Fixed Income Flashcards
Spot rates / zero-coupon rates
Effective annual rates which pay no interim interest. Geometric averages of forward rates. Not fully observable but required to fully disocunt the cashflows of any bond.
Spot curve
Annualized return on a risk-free zero coupon bond with a single payment of principal at maturity
YTM
Annual return an investor would achieve by investing in a particular coupon-paying bond to maturity and reinvesting all cash flows at the yield itself. Weighted average of spot rates.
YTM will only be expected return if:
- Bond is held to maturity 2. Cash flows are made in full and on time 3. All cash flows are reinvested at the original YTM
Forward rates
Spot rates starting in the future. They can be derived from Spot rates.
Price of a bond is
PV of all cash flows discounted at each cash flow’s zero coupon rate aka spot rate
Bootstrapping a yield curve
In reality we cannot observe zero rates directly. We need to extract them from bond yields = bootstrapping. We generally look at par yield curves where there is no tax distortion.
On-the-run bonds
Most recently issued! Similar to par yield curves (Y=C // P=Par).
Par Yield Curve
A par yield curve is a graphical representation of the yields of hypothetical Treasury securities with prices at par. On the par yield curve, the coupon rate will equal the yield to maturity (YTM) of the security, which is why the Treasury bond will trade at par.
Valuing a bond / discounting the cash flows with YTM when
Coupon = YTM (par) yield
Valuing a bond / discounting the cash flows with Spot rates and forward rates when
If coupon doesn’t equal YTM (par) yield
Forwards haben keine Potenzen
Maturity / Tenor
refers to the length of time remaining before a financial contract expires
Riding the Yield Curve
Purchase long-term bonds with a maturity date longer than their investment time horizon. Sell at the end of their time horizon, profiting from the declining yield that occurs over the life of the bond (profits from the higher six-month yield altough he is only holding 3 months)
Works best in a stable interest rate environment where interest rates are not increasing. Additionally, the strategy only produces excess gains when longer-term interest rates are higher than shorter-term rates (upward sloping yield curve)
Swap
Derivative contract through which two parties exchange the cash flows or liabilities from two different financial instruments
Like all derivatives the value of an interest rate swap at inception is zero
Swap curve
Is the yield curve of swap rates
Why use a swap curve as a benchmark?
- Liquidity – if the swap market is more liquid than the government bond market
- It contains more maturities than the government spot curve
- If a bank or client of a bank uses swaps to hedge interest rate risk then it makessense (for hedging purposes) to value their assets and liabilities using the swap curve
Par yield
The yield on a bond priced at par
(Par) Swap Rate
Swap spread
Is defined as the spread paid by the fixed-rate payer of an interest rate swap over the rate of the on-the-run (most recently issued) government security with the same maturity as the swap
It It represents the extra return above the equivalent equivalentrisk -free return which compensates the investor for additional time value, credit and liquidity risk. It is conceptually equivalent to a Z-spread.
The Z-spread, ZSPRD
Spread over the default-free spot curve
The static spread required to be added to each implied government spot rate such that the present value of cash flows equals the bond price. Reflects compensation for credit, liquidity and option risk. Reflects compensation for credit, liquidity and option risk.
TED spread
Difference between the interest rates on interbank loans and on short-term U.S. government debt (“T-bills”). TED is an acronym formed from T-Bill and ED, the ticker symbol for the Eurodollar futures contract.
Indicator of perceived credit risk in the general economy. A widening of the TED spread indicates higher concerns about the wider wider economy (T-Bills Flight to safety > Prices up > Yields go down = ED Yields go up as people sell)
Credit (G) spread
A credit spread is the difference in yield between a U.S. Treasury bond and another debt security of the same maturity but different credit quality
G-spread (also called nominal spread) is the difference between yield on Treasury Bonds and yield on corporate bonds of same maturity
- Compensation to the investor for bearing the default risk of the issuer ( investors wants more RETURN)
- Considers the possibility that the issuer fails to make a scheduled payment in full on the due date // losses incurred in the event of default
LIBOR-OIS Spread
The spread between LIBOR for a given maturity (typically three months) and the
Overnight Indexed Swap (OIS) rate for the equivalent maturity.
