Level I · Fixed Income

Learning Module 9
The Term Structure of Interest Rates

Key Outcomes Summary & Practice Problems

Learning Outcomes

What you must be able to do

Curriculum Year: 2026

LOS 1

Define spot rates and the spot curve, and calculate the price of a bond using spot rates — no‑arbitrage pricing using a sequence of default‑risk‑free zero‑coupon rates.

LOS 2

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.

LOS 3

Compare the spot curve, par curve, and forward curve — relationships between their shapes and the interpretation of forward rates as breakeven reinvestment rates.

1 · Spot Rates and the Spot Curve

Spot rate (Z): The yield‑to‑maturity on a default‑risk‑free zero‑coupon bond. Also called the zero rate or zero‑coupon yield.
Spot curve: A graph of spot rates against time‑to‑maturity for default‑risk‑free bonds. Ideal for term structure analysis because zero‑coupon bonds have no reinvestment risk.
Bond pricing using spot rates: PV = Σ PMT/(1+Zn)n + FV/(1+ZN)N
This establishes no‑arbitrage prices — if a bond's price differs, an arbitrage opportunity exists (absent transaction costs).
In practice, spot curves are constructed from recently issued coupon‑paying government bonds with interpolation between observed yields.

BOND PRICE WITH SPOT RATES

PV = Σ_n=1^N [PMT / (1 + Z_n)ⁿ] + FV / (1 + Z_N)^N
where Z_n = spot rate for period n

2 · Par and Forward Rates

Par Rates

Par rate: The coupon rate (and YTM) that would make a bond priced at par (100) given the spot rate curve.
Derived from spot rates: 100 = Σ PMT/(1+Zn)n + 100/(1+ZN)N
Par rates are used for the widely cited US Treasury yield curve (which is composed of par rates).

Forward Rates

Implied forward rate (IFR): The breakeven reinvestment rate that links a shorter‑term investment to a longer‑term investment.
Naming convention: "A y B" or "A y B" where A = number of years from today, B = tenor (e.g., 3y1y = one‑year rate starting in 3 years).
Formula: (1 + ZA)A × (1 + IFRA,B‑A)B‑A = (1 + ZB)B
If an investor expects future rates to be higher than the implied forward rate, they should prefer the shorter‑term bond and reinvest at the expected higher rate.
Spot rates can be calculated from forward rates by taking the geometric average of forward rates.

FORWARD RATE

IFR_A,B‑A = [(1 + Z_B)^B / (1 + Z_A)^A]^1/(B‑A) − 1

SPOT RATE FROM FORWARDS

(1 + Z_N)^N = ∏_t=1^N (1 + IFR_t‑1,1)

3 · Spot, Par, and Forward Curve Relationships

📈 Upward Sloping

Normal yield curve: longer maturities have higher yields.

Par < Spot < Forward

📉 Downward Sloping

Inverted yield curve: shorter maturities have higher yields.

Forward < Spot < Par

➡️ Flat

All maturities have the same yield.

Spot = Par = Forward

When the spot curve is upward sloping: Par rates are below spot rates (especially at long maturities). Forward rates are above spot rates.
When the spot curve is downward sloping: Par rates are above spot rates. Forward rates are below spot rates.
When the spot curve is flat: Par rates and forward rates equal spot rates.
Forward rates are incremental rates of return — they represent the marginal return for extending maturity.
Par rates are near spot rates but are lower in an upward‑sloping environment because low short‑term spot rates result in higher bond prices, which translates to lower par rates.

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

4 · Quick Reference — Key Formulas

Concept	Formula
Bond price with spot rates	`PV = Σ PMT/(1+Zn)n + FV/(1+ZN)N`
Par rate from spot rates	`100 = PMT × Σ 1/(1+Zn)n + 100/(1+ZN)N`
Forward rate from spot rates	`IFRA,B‑A = [(1+ZB)B/(1+ZA)A]1/(B‑A) − 1`
Spot rate from forward rates	`(1+ZN)N = ∏(1+IFRt‑1,1)`
Bond price with forward rates	`PV = Σ PMT / ∏(1+IFR) + FV / ∏(1+IFR)`