1. Fixed Income Portfolios

2. Overview Setting Investment Objectives Establishing investment policy
Selecting a portfolio strategy
Selecting assets
Measuring and Evaluating performance

3. 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

4. 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

5. 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

6. 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

7. 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)

8. Selecting Assets
Identifying individual securities (identifying mispriced securities if not indexing or matching cash flows
Identifying cash flow characteristics

9. Measuring and Evaluating Performance
Measuring against a benchmark
Meeting liability constraints

10. 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

11. 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.

12. 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

13. Yield Curve Strategies
Positioning your portfolio to capitalize on expected changes in the shape of the Treasury Yield Curve

14. Short Intermediate Long Short Intermediate Long
Parallel Shifts
Short Intermediate Long
Maturity
Short Intermediate Long
Maturity

15. Short Intermediate Long Short Intermediate Long
Twists
Flattening Twist
Short Intermediate Long
Maturity
Steepening Twist
Short Intermediate Long
Maturity

16. Short Intermediate Long Short Intermediate Long
Butterfly Shifts
Positive Butterfly
Short Intermediate Long
Maturity
Negative Butterfly
Short Intermediate Long
Maturity

17. 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

18. 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

19. 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.
20%
20%
20%
20%
20% 1 2 3 4 5

20. 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

21. Front End Load (bullet) Strategy
Place all of securities in a short period of time
20%
80% 1 5 2 3 4

22. 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.

23. Back End (bullet) Strategy
Places all of portfolio at the upper end of the maturity
60%
20%
20% 9 10 1 2 3 4 5 6 7 8

24. 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.

25. Barbell Strategy
Combination of front end and back end load. The goal is to balance the desire for liquidity and income.
25%
25%
15%
15%
10%
10% 1 2 3 4 5 6 7 8 9 10

26. Barbell 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.

27. Rate Expectations Aggressive strategy based on expected rates
Shift if rates are expected to decrease
Shift if rates are expected to increase 1 2 3 4 5 6 7 8 9 10

28. 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.

29. 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.

30. Want to look at total return
The best way to compare across portfolios is to compare total return if a shift actually occurs.
Bond
Coup
Mat
Price
YTM
$Dur
$Conv A
8.5 5
100
4.005
19.816 B
9.5 20
8.882
124.17 C
9.25 10
6.434
55.45

31. Compare two portfolios
Bullet: 100% in Bond C
$ duration = 6.434
$ convexity =
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)( )+(.498)(124.17) =
YTM = .502(.0850)+.498(.0950) = 8.998%

32. 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)

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

34. 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.

35. 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.

36. 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 p

37. Consider 3 portfolios Portfolio 1 (Barbell) Portfolio 2 (Bullet)
$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.

38. 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.

39. 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.

40. Portfolio Key Rate Durations

41. 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

42. 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%

43. 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

44. Comparison of shifts Portfolio Scenario1 Scenario 2 Scenario3 I +1.6%
+1.4%
-1.4% II 0%
III
+0.94%
-0.94%

45. 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

46. 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.

47. 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.

48. Example: Credit Spreads Expected to Widen
10% BBB rated Corp,
5 yrs to mat, YTM = .10
8% Treasury, 5 yrs to Mat, YTM =
Yield Spread = = ( Bp)
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

49. Example continued
Assume that you hold the positions for 1day. At that time the
treasury yield has decreased to 8.70%
Corp
Time 0
Receive $100
Next Day
Pay $100
Total = $100
Treasury
Time 0
Buy of Treas =$100
Next Day
Sell 97.21
Total =

50. 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.

51. Example: Credit Spreads Expected to Widen
8% Treasury
5 yrs to Mat, YTM =
Mac Duration = 4.19
Modified Duration =
4.19/ =
$ duration = ( ) =
10% A rated Corp,
5 yrs to mat, YTM = .10
Mac Duration =
Modified Duration = /1.10 =
$ duration =
(100) =

52. Example continued
Assume that you hold the positions for 1day. At that time both
yields increased by 1 basis point
Corp
Time 0
Receive $100
Next Day
Pay $
Total = $99.961
Treasury
Time 0
Buy of Treas =$100
Next Day
Sell
Total =
The price change was basically the same for both!

53. 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.

54. 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.

55. Rating Transition

56. 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

57. 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.

58. 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)

59. 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

60. 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).

61. Repo Interest
The dollar value of interest is calculated using a 360 day convention.

62. 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).

63. 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

64. 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.

65. 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.

66. Structured Portfolios
Structured portfolios are intended to satisfy an investment objective, and are not based upon interest rate expectations.
Indexing
Match Liabilities and Assets

67. Indexing 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….)

68. 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).

69. 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

70. 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)

71. Selecting an Index
Return objectives -- look at both return and variability
Risk factors -- match index to acceptable risk levels

72. 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

73. 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.

74. 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.

75. 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.

76. 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

77. 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.

78. 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.

79. 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.

80. 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

81. 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)

82. Optimization
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

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

84. 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.

85. 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)

86. 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.

87. 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.

88. Liability Classification

89. 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.

90. 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

91. 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.

92. 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

93. 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.

94. 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.

95. FASB 115

96. Regulatory Surplus
Regulators require reports based upon separate accounting principles (RAP).
Often the regulatory surplus will differ significantly from the accounting or economic surplus.

97. Immunization
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”

98. 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.

99. 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

100. 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.

101. 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.

102. Immunization attempt 4
The life insurance firm uses the $8,829,262 to purchase a % 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.

103. Duration
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 % coupon bond is 5.18% (Macaulay duration of 5.5).

104. Immunization 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.

105. 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.

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

107. 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.

108. 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.

109. Measuring Immunization Risk

110. 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

111. 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.

112. 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.

113. 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.

114. 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.

115. 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.

116. 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.