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Term Structure & Interest Rate Dynamics

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Spot Rate
Exchange rate requiring delivery of the traded currency within two business days.
Forward Curve
When the spot curve is flat, forward rates will equal spot rates and yields. When the spot curve is upward sloping (downward sloping), forward rate curves will be above (below) the spot curve and…
Forward Pricing Model
The forward pricing model values forward contracts by using an arbitrage-free framework that equates buying a zero-coupon bond to entering into a forward contract to buy a zero-coupon bond in the future that matures at the same time:
Forward Rate Model
The forward rate model tells us that the investors will be indifferent between buying a long-maturity zero-coupon bond versus buying a shorter-maturity zero-coupon bond and reinvesting the principal at the locked in forward rate f(j,k)
One-Period Return
If spot rates evolve as predicted by forward rates, bonds of all maturities will realize a one-period return equal to the one-period spot rate and the forward price will remain unchanged.
Active Bond Portfolio Management
Active bond portfolio management is built on the presumption that the current forward curve may not accurately predict spot rates. Managers attempt to outperform the market by making predictions about how spot rates will ch…
Swap Rate Curve
The swap rate curve provides a benchmark measure of interest rates. It is similar to the yield curve except that the rates used represent the interest rates of the fixed-rate leg in an interest rate swap.
Swap Curve Differences
Swap curves differ from treasury curves due to differences in liquidity, credit exposures, and supply versus demand. Institutions like wholesale banks are familiar with swaps and thus often use swap curves (rather than other inte…
The Z-spread is the spread that when added to each spot rate on the yield curve makes the present value of a bond's cash flows equal to the bond's market price. The Z refer…
TED spreads
TED = T-bill + ED (the ticker symbol for the Eurodollar futures contract)
The LIBOR-OIS spread is the amount by which the LIBOR rate (which includes credit risk) exceeds the overnight indexed swap (OIS) rate (which includes only minimal credit risk). The LIBOR-OIS spread is a useful mea…
Unbiased Expectations Theory
Forward rates are an unbiased predictor of future spot rates. Also known as the pure expectations theory.
Local Expectations Theory
Bond maturity does not influence returns for short holding periods.
Liquidity Preference Theory
Investors demand a liquidity premium that is positively related to a bond's maturity.
Segmented Markets Theory
The shape of the yield curve is the result of the interactions of supply and demand for funds in different market (i.e. maturity) segments.
Preferred Habitat Theory
Similar to the segmented markets theory, but recognizes that market participants will deviate from their preferred maturity habitat if compensated adequately.
Modern Term Structure Model
Modern term structure models are used to predict the shape of the yield curve in order to value bonds and fixed-income derivatives.
Equilibrium Term Structure Models
Attempt to model the term structure using fundamental equilibrium variables that are thought to determine interest rates.
Cox-Ingersoll-Ross Model
Type of equilibrium term structure model. Assumes the economy has a natural long-run interest rate (b) that the short-term rate (r) converges to.
Vasicek Model
Similar to the CIR model, but assumes that interest rate volatility level is independent of the level of short-term interest rates. Main disadvantage is that the model does not force interest rates to be non-negative.
Arbitrage Free Models (Ho-Lee Model)
Begins with observed market prices and the assumption that securities are correctly priced.
Effective Duration
Measures the sensitivity of a bond's price to parallel shifts in the benchmark yield curve.
Key rate duration
Measures bond price sensitivity to a change in a specific spot rate keeping everything else constant.
Sensitivity to parallel, steepness, and curvature movements
Measures sensitivity to three distinct categories of changes in the shape of the benchmark yield curve.
Maturity Structure of Yield Volatility
The maturity structure of yield volatilities indicates the level of yield volatilities at different maturities. This term structure thus provides an indication of yield curve risk. The volatility term structure usually indicates that short-term r…
Spot Rates, define
annualized market interest rates for a single payment to be received in the future.
Forward Rate, define
Annualized interest rate on a loan to be initiated at a future period
Spot Rate, equation
Pt = 1 / (t+St)^T
the graph of the spot rate versus maturity.
Spot Yiel Curve or Spot Curve, what is it?
