Fixed Income Flashcards
Fixed Income Spot Rates
are the yields on zero-coupon bonds
No coupons –> no reinvestment risk
Zero Coupon Bond Price
Pt = 1 / (1 + St)^T
YTM
Price today: Calculate the PV of the cash flows
Yield: Put everything into the calculator and solve
Example:
Calculate the price and YTM on a three-year, 4% annual-pay, $1,000 face. S1 = 5%, S2 = 6%, S3 = 7%
Price = 40 / (1.05) + 40 / (1.06)² + 1040 / (1.07)^3 = $922.64
Then use calculator
N = 3, PV = -922.64, PMT = 40, FV = 1,000 CPT I/Y = 6.94%
Return will be equal to the bond’s yield only when:
- Held to Maturity
- All payments are made on time
- All coupons are reinvested at the original YTM (least realistic since the YTM changes)
Forward Pricing Model (zero-coupon)
Based on arbitrage-free pricing; basically says that the same timeframe yields the same
Step One: Calculate discount factors for each
1 / (1 + r)^t
Step Two: Calculate Forward Price
F = Discount(long) / Discount(short)
Fixed Income: Implied Forward Rate
[(Rlong * t) - (Rshort * t)] / timplied
Example: S2 = 4%, S5 = 6%, calculate 3 year implied
Long: 6 * 5 = 30 Short: 4 * 2 = 8
30 - 8 = 22
22 / 3 = 7.33
Riding the Yield Curve
Purpose: Purchasing bonds with maturities longer than your investment horizon.
Think: I want a bond for 5 years, so I purchase a 30 year bond and then sell it in 5 years
Increases interest rate risk
AKA “rolling down the yield curve”:
What is a Swap Rate Curve?
Why do people use them?
Definition: One party makes a payment based on a fixed rate, the other makes a payment based on a floating rate
Formula: 1 = SFR / (1 + S1) + 1 / (1 + S1) Plug in for SFR
Used because:
- Reflects credit risk of banks (not governments)
- Not regulated (makes it easier to compare across countries)
- More maturity ranges
Swap Spread
Swap Spread = swap rate - treasury yield
I-spread
Definition: amount by which the yield on the risky bond exceeds the swap rate for the same maturity
Interpolation
Definition: finding a rate between 2 listed spot rates
Formula: Rshort + (Rlong - Rshort) * [(Xt - Tshort) / (Tlong - Tshort)]
Example: 2 year swap at 2.0% and 2.5 year swap at 2.3%
Calculate 2.2 year
.02 + (.023 - .02) * [(2.2 - 2.0) / (2.5 - 2.0)]
So = 2.12%
Z-spread
Definition: the spread over the entire spot rate curve. Should be added to each spot rate
Think: I need to add 50 bps to each spot to equal the risk for this particular bond
Cannot be used when a bond has options,
Used for corporate bonds and ABS
TED Spread
Formula: 3 Month LIBOR - 3 Month T Bill
- Seen as indication of the level of credit risk in the economy
Analysis: Higher = banks more likely to default on loans and lower liquidity
LIBOR-OIS Spread
OIS is the overnight indexed swap
Formula: LIBOR - OIS rate
LIBOR includes credit risk, OIS does not
Traditional Theories
- Unbiased (Pure) Expectations Theory
- Local Expectations Theory
- Liquidity (biased) Preference Theory
- Segmented Markets Theory
- Preferred Habitat Theory
Unbiased (Pure) Expectations Theory
- Investors expectations determine the shape of the yield curve
- Forward rates = expected future spot rates
- If yield curve is upward sloping, short-term rates are expected to rise
Local Expectations Theory
- Not every maturity strategy should have the same return
- Risk-neutrality is preserved for only short-term
Remember: only applies to short-term, does not work
Liquidity (biased) Preference Theory
- Forward rates reflect expectations plus a liquidity premium
a. If YC is flat, the liquidity premium would push it upward sloping - Longer-term bonds more sensitive to rate changes
Segmented Markets Theory
- Yield at each maturity is determined independently
(e.g. Banks have to play in short-term bonds
HF may like to play in the l/t bonds) - The supply and demand of each segment effects rates.
Preferred Habitat Theory
- Forward rates are expected future spot rates plus a premium
- Investors prefer a particular maturity but will jump to different segments
Modern Theory: Vasicek
- Vasicek Model: suggest that interest rates mean revert over time
a. Interest rates can become negative
b. Volatility does not increase as interest rates does
Duration
approximate change in value based on a 1% increase/decrease in interest rates
Effective Duration Definition
Approximate Percent Price Change Formula
Definition: measures the sensitivity of a bond’s price
Formula: -D x change in Yield
Example: Duration of 5.0 and change of .5 would be:
-2.5% change
Yield Curve shifts
In order of importance:
- Change in level: a parallel increase or decreases of interest rates
- Change in steepness: long maturity interest rates increases, short rate decrease
- Change in Curvature: short and long rates increase, intermediate rates don’t change
Important: S/t rates change more
Value additivity and dominance
Additivity: when the value of the whole differs from the sum of the parts
Dominance: when one asset trades at a lower price than another with identical characteristics
Stripping and Reconstitution
Stripping: If a bond is worth less than its component parts, one could purchase the bond, break it into a portfolio of strips then sell
Reconstitution: the opposite - buy all the strips and combing (reconstitute) into a full bond
Binomial Interest Rate Tree
Valuation is done backwards
Formula at each ending node:
(Par + Coupon) / (1 + r)
Formula for every other node:
(Previous Value Upper + Coupon) / (1 + r)
+
(Previous Value Lower + Coupon) / (1 + r)
Then divide by 2
Pathwise Valuation
Just go down each path:
Coupon / (1 + r) + Coupon / (1 + r)(1 + r2) + Coupon / (1 + r)(1 + r2)(1 + r3)
Types of Embedded Options
Callable Bonds: issuer can call back the bond
European style option: option can only be exercised on a single day
American style option: anytime after lockout period
Bermudan-style option: option can be exercised at fixed dates after lockout period
Putable bonds; allow the investor to put back the bond to the issuer
Valuation of Embedded options
Call rule: the issuer will call the bond if the price is OVER 100.
