Fixed Income Flashcards
3 key roles of fixed income in an investment portfolio
Diversification
Regular cash flows
Inflation protection (if floating or inflation linked)
Two major types of fixed income mandates
Liability based (invest to meet future liabilities)
Total Return based (invest to track or beat a benchmark)
3 types of Liability based mandates
Cash flow matching
Duration matching
Contingent Immunisation
Describe a cash flow matching investment strategy
Coupon and par amounts to be received on dates liabilities are due
Describe a duration matching strategy, and what is a benefit (vs cash flow matching)?
Matches asset and liability duration to achieve comparable results
More flexibility in asset selection, so can be done at a lower cost
Describe a contingent immunisation strategy
A hybrid of active management and potential immunisation
The portfolio is overfunded and actively managed
The surplus can grow and ultimate cost can end up being lower than from immunisation
3 types of total return mandates
Pure indexing - matches holdings of index exactly
Enhanced indexing - allows modest deviations, but must maintain the same duration
Active management - no restrictions
Define Macaulay duration
Weighted average time to receive cash flows
Define;
Modified duration
Effective duration
Key rate duration
Estimated % change change in bond price given a 1% change in yield [Macaulay duration/ (1+periodic yield)]
% change in bond price given a 1% change in a benchmark curve [used for bonds with embedded options]
% change in bond price given a 1% change in a key benchmark maturity yield [holding other yields constant]
What is empirical duration? And Money duration?
Actual sensitivity of a bond’s price relative to movements in a benchmark rate from a linear regression
Modified duration x market value [gives a sense of size]
Define price value of a basis point
money duration x 0.0001
Measures absolute currency sensitivity to a basis point shift in rates
Define convexity and effective convexity
Curvature of the relationship between price and yields. More convex bonds outperform less convex when yields shift
Effective convexity measures convexity when cash flows are uncertain [used for option embedded bonds]
How does Macaulay duration change with maturity?
Increases linearly with maturity
What is an alternative approximate expression for Convexity?
Duration squared
How is convexity broadly related to Macaulay duration
Directly related to dispersion of cash flows in time around Macaulay duration
How would you typically calculate aggregate duration and convexity measures for a fixed income portfolio?
Cash weighted average of durations and convexities of individual instruments
What is a bond’s spread duration?
Sensitivity of a bond’s price to a unit change in spreads
How is duration x spread calculated? What does it account for?
spread duration x credit spread
Bonds with larger spreads tend to have larger movements in spread
What is the play if a manager expects interest rates to fall?
And one who expects credit spreads to widen?
Increase duration
Lower spread duration
What is relative value analysis?
Ranking individual bonds according to fundamental value drivers in order to pick best securities to express a top-down view on markets
How does fixed income liquidity compare to that of equities?
Generally less liquid
How are bonds primarily traded on the secondary market?
OTC. With no key features of the trade publicly reported
Which types of bonds tend to be the most liquid?
- Sovereign/ Government issued
- Higher credit quality
- Recently issued (on the run)
Two impacts of bond market illiquidity
- Pricing data is difficult to obtain
- Derivatives/ ETFs tend to be more liquid and offer an alternative investment to bonds
Break fixed income return down into 5 components
- Coupon income
[Coupon amount/ current bond price] - Rolldown return (assuming no curve shift)
[(projected ending bond price (BP) - beginning bond price (BP))/ beginning bond price (BP)] - Price change due to investor yield change predictions
[(-MD x ΔY) + (1/2C x ΔY^2)] - Price change due to investor yield change predictions
[(-MD x ΔS) + (1/2C x ΔS^2)] - Currency G/L
[Projected change in value of foreign currencies weighted for exposure]
What is the rolling yield?
Coupon income + rolldown return
How can you express leveraged portfolio return?
rl + [(VB/ VE) x (rl - rB)]
{if rl > rB then the leverage enhances portfolio return}
What are three tools through which to leverage fixed income return?
- Repurchase contracts (securities lending)
- Futures contracts
- Swaps
What’s an additional risk of leverage (additional to lowered return)?
Lender can demand repayment, forcing asset liquidation
How are assets managed for an immunisation strategy?
Dedicated to the purpose of repaying defined liabilities, all cash flows reinvested for this purpose
What risks does matching of asset/ liability Macaulay duration balance?
