5 June - LIABILITY-DRIVEN AND INDEX-BASED STRATEGIES
LIABILITY-DRIVEN AND INDEX-BASED STRATEGIES
by James F. Adams, PhD, CFA and Donald J. Smith, PhD
James F. Adams, PhD, CFA, is at New York University (USA). Donald J. Smith, PhD, is at Boston University Questrom School of Business (USA).
LEARNING OUTCOMES
The candidate should be able to:
describe liability-driven investing
evaluate strategies for managing a single liability
compare strategies for a single liability and for multiple liabilities, including alternative means of implementation
describe construction, benefits, limitations, and risk–return characteristics of a laddered bond portfolio
evaluate liability-based strategies under various interest rate scenarios and select a strategy to achieve a portfolio’s objectives
explain risks associated with managing a portfolio against a liability structure
discuss bond indexes and the challenges of managing a fixed-income portfolio to mimic the characteristics of a bond index
compare alternative methods for establishing bond market exposure passively
discuss criteria for selecting a benchmark and justify the selection of a benchmark
SUMMARY
Structured fixed-income investing requires a frame of reference, such as a balance sheet, to structure the bond portfolio. This frame of reference can be as simple as the time to retirement for an individual or as complex as a balance sheet of rate-sensitive assets and liabilities for a company. 💡
Assets and liabilities can be categorized by the degree of certainty surrounding the amount and timing of cash flows. Type I assets and liabilities, such as traditional fixed-rate bonds with no embedded options, have known amounts and payment dates. For Type I assets and liabilities, such yield duration statistics as Macaulay, modified, and money duration apply. 📍
Type II, III, and IV assets and liabilities have uncertain amounts and/or uncertain timing of payment. For Type II, III, and IV assets and liabilities, curve duration statistics, such as effective duration, are needed. 💡 A model is used to obtain the estimated values when the yield curve shifts up and down by the same amount.
Immunization is the process of structuring and managing a fixed-income portfolio to minimize the variance in the realized rate of return over a known investment horizon.📍
In the case of a single liability, immunization is achieved by matching the Macaulay duration of the bond portfolio to the horizon date. 📍 As time passes and bond yields change, the duration of the bonds changes and the portfolio needs to be rebalanced. This rebalancing can be accomplished by buying and selling bonds or using interest rate derivatives, such as futures contracts and interest rate swaps. 💡
An immunization strategy aims to lock in the cash flow yield on the portfolio, which is the internal rate of return on the cash flows. 💡 It is not the weighted average of the yields to maturity on the bonds that constitute the portfolio.
The risk to immunization is that as the yield curve shifts and twists, the cash flow yield on the bond portfolio does not match the change in the yield on the zero-coupon bond that would provide for perfect immunization.💡
A sufficient, but not necessary, condition for immunization is a parallel (or shape-preserving) shift whereby all yields change by the same amount in the same direction. 💡 If the change in the cash flow yield is the same as that on the zero-coupon bond being replicated, immunization can be achieved even with a non-parallel shift to the yield curve.💡
Structural risk to immunization arises from some non-parallel shifts and twists to the yield curve. 💡 This risk is reduced by minimizing the dispersion of cash flows in the portfolio, which can be accomplished by minimizing the convexity statistic for the portfolio 💡. Concentrating the cash flows around the horizon date makes the immunizing portfolio closely track the zero-coupon bond that provides for perfect immunization.
For multiple liabilities, one method of immunization is cash flow matching.💡 A portfolio of high-quality zero-coupon or fixed-income bonds is purchased to match as closely as possible the amount and timing of the liabilities. 📍
A motive for cash flow matching can be accounting defeasance, whereby both the assets and liabilities are removed from the balance sheet. 📍
A laddered bond portfolio is a common investment strategy in the wealth management industry. The laddered portfolio offers “diversification” over the yield curve compared with “bullet” or “barbell” portfolios. 💡 This structure is especially attractive in stable, upwardly sloped yield curve environments as maturing short-term debt is replaced with higher-yielding long-term debt at the back of the ladder. 💡
A laddered portfolio offers an increase in convexity💡 because the cash flows have greater dispersions than a more concentrated (bullet) portfolio.
A laddered portfolio provides liquidity💡 in that it always contains a soon-to-mature bond that could provide high-quality, low-duration collateral on a repo contract if needed.