Put rule: the investor will call the bond if the price is LOWER than 100
Interest rate volatility
Straight bonds unaffected
Option Adjusted Spread (OAS)
Comparison to Z spread
Constant spread added to all rates in binomial tree so model price = market price
Accounts for credit and liquidity risk
Z spread = OAS spread it is a straight bond
Z spread > OAS spread callable bond
Z spread < OAS spread putable bond
Effective Duration
Duration of call or put bonds < Duration of straight bond
Duration zero = bond maturity
Duration fixed-rate < Bond maturity
Duration floater = time (years) to next reset
Effective convexity
Convexity
Straight Bond Positive
Putable Bond Positive
Callable Bond Negative
Key Rate Duration
Purpose: capture the interest rate sensitivity of a bond to changes in yields
- Captures shaping risk
- However the key rate will change if an embedded option is near the money
Formula: -Rate Duration * change in yield
When the underlying option is near money
Callable bonds will have a lower one-sided down-duration
Putable bonds will have higher one-side down duration
Market conversion price
Market price of bond/conversion ratio
Conversion value
market price of stock after conversion * conversion ratio
Minimum value of a convertible bond: Greater of conversion value and straight value
Market conversion premium
Market conversion premium: market conversion price - market price of stock
Market conversion premium ratio: market conversion premium / market price
Recovery rate
Loss given Default
Expected Loss
Recovery rate: % of money received upon default
Loss given default (%) = 100 - recovery rate
Expected Loss = probability of default * loss given default
Ordinal rankings
Purpose: categories of borrowers from highest to lowest risk
e. g. FICO Score
1. Does not explain degree of risk, just order
Credit Rating Types, Strengths/Weaknesses
Types:
- Standard & Poors: BBB and above is investment grade
- Moody’s: Baa3 and above is investment grade
Strengths
- Simple
- Stable over time (this reduces volatility)
Weaknesses
- Stability means reduced correlation with default
- Ratings do not adjust with the business cycle (i.e. the economy tanks)
- Conflicts of interest
Structural Model Assumptions
- Company assets are tradable considering a perfect world (arbitrage free, etc.)
- Rf and volatility are constant over time
- Simple balance sheet that includes zero-coupon debt
Structural Models Weaknesses
- Balance sheet cannot be modeled realistically using a single zero-coupon bond
- Company assets are not traded
- Assumes constant volatility and risk-free rate
Reduced Form Model Differences from Strucural
- No assumption on structure of balance sheet
- Volatility and risk-free rate vary
- Can use historical data
Calculate the PV of expected loss
Formula: Coupon * log^ minus time * rate
Example:
25 * e^ -.5 * 0.0112 = 24.86
25 = coupon
-.5 is 6 months
.0112 is the rate (1.12%)
Modern Theory: Cox-Ingersoll-Ross (CIR)
- Cox-Ingersoll-Ross (CIR): same as Vasicek BUT can’t be negative
a. Volatility IS related to interest rates
Modern Theory: Ho-Lee Model
- Ho-Lee Model
a. Must be calibrated using market prices
b. Can be used to price zero-coupon bonds
c. Produces a normal distribution of rates
Bootstrapping
Purpose: Build up method for interest rates
Formula: 100 = Coupon / (1 + r) + (Par + coupon) / (1 + x)²
Think: Solve for X to get the bond to 100
Calculate the forward rates based on spot rates
[(1 + Slong)^t / (1 + Sshort)^t] - 1
Calculate Swap Fixed Rates
Formula: (1 - P3) / (P1 + P2 + P3)
Busted Convertibles
embedded call option is so far out of the money that a small change in the price of common stock will have a minimum impact on the value of the convertible bonds.
Calculate Forward Interest Rate
[(1 + Rlong)^t / (1 + Rshort)^t] - 1
Example:
Forward rate starts in 2 years, and last 1 year
[1 + r(3)]³ / [1 + r(2)]² - 1
Convertible Bonds
- Conversion option is NOT interest-rate sensitive
- Minimum value of a convertible bond: great of conversion value and straight value
Stocks vs. Convertible Bonds
- Stock price lower –> CV bond outperforms
- Stock price higher –> CV underperforms stock
- Stock flat –> CV bond outperforms (b/c of coupons)
