Fixed Income Flashcards
Conversation Ratio
Bond face value / bond conversion price
Measurement of credit risk
Difference between yield to maturity on a corporate bond and a government bond with the same maturity
This is known as the credit spread and as G-spread
Considers both default probability and expected loss given default
Expected exposure to default loss
Projected amount of money the investor could lose if default occurs before factoring in possible recovery - i.e. PV of bond
Recovery rate
Percentage of the loss recovered from a bond in default
Loss given default (LGD)
Amount of loss if a default occurs
Notional principle x (1 - recovery rate)
(1 - recovery rate) = also known as loss severity
RIsk-neutral Probability of default (a.k.a. Hazard rate)
Probability that a bond issuer will not meet its contractual obligations on schedule
Risk-neutral probability of default should be used to value corporate bonds (i.e. using risk-free interest rate)
P(bond) = [Par value + coupon * prob of survival] + (value recovered given default * risk-neutral prob of default) / 1+ RFR
Difference between using actual vs. risk-neutral default probabilities
Actual default probabilities do not include the default risk premium associated with uncertainty over the timing of possible default loss
The observed spread of the yield on a risk-free bond includes liquidity and tax considerations in addition to credit risk
Credit valuation adjustment
Value of the credit risk in present value terms. Allows us to calculate the fair value of the bond
8-Steps to Calculate FV of a bond adjusted for credit risk
- Exposure = Par Value / (1+RFR)^(T-t)
- Recovery = Exposure x Recovery Rate
- LGD = Exposure - Recovery
- POD = Hazard rate - POS(t-1)
- POS = (100% - Hazard Rate)^(T-t)
- Expected Loss = LGD x POD
- Discount Factor = 1 / (1+RFR)^(T-t)
- PV of expected loss - Expected loss x DF
- CVA = sum of PV of expected loss
Bond Duration
Measures the sensitivity of the bonds full price (including accrued interest) to SMALL changes in the bond’s YTM or benchmark rates
Modified Duration - used only for option-free bonds; assumes bond’s expected cash flows do not change when yield changes
Effective Duration - used for straight and embedded option bonds
Change in P = -D x (change in R / 1 + R)
Effective Duration
Indicates the sensitivity of the bond’s price to a 100 bps PARALLEL shift of the benchmark yield curve (govt. par curve)
= (PV_) - (PV+) / [2 * (change in rate) x (PV of bond)]
Effective Duration (Callable Bonds)
Effective duration of a callable bond cannot exceed that of a straight bond
If R > Coupon rate, call option is out of the money
Effect of an interest rate change on price of a callable bond is similar to otherwise identical option-free bond
When interest rate decreases, price increases and call option is in the money; issue will limit the price appreciation by retiring the bond and therefore, call option reduces the effective duration relative to straight bond
Effective Duration (Putable Bond)
Effective duration of a putable bond cannot exceed that of straight bond
If R < Coupon R, put option is OTM; effective duration of put is similar to straight bond
Put option reduces the effective duration of the putable bond relative to straight bond
Credit Spread Migration
Typically reduces the expected return for 2 reasons:
- Probabilities for change are not symmetrically distributed around the current rating. They are skewed toward a downgrade rather than upgrade
- Increase in credit spread is much larger for downgrades than the decrease in the spread for upgrades
Convexity
Measures the sensitivity of a bond’s duration to large changes in interest rates and long holding periods
Bond’s duration + Convexity =
-D x (change in rate / 1 + rate) + [C x change in rate^2 / (1+rate)^2] / 2
Market conversion premium ratio
allows investors to identify the premium or discount payable when buying the convertible bond rather than the underlying common stock
(Market conversion price / share price of common share) - 1
Change of control price
Price at which the bond holder can convert the bond if the firm is merged with or acquired by another firm and is unaffected by a cash dividend to shareholders
Convertible bond
Hybrid security that presents the characteristics of an option-free bond and an embedded conversion option
The conversion option is a call option on the issuer’s common stock
Conversion ratio
The number of shares of common stock that the bondholder receives from converting the bonds into shares
Conversion value
This value of a convertible bond indicates the value of the bond if it is converted at the market price of the shares
Underlying share price * conversion ratio
Market conversion price
Represents the price that investors effectively pay for the underlying common stock if they buy the convertible bond and then convert it into shares
Convertible bond price / conversion ratio
Value of a convertible bond
Value of a straight bond + call option on issuer stock
Value of a callable bond
Value of straight bond - call option
Value of a callable convertible bond
Value of straight bond + value of call option on issuer stock - value of call option
Value of a callable putable convertible bond
Value of straight bond + value of call option on issuer stock - value of call option + value of investor put option
Busted Convertible
When underlying share price is well below the conversion price, it exhibits mostly bond risk-return characteristics
share price movements do not significantly affect the price of the call option or price of the convertible bond
When does convertible bond exhibit mostly stock risk-return characteristics?