Immunization of multiple liabilities can be achieved by structuring and managing a portfolio of fixed-income bonds. Because the market values of the assets and liabilities differ, the strategy is to match the money durations. 💡 The money duration is the modified duration multiplied by the market value. 📍 The basis point value is a measure of money duration calculated by multiplying the money duration by 0.0001. 📍
The conditions to immunize multiple liabilities are that (1) the market value of assets is greater than or equal to the market value of the liabilities, (2) the asset basis point value (BPV) equals the liability BPV, and (3) the dispersion of cash flows and the convexity of assets are greater than those of the liabilities. 📍
A derivatives overlay—for example, interest rate futures contracts—can be used to immunize single or multiple liabilities. 💡
The number of futures contracts needed to immunize is the liability BPV minus the asset BPV, divided by the futures BPV. If the result is a positive number, the entity buys, or goes long, futures contracts. If the result is a negative number, the entity sells, or goes short, futures contracts. 📍 The futures BPV can be approximated by the BPV for the cheapest-to-deliver security divided by the conversion factor for the cheapest-to-deliver security. 📍
Contingent immunization adds active management of the surplus 📍, which is the difference between the asset and liability market values📍, with the intent to reduce the overall cost of retiring the liabilities💡. In principle, any asset classes can be used for the active investment.💡 The entity can choose to over-hedge or under-hedge the number of futures contracts needed for passive immunization. 💡
Liability-driven investing (LDI) often is used for complex rate-sensitive liabilities, such as those for a defined benefit pension plan. The retirement benefits for covered employees depend on many variables, such as years of employment, age at retirement, wage level at retirement, and expected lifetime. There are different measures for the liabilities: for instance, the accumulated benefit obligation (ABO) that is based on current wages💡 and the projected benefit obligation (PBO) that is based on expected future wages. 💡For each liability measure (ABO or PBO), a model is used to extract the effective duration and BPV. 💡
Interest rate swap overlays can be used to reduce the duration gap as measured by the asset and liability BPVs. 💡There often is a large gap because pension funds hold sizable asset positions in equities that have low or zero effective durations and their liability durations are high.💡
The hedging ratio is the percentage of the duration gap that is closed with the derivatives. 📍 A hedging ratio of zero implies no hedging. A hedging ratio of 100% implies immunization—that is, complete removal of interest rate risk.💡
Strategic hedging is the active management of the hedging ratio.📍 Because asset BPVs are less than liability BPVs in typical pension funds, the derivatives overlay requires the use of receive-fixed interest rate swaps 💡. Because receive-fixed swaps gain value as current swap market rates fall, the fund manager could choose to raise the hedging ratio when lower rates are anticipated. If rates are expected to go up, the manager could strategically reduce the hedging ratio.
An alternative to the receive-fixed interest rate swap is a purchased receiver swaption 💡. This swaption confers to the buyer the right to enter the swap as the fixed-rate receiver. Because of its negative duration gap (asset BPV is less than liability BPV), the typical pension plan suffers when interest rates fall and could become underfunded. The gain on the receiver swaption as rates decline offsets the losses on the balance sheet.
Another alternative is a swaption collar, the combination of buying the receiver swaption and writing a payer swaption. The premium received on the payer swaption that is written offsets the premium needed to buy the receiver swaption.
The choice among hedging with the receive-fixed swap, the purchased receiver swaption, and the swaption collar depends in part on the pension fund manager’s view on future interest rates. If rates are expected to be low, the receive-fixed swap typically is the preferred derivative. If rates are expected to go up, the receiver swaption can become attractive. And if rates are projected to reach a certain threshold that depends on the option costs and the strike rates, the swaption collar can become the favored choice.
Model risks arise in LDI strategies because of the many assumptions in the models and approximations used to measure key parameters. For example, the liability BPV for the defined benefit pension plan depends on the choice of measure (ABO or PBO) and the assumptions that go into the model regarding future events (e.g., wage levels, time of retirement, and time of death).
Spread risk in LDI strategies arises because it is common to assume equal changes in asset, liability, and hedging instrument yields when calculating the number of futures contracts, or the notional principal on an interest rate swap, to attain a particular hedging ratio. The assets and liabilities are often on corporate securities, however, and their spreads to benchmark yields can vary over time.
Investing in a fund that tracks a bond market index offers the benefits of both diversification and low administrative costs. Tracking risk arises when the fund manager chooses to buy only a subset of the index, a strategy called enhanced indexing, because fully replicating the index can be impractical as a result of the large number of bonds in the fixed-income universe. ✅
Corporate bonds are often illiquid. Matrix pricing uses available data on comparable securities to estimate the fair value of the illiquid bonds. 💡
The primary risk factors💡 encountered by an investor tracking a bond index include decisions regarding duration (option-adjusted duration for callable bonds, convexity for possible large yield shifts, and key rate durations for non-parallel shifts) and portfolio weights (assigned by sector, credit quality, maturity, coupon rate, and issuer).