Forward rate, equation
F(j,k) = 1 / [1 + f(j,k)]^k
Yield to maturity, define
is the spot interest rate for a maturity of T. However, if the spot rate curve is not flat, the YTM will not be the same as the spot rate
When all three of these hold:
When will the expected return be equal to the bonds yield?
Forward Pricing Model, define
values forward contracts based on arbitrage-free pricing
What does F(j,k) mean
Forward price of a $1 par zero-coupon bond applicable on a k-year loan starting in j years.
Forward Pricing Model equation
F(j,k) = P(j,k) / Pj
So, how do you find Pj ?
Pj equals the price when the forward bond starts, so you discount the two year spot rate.
Finding Pj, example
assume S2 (two-year spot rate) is 4%,
Finding P(j+k) ?
exact same, but ow you are discounting the spot rate for j+k, so if it is a three year bond starting in two years, J=k = 5. If five-year spot - 6%,
F(j,k) = P(j+k) / {j
Then how do you find the F(j,k)?
Forward Rate Model, equation
[1 + S(j+k)] ^ (j+k) = [(1+Sj)^j] * [1+f(j,k)]^k
Forward Rate Model, theory
suggests that the forward rate f(2,3) should make investors indifferent b.n buying a 5-year zero coupon bond versus buying a two-year zero coupon bond and reinvesting the principal for three years when it matures
Forward Rate Model, example
given: S2 = 4%, s5 = 6%. Calc implied three-year forward rate for a loan starting two years from now (i.e. f(2,3)
Upward/downward-sloping spot curve: slope v j
Upward: forward rate rises as j increases
Riding the Yield Curve, define
Deployed in an upward-sloping yield curve environment. Investors purchase bonds with maturities longer than the investment horizon bc longer maturity bonds have higher yields. As the bond approaches maturity (ie rolls down the yield cur…
Plain vanilla interest rate swap
one party makes payments based on a fixed rate while the counterparty makes payments based on a floating rate
Fixed rate term in plain vanilla interest rate swap
Is called the swap fixed rate, or swap rate
swap rate curve, define
if we consider how swap rates vary for various maturities, we get the swap rate curve
Swap Fixed Rate, equation
Sum T, (t=1) (the sum E thing..) =
Construct the swap rate curve
SFR1: [SFR1/(1+1.03)] + [1/1.03)] = 1 --> SFR1 = 3%
Swap Spreads, define
refers to the amount by which the swap rate exceeds the yield of a gov bond with the same maturity
Swap Spread, equation
Swap Spread = Swap Rate - Treasury Yield
I Spread, define
for a credit risky bond, amount by which the yield on the risky bond exceeds the swap rate for the same maturity
I Spread equation
I Spread = Yield on the bond - swap rate
Z-spread, define
spread that, when added to each spot rate on the default-free spot curve, makes the PV of a bonds CFs equal to the bonds market price.
Zero Volatility in the Z Spread, define
Refers to the assumption of zero interest rate volatility. the Z spread is not appropriate to use to value bod with embedded options.
Z Spread, equation
Price = [Coupon pmt / (1+S1 +Z)] + [(princ + coupon) / (1+S2+Z)^2]
TED Spread, define
the amount by which the interest rate on loans b/n banks exceeds the interest rate on short-term us gov debt (three month libor v 3 month T-bills)
LIBOR OIS Spread, What does OIS stand for?
Overnight indexed swap. The OIS rate roughly reflects the fed funds rate and included minimal counterparty risk
LIBOR OIS Spread, define
amount by which the LIBOR rate exceeds the OIS rate (Libor typically includes credit risk, OIS includes minimal credit risk).
LIBOR OIS Spread, Use
useful measure of credit risk and an indication of the overall well-being o the banking system.
Unbiased Expectations Theory, or Pure Expectations Theory, define
we hypothesize that is the investors' expectations that determine the shape of the interest rate structure.
Unbiased Expectations Theory, or Pure Expectations Theory, in normal terms
basically long-term interest rates equal the mean of future expected short-term rates.
Upward Sloping: short-term rates are expected to rise
Unbiased Expectations Theory, or Pure Expectations Theory, meaning of yield curve slope
Local Expectations Theory, define
similar to unbiased expectations theory, main difference = the local expectations theory preserves the risk-neutrality assumption only for short holding periods.
Does the Local Expectations Theory hold?