Price and Reinvestment risk
With duration matching, what should Duration/ portfolio statistics be based on?
Express the steps to immunising a single period liability
Portfolio yield (IRR)
- Initial PVA = (or greater than) initial PVL
- Macaulay durations match
- Portfolio convexity matched
- Rebalance portfolio to maintain duration match
What are two primary immunisation issues to consider?
- Assets have greater convexity than single date liability, therefore while benefitting from curve shifts, the portfolio is at risk from curve twists
- Immunisation can be interpreted as ‘zero replication’, so if done properly will match the price/ yield path of a zero coupon bond which could have been used for perfect cash flow immunisation
How can you apply basis point value to duration matching?
- Initial PVA = or exceeds PVL
- BPVA = BPVL
- Asset dispersion of cash flows/ convexity exceeds those of liabilities (but not by too much)
- Regularly rebalance portfolio to maintain BPV match
How can derivatives be used to adjust BPV of assets (hedge duration gap)?
- Buying futures or receive fixed swaps increases asset duration and BPV
- Futures BPV ≈ BPVctd/ CFctd
- BPV = MD x V x 0.0001 [modified duration]
- Nf = (BPV liability - BPV current portfolio)/ BPV of futures [number of futures needed]
- NP for swap = (BPV liability - BPV current portfolio)/ BPV of 1 NP for the swap
- BPV of the swap is the difference in BPV between fixed and floating side
[NP-Notional Principal]
What does Contingent Immunisation require?
How would a successful strategy be defined?
Overfunded portfolio with a positive surplus (PVA > PVL), actively managed through surplus
Return will exceed initial immunisation rate, surplus will grow and cost of strategy will be less than immunising
4 features of laddered portfolios
- Useful in cash flow matching multiple liabilities
- Diversification across the yield curve, natural liquidity. Each is rolled into highest yielding maturity once it matures
- More convexity than bullet portfolio because cash flows are more distributed
- Could be constructed with a sequence of target-date ETFs as an alternative to individual bonds
Where BPVA < BPVL, what would a manager do when they expect rates to increase/ decrease?
- Increase: Reduce hedge size, leaving BPV of assets lower than that of a fully hedged duration gap
- Decrease: Increase hedge size, increasing BPV of assets above that of a fully hedged duration gap
How would you reduce a negative duration gap?
Increase BPV of assets
Comment on the 3 swap methods to reduce a negative duration gap
- Receive fixed swap: optimal if rates in future are below swap’s SFR
- Zero-cost collar: optimal if interest rates in future are moderately higher (between swap and payer swaption SFRs)
- Buying a receiver swaption: optimal if interest rates in future exceed payer swaption SFR by some amount
Give 5 risks in fixed income portfolio hedging/ matching
- Hedge amounts are approximate and based on assumed duration NOT convexity
- Twists in yield curve can create structural risk
- Futures base calculations are approximations based on Cheapest To Deliver (which can change)
- Measurement error is possible when weighted averages are used and not portfolio statistics
- Cash flow risk for exchange traded/ OTC derivatives require cash settling of gain/ loss on margin
What are 4 challenges with bond index funds vs equity equivalents?
- A much larger volume of bond issues exist. Full replication is impractical
- Liquidity is low for bonds
- Most individual bonds rarely trade, so pricing must be estimated
- Bond index composition and characteristics can change
What does enhanced indexing do?
Matches primary risk factors of the index
- Match modified duration (Effective if there are options)
- Match key rate durations
- Match weighting exposure to various bond sectors/ issuers
3 ways to achieve passive bond market exposure
- Separately managed account that replicates the index
- Index mutual funds (open ended or ETFs)
- Synthetic strategies (e.g. total return swap receiving bond index return)
What are two complicating factors in selecting a suitable bond index?
- Possible decline in duration as bonds age
- Changing characteristics of many indices over time as holdings change
How is yield curve curvature measured?
What is this known as?
- (short-term yield) + (2 x medium-term yield) - (long-term yield)
Butterfly spread
Give 5 Active management strategies when the upward sloping yield curve is expected to be stable
- Buy and hold (extend duration to increase yield)
- Roll down yield curve (weight portfolio highest for securities at long end of steepest section, to benefit most from time to maturity)
- Repo carry trade (buy long term bond using short term repo financing)
- Long futures (increase leverage of portfolio using futures contracts)
- Receive fixed swap (earn swap carry of swap fixed rate)
A manager expecting a downward parallel shift of yield curve should do what to the duration of the portfolio?