Index replication is one method to establish a passive exposure to the bond market 💡. The manager buys or sells bonds only when there are changes to the index. Full replication can be expensive, however, as well as infeasible for broad-based fixed-income indexes that include many illiquid bonds.
Several enhancement strategies can reduce the costs to track a bond index 💡: lowering trading costs, using models to identify undervalued bonds and to gauge relative value at varying points along the yield curve, over/under weighting specific credit sectors over the business cycle, and evaluating specific call features to identify value given large yield changes.
Investors can obtain passive exposure to the bond market using ETFs or mutual funds. Exchange-traded fund (ETF) shares have the advantage of trading on an exchange throughout the day. ✅
A total return swap, an over-the-counter derivative, allows an institutional investor to transform an asset or liability from one asset category to another—for instance, from variable-rate cash flows referencing the MRR to the total return on a particular bond index.
A total return swap (TRS) can have some advantages over a direct investment in a bond mutual fund or ETF. As a derivative, it requires less initial cash outlay 💡 than direct investment in the bond portfolio for similar performance. A TRS also carries counterparty credit risk, however. As a customized over-the-counter product, a TRS can offer exposure to assets that are difficult to access directly, such as some high-yield and commercial loan investments.
Selecting a particular bond index is a major decision for a fixed-income investment manager. Selection is guided by the specified goals and objectives for the investment. The decision should recognize several features of bond indexes: (1) Given that bonds have finite maturities, the duration of the index drifts down over time; (2) the composition of the index changes over time with the business cycle and maturity preferences of issuers.
INTRODUCTION
Fixed-income instruments make up nearly three-quarters of all global financial assets available to investors. It is thus not surprising that bonds are a critical component of most investment portfolios. In our coverage of structured and passive total return fixed-income investment strategies, we explain that “passive” does not simply mean “buy and hold.” The primary strategies discussed—immunization and indexation—can entail frequent rebalancing of the bond portfolio. We also note that “passive” stands in contrast to “active” fixed-income strategies that are based on the asset manager’s particular view on interest rate and credit market conditions.
We explain liability-driven investing by demonstrating how to best structure a fixed-income portfolio when considering both the asset and liability sides of the investor’s balance sheet. It is first important to have a thorough understanding of both the timing and relative certainty of future financial obligations. Because it is rare to find a bond investment whose characteristics perfectly match one’s obligations, we introduce the idea of structuring a bond portfolio to match the future cash flows of one or more liabilities that have bond-like characteristics. Asset–liability management (ALM) strategies are based on the concept that investors incorporate both rate-sensitive assets and liabilities into the portfolio decision-making process. When the liabilities are given and assets are managed, liability-driven investing (LDI), a common type of ALM strategy, may be used to ensure adequate funding for an insurance portfolio, a pension plan, or an individual’s budget after retirement. The techniques and risks associated with LDI are introduced using a single liability and then are expanded to cover both cash flow and duration-matching techniques and multiple liabilities. This strategy, known as immunization, may be viewed simply as a special case of interest rate hedging.
We then turn our attention to index-based investment strategies, through which investors gain a broader exposure to fixed-income markets, rather than tailoring investments to match a specific liability profile. We explain the advantages of index-based investing, such as diversification, but we also note that the depth and breadth of bond markets make both creating and tracking an index more challenging than in the equity markets. We also explore a variety of alternatives in matching a bond index, from full replication to enhanced indexing using primary risk factors. Finally, we explain that it is critical to select a benchmark that is most relevant to a specific investor based on factors such as the targeted duration profile and risk appetite.
LIABILITY-DRIVEN INVESTING
Learning Outcome
describe liability-driven investing
Let us start with the example of a 45-year-old investor who plans to retire at age 65 and who would like to secure a stable stream of income thereafter. It is quite probable that he currently has a diversified portfolio that includes bonds, equities, and possibly other asset classes. Our focus here is on the fixed-income portion of his overall portfolio. We will assume that the investor builds the bond portfolio (immediately) and will add to it each year. Upon retirement, he plans to sell the bonds and buy an annuity that will pay a fixed benefit for his remaining lifetime. This investor’s initial 20-year time horizon is critical to identifying and measuring the impact on retirement income arising from future interest rate volatility, and it forms the initial frame of reference for understanding and dealing with interest rate risk.
More generally, the frame of reference is in the form of a balance sheet of rate-sensitive assets and liabilities. In the example of the 45-year-old investor, the asset is the growing bond portfolio and the liability is the present value of the annuity that the investor requires to satisfy the fixed lifetime benefit.
