Concepts Description
The Term Structure and Interest Rate Dynamics
Spot rate / Forward rate / 1. Spot rate : annualised market interest rates for a single payments to be received in the future
Spot curve / Forward curve / 2. Forward rate : interest eate (as agreed to today) for a loan to be made in some future date
Yield to maturity / 3. Spot yield curve / Spot curve : Graph of Spot rate vs. Maturity
Expected returns / 4. Forward curve : Term structure of forward rates
Realised returns 5. Yield to maturity : Spot interest rate of a zero‐coupon bond with maturity T
6. Expected returns : holding period return that bond investor expect to earn. Expected return = Bond yield if:
‐ Bond is held to maturity
‐ Coupon and interest payments are made on time and in full
‐ All coupons are reinvested @ original YTM
7. Realised returns : actual return over the holding period
Forward pricing model Forward pricing model : value forward contracts based on arbitrage‐free pricing
,
in which:
, ,
Forward rate model
1 1 1 ,
Par rate Par rate = YTM of bond trading @ par
Par rate curve / Par curve : Par rates for bonds with different maturities
Bootstrapping : method for calculating spot rate from par rate
‐ Step 1 : 1‐year spot rate = 1‐year par rate
‐ Step 2 : Calculate 2‐year spot rate, using:
2 1 2 2 1 2
1 2 1 2 1 1
‐ Step 3 : calculate 3‐year spot rate, using 1‐year spot rate and 2‐year spot rate from above (similar to step 2)
Relationship between spot and ‐ Upward‐sloping spot curve : time increase → forward rate increase
forward rates ‐ Downward‐sloping spot curve : time increase → forward rate decrease
‐ Upward‐sloping spot curve : forward curve is above spot curve
‐ Downward‐sloping spot curve : forward curve is below spot curve
Forward price evolution ‐ Future spot rates involve as forecasted by forward curve → forward price remain unchanged.
‐ Change in forward price → future spot rate do not confirm to the forward curve
‐ Spot rate lower than implied by forward curve → increase forward price
‐ Spot rate higher than implied by forward curve → decrease forward price
Strategy of riding the yield curve 1. Maturity matching : purchase bonds with maturity = investment horizon
2. Riding the yield curve : purchase bonds with maturities > investment horizon, and sell @ the end of the investment horizon.
‐ At the end of the investment horizon, bond's price is calculated using lower yields → higher price
‐ If yield curve remains unchanged over the investment horizon → higher returns than maturity matching
‐ Possible risk : possibility of increase in spot rate
Swap rate curve Interest rate swap : 1 party make fixed rate payments , while the counterparty make floating rate payments
Swap rate : Fixed rate in an interest rate swap
Swap rate is more preferable than G‐bond yield curve, due to :
‐ Swap rate reflect the credit risk of commercial banks, rather than credit risk of governments
‐ Swap rate is not regulated by governments → more comparable in different countries
‐ Has yield quotes at many maturities, while G‐bond only has several maturities
Calculation of Swap fixed rate :
1
1
1 1
In which :
Swap spread / Swap spread :
I‐spread /
I‐spread :
‐ I‐spread reflect compensation for credit and liquidity risk. Higher I‐spread → higher compensa on
Z‐spread Z‐spread : spread, that when added to each spot rate on the spot rate, make PV of bond's CF = Bond's market price
⋯
1 1 1
‐ Assumptions of Z‐spread : zero interest rate volatility→ not appropriate to value bonds with embedded op ons
, TED spread TED spread = Interest rate on interbank loans ‐ Interest rate on ST US government debt
‐ Indication of risk of interbank loans
‐ Higher TED spread → banks are more likely default on loans ; T‐bills are more valuable
LIBOR‐OIS spread OIS : overnight indexed swap
OIS rate reflects the federal funds rate, include minimal counterparty risk
LIBOR‐OIS spread = LIBOR rate ‐ OIS rate
LIBOR‐OIS rate measure thecredit risk, amd indicate the overall wellbeing of banking system
Low LIBOR‐OIS → high liquidity market
High LIBOR‐OIS → banks are unwilling to lend, due to creditworthness concerns
Traditional theories of term 1. Unbiased expectations theory / Pure expectations theory : Investors' expectations determine the shape of interest rate term structure
structure of interest rates ‐ Forward rate = expected future spot rates
‐ Investors don't demand a risk premium for maturity strategies that differ from their investment horizon
‐ Implications for yield curve shape :
+ Upward sloping yield curve → ST rates are expected to rise
+ Downward sloping yield curve → ST rates are expected to fall
+ Flat yield curve → ST rates are expected to remain constant
2. Liquidity preference theory : Forward rates = Investors' expectations of future spot rates + Liquidity premium fo exposure to interest rate risk
‐ Liquidity premium : positively related to maturity
‐ Implications for yield curve shape :
+ Upward sloping yield curve → (1) future interest rate is expected to rise; or (2) rates are expected to remain, and upward sloping curve is solely due to liquidity premium