Increase it
How would you increase duration for each of the below?
- Cash bond
- Swap
- Futures
Cash bond: Overweight longer dated bonds
Swap: Receive fixed
Futures: Long contracts
How would you decrease duration for each of the below?
- Cash bond
- Swap
- Futures
Cash bond: Short sell bonds/ overweight shorter dated
Swap: Pay fixed
Futures: Short contracts
How should a manager react to
- a steepening curve
- a flattening curve
- Short sell long dated bonds and buy shorter dated bonds
- Buy long dated bonds and short sell short dated bonds
How should a manager ‘bullish’ on the yield level structure duration?
Should position the portfolio to have positive duration
What is the strategy for increasing curvature (negative butterfly twist) and decreasing curvature (positive butterfly twist)?
- Short sell bullet and buy barbell
- Buy bullet and short sell barbell
How does a callable bond react when yields fall? What does this exhibit?
It rises at a slower rate. This is negative convexity
How does a putable bond react when yields rise? What does this exhibit?
A putable bond will fall at a slower rate than an option free bond as yields rise. This is positive convexity
What is the impact on duration of the below strategies?
1. Long call on bond prices/ bond futures prices
2. Long put on bond prices/ bond futures prices
3. Long payer swaption
4. Long receiver swaption
- Increase duration
- Decrease duration
- Decrease duration
- Increase duration
What is the formula for key rate duration?
Key rate duration = Change in portfolio value / (portfolio value x change in key rate)
Express the domestic return of a foreign bond with a foreign currency return, RFC, and a return from the DC/FC exchange rate of RFX is as follows:
RDC = (1+RFC)(1+RFX) - 1
What does covered interest parity imply and what does this mean for an investor?
High interest rate currencies trade at a forward discount
Hedged foreign bond position cannot earn excess return, as interest rate excess is offset by currency discount
Assumes efficient markets
What does Uncovered Interest Parity theorise? What is notable about this and what does this allow?
High interest rate currencies should weaken over time
This tends not to hold, and allows unhedged bond managers to earn excess returns through the carry trade
Break down the two primary components of credit risk
Probability of Default
Loss Given Default (1 - Recovery Rate)
How would you calculate the CVA for a bond?
What is then its fair value?
Sum of POD x LGD x expected exposure across the life of the bond
The fair value of a credit risk free equivalent bond - CVA
How might you estimate credit spread for the next period?
Spread ≈ POD x LGD
Comment on the credit spread level at each of the following economic cycle stages
- Early Expansion
- Late Expansion
- Peak
- Contraction
- Stable
- Falling
- Rising
- Peak
Comment on the credit spread slope for both IG and HY at each of the following economic cycle stages
- Early Expansion (Recovery)
- Late Expansion
- Peak
- Contraction
- IG: Stable, HY: Inverted
- Upward sloping for both
- Upward sloping for both
- IG: Flat, HY: Inverted
What can be said of empirical vs analytical duration for low quality securities? Why?
Empirical duration will likely be lower. This is due to the impact of credit spreads
Describe or express each of
- g-spread
- i-spread
- asset swap spread
- zero-volatility spread
- CDS spread
- Bond’s YTM - interpolated YTM of the two adjacent maturity on-the-run government bonds
- Bond’s YTM - the maturity interpolated swap fixed rate
- Bond’s fixed coupon - the maturity interpolated swap fixed rate
- Bond’s spread over risk-free spot rates
- CDS basis = CDS spread - z-spread
[CDS spread is fair value of protection bought under a CDS contract as a periodic % of notional exposure]
Describe the Option Adjusted Spread
Bond’s spread over an interest rate tree of potential future risk-free forward rate paths
Used to compare option embedded bonds with vanilla
Describe Quoted and Discount Margin in the context of floating rate notes
What can be said if DM < QM?
QM: Fixed margin above a floating market reference rate (MRR) making up an FRN coupon
DM: Constant spread above current MRR offered by an FRN
If DM is less than QM then the FRN will trade above par
Give the effective rate duration and effective spread duration for a FRN
EffRateDurFRN = ((PV-) - (PV+))/ (2(ΔMRR)(PV0)
EffSpreadDurFRN = ((PV-) - (PV+))/ (2(ΔDM)(PV0))
What is the zero discount margin in the context of an FRN?