Liability-Driven Investing vs. Asset-Driven Liabilities
Liability-driven investing (LDI) and asset-driven liabilities (ADL) are special cases of ALM. The key difference is that with ADL, the assets are given and the liabilities are structured to manage interest rate risk; whereas with LDI, which is much more common, the liabilities are given and the assets are managed. As an example of LDI, a life insurance company acquires a liability portfolio based on the insurance policies underwritten by its sales force.
Another example involves the future employee benefits promised by a defined benefit pension plan, which create a portfolio of rate-sensitive liabilities. In each circumstance, the liabilities are defined and result from routine business and financial management decisions. The present value of those liabilities depends on current interest rates (as well as other factors). A life insurance or pension manager will use the estimated interest rate sensitivity of plan liabilities as a starting point when making investment portfolio decisions. This process often requires building a model for the liabilities.
With ADL, the asset side of the balance sheet results from a company’s underlying businesses, and the debt manager seeks a liability structure to reduce interest rate risk. One example might be a leasing company with short-term contracts that chooses to finance itself with short-term debt. The company is aiming to match the maturities of its assets and liabilities to minimize risk. Alternatively, a manufacturing company might identify that its operating revenues are highly correlated with the business cycle. Monetary policy is typically managed so there is positive correlation between interest rates and the business cycle. Central banks lower policy rates when the economy is weak and raise them when it is strong. Therefore, this company has a natural preference for variable-rate liabilities so that operating revenue and interest expense rise and fall together.
Types of Liabilities
Exhibit 1:
Classification of Liabilities
Note that effective duration is needed with Types II, III, and IV liabilities, based on initial assumptions about the yield curve. Then, the yield curve is shifted up and down to obtain new estimates for the present value of the liabilities. We demonstrate this process later for the sponsor of a defined benefit pension plan, which is another example of an entity with Type IV liabilities.
EXAMPLE 1
Modern Mortgage, a savings bank, decides to establish an ALCO (asset–liability committee) to improve risk management and coordination of its loan and deposit rate-setting processes. Modern’s primary assets are long-term, fixed-rate, monthly payment, fully amortizing residential mortgage loans. The mortgage loans are prime quality and have loan-to-value ratios that average 80%. The loans are pre-payable at par value by the homeowners at no fee. Modern also holds a portfolio of non-callable, fixed-income government bonds (considered free of default risk) of varying maturities to manage its liquidity needs. The primary liabilities are demand and time deposits that are fully guaranteed by a government deposit insurance fund. The demand deposits are redeemable by check or debit card. The time deposits have fixed rates and maturities ranging from 90 days to three years and are redeemable before maturity at a small fee. The banking-sector regulator in the country in which Modern operates has introduced a new capital requirement for savings banks. In accordance with the requirement, contingent convertible long-term bonds are issued by the savings bank and sold to institutional investors. The key feature is that if defaults on the mortgage loans reach a certain level or the savings bank’s capital ratio drops below a certain level, as determined by the regulator, the bonds convert to equity at a specified price per share.
Specify and explain the classification scheme for the following:
Residential mortgage loans
Solution to 1:
Residential mortgage loans are Type IV assets to the savings bank. The timing of interest and principal cash flows is uncertain because of the prepayment option held by the homeowner. This type of call option is complex. Homeowners might elect to prepay for many reasons, including sale of the property as well as the opportunity to refinance if interest rates come down. Therefore, a prepayment model is needed to project the timing of future cash flows. Default risk also affects the projected amount of the cash flow for each date. Even if the average loan-to-value ratio is 80%, indicating high-quality mortgages, some loans could have higher ratios and be more subject to default, especially if home prices decline.
Government bonds
Solution to 2:
Fixed-rate government bonds are Type I assets because the coupon and principal payment dates and amounts are determined at issuance.
Demand and time deposits
Solution to 3:
Demand and time deposits are Type II liabilities from the savings bank’s perspective. The deposit amounts are known, but the depositor can redeem the deposits prior to maturity, creating uncertainty about timing.
Contingent convertible bonds
Solution to 4:
The contingent convertible bonds are Type IV liabilities. The presence of the conversion option makes both the amount and timing of cash flows uncertain.
MANAGING THE INTEREST RATE RISK OF A SINGLE LIABILITY
Learning Outcome
evaluate strategies for managing a single liability
Liability-driven investing in most circumstances is used to manage the interest rate risk on multiple liabilities. In this section, we focus on only a single liability to demonstrate the techniques and risks of the classic investment strategy known as interest rate immunization.