+ Downward sloping yield curve → steep fall in ST rate
3. Segmented markets theory : yield at each maturity is determined independently, by the preferences of borrower and lenders, which drive the balance between demand and supply
for loans at each maturities
4. Preferred habitat theory : forward rate = expected future spot rate + premium
‐ Imbalance between supply/demand for funds in a given maturity rate → investors switch from preferred habitats (maturity range) to range with opposite imbalance
‐ Investors must be compensated for price and/or reinvestment rate risk in the less‐than‐preferred habitat (borrowers require lower yield; lenders require higher yield)
Modern models of term structure 1. Equilibrium term structure models : describe changes in term structure through fundamental economic variables that drive interest rates
of interest rates a. Cox‐Ingersoll‐Ross model : interest rate movements are drivien by choosing between consumption today vs. investing and consuming later
∆ ∆ ∆
b. Vasicek model : interest rate should revert to some long‐run value
∆ ∆ ∆
‐ Disadvantage : Vasicek model does not force interest rate to be non‐negative
2. Arbitrage‐free model : assume that bonds trading in the market are correctly priced @ market price (bond value should equal its market price)
‐ Ho‐Lee model : assume changes in yield curve are consistent with no‐arbitrage condition
∆ ∆ ∆ ∆
→
∆
∆
Managing yield curve risks Yield curve risk : risk to the value of a bond portfolio, due to unexpected changes in yield curve
Counter yield curve risk by : identify portfolio's sensitivity to yield curve changes
‐ Effective duration : price sensitivity to small parallel shifts in yield curve (not accurately measure price sensitivity to non‐parallel shift)
+ Non‐parallel shift → shaping risk (change in por olio value due to change in shape of the benchmark curve)
‐ Key rate duration : sensitivity of portfolio value to change in a single par rate, holding all other spot rates constant
∆
∆
∆
‐ Sensitivity to Parallel, Steepness and Curvature movements : decomposing yield curve risk into sensitivity to changes at various maturities :
+ Level : parallel increase / decrease of interest rates
+ Steepness : LT interest rates increase + ST interest rates decrease
+ Curvature : Increase curvature → ST and LT interest rates increases ; intermediate rate do not change
∆ ∆
∆ ∆ ∆
∆
∆
Maturity structure of yield curve Interest rate volatility drives price volatility of fixed income portfolio, especially securities with embedded options
volatilities ST interest rates are more volatile than LT interest rates
LT interest rate : associated with uncertainty in real economy and inflation
ST interest rate : associated with risks in monetary policy
The Term Structure and Interest Rate Dynamics
Spot rate / Forward rate / 1. Spot rate : annualised market interest rates for a single payments to be received in the future
Spot curve / Forward curve / 2. Forward rate : interest eate (as agreed to today) for a loan to be made in some future date
Yield to maturity / 3. Spot yield curve / Spot curve : Graph of Spot rate vs. Maturity
Expected returns / 4. Forward curve : Term structure of forward rates
Realised returns 5. Yield to maturity : Spot interest rate of a zero‐coupon bond with maturity T
6. Expected returns : holding period return that bond investor expect to earn. Expected return = Bond yield if:
‐ Bond is held to maturity
‐ Coupon and interest payments are made on time and in full
‐ All coupons are reinvested @ original YTM
7. Realised returns : actual return over the holding period
Forward pricing model Forward pricing model : value forward contracts based on arbitrage‐free pricing
,
in which:
, ,
Forward rate model
1 1 1 ,
Par rate Par rate = YTM of bond trading @ par
Par rate curve / Par curve : Par rates for bonds with different maturities
Bootstrapping : method for calculating spot rate from par rate
‐ Step 1 : 1‐year spot rate = 1‐year par rate
‐ Step 2 : Calculate 2‐year spot rate, using:
2 1 2 2 1 2
1 2 1 2 1 1
‐ Step 3 : calculate 3‐year spot rate, using 1‐year spot rate and 2‐year spot rate from above (similar to step 2)
Relationship between spot and ‐ Upward‐sloping spot curve : time increase → forward rate increase
forward rates ‐ Downward‐sloping spot curve : time increase → forward rate decrease
‐ Upward‐sloping spot curve : forward curve is above spot curve
‐ Downward‐sloping spot curve : forward curve is below spot curve
Forward price evolution ‐ Future spot rates involve as forecasted by forward curve → forward price remain unchanged.
‐ Change in forward price → future spot rate do not confirm to the forward curve
‐ Spot rate lower than implied by forward curve → increase forward price
‐ Spot rate higher than implied by forward curve → decrease forward price
Strategy of riding the yield curve 1. Maturity matching : purchase bonds with maturity = investment horizon
2. Riding the yield curve : purchase bonds with maturities > investment horizon, and sell @ the end of the investment horizon.
