CFA 2018 level 3 schweser practice exam CFA 2018 level 3 question bank CFA 2018 CFA 2018 r22 liability driven and index based strategies IFT notes

However, if a suitable zero-coupon bond is not available then in order to immunize a single liability, we need to create a portfolio of coupon paying bonds with the following characteris

Table of Contents

1 Introduction 3

2 Liability-Driven Investing 3

3 Interest Rate Immunization—Managing the Interest Rate Risk of a Single Liability 4

4 Interest Rate Immunization—Managing the Interest Rate Risk of Multiple Liabilities 12

4.1 Cash Flow Matching 12

4.2 Duration Matching 13

4.3 Derivatives Overlay 15

4.4 Contingent Immunization 16

5 Liability-Driven Investing—An Example of a Defined Benefit Pension Plan 17

6 Risks in Liability-Driven Investing 21

7 Bond Indexes and the Challenges of Matching a Fixed-Income Portfolio to an Index 22

8 Alternative Methods for Establishing Passive Bond Market Exposure 24

9 Benchmark Selection 26

10 Laddered Bond Portfolios 27

Summary 28

Examples from the Curriculum 32

Example 1 32

Example 2 33

Example 3 34

Example 4 36

Example 5 37

Example 6 38

Example 7 39

Example 8 40

Example 9 42

Example 10 43

Example 11 44

Example 12 45 This document should be read in conjunction with the corresponding reading in the 2018 Level III CFA® Program curriculum Some of the graphs, charts, tables, examples, and figures are copyright

2017, CFA Institute Reproduced and republished with permission from CFA Institute All rights reserved

Required disclaimer: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by IFT CFA Institute, CFA®, and Chartered Financial Analyst® are trademarks owned by CFA Institute

1 Introduction

This reading focuses on structured and passive total return fixed-income investment strategies Sections

2 through 6 explain how to construct fixed income portfolios after considering both the asset and liabilities on the balance sheet, i.e liability-driven investing Sections 7 through 9 explain how to construct fixed income portfolios that replicate an index, i.e index-based strategies Finally, section 10 covers the concept of laddered bond portfolios

This section addresses LO.a:

LO.a: describe liability-driven investing;

2 Liability-Driven Investing

Asset–liability management (ALM) strategies consider both assets and liabilities in the portfolio making process ALM strategies include Liability-driven investing (LDI) and asset-driven liabilities (ADL)

decision-In ADL, assets are given and the liabilities are selected so as to minimize the mismatch between assets

and liabilities in order to manage interest rate risk Whereas in LDI, liabilities are given and the assets are structured to minimize the mismatch between assets and liabilities An example of LDI is a life insurance company which has a liability portfolio comprising of insurance policies

In order to structure the portfolio of liabilities, it is important to understand the nature of liability

depending on the amount of cash outlay and the timing of cash outlay Exhibit 1 below shows four

different types of liability based on the amount of cash outlay and the timing of cash outlay

Exhibit 1 Liability

Type

Amount of

Cash Outlay

Timing of Cash Outlay Example

Measure for Interest rate sensitivity

II Known Uncertain Callable and

putable bonds, and

a term life insurance policy

A curve duration statistic known as effective duration is needed to estimate interest rate sensitivity

III Uncertain Known Floating rate

notes, indexed bonds

Inflation-A curve duration statistic known as effective duration is needed to estimate interest rate sensitivity

IV Uncertain Uncertain Defined benefit

plan obligations

A curve duration statistic known as effective duration is needed to estimate interest rate sensitivity

Refer to Example 1 from the curriculum

This section addresses LO.b:

b evaluate strategies for managing a single liability;

3 Interest Rate Immunization—Managing the Interest Rate Risk of a Single Liability

Immunization is the process of structuring and managing a fixed-income bond portfolio to minimize the variance in the realized rate of return, arising due to the volatility of future interest rates, over a known time horizon

The interest rate risk on a single liability can be immunized by buying a zero-coupon bond that matures

on the obligation’s due date In this case,

 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 till maturity

However, if a suitable zero-coupon bond is not available then in order to immunize a single liability, we need to create a portfolio of coupon paying bonds with the following characteristics:

