STAT2032 Chap.7 Term Structure of Interest Rates
Term Structure of Interest Rates
The term structure replaces the shortcut of one yield for all maturities. A spot rate discounts one cash flow from today to its particular maturity, while a forward rate describes a future interval implied by current spot rates. No-arbitrage links them by requiring alternative accumulation paths to the same date to have equal value.
The chapter prices securities one cash flow at a time, derives forward rates from successive spot maturities, and interprets normal, flat, inverted and humped curve shapes without treating any one shape as automatic.
A spot rate is indexed by its complete interval from today to maturity. A coupon security therefore becomes a package of dated zero-coupon payments, each discounted with the spot rate matching its own date.
Bootstrapping reverses this process: discount earlier known payments first, subtract them from price, and solve the next maturity rate from the remaining cash flow. One redemption yield can summarize a bond, but it does not replace a maturity-specific discount schedule when the curve is not flat.
Forward rates are obtained by equating direct and staged accumulation.
An investment carried directly to a later date must match an investment carried to an earlier date and then rolled across the future interval. The exponent on each factor is its interval length. Substitution back into this path equality is the most reliable check. Curve shape is then a separate interpretive layer.
Normal, flat, inverted and humped curves describe observed yield against maturity, but no shape alone proves a future path. Implied forwards are algebraic consequences of today's prices, not guaranteed future spot rates.
What this chapter covers
- 01
Spot rates by maturity
- 02
Discount factors from spots
- 03
Forward rates for future intervals
- 04
No-arbitrage accumulation paths
- 05
Bond pricing on a spot curve
- 06
Normal, flat, inverted and humped curves
Worked example · free
Extract forward rates from spot rates
- 1Equate the two-year paths: 1.05(1+f12) = (1.055)^2.
- 2Therefore f12 = (1.055)^2/1.05 - 1 = 6.0095%.
- 1Equate the three-year paths: (1.055)^2(1+f23) = (1.058)^3.
- 2Therefore f23 = (1.058)^3/(1.055)^2 - 1 = 6.5072%.
Key terms
- Term structure
- The relationship between yields or spot rates and time to maturity at one valuation date.
- Spot rate
- The rate from today to one stated maturity for discounting a single maturity cash flow.
- Forward rate
- A future-interval rate implied by current spot rates through equality of accumulation paths.
- Discount factor
- The present value of one unit payable at a stated maturity under the current term structure.
Term Structure of Interest Rates FAQ
Why does each cash flow use a different spot rate?
A spot curve assigns a market discount rate to each maturity. A coupon at year one and a redemption at year five belong to different maturities, so they use different discount factors. Summing their separately discounted values prices the security without imposing the extra assumption that one flat yield applies at every date.
What makes a forward rate a no-arbitrage rate?
Two strategies that start today and finish at the same future date should have the same accumulated value when their intermediate cash flows and risk basis match. Investing for the long spot maturity must equal investing to an earlier date and then over the implied forward interval. Solving that equality gives the forward rate.
Does an upward yield curve guarantee future short rates will rise?
An upward curve produces forward rates that may exceed current short spot rates, but interpretation requires care. The curve reflects current prices and can include expectations, risk premiums and market conditions. The safe mathematical claim is the no-arbitrage implication among current spot and forward rates, not a guaranteed future realized path.
How can I check the indices on a forward rate?
Translate them into words before calculating. A forward from year one to year two covers one future year, while a two-year spot covers the full interval from today to year two. Multiply the earlier spot factor by the forward factor and confirm that it reproduces the later spot accumulation factor.
Exam move
Create a spot-rate table with maturity, rate, accumulation factor and discount factor. Price a short coupon bond by matching each flow to its maturity row, then compare the result with a flat-yield calculation to see where the shortcut differs. Derive every forward rate from an equality of paths and label it with its start and end dates. Practise rebuilding a missing spot rate from forward information as the reverse exercise.
Sketch normal, flat, inverted and humped curves with yield on the vertical axis and maturity on the horizontal axis, but keep curve description separate from prediction. Check exponents carefully: a two-year spot factor covers two years, whereas a one-year forward factor covers only its single future interval.
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