.

  • The net effect of a change in interest rates will depend upon the duration of both the assets and liabilities.

  • In both cases an increase in rates will decrease the value and vice versa.

Economic surplus1

Economic Surplus

  • Assuming that the $ value of assets is greater than liabilities whether the surplus increases or decreases will depend on duration and the direction of an interest rate change.

  • If duration of assets > duration of liabilities:

    An increase in rates implies an decrease in surplus

    A decrease in interest rates implies an increase in surplus

Economic surplus2

Economic Surplus

  • What if the duration is the same?

  • If the market value of assets is greater than market value of liabilities then then a decrease in rates still will increase the surplus and vice versa.

Accounting surplus

Accounting Surplus

  • Three methods for reporting the value of assets

    • Amortized cost (historical cost)

    • Market value

    • Lower of cost or market value

  • FASB specifies how different types of assets must be valued.

Fasb 115

FASB 115

Regulatory surplus

Regulatory Surplus

  • Regulators require reports based upon separate accounting principles (RAP).

  • Often the regulatory surplus will differ significantly from the accounting or economic surplus.



  • F.M. Reddington (1952): “The investment in assets in such a way that the existing business is immune to a general change in interest rates”

Immunization of a single liability

Immunization of a single liability

  • Assume that an insurance co has offered a guaranteed investment contract.

  • The guarantee is to pay a 6.25% return each 6 months (12.5% bond equivalent yield) for 5.5 years.

  • Invest $8,829,262 today and the buyer will have $8,829,262(1.0625)11=$17,183,033 which is also a liability for the insurance co.

Immunization attempt 1

Immunization attempt 1

  • The life insurance firm uses the $8,829,262 to purchase a 12.5% coupon bond selling at par that matures in 5.5 years.

  • Will this immunize the portfolio?

  • NO -- you will only have the required $17,183,033 if the coupons can be reinvested at 6.25% each six months until the maturity of the bond.

  • If rates increase (decrease) immediately total value will be above (below) $17,183,033

Immunization attempt 2

Immunization attempt 2

  • The life insurance firm uses the $8,829,262 to purchase a 12.5% coupon bond selling at par that matures in 15 years.

  • Will this immunize the portfolio?

  • NO -- you will only have the required $17,183,033 if the coupons can be reinvested at 6.25% each six months until 5.5 years have passed

  • If rates increase immediately total value will be below $17,183,033, and vice versa.

Immunization attempt 3

Immunization attempt 3

  • The life insurance firm uses the $8,829,262 to purchase a 12.5% coupon bond selling at par that matures in 6 months.

  • Will this immunize the portfolio?

  • NO -- you will only have the required $17,183,033 if the bond can be reinvested at 6.25% each six months until 5.5 years have passed

  • If rates increase (decrease) immediately total value will be above (below) $17,183,033.

Immunization attempt 4

Immunization attempt 4

  • The life insurance firm uses the $8,829,262 to purchase a 10.125% coupon bond selling to yield 12.5% that matures in 8 years ($10,000,000 par value)

  • Will this immunize the portfolio?

  • Yes -- you will have the required $17,183,033 regardless of an immediate change in yield.

  • If rates increase the interest on interest offsets the decline in value

  • If rates decrease the increase in value offsets the decline in interest on interest.



  • The Macaulay duration of the liability is simply the 5.5 years (a modified duration of 5.18).

  • The modified duration of the 8 year 10.125% coupon bond is 5.18% (Macaulay duration of 5.5).



  • Two things to satisfy:

  • The Macaulay Duration of the portfolio is the same as the liability.

  • The PV of the cash flows of the portfolio is the same as the PV of the liability.

  • Note this assumes option free bond (if the Macaulay duration is the same for both then the modified duration will also be the same.) If embedded options exist effective duration must be used.

Changes over time

Changes over time

  • The duration of the portfolio and of the liability will change over time.

  • Fro immunization to remain intact the portfolio should be rebalanced to keep the duration equal to that of the liability.

  • Frequent rebalancing causes an increase in transaction costs. Infrequent rebalancing causes an increased risk of failing to meet the target.

Other complications

Other complications

  • Our example assumed that the yield curve is flat and that any shifts in the yield curve are parallel shifts.

Immunization risk

Immunization Risk

  • There are multiple portfolios that can be created that satisfy the duration criteria.

  • Which one should be chosen?

  • Bierwag, Kaufman, and Toves (1981) If the portfolio cash flows are more concentrated around the liability due date it is less risky. Immunization risk is closely tied to reinvestment rate risk.

Measuring immunization risk

Measuring Immunization risk

  • The product of two terms determine the impact of a change in the shape of the yield curve.

  • The first term is based upon the characteristics of the cash flows

  • The second term is based upon the change in the shape of the yield curve which cannot be predicted.

  • Therefore the first term can be used to measure risk.

Measuring immunization risk1

Measuring Immunization Risk

Immunizing with zero coupon bonds

Immunizing with Zero Coupon Bonds

  • An alternative possibility is to invest in zero coupon bonds that mature at the same time as the investment horizon of the liability.

  • This satisfies the duration requirement however the yield on the zero coupon is usually less than on coupon instruments so it requires a greater investment today

Portfolio construction

Portfolio Construction

  • Credit Risk -- if a bond defaults the target yield may not be reached

  • Call risk -- If callable issues are included then there is a risk of the call being exercised and the target yield not being reached.

Contingent immunization

Contingent Immunization

  • Actively managing the portfolio until a negative outcome puts the potential total return (Realized and immunized) down to a safety level. The manager is then required to immunize the entire portfolio to ensure the safety net level.

Satisfying multiple liabilities

Satisfying Multiple Liabilities

  • Multiperiod immunization. Just matching duration will not guarantee matching the multiple future liabilities.

    Each liability must be immunized by a separate cash flow stream of the portfolio.

    Note: this requires decomposition of the portfolios combined cash flow stream, not the assets in the portfolio.

Satisfying multiperiod liabilities

Satisfying Multiperiod Liabilities

  • Cash Flow Matching - working backward through the multiple cash flows.

    Starting with the final liability, using a bond with the same maturity as the final liability, an amount is invested that will produce a final payment and coupon equal to the liability.

    The other cash flows are reduced by the coupons on the bond and the process is repeated for the next to last liability and so on.

Satisfying multiperiod liabilities1

Satisfying Multiperiod Liabilities

  • Symmetric Cash Matching -- allows short term borrowing of funds to satisfy a liability prior to the liability due date, reducing the cost of funding.

Active immunization combination

Active / Immunization Combination

  • Combining the two strategies (contingent immunization is one or the other..).

  • A combined strategy might include immunizing a portion of the portfolio and actively managing the remainder of the portfolio.

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