 Market value of bond portfolio should be ≥ present value of liability

 Macaulay duration of bond portfolio = liability’s due date

 The convexity of bond portfolio should be minimized

Further, we need to rebalance the bond portfolio as duration of bonds change to maintain the target duration, because the portfolio Macaulay duration changes as time passes and as yields change

Refer to exhibit 2 below Assume that the bond is currently priced at par value If a one-time, upward shift occurs in the yield curve, this cause the bond’s price to fall, as shown below The decrease in value

of bond is estimated by the money duration of the bond (i.e bond’s modified duration x Bond’s price)

 Subsequently, as the bond reaches it maturity date (assuming no default), the bond price will be

“pulled to par

 However, at the same time, if interest rates remain higher, the future value of reinvested

coupon payments will also increase

 As reflected in Exhibit 2 below, the price effect and the coupon reinvestment effect will offset each other at some point in time (point circled in red)

The lower panel shows the effect of a downward shift in interest rates

Zero Replication: An immunization strategy is essentially “zero replication.” Zero replication implies that

a liability obligation of EUR 250 million at the end of 6 years can be perfectly hedged by six-year coupon bond with a face value that matches the EUR 250 million liability However, if no such zero-coupon bond exists, then we can structure and manage a portfolio of coupon-bearing bonds that replicates the period-to-period performance of the zero-coupon bond This strategy of replicating a performance of a zero-coupon bond is referred to as “zero-replication” In this strategy,

zero- the portfolio’s initial market value must match or exceed PV of zero-coupon bond;

 immunization 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

 the portfolio Macaulay duration is continuously matched with Macaulay duration of

zero-coupon bond

Exhibit 6 shows movement of a zero-coupon bond’s value below and above the constant-yield price trajectory Two paths for the zero-coupon yield are presented: Path A for generally lower rates (and higher values) and Path B for higher rates (and lower values) Note that the market value of the zero-coupon bond will be “pulled to par” as maturity nears

Impact of Yield Curve Movements:

Immunization is achieved if the change in cash flow yield is the same as that on a zero-coupon bond being replicated In general, the immunization can be achieved in case of the following yield curve movements

 Parallel Shifts

 Bear Steepener  an upward and steepening shift

 Bear Flattener  an upward and flattening shift

 Bull Steepener  a downward and steepening shift

 Bull Flattener  downward and flattening shift

However, immunization is not achieved in case of twists to the yield curve as reflected in the Exhibit 8 below Assume that the immunizing portfolio has a “barbell” structure, comprising half short-term bonds and half long-term bonds The portfolio Macaulay duration for the barbell is six years The zero-coupon bond that provides perfect immunization has a maturity (and Macaulay duration) also of six years

 The above chart shows that short-term yields go down and long-term yields go up by approximately the same amount The value of the barbell portfolio goes down because the losses on the long-term positions exceed the gains on the short-term holdings as a result of the difference in duration between the holdings Therefore, this portfolio does not track the value of the zero-coupon bond Now, refer to following chart The short-term and long-term yields go up while the six-year yields go down This type of twist is referred to as “positive butterfly.” (In a “negative butterfly” twist, short-term and long-term yields go down and intermediate-term yields go up.) The value of immunizing portfolio decreases as its yields go up and the value of zero-coupon bond goes up As a result, the portfolio does not track the value of the zero-coupon bond

Structural risk: 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 Structural risk can be reduced by

i selecting a portfolio with the lower dispersion of cash flows;

ii selecting a portfolio with the lower convexity;

iii selecting a portfolio with concentrated cash flows around horizon date

Embedded Example: Suppose that an entity has a single liability of EUR 250 million due 15 February

2023 The current date is 15 February 2017, 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

Exhibit 3 reports the prices, yields, risk statistics (Macaulay duration and convexity), and par values for the chosen portfolio The portfolio’s current market value is EUR 200,052,250 (= EUR 47,117,500 + EUR 97,056,750 + EUR 55,878,000) The semi-annual coupon payments on the bonds occur on 15 February and 15 August of each year The price is per 100 of par value, and the yield to maturity is based on semi-annual bond basis Both the Macaulay duration and the convexity are annualized

The following table shows the cash flows and calculations used to obtain the relevant portfolio statistics

