Fixed-Income Portfolio Management, Liability-Driven and Index-Based Strategies Flashcards
Immunization
- Is the process of structuring and managing a fixed-income portfolio to minimize the variance in the realized rate of return and to lock in the cash flow yield (internal rate of return) on the portfolio
- The cash flow yield is not the weighted average of the yields to maturity on the bonds that constitute the portfolio
- Asset–liability management
- Liability-driven investing
- Asset-driven liabilities
- Asset–liability management strategies consider both assets and liabilities
- Liability-driven investing takes the liabilities as given and builds the asset portfolio in accordance with the interest rate risk characteristics of the liabilities
- Asset-driven liabilities take the assets as given and structures debt liabilities in accordance with the interest rate characteristics of the assets
Approaches to liability-based mandates
- Cash flow matching
- Duration matching
- Contingent immunization
- Horizon matching (also called combination matching)
- Derivative overlay
Liability-based approaches - key features
Total return approaches - key features
Zero replication
In reference to a zero-coupon bond, means that a liability has been immunized
Contingent immunization
Combines immunization with an active management approach when the asset portfolio’s value exceeds the present value of the liability portfolio
Horizon matching (also called combination matching)
Short-term liabilities are covered by a cash flow matching approach while long-term liabilities are covered by a duration matching approach
Symmetric cash flow matching
- A cash flow matching technique that allows cash flows occurring both before and after the liability date to be used to meet a liability
- Allows for the short-term borrowing of funds to satisfy a liability prior to the liability due date
Expected fixed-income return
E(R) ≈
- Yield income +
- Rolldown return +
- E(Change in price based on investor’s views of yields and yield spreads) −
- E(Credit losses) +
- E(Currency gains or losses)
Yield income (or Current yield)
= Annual coupon payment/Current bond price
Rolldown return
The expected change in price based on investor’s views of yields and yield spreads
- ∆Yield input (55 bps is to be input as 0.0055)
Rolling yield
= yield income + rolldown return
Leveraged portfolio return
- VE = value of the portfolio’s equity
- VB = borrowed funds
- rB = borrowing rate (cost of borrowing)
- rI = return on the invested funds (investment returns)
- rp = return on the levered portfolio
The futures leverage
REPO dollar interest
Rebate rate on securities lending
Convexity and duration with and without embedded options
- With embedded options ⇒ use effective duration and effective convexity
- Without embedded options ⇒ use modified duration and convexity
Types of fixed-income risks
- Non MBS securities
- Interest rate
- Yield curve
- Spread
- Credit
- Optionality
- MBS securities
- Sector
- Prepayment
- Convexity
Tracking risk (also called active risk or tracking error)
- The standard deviation of the portfolio’s active return
- Active return = Portfolio’s return – Benchmark index’s return
- Tracking risk = Standard deviation of the active returns
Total return
The rate of return that equates the future value of the bond’s cash flows with the full price of the bond. The total return takes into account all three sources of potential return: coupon income, reinvestment income, and change in price
The bond equivalent yield (BEY)
- The BEY is a calculation for restating semi-annual, quarterly or monthly discount bond or note yields into an annual yield
- Example: 7.5% on a BEY is 3.75% every six months
Effective duration definition
Measures the sensitivity of the price to a relatively small parallel shift in interest rates (interest rate risk)
Key rate duration definition (also called multifunctional or functional duration)
Measures the sensitivity of the price to nonparallel shift in interest rates. Takes into account rate changes in a specific maturity along the yield curve (yield curve risk)
Macaulay duration formula
- t = the number of days from the last coupon payment to the settlement date
- T = the number of days in the coupon period
- t/T = the fraction of the coupon period that has gone by since the last payment
- PMT = the coupon payment per period
- FV = the future value paid at maturity, or the par value of the bond
- r = the yield-to-maturity, or the market discount rate, per period
- N = the number of evenly spaced periods to maturity as of the beginning of the current period
Macaulay duration for a zero-coupon bond
Is equal to its maturity
Modified duration
= Macaulay duration / (1 + cash flow yield)
Effective duration formula
- PV0 is the initial value
- PV– is the new value after the yield curve is lowered by ΔCurve
- PV+ is the value after the yield curve is raised by ΔCurve
Dollar duration (also called money duration)
= price * modified duration
Spread duration
Refers to the change in a non-Treasury security’s price given a widening or narrowing of the spread compared with the benchmark
Basis Point Value (BPV)
Is a measure of money duration calculated by multiplying the money duration by 0.0001
Portfolio modified adjusted duration