📍 Immunization is the process of structuring and managing a fixed-income bond portfolio to minimize the variance in the realized rate of return over a known time horizon. This variance arises from the volatility of future interest rates. Default risk is neglected at this point because the portfolio bonds are assumed to have default probabilities that approach zero.
The most obvious way to immunize the interest rate risk on a single liability is to buy a zero-coupon bond that matures on the obligation’s due date.
The bond’s face value matches the liability amount. There is no cash flow reinvestment risk because there are no coupon payments to reinvest, and there is no price risk because the bond is held to maturity. Any interest rate volatility over the bond’s lifetime is irrelevant in terms of the asset’s ability to pay off the liability. The problem is that in many financial markets, zero-coupon bonds are not available. Nevertheless, the perfect immunization provided by a zero-coupon bond sets a standard to measure the performance of immunizing strategies using coupon-bearing bonds.
Exhibit 2:
Immunization with a Single Bond: Rate Rise Scenario
Assume that the bond is currently priced at par value. Then, an instantaneous, one-time, upward (parallel) shift occurs in the yield curve. The bond’s value falls. That drop in value is estimated by the money duration of the bond. Recall that the money duration is the bond’s modified duration statistic multiplied by the price. Subsequently, the bond price will be “pulled to par” as the maturity date nears (assuming no default, of course). But another factor is at work. Assuming interest rates remain higher, the future value of reinvested coupon payments goes up. It is shown by the rising line as more and more payments are received and reinvested at the higher interest rates.
Exhibit 3:
Immunization: Interest Rate Fall Scenario
A Numerical Example of Immunization
We now show that the strategy of matching the Macaulay duration to the investment horizon works for a bond portfolio as well as for an individual security. Suppose that some entity has a single liability of EUR 250 million due 15 February 2027. Further assume that the current date is 15 February 2021, so the investment horizon is six years. The asset manager for the entity seeks to build a three-bond portfolio to earn a rate of return sufficient to pay off the obligation.
Portfolio Features
Exhibit 4:
The Bond Portfolio to Immunize the Single Liability
2.5-Year Bond
7-Year Bond
10-Year Bond
Coupon rate
1.50%
3.25%
5.00%
Maturity date
15 August 2023
15 February 2028
15 February 2031
Price
100.25
99.75
100.50
Yield to maturity
1.3979%
3.2903%
4.9360%
Par value
47,000,000
97,300,000
55,600,000
Market value
47,117,500
97,056,750
55,878,000
Macaulay duration
2.463
6.316
7.995
Convexity
7.253
44.257
73.747
Allocation
23.55%
48.52%
27.93%
Exhibit 5:
Portfolio Statistics
Time
Date
Cash Flow
PV of Cash Flow
Weight
Time × Weight
Dispersion
Convexity
0
15-Feb-21
–200,052,250
1
15-Aug-21
3,323,625
3,262,282
0.0163
0.0163
1.9735
0.0326
2
15-Feb-22
3,323,625
3,202,071
0.0160
0.0320
1.6009
0.0960
3
15-Aug-22
3,323,625
3,142,971
0.0157
0.0471
1.2728
0.1885
4
15-Feb-23
3,323,625
3,084,962
0.0154
0.0617
0.9871
0.3084
5
15-Aug-23
50,323,625
45,847,871
0.2292
1.1459
11.2324
6.8754
6
15-Feb-24
2,971,125
2,656,915
0.0133
0.0797
0.4782
0.5578
7
15-Aug-24
2,971,125
2,607,877
0.0130
0.0913
0.3260
0.7300
8
15-Feb-25
2,971,125
2,559,744
0.0128
0.1024
0.2048
0.9213
9
15-Aug-25
2,971,125
2,512,500
0.0126
0.1130
0.1131
1.1303
10
15-Feb-26
2,971,125
2,466,127
0.0123
0.1233
0.0493
1.3560
11
15-Aug-26
2,971,125
2,420,610
0.0121
0.1331
0.0121
1.5972
12
15-Feb-27
2,971,125
2,375,934
0.0119
0.1425
0.0000
1.8527
13
15-Aug-27
2,971,125
2,332,082
0.0117
0.1515
0.0116
2.1216
14
15-Feb-28
100,271,125
77,251,729
0.3862
5.4062
1.5434
81.0931
15
15-Aug-28
1,390,000
1,051,130
0.0053
0.0788
0.0473
1.2610
16
15-Feb-29
1,390,000
1,031,730
0.0052
0.0825
0.0825
1.4028
17
15-Aug-29
1,390,000
1,012,688
0.0051
0.0861
0.1265
1.5490
18
15-Feb-30
1,390,000
993,997
0.0050
0.0894
0.1788
1.6993
19
15-Aug-30
1,390,000
975,651
0.0049
0.0927
0.2389
1.8533
20
15-Feb-31
56,990,000
39,263,380
0.1963
3.9253
12.5585
82.4316
200,052,250
1.0000
12.0008
33.0378
189.0580
For instance, EUR 3,323,625 is the sum of the coupon payments for the first four dates:(1.50% × 0.5 × EUR 47,000,000) + (3.25% × 0.5 × EUR 97,300,000) + (5.00% × 0.5 × EUR 55,600,000) = EUR 352,500 + EUR 1,581,125 + EUR 1,390,000 = EUR 3,323,625On 15 August 2023, the principal of EUR 47,000,000 is redeemed so that the total cash flow is EUR 50,323,625. The next eight cash flows represent the coupon payments on the second and third bonds, and so forth.