Time Date Cash Flow PV of Cash Flow Weight Time × Weight =

portfolio’s Macaulay duration

352,500 + EUR 1,581,125 + EUR 1,390,000 = EUR 3,323,625

 On 15 August 2019, 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

 The internal rate of return (IRR) on the cash flows in column 3 for the 20 semi-annual periods,

including the portfolio’s initial market value on 15 February 2017, is 1.8804%

 Annualized on a semi-annual bond basis, the portfolio’s cash flow yield is 3.7608% (= 2 × 1.8804%)

 Since the yield curve is not flat, the portfolio’s cash flow yield is significantly higher than the market value weighted average of the individual bond yields presented in Exhibit 3, which is estimated as (1.3979% × 0.2355) + (3.2903% × 0.4852) + (4.9360% × 0.2793) = 3.3043%

Note: It is important to note that the goal of the immunization strategy is to achieve a rate of return

close to the portfolio’s cash flow yield (i.e 3.76%), not the market value weighted average of the individual bond yields (i.e 3.30%)

 The fourth column in the table above shows the present values for each of the aggregate cash flows, calculated using the internal rate of return per period (1.8804%) as the discount rate

For example, the PV of combined payment of EUR 100,271,125 due on 15 February 2024 is calculated as follows:

100,271,125(1.018804)14= 77,251,729

 The sum of the present values in column 4 is EUR 200,052,250 This reflects the current market

value for the bond portfolio

 Column 5 shows the weights, which are the PV of each cash flow divided by the total PV of EUR

200,052,250

 The sixth column reflects the portfolio’s Macaulay duration For example, the contribution to total portfolio duration for the second cash flow on 15 February 2018 is 0.0320 (= 2 × 0.0160) The sum of column 6 is 12.0008, which represents the Macaulay duration for the portfolio in terms of semi-

annual periods The annualized Macaulay duration for the portfolio is 6.0004 (= 12.0008/2) Note

that the portfolio Macaulay duration matches the investment horizon of six years

 The average Macaulay duration = (2.463 × 0.2355) + (6.316 × 0.4852) + (7.995 × 0.2793) = 5.8776

Effect of shape of yield curve on cash flow yield and market value weighted average of the individual bond yields: The difference between the portfolio’s cash flow yield and the market value weighted

average of the individual bond yields is due to the steepness in the yield curve

 When the yield curve is upwardly sloped, average duration (5.8776) is less than the portfolio duration (6.0004) Use of average duration in building the immunizing portfolio instead of the portfolio duration would introduce model risk to the strategy

The sum of the seventh column in the above table shows the portfolio dispersion statistic, which is the weighted variance It measures the extent to which the payments are spread out around the duration For example, the contribution to total portfolio dispersion for the fifth cash flow on 15 August 2019 = (5 – 12.0008)2 × 0.2292 = 11.2324 The total portfolio’s dispersion is 33.0378 in terms of semi-annual periods and annualized dispersion is 8.2594 (= 33.0378/4)

Note: 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

 The eighth column shows the portfolio convexity, which is the sum of the times to the receipt of cash flow, multiplied by those times plus one, multiplied by the shares of market value for each date (weight), and all divided by one plus the cash flow yield squared

For example, the contribution to the sum for the 14th payment on 15 February 2024 (= 14 × 15 × 0.3862) The sum of the column is 189.0580 The convexity in semi-annual periods is 182.1437:

189.0580(1.018804)2= 182.1437 Annualized portfolio convexity =182.1437

The above equation shows that for a given Macaulay duration and cash flow yield, we can minimize portfolio convexity by minimizing portfolio dispersion

In 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%

Advantage of using portfolio convexity to measure the extent of structural risk: The portfolio convexity

statistic can be approximated by the market value weighted average of the individual bonds’

convexities

Disadvantage of using portfolio dispersion to measure the extent of structural risk: The portfolio

dispersion statistics for individual bonds can be misleading because, in a portfolio of all zero-coupon

bonds of varying maturities, each individual bond has zero dispersion (because it has only one payment),

so the market value weighted average is also zero However, the portfolio overall can have significant

(non-zero) dispersion

Now, refer to the table below

 The fourth column shows the values of the cash flows as of the horizon date of 15 February 2023,

assuming that the cash flow yield remains unchanged at 3.7608% For instance, the future value of the EUR 3,323,625 in coupon payments received on 15 August 2017 is