Takes into account option-adjusted duration
Convexity relation to the Macaulay duration
Annualizing Macaulay duration, dipersion and convexity
- Macaulay duration ⇒ divide by the periodicity of the bond
- Dispersion and convexity ⇒ divide by the periodicity2 of the bond
Dispersion statistic
Dispersion is the weighted variance. It measures the extent to which the payments are spread out around the duration
Dispersion statistic individual cash flow contribution
- = (T - Macaulay duration)2 * weight
- T is the integer position of the cash flow (1, 2, 3, 4…)
Convexity statistic individual cash flow contribution
- = (N * (N + 1) * weight)
- N is the integer number of the cash flow (1, 2, 3, 4…)
Convexity from the sum of individual cash flow contribution
= [Σ of individual cash flow contribution to convexity / (1 + effective annual cash flow yield/n)n] / 4
- n = number of period per year
Futures BPV
- CTD = Cheapest-to-deliver
- CF = Conversion factor
- In interest futures markets that do not have a CTD security, the Futures BPV is simply the BPV of the deliverable bond
Number of futures required to rebalance the BPV of the portfolio
3 major types of spread
- Nominal spread ⇒ the spread of a bond or portfolio above the yield of a certain maturity Treasury
- Static spread or zero-volatility spread ⇒ the yield spread that must be added to each point of the implied spot yield curve to make the present value of a bond’s cash flows equal its current market price
- Option-adjusted spread (OAS) ⇒ the constant spread that, when added to all the one-period forward rates on the interest rate tree, makes the arbitrage-free value of the bond equal to its market price
Portfolio duration versus spread duration
- For a portfolio of non-Treasury securities, spread duration equals portfolio duration
- For a portfolio that includes both Treasury and non-Treasury securities, spread duration is different from the portfolio duration because the spread duration of Treasury securities is zero
Economic surplus of the portfolio
The market value of assets minus the present value of liabilities
Accumulated Benefits Obligation (ABO)
- G is the number of years worked
- m is a multiplier
- The term in brackets is the value of the Z-year annuity as of year T, and that sum is discounted back over T years to Time 0 (w0)
Projected Benefits Obligation (PBO)
- G is the number of years worked
- m is a multiplier
- WT = W0 × (1 + w)T
- w is the average annual wage growth rate for the employee’s remaining work life of T years
The notional principal (NP) on an interest rate swap needed to close the duration gap to zero
- When the Swap BVP is quoted per $100 of notional principal
- Barbell portfolio
- Bullet portfolio
- A portfolio made up of short and long maturities relative to the horizon date and interim coupon payments
- A portfolio made up of bond with maturities that are very close to the investment horizon
Swaption
- Option to enter a fixed-rate swap
- The receiver swaption receives the fixed rate
- The payer swaption pays the fixed rate
Laddered, bullet and barbell portfolio visual depiction
Payoffs on Received-Fixed Swap, Receiver Swaption, and Swaption Collar
- If the plan manager expects the swap rate to be at or below 4.16%, the receive-fixed swap is preferred
- If the manager expects the swap rate to be above 4.16% the swaption collar is preferred
- At some point above 5.00%, the purchased receiver swaption is better because it limits the loss. That breakeven rate can be found by trial-and-error search
Full interest rate hedging relation
Multiple liability immunization versus cash flow matching
- Multiple liability immunization and cash flow matching approaches do not have the same risks and costs
- Cash flow matching generally has less risk of not satisfying future liabilities
- Multiple liability immunization generally costs less
The conditions to immunize multiple liabilities
- The market value of assets is greater than or equal to the market value of the liabilities
- The assets BPV equals the liabilities BPV
- The dispersion of cash flows and the convexity of assets are greater than those of the liabilities
The conditions to immunize a single liability
- The immunization strategy is to match the portfolio Macaulay duration with the investment horizon
- The initial investment needs to match (or exceed) the present value of the liability
*The portfolio should minimize the convexity statistic
Type of products which have negative convexity
- Callable bonds
- Mortgage-backed securities
- High-coupon issues have less convexity than low-coupon issues
Duration matching when the market values of the assets and liabilities differ
Should match the money duration, in particular the BPV
Duration of an option
Dollar duration of a swap
Total Return Analysis
Analysis of the expected effect of a trade on the portfolio’s total return, given an interest rate forecast
Portfolio rate of return using borrowed funds
- k = borrowing rate
- rf = portfolio rate of return
Immunize a portfolio to nonparallel shift in the yield curve
Applying functional duration or key rate durations allows durations along the yield curve to match those of the liabilities. A nonparallel shift in the yield curve will affect assets and liabilities in an offsetting manner.