Portfolio Duration
The sixth column of Exhibit 5 is used to obtain the portfolio’s Macaulay duration. This duration statistic is the weighted average of the times to the receipt of cash flow, whereby the share of total market value for each date is the weight. Column 5 shows the weights, which are the PV of each cash flow divided by the total PV of EUR200,052,250. The times to receipt of cash flow (the times from column 1) are multiplied by the weights and then summed. For example, the contribution to total portfolio duration for the second cash flow on 15 February 2022 is 0.0320 (= 2 × 0.0160). The sum of column 6 is 12.0008. That is the Macaulay duration for the portfolio in terms of semi-annual periods. Annualized, it is 6.0004 (= 12.0008/2). It is now clear why the asset manager for the entity chose this portfolio: The portfolio Macaulay duration matches the investment horizon of six years.
The difference, as with the cash flow yield and the market value-weighted average yield, arises because the yield curve is not flat. When the yield curve is upwardly sloped, average duration (5.8776) is less than the portfolio duration (6.0004). This difference in duration statistics is important because using the average duration in building the immunizing portfolio instead of the portfolio duration would introduce model risk to the strategy, as we will see later.
Portfolio Dispersion
This portfolio’s dispersion is 33.0378 in terms of semi-annual periods. Annualized, it is 8.2594 (= 33.0378/4). The Macaulay duration statistic is annualized by dividing by the periodicity of the bonds (two payments per year); dispersion (and convexity, which follows) is annualized by dividing by the periodicity squared (i.e., 2^2 = 4 for semi-annual payment bonds).
Portfolio Convexity
There is an interesting connection among the portfolio convexity, Macaulay duration, dispersion, and cash flow yield in immunized portfolio convexity, also known as the “portfolio convexity statistic”:Immunized Portfolio Convexity=MacDur2+MacDur+Dispersion(1+Cash flow yield)21In terms of semi-annual periods, the Macaulay duration for this portfolio is 12.0008, the dispersion is 33.0378, and the cash flow yield is 1.8804%.Immunized Portfolio Convexity=12.00082+12.0008+33.0378(1.018804)2=182.1437.The portfolio dispersion and convexity statistics are used to assess the structural risk to the interest rate immunization strategy. Structural risk arises from the potential for shifts and twists to the yield curve. This risk is discussed later.
Investment Horizon and Immunization
We now demonstrate how matching the Macaulay duration for the portfolio to the investment horizon leads to interest rate immunization. The first three columns of Exhibit 6 are identical to the ones in Exhibit 5.
The fourth column shows the values of the cash flows as of the horizon date of 15 February 2027, assuming that the cash flow yield remains unchanged at 3.7608%. For instance, the future value of the EUR3,323,625 in coupon payments received on 15 August 2021 is EUR4,079,520:3,323,625×(1+0.0376082)11=4,079,520The value of the last cash flow for EUR56,990,000 on 15 February 2031 is EUR49,099,099 as of the horizon date of 15 February 2027:56,990,000(1+0.0376082)8=49,099,099We assume that all of the payments received before the horizon date are reinvested at the cash flow yield. All of the payments received after the horizon date are sold at their discounted values. The sum of the fourth column in Exhibit 6 is EUR250,167,000, which is more than enough to pay off the EUR250 million liability. The six-year holding period rate of return (ROR), also called the horizon yield, is 3.7608%. It is based on the original market value and the total return and is the solution for ROR:200,052,250=250,167,000(1+ROR2)12, ROR = 0.037608The holding period rate of return equals the cash flow yield for the portfolio. This equivalence is the multi-bond version of the well-known result for a single bond: The realized rate of return matches the yield to maturity only if coupon payments are reinvested at that same yield and if the bond is held to maturity or sold at a point on the constant-yield price trajectory.