Total Return at 4.7608%

2 )12 ==> ROR = 0.037608 = 3.7608%  this equals the cash flow yield

for the portfolio

 The fifth column in the above table repeats the calculations for one-time, 100 bp drop in the cash flow yield on 15 February 2017 The future values of all cash flows received are now lower because they are reinvested at 2.7608% instead of 3.7608% For example, the payment of EUR 50,323,625 on

15 August 2019, which contains the principal redemption on the 2.5-year bond, grows to only EUR 55,392,367 as calculated below

2 )8

= 51,070,094

 The total return as of the horizon date is EUR 250,267,858, demonstrating that the cash flow

reinvestment effect is balanced by the price effect The holding-period rate of return is

200,052,250 = 250,267,858

(1 + 𝑅𝑂𝑅

2 )12

==> ROR = 0.037676 = 3.7676%

The sixth column shows 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 EUR 250,265,241 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 + 𝑅𝑂𝑅

2 )12

==> ROR = 0.037674 = 3.7674%

Refer to Example 2 from the curriculum

This section addresses LO.c:

LO c compare strategies for a single liability and for multiple liabilities, including alternative means of implementation;

4 Interest Rate Immunization—Managing the Interest Rate Risk of Multiple Liabilities

Following are the approaches used to manage the interest rate risk of multiple liabilities

1) Cash flow matching

2) Duration matching

3) Derivatives overlay

4) Contingent immunization

4.1 Cash Flow Matching

Cash flow matching involves building a dedicated portfolio of zero-coupon or fixed-income bonds to ensure that there are sufficient cash inflows to pay the scheduled cash outflows “Dedicated” means that the bonds are placed in a held-to-maturity portfolio

Panel A in Exhibit 9 illustrates the dedicated cash flow matching asset portfolio These assets could be zero-coupon bonds or traditional fixed-income securities

Panel B represents the amount and timing of the debt liabilities

Accounting defeasance: Accounting defeasance is a process under which a company can remove both

the dedicated asset portfolio and the debt liabilities from its balance sheet, under some circumstances This strategy suffers from cash-in-advance constraint, which implies that sufficient funds must be

available on or before each liability payment date to meet the obligation If a portfolio is made up of traditional bonds, with a level payment annuity, then it is difficult to implement cash-flow matching strategy because annuity-like liability could lead to large cash holdings between payment dates and, therefore, cash flow reinvestment risk, especially if yields on high-quality, short-term investments are low (or worse, negative)

Refer to Example 3 from the curriculum

4.2 Duration Matching

In the case of multiple liabilities, the immunization strategy is to match the money duration of the immunizing portfolio with the money duration of the debt liabilities Money duration (also known as dollar duration) is the portfolio modified duration multiplied by the market value

Modified duration = Portfolio Macaulay Duration

(1+ cash flow yield per period)For example, in the previous example, Macaulay duration was 6.004 and cash flow yield was 3.7608%

So, the money duration for the debt liabilities will be:

[ 6.0004(1 + 0.0376082 )] × 200,052,250 = 1,178,237,935

To keep the numbers manageable, we use the basis point value (BPV) measure for money duration This measure is the money duration multiplied by 1 bp The BPV is EUR 117,824 (= EUR 1,178,237,935 × 0.0001) For each 1 bp change in the cash flow yield, the market value changes by approximately EUR 117,824

Example: Hedging Multiple Liabilities

Liabilities:

Total market value of the liabilities = 200,052,250

Cash flow yield = 0.037608

Macaulay duration = 6.0004

BPV = 117,824

Assets:

Total market value of the immunizing fixed-income bonds = 202,224,094

Cash flow yield = 0.035822

Downward Parallel Shift Immunizing Assets Debt Liabilities Difference

Asset portfolio BPV + (Nf × Futures BPV) = Liability portfolio BPV

𝑁𝑓 = Liability portfolio BPV − Asset portfolio BPV

CFCTD is the conversion factor for the Cheapest to deliver security (In interest futures markets that do

not have a CTD security, the Futures BPV is simply the BPV of the deliverable bond.)