Cash flow matching rate of return requirements
Cash flow matching requires a relatively conservative rate-of-return assumption for short-term cash, and cash balances may occasionally be substantial
Hedging ratio
The percentage of the duration gap that is closed with the derivatives
- Basis
- Basis risk
- The difference between the cash price and the futures price
- The risk that the basis will change in an unpredictable way
Model risk
Arise in LDI strategies because of the many assumptions in the models and approximations used to measure key parameters
Spread risk
Arises in LDI strategies because it is common to assume equal changes in asset, liability, and hedging instrument yields
Counterparty credit risk
The risk that the counterparty defaults when time comes to meet its obligation
Collateral risk
The risk the the counterparty has exhausted its availble collateral and is unable to mark-to-market its position
Structural risk
- 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
Primary indexing risk factors
- Portfolio modified adjusted duration
- Key rate duration
- Percent in sector and quality
- Sector and quality spread duration contribution
- Sector/coupon/maturity cell weights
- Issuer exposure
Pure bond indexing (or full replication approach)
The pure bond indexing approach attempts to duplicate the index by owning all the bonds in the index in the same percentage as the index
Enhanced indexing by matching primary risk factors
This management style uses a sampling approach in an attempt to match the primary index risk factors and achieve a higher return than under full replication (also called stratified or cell approach)
Enhanced indexing by small risk factor mismatches
While matching duration (interest rate sensitivity), this style allows the manager to tilt the portfolio in favor of any of the other risk factors. The manager may try to marginally increase the return by pursuing relative value in certain sectors, quality, term structure, and so on
Alpha of fixed income managers
For long periods, when fund fees and expenses are factored in, the realized alpha of fixed-income managers has averaged close to zero with little evidence of persistence
Convexity in duration matching
- Higher convexity is better
- To immunize multiple liabilities, the convexity (and dispersion of cash flows) of the assets needs to be greater than the liabilities
Matrix pricing (also called evaluated pricing)
Matrix pricing makes use of observable liquid benchmark yields of similar maturity and duration as well as the benchmark spreads of bonds with comparable times to maturity, credit quality, and sector or security type in order to estimate the current market yield and price
Present value of distribution of cash flows methodology
An approach that seeks to approximate and match the yield curve risk of an index over discrete time periods referred to as cash flow vertices
Enhancement strategies seeking to reduce the component of tracking error associated with the expenses and transactions costs of portfolio
- Lower cost enhancements
- Issue selection enhancements
- Yield curve enhancements
- Sector/quality enhancements
- Call exposure enhancements
Total return swap (TRS) mechanic
Highest to lowest convexity for portfolio structure
- Barbell
- Laddered
- Bullet
Accounting defeasance (also called in-substance defeasance)
Is a way of extinguishing a debt obligation by setting aside sufficient high-quality securities to repay the liability
Classification of Liabilities
The “bums” problem
As a particular issuer or sector of the economy borrows more, investors tracking a value-weighted index will automatically increase their fixed-income exposure to these borrowers
Requirements for an appropriate benchmark portfolio
- Clear, transparent rules for security inclusion and weighting
- Investability
- Daily valuation
- Availability of past returns and turnover
Laddered portfolio advantages
- Offers diversification over the yield curve
- Provides liquidity as it always contains soon-to-mature bonds
Bond tender offer
A corporate finance term denoting the process of a firm retiring its debt by making an offer to its bondholders to repurchase a specific number of bonds at a specified price and specified time. Firms use these offers to refinance or restructure their current capital structure
Most relevant considerations for:
- Investment-grade bonds
- High-yield bonds
- Investment-grade bonds
- Credit migration (or credit downgrade) risk
- Spread risk
- Interest rate risk
- High-yield bonds
- Credit risk
Causes of liability noise
- Plan demographic experience differing from the actuary’s model even if the underlying probabilities were certain
- Model uncertainty—the fact that the underlying probabilities are not certain (e.g., mortality rate change due to medical innovations)