Exhibit 6:
Interest Rate Immunization
Time
Date
Cash Flow
Total Return at 3.7608%
Total Return at 2.7608%
Total Return at 4.7608%
0
15-Feb-21
–200,052,250
1
15-Aug-21
3,323,625
4,079,520
3,864,613
4,305,237
2
15-Feb-22
3,323,625
4,004,225
3,811,992
4,205,138
3
15-Aug-22
3,323,625
3,930,319
3,760,088
4,107,366
4
15-Feb-23
3,323,625
3,857,777
3,708,891
4,011,868
5
15-Aug-23
50,323,625
57,333,230
55,392,367
59,332,093
6
15-Feb-24
2,971,125
3,322,498
3,225,856
3,421,542
7
15-Aug-24
2,971,125
3,261,175
3,181,932
3,341,989
8
15-Feb-25
2,971,125
3,200,984
3,138,607
3,264,286
9
15-Aug-25
2,971,125
3,141,904
3,095,871
3,188,390
10
15-Feb-26
2,971,125
3,083,914
3,053,718
3,114,258
11
15-Aug-26
2,971,125
3,026,994
3,012,138
3,041,850
12
15-Feb-27
2,971,125
2,971,125
2,971,125
2,971,125
13
15-Aug-27
2,971,125
2,916,287
2,930,670
2,902,045
14
15-Feb-28
100,271,125
96,603,888
97,559,123
95,662,614
15
15-Aug-28
1,390,000
1,314,446
1,333,991
1,295,282
16
15-Feb-29
1,390,000
1,290,186
1,315,827
1,265,166
17
15-Aug-29
1,390,000
1,266,373
1,297,911
1,235,750
18
15-Feb-30
1,390,000
1,242,999
1,280,238
1,207,018
19
15-Aug-30
1,390,000
1,220,058
1,262,806
1,178,955
20
15-Feb-31
56,990,000
49,099,099
51,070,094
47,213,270
250,167,000
250,267,858
250,265,241
A Drop in the Cash Flow Yield Scenario
200,052,250=250,267,858(1+ROR2)12, ROR = 0.037676
An Increase in the Cash Flow Yield Scenario
To complete the example, the sixth column in Exhibit 6 reports the results for an instantaneous, one-time, 100 bp jump in the cash flow yield, up to 4.7608% from 3.7608%. In this case, the future values of the reinvested cash flows are higher and the discounted values of cash flows due after the horizon date are lower. Nevertheless, the total return of EUR250,265,241 for the six-year investment horizon is enough to pay off the liability. The horizon yield is 3.7674%:
200,052,250=250,265,241(1+ROR2)12, ROR = 0.037674
This numerical exercise demonstrates interest rate immunization using a portfolio of fixed-income bonds. The total returns and holding period rates of return are virtually the same—in fact, slightly higher because of convexity—whether the cash flow yield goes up or down.
Immunization and Rebalancing
Exhibit 7:
Interest Rate Immunization as Zero Replication
Immunizing with coupon-bearing bonds entails continuously matching the portfolio Macaulay duration with the Macaulay duration of the zero-coupon bond over time and as the yield curve shifts, even though the zero-coupon bond could be hypothetical and not exist in reality. Also, in order to fully match the liability, the bond portfolio’s initial market value has to match or exceed the present value of the zero-coupon bond. The Macaulay duration of that, perhaps hypothetical, zero-coupon bond always matches the investment horizon. Immunization will be achieved if any ensuing change in the cash flow yield on the bond portfolio is equal to the change in the yield to maturity on the zero-coupon bond. That equivalence will ensure that the change in the bond portfolio’s market value is close to the change in the market value of the zero-coupon bond. Therefore, at the end of the six-year investment horizon, the bond portfolio’s market value should meet or exceed the face value of the zero-coupon bond, regardless of the path for interest rates over the six years.
Immunization and Shifts in the Yield Curve
The key assumption to achieve immunization is the statement that “any ensuing change in the cash flow yield on the bond portfolio is equal to the change in the yield to maturity on the zero-coupon bond.” A sufficient, but not necessary, condition for that statement is a parallel (or shape-preserving) shift to the yield curve whereby all yields change by the same amount. Sufficient means that if the yield curve shift is parallel, the change in the bond portfolio’s cash flow yield will equal the change in yield to maturity of the zero-coupon bond, which is enough to ensure immunization. To achieve immunization, however, it is not necessary that the yield curve shifts in a parallel manner. That is, in some cases, the immunization property can prevail even with non-parallel yield curve movements, such as an upward and steepening shift (sometimes called a “bear steepener”), an upward and flattening shift (a “bear flattener”), a downward and steepening shift (a “bull steepener”), or a downward and flattening shift (a “bull flattener”).