Refer to Exhibit 13 below

If the CTD security is the 6.5-year T-note, the Futures BPV is estimated as:

= 56.8727/0.8226 = USD 69.1377

Then the required number of contracts is approximately 1,432:

117,824 − 18,81969.1377 = 1,432 But, if the CTD security is the 10-year T-note, the Futures BPV is USD 95.8909 (= 81.6607/0.8516) Then the required number of contracts is approximately 1,032:

117,824 − 18,81995.8909 = 1,032 Refer to Example 5 from the curriculum

4.4 Contingent Immunization

Contingent immunization is a hybrid approach, which combines immunization with an active

management approach when the value of asset portfolio is greater than the present value of the liability portfolio The portfolio manager is allowed to actively manage the asset portfolio by investing the surplus in any asset category, including equity, fixed-income, and alternative investments until a

minimum threshold is reached and that threshold is identified by the interest rate immunization

strategy It is important to consider liquidity of the investments to be selected because the positions would need to be liquidated if losses cause the surplus to near the threshold

If the actively managed assets perform poorly and the surplus vanishes, active management ceases and

a conventional duration matching or a cash flow matching approach is used

A contingent immunization strategy can be implemented using the fixed-income derivatives overlay strategy We can over-hedge when yields are expected to fall and under-hedge when they are expected

to rise

Refer to Example 6 from the curriculum

This section addresses LO.d:

LO d evaluate liability-based strategies under various interest rate scenarios and select a strategy to achieve a portfolio’s objectives;

5 Liability-Driven Investing—An Example of a Defined Benefit Pension Plan

Let us discuss an example of a final-pay US defined benefit plan Defined benefit pension plan

obligations represent a Type IV liability for which both the amounts and dates are uncertain Assume that the retired representative employee has worked for G years The employee is expected to work for another T years and then to retire and live for Z years Exhibit 14 illustrates this time-line

Also assume that the retired employee receives a fixed lifetime annuity based on her wage at the time

of retirement, denoted WT

Following are the two measures of the retirement obligations as of Time 0

1) Accumulated benefit obligation (ABO): ABO represents the legal liability today of the plan sponsor

if the plan were to be closed or converted to another type of plan, such as a defined contribution plan The ABO is the present value of the annuity, discounted at an annual rate r on high-quality corporate bonds, which for simplicity we assume applies for all periods (a flat yield curve) It is calculated as follows:

ABO is the present value of the annuity, discounted at an annual rate r on high-quality corporate bonds

2) Projected benefit obligation (PBO): PBO is calculated using the following equation

𝑃𝐵𝑂 = 𝑚 × 𝐺 × 𝑊0 × (1 + 𝑤)

𝑇(1 + 𝑟)𝑇 × [1

that results as long as w < r Assuming w < r is reasonable if it can be assumed that employees over time generally are compensated for price inflation and some part of real economic growth,

as well as for seniority and productivity improvements

Example: Assume that m = 0.02, G = 25, T = 10, Z = 17, W0 = USD 50,000 and r = 0.05 Also assume that wage growth rate w = 0.9 × r so that w = 0.045 (= 0.9 × 0.05) Based on these assumptions, the ABO and PBO for the representative employee are calculated as below

𝐴𝐵𝑂 = 0.02 × 25 × 50,000

(1.05)10 × [ 1

0.05 −

10.05 × (1.05)17] = 173,032

If the plan covers 10,000 similar employees, the total liability is approximately USD 1.730 billion ABO and USD 2.687 billion PBO If the market value of pension plan assets is USD 2.700 billion, then the plan

is overfunded by both measures of liability

As discussed in Exhibit 1, for type IV liabilities, we need effective duration to estimate interest rate sensitivity Effective duration is calculated using the following equation

Effective Duration = (𝑃𝑉−) − (𝑃𝑉+)

2 × ∆𝐶𝑢𝑟𝑣𝑒 × (𝑃𝑉0) Assuming a flat yield curve, we will estimate the effective duration assuming an increase in interest rate from 0.05 to 0.06 (and w from 0.045 to 0.054) and decrease in interest rate from 0.05 to 0.04 (and w from 0.045 to 0.036) Therefore, ΔCurve = 0.01 Given our assumptions, ABO0 is USD 173,032 Redoing the calculations for the higher and lower values for r and w gives USD 146,261 for ABO+ and USD

