New - Reading 42 - The Term Structure and Interest Rate Dynamics Flashcards
What does the notation P(T) represent?
The discount factor
What does the notation r(T) represent?
The spot rate
What is the equation to calculate the discount factor for a risk-free single unit payment?
What is a Forward Rate?
An interest rate that is determined today for a loan that will be initiated in a future time period.
What is the Forward Pricing Model and what is the equation to calculate it?
**using discount factors**
It describes the valuation of forward contracts
P(T*+T) = P(T*) x F(T*,T)
What is the no-arbitrage principle?
That tradable securities with identical cash flow payments must have the same price.
When the spot curve is upward(downward) sloppping, the forward curve will….
lie above(below) the spot curve
What does the Forward Rate Model tell us?
It shows how forward rates can be determined by using spot rates
What is the equation to use the Forward Rate Model?
**when using spot rates**
What does the notation f(7,1) mean?
The rate agreed to today for a 1 year loan starting in 7 years.
What are 2 ways Forward Rates can be viewed as?
- As a type of break even interest rate
- As a rate that can be locked in by extending maturity by 1 year
By re-arranging the Forward Rate Model, how can we calculate the forward rate?
What is the government par curve?
- Shows the yield to maturity on coupon paying gov’t bonds priced at par over a range of maturities
- It is important b/c it can be used to construct a zero coupon yield curve
Is the YTM the expected return of a bond?
NO
YTM = E(r) of a bond **only **if it is held to maturity, all coupon and principal payments are made _and _ all coupons can be invested at the original YTM
When can the YTM provide a poor estimate of the expected return?
**4 Reasons**
- Interest rates are volatile
- Yield curve is steeply sloped (can be upward or downward)
- There is significant risk of default
- The bond has one or more embedded option
How do active bond portfolio managers attempt to outperform the market?
By anticipating changes in interest rates relative to the projected spot rates reflected in today’s forward curve
What does it mean when a portfolio managers is “Rolling the Yield Curve” ??
**This is also known as “Riding the Yield Curve”
It is a trading strategy that involves buying bonds with a maturity longer than the intended investment horizon
In the forward rate model, what does T* stand for?
**Critical Concept**
The number of years until the loan begins
In the forward rate model, what does the term T stand for?
The length of the loan
What is a Swap Rate?
An interest rate for the fixed rate leg of an interest rate swap
What are 2 reasons why the Swap Market is so liquid?
- Significant flexibility & customization is available
- An efficient way to hedge interest rate risk
What is a Swap Spread?
- A popular way to indicate credit spreads in a market
- It equals the spread paid by the fixed rate payer of an interest rate swap over the rate of the “on the run” gov’t security with the same maturity as the swap
What is the Z-Spread?
The constant basis point spread that would need to be added to the implied spot yield curve so that the discounted cash flows are equal to the current market price
What is the **Ted Spread **?
- An indicator of perceived credit risk in the general economy
- = LIBOR - Tbill of same maturity
- An increase is a sign that the risk of default on interbank loans is increasing
What are the 4 different theories to explain the term structure of interest rates?
- Local Expectations Theory
- Liquidity Preference Theory
- Segmented Markets Theory
- Preferred Habitat Theory
Describe the Pure Expectations Theory?
That the forward rate is an unbiased predictor of the future spot rate
Describe the Local Expectations Theory
- Similar to the Pure Expectations Theory
- Says that the expected return for every bond over the short term is the risk free rate
**In reality this doesn’t hold perfectly. Short Holding Period Returns are often greater for long dated bonds than for shorter dated bonds.The need for liquidity and to hedge risk ensure that the demand for s/t bonds will exceed that for l/t bonds
Describe the Liquidity Preference Theory
Liquidity premiums exist to compensate investors for the added interest rate risk they face when lending long term and that these premiums increase with maturity
Describe the Segmented Markets Theory
- allows for lender and borrower preferences to influence the shape of the yield curve
- yields are solely a function of the supply and demand for funds of a particular maturity
Describe the Preferred Habitat Theory
- says that if the expected additional returns to be gained are large enough, institutions will be willing to deviate for their preferred maturities or habitats
- ie… accept additional risk for expected additional return
What are the 2 different types of Modern Term Structure Models?
- Equilibrium Term Structure Models
- Arbitrage Free Models
What are Equilibrium Term Structure Models?
