The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Fixed Income

Maturity Structure of Interest Rates and Spot Rates

Learning Outcome Statement:

define spot rates and the spot curve, and calculate the price of a bond using spot rates

Summary:

This LOS focuses on understanding the maturity structure of interest rates, particularly through the use of spot rates, par rates, and forward rates. It emphasizes the calculation of bond prices using spot rates and the relationships between different types of rates. The content covers the definitions, calculations, and comparisons of these rates, and how they are used to price bonds and understand the term structure of interest rates.

Key Concepts:

Spot Rates

Spot rates are the yields on zero-coupon bonds, which are bonds that do not make periodic interest payments. They are used to discount future cash flows from bonds to their present value.

Par Rates

Par rates are the yields that cause the price of a bond to equal its face value. They are derived from spot rates for bonds of different maturities.

Forward Rates

Forward rates represent the expected future interest rates between different time periods. They are calculated from spot rates and indicate the incremental return for extending the time-to-maturity.

Bond Pricing Using Spot Rates

Bond pricing using spot rates involves discounting each of the bond's cash flows by the spot rate corresponding to its period until maturity. This method ensures a no-arbitrage price, reflecting the true market value of the bond.

Formulas:

Bond Price Calculation Using Spot Rates

PV = \sum_{t=1}^{N} \frac{C_t}{(1 + Z_t)^t}

This formula calculates the present value of a bond by discounting each future cash flow back to its present value using the appropriate spot rate for each period.

Variables:

$PV$ :: Present Value or Price of the bond
$C_t$ :: Cash flow at time t (either coupon payment or coupon plus face value)
$Z_t$ :: Spot rate at time t
$N$ :: Total number of periods until maturity

Units: currency units (e.g., dollars)

Forward Rate Calculation from Spot Rates

(1 + Z_A)^A \times (1 + IFR_{A,B-A})^{B-A} = (1 + Z_B)^B

This formula is used to calculate the implied forward rate between two periods based on the spot rates for those periods. It reflects the expected rate of return for investments over that specific future period.

Variables:

$Z_A$ :: Spot rate for period A
$Z_B$ :: Spot rate for period B
$IFR_{A,B-A}$ :: Implied forward rate from period A to B
$A$ :: Starting period
$B$ :: Ending period

Units: percentage (%)

Par and Forward Rates

Learning Outcome Statement:

define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates

Summary:

This LOS focuses on understanding and calculating par rates, forward rates, and spot rates, as well as using these rates to price bonds. Par rates are yields that make the present value of bond cash flows equal to the face value. Forward rates are implied yields between different maturities, and spot rates are yields of zero-coupon bonds that can be derived from forward rates. These concepts are crucial for analyzing the term structure of interest rates and for bond pricing.

Key Concepts:

Par Rates

Par rates are the yields-to-maturity on coupon bonds that price the bonds at their face value. They are derived from spot rates and are used to construct yield curves that are free from distortions like taxes or trading costs.

Forward Rates

Forward rates, or forward yields, are the implied future rates between different time periods derived from current spot rates. They represent the expected yield on a bond if purchased at a future date and are crucial for understanding future interest rate movements.

Spot Rates

Spot rates are the yields on zero-coupon bonds that mature at different future dates. They can be derived from forward rates and are used to price bonds and construct spot rate curves.

Bond Pricing using Forward Rates

Bonds can be priced using forward rates by discounting each cash flow by the appropriate forward rate for its respective time period. This method ensures that the bond's price reflects the expected future path of interest rates.

Formulas:

Par Rate Calculation

100 = \sum_{t=1}^{N} \frac{PMT}{(1 + z_t)^t} + \frac{100}{(1 + z_N)^N}

This formula calculates the coupon payment (PMT) that would make the price of a bond equal to its face value (100) given a series of spot rates.

Variables:

$PMT$ :: Coupon payment
$z_t$ :: Spot rate at time t
$N$ :: Number of periods to maturity

Units: Percentage

Implied Forward Rate

(1 + Z_A)^A \times (1 + IFR_{A,B-A})^{B-A} = (1 + Z_B)^B

This formula is used to calculate the implied forward rate between two periods based on the spot rates at those times.

Variables:

$Z_A$ :: Spot rate at time A
$IFR_{A,B-A}$ :: Implied forward rate from time A to B
$Z_B$ :: Spot rate at time B
$A$ :: Starting period
$B$ :: Ending period

Units: Percentage

Spot, Par, and Forward Yield Curves and Interpreting Their Relationship

Learning Outcome Statement:

compare the spot curve, par curve, and forward curve

Summary:

This LOS explores the relationships and differences between spot, par, and forward yield curves. It explains how these curves are derived and their implications in financial markets, particularly in bond pricing and interest rate expectations.

Key Concepts:

Spot Rates

Spot rates are the yields available on zero-coupon bonds. A spot curve plots these rates against their respective maturities and is typically upward sloping, indicating higher yields for longer maturities.

Par Rates

Par rates are the yields on coupon-bearing bonds priced at par value. The par curve closely follows the spot curve but is slightly lower, especially at longer maturities, due to the averaging effect of lower short-term rates.

Forward Rates

Forward rates represent future interest rates agreed upon today. They are derived from spot rates and indicate expected future rates. The forward curve generally lies above the spot and par curves if the spot curve is upward sloping.

Implied Forward Rate Calculation

Implied forward rates are calculated using spot rates to ensure no arbitrage opportunities. They represent the breakeven reinvestment rates between different maturities.

Formulas:

Implied Forward Rate Formula

(1 + Z_A)^A \times (1 + IFR_{A,B-A})^{B-A} = (1 + Z_B)^B

This formula calculates the implied forward rate between two time periods based on the spot rates for those periods. It ensures that the compounded return of investing at the spot rate for period A and then reinvesting at the forward rate from A to B equals the return of directly investing at the spot rate for period B.

Variables:

$Z_A$ :: Spot rate for maturity A
$Z_B$ :: Spot rate for maturity B
$IFR_{A,B-A}$ :: Implied forward rate from time A to B
$A$ :: Time period A
$B$ :: Time period B

Units: Percentage (%)

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