205,467 for ABO– The ABO effective duration is 17.1

ABO Duration = (205,467) − (146,261)

2 × 0.01 × (173,032) = 17.1 Repeating the calculations for the PBO liability measure gives USD 247,477 for PBO+ and USD 292,644 for PBO– Given that PBO0 is 268,714, the PBO duration is 8.4

PBO Duration = (292,644) − (247,477)

2 × 0.01 × (268,714) = 8.4

 ABO BPV = USD 1.730 billion × 17.1 × 0.0001 = USD 2,958,300

 PBO BPV USD 2.687 billion × 8.4 × 0.0001 = USD 2,257,080

 At present, the surplus = 2.700 billion – 2.687 billion = USD 13 million If yields on high-quality corporate bonds that are used to discount the projected benefits drop by about 5 bps to 6 bps, then this surplus would disappear

Given these assumptions, we conclude that the asset BPV = USD 2.700 billion × [(0.50 × 0) + (0.40 × 5.5) + (0.10 × 0)] × 0.0001 = =USD 594,000 This represents a duration gap because the asset BPV of USD 594,000 is much lower than the liability BPV of USD 2,257,080, using the PBO measure

 The term in brackets is the estimated effective duration for the asset portfolio, calculated using the shares of market value as the weights

Managing Duration Gap: The duration gap can be reduced or even eliminated using a derivatives

overlay For example, the pension plan would need to buy, or go long, 17,343 contracts to fully hedge the interest rate risk created by the duration gap as expressed below

2,257,080 − 594,000

If Futures BPV is USD 95.8909 per contract and 10-year Treasury yields go up by 5 bps, then the pension plan would realize a loss of USD 95.8909 × 5 × 17,343 = 8,315,179 At the same time, present value of liabilities will also decrease by the same amount

This implies that a large position in futures contracts may lead to significant daily cash inflows and outflows Therefore, for hedging duration gap, it is preferred to use over-the-counter interest rate swaps rather than exchange-traded futures contracts

Interest rate swap example: Suppose that the pension fund manager can enter a 30-year, receive-fixed,

interest rate swap against three-month Libor The fixed rate on the swap is 4.16% Its effective duration

is +16.73, and its BPV is +0.1673 per USD 100 of notional principal Exhibit 15 illustrates this swap From the pension fund’s perspective, the swap is viewed as buying a 30-year, 4.16% fixed-rate bond from the swap dealer and financing that purchase by issuing a 30-year floating-rate note (FRN) that pays three-month Libor

Swap’s duration = (high) duration of the fixed-rate bond - (low) duration of the FRN  reflecting a

receive-fixed swap has positive duration

The notional principal (NP) on the interest rate swap needed to close the duration gap to zero can be calculated with this expression:

Asset BPV + [NP × Swap BPV

100 ] = Liability BPV Given that the Asset BPV is USD 594,000 and the Liability BPV is USD 2,257,080 using the PBO measure, the required notional principal for the receive-fixed swap having a BPV of 0.1673 is as follows:

594,000 + [NP × 0.1673

100 ] = 2,257,080, ==> NP = 994,070,532

Hedging ratio and Strategic hedging:

Hedging ratio is the percentage of the duration gap that is reduced using the derivatives

 0% hedging ratio represents no hedging

 100% hedging ratio indicates fully hedged position

Practically, partial hedges are preferred, that is, between 0% and 100% For example, a manager may be allowed to have a hedging ratio of 25% to 75%

 If market (swap) rates are expected to decrease and manager anticipates gains on receive-fixed

interest rate swaps, the manager may increase the hedging ratio to the higher end of the range

 On the other hand, if market (swap) rates are expected to increase, the manager could reduce the hedging ratio to the lower end of the range

Option-based derivatives overlay strategy: It is another overlay strategy that can be used by pension

plan manager to manage duration gap risk For example, a pension fund can purchase an option to enter

a similar receive-fixed swap This contract is called a receiver swaption

Suppose that the strike rate on the swaption is 3.50% Given that the current 30-year swap fixed rate is assumed to be 4.16%, this receiver swaption is out of the money Suppose that the swaption premium is

100 bps Given a notional principal of USD 497 million, the pension plan pays USD 4.97 million (= USD