When the underlying share price is above the conversion price
The call option on the issue is in the money and the price of the call option and convertible bond is significantly affected by share price movements but unaffected by interest rates
Interest rate volatility impact on a bond with an embedded call option
Call option increases in value with interest rate volatility (positive correlation)
As interest rate volatility increases, the value of the callable bond decreases because:
Callable bond = OFB - call option
Interest rate volatility impact on a bond with an embedded put option
The put option increases in value with interest rate volatility
As interest rate volatility increases, the value of the putable bond increases because:
Putable bond = OFB + put option
Effects of Yield Curve Level and Shape on Bonds with Embedded Call Options
Yield Curve Level:
As interest rates decline (increase), value of a call option increases (decreases) and the value of the callable bond decreases (increases)
Call option limits the upside potential for investor when rates decline
Yield Curve Shape:
As the yield curve moves from upward sloping -> flat -> Downward sloping, value of call option increases (more opportunity for issue to call back the bond to refinance at lower rate)
Effects of Yield Curve Level and Shape on Bonds with Embedded Put Options
Yield Curve Level:
Put option is considered a hedge against rising interest rates
Value of straight bond will decline as rates increase but will be partially offset by rising value of put option (sell back to issuer and lend at the higher rate)
Yield Curve Shape:
As the slope of the yield curve moves from upward sloping -> flat -> downward sloping, value of put option will decrease
Z-spread
A fixed spread that is estimated from the market prices of suitable bonds of similar credit quality
This is added to the forward rates derived from the default-free benchmark yield curve
Define Option Adjusted Spread (OAS)
A constant spread that is added to all the one-period forward rates on interest rate tree when valuing risky bonds with embedded options
This makes the arbitrage-free value of the bond equal to its market price
Often used a measure of value relative to the benchmark
What is the consideration of the bond’s value if the OAS on the bond is greater than the OAS on a bond with similar characteristics and credit quality?
The bond would be considered underpriced
A higher OAS results in a lower valuation for the bond because the purpose of the OAS in the model is to match the market price and
What is the effect of a decline in interest rate volatility on OAS and a callable bond?
Decline in volatility of interest rate means that the value of the call option will decrease and the value of the callable bond will increase (OFB-call option)
The higher valuation of bond will require a higher OAS because less adjustment is required to match the market price of the bond
Define One-sided durations
They are better at capturing the interest rate sensitivity of a callable or putable bond than two-sided effective duration, particularly when the embedded option is near the money
What does one-side durations explain about sensitivity of bonds with embedded options to interest rate changes?
Callable bonds are more sensitive to interest rate increases than decreases
When rates are declining, a call option is ITM and there is limited upside potential because the issuer will recall the bond. This results in a lower one-sided down duration
When rates are increasing, call option is OTM and there is no downside protection. This results in a higher one-sided up-duration
Define Key Durations (aka partial durations)
Reflects the sensitivity of a bond’s price to changes in specific maturities on the benchmark yield curve
Helps to identify shaping risk - i.e. whether the yield curve is steepening or flattening
How to assess the fair value for the bond under provided assumptions
- Determine the value for the corporate bond assuming no default
- Calculate the credit value adjustment
FV of bond =
VND - CVA
How to determine Value no Default (VND)
Option 1:
Using the binomial tree
Option 2:
Sum (coupon x benchmark discount factors for each year) + (principal + coupon * DF at maturity)
How to calculate the expected return on a bond that is expected to have its credit rating notched?
[-Duration * (new - old spread)] + annual coupon (if applicable)
Explain Structural model of Credit Risk and model components
When a company defaults (value of assets < liabilities), the probability of default has features of an option
Equity = call option purchases on the assets of the company
Strike price = face value of debt
Assumes assets on which the options are written are actively traded
Aims to explain WHY default occurs
Best used for internal risk management by managers of the company
Explain reduced form model and its components
Default is an external variable that occurs randomly
Aims to explain WHEN default occurs statistically
Key parameter is default intensity, which is the probability of default over the next time increment
Should be used to value risky debt securities and credit derivatives
Advantage of Structural Model
Provides insight into the nature of credit risk
Disadvantages of Structural Model
Burdensome to implement
Difficult to determine default barrier due to limitations in available data
Assets of company typically do not trade in market
List the advantages of reduced-form model
Inputs are observable variables and includes historical data
Default intensity can be estimated using regression analysis on company-specific and macroeconomic variables
Allows the model to reflect the business cycle in the credit risk measure
List the disadvantages of reduced-form model
Doesn’t explain the economic reason(s) for default
Model assumes default comes at a surprise and can occur at any time, which is not realistic