Models that seek to describe the dynamics of the term structure using fundamental economic variables that are assumed to affect interest rates
What are the 2 best known Equilibrium models?
- Cox-Ingersoll-Ross Model (CIR)
- Vasicek Model
What are some characteristics that Equilibrium term structure models share?
- They are one factor or multi factor
- They make assumptions about the behavior of factors
- They are more sparing with respect to the # of parameters that must be estimated than other models
What is the single factor that the CIR and Vasicek models use?
The Short Term Interest Rate
Describe the CIR model….
It assumes that every individual has to make consumption and investment decisions with their limited capital
What is the equation for the CIR Model?
dr = a(b-r)dt+σ√rdz
**is an instance of a so called continuous time finance model
Breaks down into two parts
- Drift Term
- Random Part
a= speed adjust to the long run value of the rate
b = long run value of the short term rate
r = current short term rate
What is the equation for the Vasicek Model?
dr = a(b-r)dt+σdz
**disadvantage is that it is theoretically possible for the interest rates to become negative**
What is the reasoning for Arbitrage Free Models?
The underlying assumption is that the reference set is correctly priced b/c it uses observed market prices
What is the most famous Arbitrage Free Model?
Ho-Lee Model
Describe the Ho-Lee Model
- an arbitrage free model
- used the relative valuation concept of Black Scholes option pricing model
- assumes that the yield curve moves in a way that is consistent with a no arbitrage condition
What is the equation for the Ho-Lee model?
What is Shaping Risk?
The sensitivity of a bond’s price to the changing shape of the yield curve
What is a Yield Curve Factor Model?
a model for description of yield curve movements that can be considered realistic when compared with historical data
What are the 3 factors that describe yield curve movement?
- Level- upward or downward shift to the yield curve
- Steepness- a non-paralle shift in the yield curve
- Curvature- movements of short, middle and long term segments
What are some ways to measure yield curve sensitivity?
- Duration
- Key Rate Duration
A 2 year loan (T=2) beginning in 1 year (T* = 1).
The one year spot rate is r(T*)=r(1)= 7%
The three year spot rate is r(T*+T) = r(1+2) =r(3) = 9%
Calculate the 1 yr discount factor….
A 2 year loan (T=2) beginning in 1 year (T* = 1).
The one year spot rate is r(T*)=r(1)= 7%
The three year spot rate is r(T*+T) = r(1+2) =r(3) = 9%
Calculate the 3 yr discount factor…..
A 2 year loan (T=2) beginning in 1 year (T* = 1).
The one year spot rate is r(T*)=r(1)= 7%
The three year spot rate is r(T*+T) = r(1+2) =r(3) = 9%
Calculate the forward price of a 2 yr bond to be issued in 1 yr….
To calculate this we need to remember we already have the P(1) and P(3) discount factors.
P(1) = 0.9346
P(3) = 0.7722
Using the equation : P(T*+T) = P(T*) * F(T*,T)
So 0.7722 = 0.9346 * f(T*,T)
solve for f(T*,T) = 0.7722 / 0.9346
The spot rates for 3 hypothetical zero coupon bonds are below:
Maturity(T) 1 2 3
Spot Rate 9% 10% 11%
Calculate the forward rate for a one yr bond issued one yr from today….
So we are looking for f(1,1)
(1+0.10)2 = (1+0.09)1 (1+f(1,1))1
f(1,1) =[(1.1)2 / 1.09] -1
= 11.01%
Given the below data, what is the rate for a 1 Year loan beginning in 1 Year?
1 Yr Zero-Coupon Bond = 4.%
2 Yr Zero-Coupon Bond = 5%
3 Yr Zero-Coupon Bond = 6%
Using the Forward Rate Model were a solving the following:
[1+r(2)]2 = [1+r(1)]1 * [1+f(1,1)1
[1+.05]2 = [1+.04]1 *[1+f(1,1)]
Solve for f(1,1)
How do you convert spot rates to spot prices?
If the expected future spot rate is lower than the current forward rate, a bond is most likely:
A). overvalued
B). fully valued
C). undervalued
C.
When the expected spot rates are lower than the forward rate,the market is discounting the bond’s payments at a higher rate than the investor and the bond’s market price is lower than the intrinsic value perceived by the investor
