fixed income 9: The Term Structure of Interest Rates: Spot, Par, and Forward Curves

What is the maturity (term) structure of interest rates?

The relationship between interest rates (yields) and different times-to-maturity for bonds that differ only by maturity.

What is the yield curve?

A graphical representation of yields (or spot rates) across different maturities for similar-risk bonds.

What is a spot rate?

The yield on a zero-coupon bond for a specific maturity.

Example:

• 1-year spot rate = discount rate for cash flows in 1 year

What is a spot curve?

A sequence of spot rates across different maturities (1Y, 2Y, 3Y, etc.).

Also called:

• Zero curve

• Strip curve

Why do government bonds form the basis of spot curves?

• Low default risk

• High liquidity

• Same currency and tax environment

Why are interpolation methods used in yield curves?

Because government bonds are not available for every maturity, so missing points are estimated.

What is bond pricing using spot rates?

Discounting each bond cash flow using the spot rate that matches its maturity.

General bond pricing formula using spot rates?

$$PV=\sum_{t}^{N}\frac{PMT_{t}}{\left(1+Z_{t}\right)^{t}}+\frac{FV}{\left(1+Z_{t}\right)^{t}}$$

z = spot rate

What is a par rate?

The coupon rate that makes a bond’s price equal to par (100), given spot rates.

• yield-to-maturity of a bond priced at 100.

• remove distortions from actual bond prices (credit, liquidity, premiums/discounts), making the yield curve cleaner.

How do you calculate a par rate?

$$100=\sum_{t}^{N}\frac{PMT_{t}}{\left(1+Z_{t}\right)^{t}}+\frac{100}{\left(1+Z_{t}\right)^{t}}$$

solve for PMT

z = spot rate

What is a forward rate?

The implied future interest rate between two future time periods.

• breakeven reinvestment rate.

What does a 3y1y rate mean?

The implied 1-year rate starting 3 years from today.

What is the key forward rate relationship?

$$\left(1+z_{A}\right)^{A}\cdot\left(1+IFR_{A,B-A}\right)^{B-A}=\left(1+Z_{B}\right)^{B}$$

$z_A$ = spot rate

$IFR_{A,B-A}$ = implied forward rate

$z_B$ = longer term spot rate

when the spot curve is upward, flat, downward, are the

1. par curve

2. forward curve

below, equal or above the spot curve?

spot Curve Shape	Par Curve	Forward Curve
Upward Sloping	Below spot curve	Above spot curve
Flat	Equal to spot curve	Equal to spot curve
Downward Sloping (Inverted)	Above spot curve	Below spot curve