Spot rate is the rate of interest on a security that makes a single payment at a future point in time
The annualized return (also can be interpreted as the yield to maturity) on an option-free and default-risk-free zero-coupon bond
Also known as zero-coupon rate or zero rate
Spot Curve
Spot curve: the term structure of spot rates, the graph of the spot rate versus maturity
The shape and level of the spot yield curve are dynamic
Discount Factor
It is the price of a risk-free single-unit payment at time j
Forward Rate
Forward rate is the rate of interest set today for a single-payment security to be issued at a future date
f(j, k): annualized k-year forward rate starting at time j
Discount Factor
The price at time j years from today for a zero-coupon bond with maturity (j+k) years and unit principal
Forward Rate Curve
The term structure of forward rates, the graph of the forward rate versus maturity
It represents the term structure of forward rates for a loan made on a specific initiation date
Forward Rate Models
Forward rate models show how forward rates can be extrapolated from spot rates
Forward Pricing Model
The forward pricing model describes the valuation of forward contracts
Spot Curve vs. Forward Curve
If the spot curve is upward sloping, forward curves lie above the spot curve
The later the initiation date, the higher the forward curve
If the spot curve is downward sloping, forward curves lie below the spot curve
The later the initiation date, the lower the forward curves
Yield Curve Shapes
In developed markets, yield curves are most commonly upward sloping with diminishing marginal increases in yield for identical changes in maturity
An inverted yield curve is uncommon and such term structure may reflect a market expectation of declining future inflation rates
A flat yield curve typically occurs briefly in the transition from an upward-sloping to a downward-sloping yield curve, or vice versa.
A humped驼峰 yield curve, which is relatively rare, occurs when intermediate-term interest rates are higher than short- and long-term rates
Par Rate
Par Rate and Par Curve
Par rate is the coupon rate for bond that would result in bond price equals to its par value
Par curve represents the yields to maturity/coupon rate/par rate on coupon-paying government bonds, priced at par, over a range of maturities
Recently issued("on the run") bonds are typically used to create the par curve because new issues are typically priced at or close to par
Bootstrapping
Spot rates may be obtained from the par curve by bootstrapping
YTM
YTM is the internal rate of return of on the cash flow, it reflects the implied single market discount rate
For coupon bonds, if the spot curve is not flat, the YTM will not be the same as the spot rate
YTM is the same as the spot rate for zero-coupon bonds
YTM is some weighted average of spot rates
Expected Return vs. Realized Return
Expected return is the ex-ante return that a bondholder expects to earn
The YTM is the expected return and will be realized only if all the three critical assumptions for YTM are met
Realized return is the actual return on the bond during the time an investor holds the bond
It is based on actual reinvestment rates and the yield curve at the end of the holding period
The YTM can provide a poor estimate of expected return if:
interest rates are volatile
the yield curve is steeply sloped, either upward or downward
there is significant risk of default
the bond has one or more embedded options (e.g., put, call, or conversion)
Implicit in the determination of the yield to maturity as a potentially realistic estimate of expected return is a flat yield curve
Swap Rate
Swap rate is the interest rate for the fixed-rate leg of an interest rate swap
Determining swap rate
The right side is the value of the floating leg, which is 1 at origination
The swap rate is determined by equating the value of the fixed leg, on the left-hand side to the value of the floating rate
Swap Curve
The yield curve of swap rates is called the swap rate curve (swap curve)
Because it is based on so-called par swaps, the swap curve is a type of par curve
Swap Curves vs Government Spot Curves
The swap market has more maturities
Many countries do not have a liquid government bond market with maturities longer than one year
The swap market is a highly liquid market
Unlike bonds, a swap does not have multiple borrowers or lenders, only counterparties who exchange cash flows
Swaps provide one of the most efficient ways to hedge interest rate risk
Swap rates reflect the credit risk of commercial banks rather than governments
Swap market is unregulated (not controlled by governments),so swap rates are more comparable across different countries
Spread
Swap Spread
Swap spread is the amount by which the swap rate exceeds the rate of the "on-the-run" government security with the same maturity
Swap spread = swap rate - government security rate
Swap spread shows the Treasury rate can differ from the swap rate for the same term