497 million × 0.0100) up front to buy the swaption

When the expiration date arrives, the plan exercises the swaption if 30-year swap rates are below 3.50% If 30-year swap rates are equal to or above 3.50% at expiration, the plan lets the swaption expire

Swaption collar: Swaption collar is another derivatives overlay strategy In swaption collar, the manager

buys the same receiver swaption, but instead of paying the premium of USD 4.97 million in cash, the plan writes a payer swaption

Suppose that a strike rate of 5.00% on the payer swaption generates an upfront premium of 100 bps

 If 30-year swap rates are below 3.50% at expiration, the purchased receiver swaption is in the money and the option is exercised

 If the swap rate is between 3.50% and 5.00%, both swaptions are out of the money

 If the swap rate exceeds 5.00%, the payer swaption is in the money to the counterparty

As the writer of the contract, the pension plan is obligated to receive a fixed rate of only 5.00% when the going market rate is higher

Exhibit 16 illustrates the payoffs on the three derivatives and the breakeven rates that facilitate the choice of contract

 If the plan manager expects the swap rate to be at or below 4.16%, the receive-fixed swap is preferred Its gains are higher than the other two derivative overlays

 If the manager expects the swap rate to be above 4.16%, the swaption collar is attractive because the swap would be incurring a loss

 At some point above 5.00%, the purchased receiver swaption is better because it limits the loss

In summary, 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 should be used

 If rates are expected to go up, the swaption collar is preferred

 If rates are anticipated to reach a certain threshold that depends on the option costs and the strike rates, the purchased receiver swaption can be used

Refer to Example 7 from the curriculum

This section addresses LO.e:

LO e explain risks associated with managing a portfolio against a liability structure;

6 Risks in Liability-Driven Investing

The full interest rate hedging can be expressed as follows:

Asset BPV x ∆ Asset yields + Hedge BPV x ∆ Hedge yields ≈ Liability BPV x ∆ Liability yields

Where, ∆ Asset yields, ∆ Hedge yields, and ∆ Liability yields are measured in basis points

In the above equation, it is assumed that all yields change by the same number of basis points—that is, ΔAsset Yields, ΔHedge Yields, and ΔLiability Yields are equal Non-parallel shifts as well as twists to the yield curve can result in changes to the cash flow yield on the immunizing portfolio that do not match the change in the yield on the zero-coupon bond that provides perfect immunization

There are some risks associated with liability-driven investing as discussed below

 Model risks: Model risks arise in LDI strategies because assumptions in models and approximations turn out to be inaccurate For example,

 the liability BPV for the defined benefit pension plan depends on the choice of measure (ABO or PBO) and the assumptions used in the model regarding future events (e.g., wage levels, time of retirement, and time of death)

 we use weighted average of the individual durations for the component bonds to

approximate the asset portfolio duration

 Spread risk: Spread risk: Derivatives overlay LDI strategies have spread risk which arises due to movements in the corporate–Treasury yield spread Similarly, interest rate swap overlays also have spread risk (i.e between high-quality bond yields and swap rates) Typically, the volatility of

corporate/swap spread is less than that of the corporate/Treasury spread

 Counterparty credit risk: Uncollateralized interest rates swap overlays have counterparty credit risk

 Risk associated with collateralization: Collateralization on derivatives used in an LDI strategy has

another risk that available collateral may become exhausted

 Risk associated with collateralization: Collateralization on derivatives used in an LDI strategy

introduces a new risk factor—the risk that available collateral becomes exhausted

 Liquidity risk: Asset liquidity is also a concern particularly in active investment strategies which

involve higher tunrover

Refer to Example 8 from the curriculum

This section addresses LO.f:

LO.f discuss bond indexes and the challenges of managing a fixed-income portfolio to mimic the characteristics of a bond index;

7 Bond Indexes and the Challenges of Matching a Fixed-Income Portfolio to an Index

Index-based investment is an investment strategy which involves selecting a broad bond market index to gain broad exposure to the fixed-income universe Index-based investments provide investors greater diversification benefits along with lower administrative fees

In index-based investment, the investment strategy’s success is measured based on the how closely the chosen market portfolio mirrors the return of the underlying bond market index Deviation of returns on the selected portfolio from bond market index returns are referred to as tracking risk or tracking error