A widening of the spread indicates higher concerns about banking default risk
and/or liquidity concerns in the money markets.
OIS in GBP: SONIA; OIS in EUR: EONIA; OIS in USD: FFER
Option adjusted spread (OAS)
[Option removed spread]
constant interest rate spread that must be added to all of the one period forward rates in a binomial tree so that the theoretical value of the callable bond is the same as the market price. The spread you would receive if the bond had no embedded option. Best thought of as ‘option removed spread
Reflects compensation for credit and liquidity risk (but not option risk)
decline in interest rate volatility corresponds with a higher OAS
Flight to safety
Investors buy bonds (safer investments) when they sell stocks (higher-risk investments) in order to reduce the losses that investors suffer in crises periods
Demand for T-Bills goes up, Prices go up, Yields go down
Pure expectations theory
Forwards perfectly reflect future zero rates
Assumes that the term structure of an interest contract only depends on the shorter term segments for determining the pricing and interest rate of longer maturities. Yields at higher maturities (such as that of 5,10, or 30 year bonds), correspond exactly to future realized rates, and are compounded from the yields on shorter maturities. In other words, buying a ten year bond is equal to buying two five year bonds in succession; you’re as safe in a ten-year as in a five-year bond
Local Expectations Theory
Suggests that the returns of bonds with different maturities should be the same over the short-term investment horizon
Liquidity Preference Theory
Higher duration bonds require a highe premium.
Suggests that an investor should demand a higher interest rate or premium on securities with long-term maturities that carry greater risk because, all other factors being equal, investors prefer cash or other highly liquid holdings
Market segmentation theory
Expectations theory predicts future short-term interest rates based on current long-term interest rates. The theory suggests that an investor earns the same amount of interest by investing in two consecutive one-year bond investments versus investing in one two-year bond today
MST: Yield curve is driven entirely by investor preferences (Demand up, Prices up, Yield down). States there is no relationship between the markets for bonds with different maturity lengths and that interest rates affect the supply and demand of bonds
The preferred habitat theory
Suggesting that different bond investors prefer one maturity length over another and are only willing to buy bonds outside of their maturity preference if a risk premium (term premium) for the maturity range is available (other than MST). The theory also suggests that when all else is equal, investors prefer to hold short-term bonds in place of long-term bonds and that the yields on longer-term bonds should be higher than shorter-term bonds.
Equilibrium Term Structure Models
Stochastic interest rate models used to estimate the correct theoretical term structure. Estimate the stochastic process that describes the dynamics of the yield curve (term structure).
Cox-Ingersoll-Ross Model (CIR) & Vasicek Model in CFA L2. Both model short term interest rates.
Cox-Ingersoll-Ross model (CIR)
Assumes current consumption and investment (delayed consumption) decisions are made subject to a capital constraint, i.e. individuals need to consider the trade-off between consuming now versus in the future. The short short-term interest rate rewards the individual for the risk of investing
Two key parts : Drift term incorporates a mean-reverting element towards the long-term rate; stochastic term incorporates the volatility of the interest rate process (vs. Vasicek with volatility constant)
Vasicek model
- Very similar to CIR model in that it includes the same drift element to incorporate the mean reverting property of the short short-term interest rate
- The stochastic term is simplified compared with the CIR model since it holds volatility constant
Ho-Lee Model (Arbitrage-Free Models)
Starts with observed market prices (unlike Cox/ Vasicek)
We use market prices to calibrate the model, i.e. given an assumed drift term and
assumed volatility we can generate a yield curve such that we can then derive the
observed prices. The underlying model usually uses a binomial approach and an assumption of risk risk-neutral probabilities
Factors for return for a bond holder
- Changes in the level of interest rates, i.e. parallel shifts in yield curve (Most significant factor, almost 77%; measured by duration)
- Changes in the slope of yield curve (steeper/ flater?; measured by key rate duration, 17%)
- Changes in curvature (Umgedrehtes U? very little effect, about 3%)
% are commonly referred to as R R2 (coefficient of multiple determination)
Interest rate volatility
Short-term volatility is predominantly driven by uncertainty regarding central bank policy whereas longer term volatility is mostly driven by macroeconomic uncertainty such as economic growth and inflation (never constant!)