‐ At the end of the investment horizon, bond's price is calculated using lower yields → higher price
‐ If yield curve remains unchanged over the investment horizon → higher returns than maturity matching
‐ Possible risk : possibility of increase in spot rate
Swap rate curve Interest rate swap : 1 party make fixed rate payments , while the counterparty make floating rate payments
Swap rate : Fixed rate in an interest rate swap
Swap rate is more preferable than G‐bond yield curve, due to :
‐ Swap rate reflect the credit risk of commercial banks, rather than credit risk of governments
‐ Swap rate is not regulated by governments → more comparable in different countries
‐ Has yield quotes at many maturities, while G‐bond only has several maturities
Calculation of Swap fixed rate :
1
1
1 1
In which :
Swap spread / Swap spread :
I‐spread /
I‐spread :
‐ I‐spread reflect compensation for credit and liquidity risk. Higher I‐spread → higher compensa on
Z‐spread Z‐spread : spread, that when added to each spot rate on the spot rate, make PV of bond's CF = Bond's market price
⋯
1 1 1
‐ Assumptions of Z‐spread : zero interest rate volatility→ not appropriate to value bonds with embedded op ons
, TED spread TED spread = Interest rate on interbank loans ‐ Interest rate on ST US government debt
‐ Indication of risk of interbank loans
‐ Higher TED spread → banks are more likely default on loans ; T‐bills are more valuable
LIBOR‐OIS spread OIS : overnight indexed swap
OIS rate reflects the federal funds rate, include minimal counterparty risk
LIBOR‐OIS spread = LIBOR rate ‐ OIS rate
LIBOR‐OIS rate measure thecredit risk, amd indicate the overall wellbeing of banking system
Low LIBOR‐OIS → high liquidity market
High LIBOR‐OIS → banks are unwilling to lend, due to creditworthness concerns
Traditional theories of term 1. Unbiased expectations theory / Pure expectations theory : Investors' expectations determine the shape of interest rate term structure
structure of interest rates ‐ Forward rate = expected future spot rates
‐ Investors don't demand a risk premium for maturity strategies that differ from their investment horizon
‐ Implications for yield curve shape :
+ Upward sloping yield curve → ST rates are expected to rise
+ Downward sloping yield curve → ST rates are expected to fall
+ Flat yield curve → ST rates are expected to remain constant
2. Liquidity preference theory : Forward rates = Investors' expectations of future spot rates + Liquidity premium fo exposure to interest rate risk
‐ Liquidity premium : positively related to maturity
‐ Implications for yield curve shape :
+ Upward sloping yield curve → (1) future interest rate is expected to rise; or (2) rates are expected to remain, and upward sloping curve is solely due to liquidity premium
+ Downward sloping yield curve → steep fall in ST rate
3. Segmented markets theory : yield at each maturity is determined independently, by the preferences of borrower and lenders, which drive the balance between demand and supply
for loans at each maturities
4. Preferred habitat theory : forward rate = expected future spot rate + premium
‐ Imbalance between supply/demand for funds in a given maturity rate → investors switch from preferred habitats (maturity range) to range with opposite imbalance
‐ Investors must be compensated for price and/or reinvestment rate risk in the less‐than‐preferred habitat (borrowers require lower yield; lenders require higher yield)
Modern models of term structure 1. Equilibrium term structure models : describe changes in term structure through fundamental economic variables that drive interest rates
of interest rates a. Cox‐Ingersoll‐Ross model : interest rate movements are drivien by choosing between consumption today vs. investing and consuming later
∆ ∆ ∆
b. Vasicek model : interest rate should revert to some long‐run value
∆ ∆ ∆
‐ Disadvantage : Vasicek model does not force interest rate to be non‐negative
2. Arbitrage‐free model : assume that bonds trading in the market are correctly priced @ market price (bond value should equal its market price)
‐ Ho‐Lee model : assume changes in yield curve are consistent with no‐arbitrage condition
∆ ∆ ∆ ∆
→
∆
∆
Managing yield curve risks Yield curve risk : risk to the value of a bond portfolio, due to unexpected changes in yield curve
Counter yield curve risk by : identify portfolio's sensitivity to yield curve changes
‐ Effective duration : price sensitivity to small parallel shifts in yield curve (not accurately measure price sensitivity to non‐parallel shift)
+ Non‐parallel shift → shaping risk (change in por olio value due to change in shape of the benchmark curve)
‐ Key rate duration : sensitivity of portfolio value to change in a single par rate, holding all other spot rates constant
∆
∆
∆
‐ Sensitivity to Parallel, Steepness and Curvature movements : decomposing yield curve risk into sensitivity to changes at various maturities :
+ Level : parallel increase / decrease of interest rates
+ Steepness : LT interest rates increase + ST interest rates decrease
+ Curvature : Increase curvature → ST and LT interest rates increases ; intermediate rate do not change
∆ ∆
∆ ∆ ∆
∆
∆
Maturity structure of yield curve Interest rate volatility drives price volatility of fixed income portfolio, especially securities with embedded options
volatilities ST interest rates are more volatile than LT interest rates
LT interest rate : associated with uncertainty in real economy and inflation
ST interest rate : associated with risks in monetary policy