In general, fixed-income markets are difficult to track due to the following challenges:

 Size and breadth of bond markets: Fixed income markets are much larger and broader than equity

markets As a result, it is quite challenging and costly to pursue a full replication approach with a broad fixed-income market index

 Unique issuance and trading patterns of bonds: The new debt issuance and the maturity of

outstanding bonds results in frequent changes in the fixed-income index The fixed-income indexes are also affected by changes in ratings and bond callability Due to such frequent changes, the rebalancing of bond market indexes usually occurs monthly rather than semi-annually or annually as

is the case for equity indexes As a result, the manager incurs greater transaction costs associated with maintaining a bond portfolio consistent with the index

 Over-the-counter trading: Unlike equity securities, which trade primarily over an exchange,

fixed-income markets are largely over-the-counter markets that rely on broker/dealers as principals to trade in these securities using a quote-based execution process rather than the order-based trading systems common in equity markets

 Challenging valuations: Fixed-income instruments that are not actively traded do not have an

observable price and therefore, involve pricing and valuation challenges The value of such securities can be estimated using matrix pricing or evaluated pricing In matrix pricing, we use recent

transaction prices of comparable bonds (i.e bonds which have similar features such as credit

quality, time to maturity, and coupon rate) to estimate the market discount rate or required rate of return on less frequently traded bonds

Primary Indexing Risk Factors

Since constructing a bond portfolio that fully replicates a benchmark index is challenging and involves greater costs, we can create a portfolio that matches the following primary risk factors of the

benchmark index By matching the primary risk factors, the portfolio is affected by broad market-moving events (e.g., changing interest rate levels, twists in the yield curve, spread changes) to the same degree

as the index The goal of matching these primary indexing risk factors is to minimize tracking error

Primary Indexing Risk Factors

a) Portfolio modified adjusted duration: Effective duration, or the sensitivity of a bond’s price to a

change in a benchmark yield curve, is an important primary factor to mimic index’s exposure to interest rate changes For securities with embedded call option, it is important to consider the option-adjusted duration

b) Key rate duration: The key rate duration reflects the rate changes in a specific maturity along the

yield curve while holding the remaining rates constant This risk factor can be used to mimic index’s sensitivity to non-parallel yield curve shifts

c) Percent in sector and quality: This involves matching the match the percentage weight in the

various sectors and qualities of the index

d) Sector and quality spread duration contribution: sector contribution refers to achieving the same

duration exposure to each sector as the index while quality spread duration refers to matching the amount of the index duration that comes from the various quality categories

e) Sector/coupon/maturity cell weights: This involves matching the sector, coupon, and maturity

weights of callable bonds by sector

f) Issuer exposure: This involves matching the event risk or issuer specific for a single issuer in the

index

g) Present value of distribution of cash flows methodology: This involves matching the portfolio’s

present value distribution of cash flows to that of the index Matching the percentage of the present value of cash flows (both coupons and redemptions) at certain time intervals with that of the

benchmark index helps in preventing the tracking error associated with yield curve twists The following steps are used to calculate the present value distribution of cash flows:

i The present value of the cash benchmark index for specific periods i.e every 6 month period is computed

ii Then each present value is divided by the present value of total cash flows from the benchmark

to determine the percentage of the index’s total market value attributable to cash flows falling

in each period e.g 5% of the index cash flows fall in the first 6-month period), 5.2% in the second period and so on

iii The time period is then multiplied by the present value of each cash flow Because each cash flow represents an effective zero-coupon payment in the corresponding period, the time period reflects the duration of each cash flow For example, the very first period has a duration of 0.5 The next time period has a duration of 1.0; the next 1.5, and so forth The duration of each period is multiplied by its weight to calculate the contribution of ea portfolio duration For example, the contribution of the first 6-month period is calculated as 0.05(0.5) = 0.025 The contribution of the second period is 0.052(1.0) = 0.052, and so forth

iv The duration contribution for each of the period is divided by the index duration (i.e., the sum of all the periods’ duration contributions) The resulting distribution is the benchmark’s PVD For example, suppose total index duration is 5.3 then PVD distribution is:

Period 1: 0.025 / 5.3 = 0.0047