USUALLY: ST Vola > LT Vola
The Arbitrage Free Valuation Framework
The fundamental assumption is that prices are set such that no (riskless) arbitrage opportunities exist
A central conclusion of this approach is that all assets of the same risk, have the same return: Law of One Price
To find the arbitrage free value we need to discount each cash flow using the relevant spot rate and then sum (if yield = coupon tarding at par you can use YTM to discount).
Arbitrage Opportunities:
1. Value additivity: the sum of the part should equal the whole. A bond can be considered as a portfolio of zero coupon bonds. The sum of the present values of each zero, discounted using an appropriate spot rate should equal the value of the bond.
2. Dominace: We can view an asset as being dominant to another asset when for the same risk we are receiving a higher
return.
Formula for calculating upper forward rate (Binominal interest rate tree).
Alternatively apply only one SD to the average forward rate (xfxM) (one + and one -)
For year two: 2 SD from middle (4SD from higher to lower)
Pathwise Valuation
Calculating PV of all paths cashflows (8 paths possible in a 3 year tree) and add up all PVs you get s_ame value as the backward induction_ binomial approach
Monte-Carlo Model
Primarily used for interest rate path dependent cash flows such as Mortgage-backed securities (MBS)
Use a computer driven Monte Carlo simulation to generate thousands of random interest rate paths, i.e. monthly forward Treasury rates.
No detail in CFA2 on MBS
Callable bonds
Straight bond that includes a call option held by the issuer. Issuer has right to buy (redeem early) the bond at a given price (often par), which may be at par or a premium to par.
Main motivation behind exercising the call, is to refinance the loan , i.e. buy back a high coupon bond and replace with a lower coupon bond. This normally happens when interest rates fall.
Most callables include a lock out period (hard protection; can’t be called in in first 2 years).
Price of Callable Bond will be lower => higher yield: YSB+YOption= YCallable (compensation)
Make whole provisions generally make it uneconomic for the issuer to refinance the loan (ie higher spreads). Not a call provision!!
Tax exempt municipal bonds (Muni’s) are typically callable at 100% (par),
American style but with a 10 year lock out period.
Formulas / Ratios : Convertible Bond
Probability of Default (POD);
Probability of Survival (POS)
Structural credit and reduced-form credit models
Analyzing the credit risk of a fixed coupon bond using a binomial valuation approach
- Build the binomial interest rate tree (Bootstrapping to get Spot rates > determine Forward Rates?; We then apply an interest volatility assumption around the forward curve [FM xeSD*Vol]
- Confirm the binomial tree is arbitrage-free by pricing a government bond using the backward induction process [equates to $100 which is indeed the price of the 2-year government bond (Coupon = 2 year YTM)]
- Now consider a 2-year, annual payment corporate bond (higher coupon) and calculate its value assuming no default (VND)
- Now calculate the credit valuation adjustment (CVA) on the Corp. Bond.
- Use the CVA to work out the fair value of the bond and the credit spread [The fair value of the bond is the VND less the CVA > solve for YTM given new price > calculate Credit spread: YTMCB - YTMGovB
- Discoutn margin (calc?) > Quoted margin credit situation has worsenend
CDS premium
- CDS premium is only paid when a default event occurs
- This simplified formula assumes that default can occur only on premium payment dates, i.e. we use discrete default probabilities
- For CDS standard contracts with a set premium then rather than solving for CDS we can insert the standard amount. In this case any difference between the two legs will be the upfront payment