No, because the short-holding-period returns of long-maturity bonds can be shown to be higher than short-holding period returns on short-maturity bonds due to liquidity premiums and hedging concerns
Liquidity Preference Theory, define
proposes that forward rates reflect investors expectations of future spot rates, plus a liquidity premium to compensate investors for exposure to interest rate risk.
Segmented Market Theory, define
Yields are not det by liq premium and expected spot rates. Rather, the shape of the yield curve is det by the preferences of borrowers and lenders, which drives the balance of supply and de…
Preferred Habitat Theory, define
proposes that forward rates represent expected future spot rates plus a premium, but it does not support the view that this premium is directly related to maturity
Preferred Habitat Theory, what is the premium tied to?
suggests that imbalance in supply and demand for funds with given maturity will induce lenders and borrowers to shift from their preferred habitat (maturity) to one that has the opposite imbalance - lenders wil…
Modern Term Structure Models, define
Attempt to capture the statistical properties of interest rates movements and provide us with quantitatively precise descriptions of how interest rates will change
Equilibrium Term Structure Models, define
attempt to describe changes in the term structure through the use of fundamental economic variables that drive interest rates.
Cox-Ingersoll-Ross (CIR) Model, define
based on the idea that interest rate movements are driven by individuals choosing between consumption today verses investing and consuming later
Cox-Ingersoll-Ross (CIR) Model, equation
dr=a(b−r)dt + σ * sqaure root ( rdz)
what does the a(b−r)dt term do?
forces the interest rate to mean-revert toward the long-run value, b, at a speed determined by the a parameter
Under the CIR model, volatility Inc/Dec with interest rate?
Increases, in other words, at hight interest rates, the amount of period-over-period fluctuation in rates is also high
The Vasicek Model, define
suggests that interest rates are mean reverting to some long-run value (just like the CIR model)
The Vasicek Model, equation
dr = a(b - r)dt + σdz
Vasicek Model v CIR model
Notice the Vasicek model does not have an r term in the second part. this means that volatility in the Vasicek model does not increase as the level of interest rates increase
Main disadvantage of the Vasicek Model
the model does not force interest rates to be non-negative
Arbitrage-Free Models, define
term structure of interest rates begin with the assumption that bonds trading in the market are correctly priced, and the model is calibrated to value such bonds consistent with the market price.
Ho-Lee Model, equation
dr(t) = θ(t)*dt + σ*dz(t)
Ho-Lee Model, assumption
that changes in the yield curve are consistent with a no-arbitrage condition
Ho-Lee Model, uses
price Zero-coupon bonds and to determine the spot curve.
Effective Duration, what does it measure
Measures price sensitivity to small parallel shift in the yield cuve.
Shaping Risk
(not measured by effective duration)
Key Rate Duration, what does it measure
more precise than effective duration bc superior for measuring the impact of nonparallel yield curve shifts
Level: Three categories of yield curve movement
Δx(l) : a parallel increase or decrease of interest rates
Steepness: Three categories of yield curve movement
Δx(s) : long-term interest rates increase while short term rates decrease
Curvature: Three categories of yield curve movement
Δx(c) : increasing curvature: means short and long terms interest rates increase while intermediate rates do not change
where D
ΔP / P ≈−Dl * Δx(l) −Ds * Δx(s) −Dc * Δx(c)
graph of yield volatility versus maturity
Term Structure of Interest Rate Volatility, define
P(T*+T) = P(T*)F(T*,T)
Forward Pricing Model (1)
Forward Price
F(T*,T)= 1/[1+f(T*,T)]^T
[1+r(T*+T)]^(T*+T)= [1+r(T*)]^T* [1+f(T*,T)]^T
Forward Pricing Model (2)
Spot Curve is Upward Sloping
Forward Curve will lie above Spot Curve
Spot Curve is Downward Sloping
Forward Curve will lie below Spot Curve
Par Curve
Represents yields to maturity on coupon paying government bonds, priced at par, over a range of maturities; can be used to construct zero-coupon curve (i.e. bootstrapping)
Forward Contract Price
F(T*,T) = P(T*+T)/P(T*)
P*(T) = P(t+T)/P(t)
Forward Discount Function after time t
F*(t,T*,T) = P*(T*+T-t)/P(T*-t)
Forward Contract Price at time t