Identifier: C2T2L3 | Revision: 2026-10-26

Decision and objectives

An upward curve does not mean every investor should buy the longest bond. Read the curve as dated prices of money under a specified issuer, currency and convention. A curve built from Treasury securities is not a corporate curve; a TRY curve is not a USD curve. Your task is to use comparable rates without mixing credit or compounding bases.

Objectives: Discount cash using spot rates; Calculate an implied forward rate; Distinguish a curve implication from a forecast.

Prerequisites: C2T2L2.

Mechanics

A spot rate discounts a single payment from its maturity to today. A coupon-bond YTM summarizes several payments and is not automatically the spot rate at the final maturity. A par curve describes coupon rates that would price selected bonds at par. A forward rate is a rate implied between two future dates by today's discount factors. Each is a different view of the same time structure when built consistently.

Price and return drivers

For annual effective rates, the two-year growth factor is (1 + s2)². The implied one-year forward from year one to year two f satisfies (1 + s1) × (1 + f) = (1 + s2)². Thus f = (1 + s2)² / (1 + s1) − 1. This multiplication follows two successive investments; no calculus is needed.

The implied forward includes market pricing and risk premia, not a promise about the future central bank rate. Realizing an arbitrage interpretation also requires executable borrowing/lending, identical credit, costs and matching dates. Investors usually cannot transact every curve point at model mid-prices.

Worked case

Hypothetical same-credit TRY spot curve: one-year annual effective spot 8%, two-year spot 10%. A bond pays 10 in year one and 110 in year two per 100 face. Curve price = 10 / 1.08 + 110 / 1.10² = 9.259259 + 90.909091 = 100.168350. Its price differs from discounting every cash flow at the two-year spot rate: the first payment belongs to the one-year rate.

The implied year-one-to-year-two forward = 1.10² / 1.08 − 1 = 12.0370%. TRY 100 grows to TRY 121 in two years at the two-year spot. It also grows to TRY 108 after year one and 108 × 1.12037037 = TRY 121 after year two at the implied forward. The equality is a price identity, not a forecast.

Adverse case

Suppose the one-year spot rises from 8% to 11% while the two-year spot stays 10%. Curve price becomes 10 / 1.11 + 110 / 1.10² = 99.918100. Only the early cash's discount factor changed; the price loss is 0.250250 per 100. A single parallel-shift duration estimate cannot reproduce this selective movement without allocating sensitivity by cash-flow date.

Practical use

Record observation time, issuer set, currency, instrument types, interpolation method and rate convention. An indicative curve can be useful for analysis while executable dealer quotes differ. For corporate debt, add issue-specific credit and liquidity analysis rather than assuming the government curve prices its default risk. Taxes, commissions and custody are excluded from the curve arithmetic and must be reintroduced for an investor comparison.

Decision boundaries

Do not describe an inverted curve as guaranteed falling short rates, or a steep curve as free extra return. A longer holding horizon faces more uncertainty over rate changes and exit price. If no reliable curve exists for an illiquid issue, disclose the approximation and stress the discount assumptions rather than report misleading precision.

Practice and references

Exercise: hypothetical one-year spot 5%, two-year spot 7%, annual effective compounding. Answer: implied forward = 1.07² / 1.05 − 1 = 9.0381%. A two-year zero-coupon repayment of TRY 10,000 is worth 10,000 / 1.07² = TRY 8,734.39 before costs and tax.

Takeaway: spot rates price dated cash; forwards express today's cross-date pricing.

Reference guide: domestic-government-bond, spot-curve-price, implied-forward-rate, Spot rate, Forward rate, Yield curve.

Next: C2T2L4. Topic test: C2T2.

Primary sources (checked 2026-10-26; changing terms require a publication recheck): Debt Securities Market data reporting and acceptance formats — Fields 29–36.

Shared reference guide