Constant spread over the current term structure of MRR rates offered by an FRN
Give the formulae for the effective spread duration and convexity for a fixed income portfolio
effective spread duration = ((PV-) - (PV+))/ (2(Δspread)(PV0)
effective spread convexity = ((PV-) - (PV+))/ (Δspread)^2(PV0)
What is an overarching formula for sensitivity of a bond portfolio to changes in spread?
%Δprice = (-EffSpreadDur x Δspread) + (1/2 x EffSpreadCon x Δspread^2)
Δspread is typically defined as the change in OAS
What is the expression for duration times spread?
What does this capture?
DTS ≈ EffSpreadDur x spread
Reflects the fact that spread changes tend to be proportional to spread size
How is the excess spread expressed? What does this capture?
expected excess spread = spread - (EffSpreadDur x Δspread) - (POD x LGD)
Excess return over credit losses expected on a bond portfolio
What two types of models are used to assess credit risk in bottom up analysis? Describe them
Structural models: look at probability of default as probability issuers’ assets fall below their liabilities
Reduced form models: Assess probability of default by modeling the relationship between macroeconomic variables and borrower characteristics
Give an example of a reduced form model and how would the ranges of outputs be interpreted?
Altman’s Z-score, maps key financial ratios to a z-score using a linear regression
> 3: Low chance of default
1.8 - 3: Some chance of default
< 1.8: Likely default
What is an example of a top-down management strategies set?
Factor-based strategy. Identifying factors within a portfolio which are rewarded with risk premia
(carry, defensive, momentum, value)
Regarding transaction costs, give the effective spread for a buy and sell order
Effective spread for a buy order = trade size x (trade price - midquote)
Effective spread for a sell order = trade size x (midquote - trade price)
Midquote = (bid + ask)/ 2
What does VaR measure?
Minimum expected loss occurring in a given time frame with a specified probability
3 methods to calculate VaR
- Parametric (using normal distribution
- Historical method
- Monte Carlo simulation
Describe 3 VaR extensions
- Conditional VaR. Expected loss given the fact that the portfolio is experiencing a loss in the tail
- Incremental VaR. Change in VaR from adding or removing a position in a portfolio
- Relative VaR. Measures VaR of a portfolio’s returns relative to a benchmark
What is the Upfront premium on CDS in the following 3 scenarios?
1. CDS spread = fixed coupon
2. CDS spread > fixed coupon
3. CDS spread < fixed coupon
- None
- [(CDS spread - fixed coupon) x EffSpreadDurCDS] paid to protection seller
- [(fixed coupon - CDS spread) x EffSpreadDurCDS] paid to protection buyer
How is CDS price (as a % of par) quoted?
CDS price ≈ 1 + [(fixed coupon - CDS spread) x EffSpreadDurCDS]
Describe a CDS long-short strategy
Buying protection on issuers where spreads are expected to widen
Selling protection on issuers where spreads expected to narrow
Describe a CDS curve trade
Buy protection at maturities where CDS spreads are expected to widen and sell protection at maturities where spreads are expected to narrow
Give two static strategies to be employed when the upward sloping credit spread curve is expected to be stable
- Lower credit quality of portfolio or extend spread duration
- Sell protection on lower quality issuers and sell protection over longer maturities
(Economic recovery), what is the cash and CDS strategy when
1. HY spreads narrow more than IG spreads
2. HY credit curve steepens
- Cash: Buy HY bonds. Sell IG bonds
CDS: Sell HY protection. Buy IG protection - Cash: Buy short-term HY bonds. Sell long-term HY bonds
CDS: Sell short term HY protection. Buy long term HY protection
(Economic slowdown), what is the cash and CDS strategy when
1. HY spreads widen more than IG spreads
2. HY credit curve flattens/ inverts
1.
Cash: Buy IG bonds. Sell HY bonds.
CDS: Sell IG protection. Buy HY protection
2.
Cash: Buy long-term HY bonds. Sell short-term HY bonds
CDS: Sell long-term HY protection. Buy short-term HY protection
What are some key considerations between different markets when executing a cross border management strategy
- reliance on a specific industry or commodity
- difference in accounting standards
- difference in credit cycles
Domestic return of a foreign credit investment expression
RDC = (1+RFC)(1+RFX) - 1