Unlike the cash flows from US Treasury bonds, the cash flows from swaps are subject to much higher default risk
Market liquidity for any specific maturity may differ: some parts of the term structure of interest rates may be more actively treaded with swaps than with Treasury bonds
I- SpreadZ
I-spreads are the bond rates net of the swap rates of the same maturities
I-spread = bond rate - swap rate
I-spread reflects compensation for credit and liquidity risks
TED Spread
The TED spread is calculated as the difference between Libor and the yield on a T-bill of matching maturity
TED = LIBOR - T-bill rate
TED spread is an indicator of perceived credit risk in the general economy
TED spread can also be thought of as a measure of counterparty risk
TED spread more accurately reflects risk in the banking system
An increase in the TED spread is a sign that lenders believe the risk of default on interbank loans is increasing
Libor-OIS Spread
Libor-OIS spread is the difference between Libor and the overnight indexed swap(OIS) rate
OIS is an interest rate swap in which the periodic floating rate is equal to the geometric average of an overnight index rate over every day of the payment period
The Libor-OlS spread is considered an indicator of the risk and liquidity of money market securities
Z-Spread
Z-spread is the constant spread that would need to be added to the implied spot curve so that the discounted cash flows of a bond are equal to its current market price
Implied assumption: the yield curve makes a parallel shift
Under the assumption of zero interest rate volatility, it is not appropriate for bonds with embedded options
Term Structure Theories
Pure Expectation Theory/Unbiased Expectation Theory
Pure expectation theory suggests that the forward rate is an unbiased predictor of the future spot rate
If yield curve is upward sloping, short-term rates are expected to rise
If yield curve is downward sloping, short-term rates are expected to fall
If yield curve is flat, short-term rates are expected to remain constant
Its interpretation is that bonds of any maturity are perfect substitutes for one another
The predictions of the unbiased expectations theory are consistent with the assumption of risk neutrality
This assumption is a significant shortcoming because investors are risk averse
Local Expectation Theory
Local expectation theory assumes risk-neutrality only in the short term while incorporate uncertainty in the long term
Under this theory, the expected return for every bond over short time periods is the same
But it is often observed that short-holding-period returns on long-dated bonds do exceed those on short-dated bonds
The need for liquidity and the ability to hedge risk essentially ensure that the demand for short-term securities will exceed that for long-term securities
Liquidity Preference Theory
Liquidity preference theory asserts that liquidity premiums exist to compensate investors for the added interest rate risk they face when lending long term Forward rate = expected from spot rates + liquidity premium
Liquidity premium increase with maturity
Segmented Markets Theory
Under segmented markets theory, each maturity sector can be thought of as a segmented market in which yield is determined by supply of and demand for loan, and independent from the yields in other maturity segments
Consistent with a world where there are asset-liability management constraints
Yields are not a reflection of expected spot rates or liquidity premiums
Preferred Habitat Theory
Preferred habitat theory contends that if the premium is large enough, investor will deviate from their preferred maturities or habitats
Premium is not directly related to maturity
Based on the realistic notion that investors will accept additional risk in return for additional expected returns
Yield Curve Factor Models
Yield Curve Movement
The yield curve movements can be decomposed into
parallel movement (\Delta X_L)
steepness movement (\Delta X_S)
curvature movements (\Delta X_C)
Parallel shift(level) → effective duration
Changes in Level rates 77%
Non-parallel shift → key rate duration
Steepness: Slope changes 17%
Curvature: Curvature changes 3%
Yield Curve Risk
Yield curve risk (Shaping risk) is the risk to portfolio value arising from unanticipated changes in the yield curve
Effective duration: measures the sensitivity of a bond's price to a small parallel shift in a benchmark yield curve Address risk associated with parallel yield curve changes
key rate duration: measures a bond's sensitivity to a small change in a benchmark yield curve at a specific maturity segment Allows identification and management of "shaping risk"
Decompose the Risk
The proportional change in portfolio value resulted from yield curve movement can be modeled as:
\frac{\Delta P}{P} \approx-D_L \Delta X_L-D_S \Delta X_S-D_C \Delta X_C
D_L, D_S, and D_C as the sensitivities of portfolio value to small changes in the level, steepness, and curvature, respectively