Exhibit 8:
Some Upward Yield Curve Shifts That Achieve Interest Rate Immunization
Change in 2.5-Year Yield
Change in 7-Year Yield
Change in 10-Year Yield
Change in Cash Flow Yield
Change in Market Value
Upward and parallel
+102.08 bps
+102.08 bps
+102.08 bps
+100 bps
–11,340,537
Upward and steepening
+72.19 bps
+94.96 bps
+120.82 bps
+100 bps
–11,340,195
Upward and flattening
+145.81 bps
+109.48 bps
+79.59 bps
+100 bps
–11,340,183
Exhibit 9:
Some Downward Yield Curve Shifts That Achieve Interest Rate Immunization
Change in 2.5-Year Yield
Change in 7-Year Yield
Change in 10-Year Yield
Change in Cash Flow Yield
Change in Market Value
Downward and parallel
–102.06 bps
–102.06 bps
–102.06 bps
–100 bps
12,251,212
Downward and steepening
–129.00 bps
–104.52 bps
–92.00 bps
–100 bps
12,251,333
Downward and flattening
–55.76 bps
–86.32 bps
–134.08 bps
–100 bps
12,251,484
In general, the interest rate risk to an immunization strategy is that the change in the cash flow yield on the portfolio is not the same as on the ideal zero-coupon bond. This difference can occur with twists to the shape of the yield curve, in addition to some non-parallel shifts.
Exhibit 10:
Immunization Risk and Steepening Twist
Exhibit 11:
Immunization Risk and a Butterfly Yield Curve Movement
Structural Risk in Immunization Strategy
In summary, the characteristics of a bond portfolio structured to immunize a single liability are that it:
has an initial market value that equals or exceeds the present value of the liability;
has a portfolio Macaulay duration that matches the liability’s due date;
minimizes the portfolio convexity statistic.
This portfolio must be regularly rebalanced over the horizon to maintain the target duration, because the portfolio Macaulay duration changes as time passes and as yields change. The portfolio manager needs to weigh the trade-off between incurring transaction costs from rebalancing and allowing some duration gap. This and other risks to immunization—for instance, those arising from the use of interest rate derivatives to match the duration of assets to the investment horizon—are covered later.
EXAMPLE 2
An institutional client asks a fixed-income investment adviser to recommend a portfolio to immunize a single 10-year liability. It is understood that the chosen portfolio will need to be rebalanced over time to maintain its target duration. The adviser proposes two portfolios of coupon-bearing government bonds because zero-coupon bonds are not available. The portfolios have the same market value. The institutional client’s objective is to minimize the variance in the realized rate of return over the 10-year horizon. The two portfolios have the following risk and return statistics:
Portfolio A
Portfolio B
Cash flow yield
7.64%
7.65%
Macaulay duration
9.98
10.01
Convexity
107.88
129.43
These statistics are based on aggregating the interest and principal cash flows for the bonds that constitute the portfolios; they are not market value-weighted averages of the yields, durations, and convexities of the individual bonds. The cash flow yield is stated on a semi-annual bond basis, meaning an annual percentage rate having a periodicity of two; the Macaulay durations and convexities are annualized.
Indicate the portfolio that the investment adviser should recommend, and explain the reasoning.
Solution:
The adviser should recommend Portfolio A. First, notice that the cash flow yields of both portfolios are virtually the same and that both portfolios have Macaulay durations very close to 10, the horizon for the liability. It would be wrong and misleading to recommend Portfolio B because it has a “higher yield” and a “duration closer to the investment horizon of 10 years.” In practical terms, a difference of 1 bp in yield is not likely to be significant, nor is the difference of 0.03 in annual duration.
Given the fact that the portfolio yields and durations are essentially the same, the choice depends on the difference in convexity. The difference between 129.43 and 107.88, however, is meaningful. In general, convexity is a desirable property of fixed-income bonds. All else being equal (meaning the same yield and duration), a more convex bond gains more if the yield goes down and loses less if the yield goes up than a less convex bond.
The structural risk to the immunization strategy is the potential for non-parallel shifts and twists to the yield curve, which lead to changes in the cash flow yield that do not track the change in the yield on the zero-coupon bond. This risk is minimized by selecting the portfolio with the lower convexity (and dispersion of cash flows).
Note that default risk is neglected in this discussion because the portfolio consists of government bonds that presumably have default probabilities approaching zero.
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