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IBBC CAPITAL ACADEMY / 2027

Spot, par and forward curves

Independent IBBC Capital draft content for self-study. Not affiliated with or endorsed by CFA Institute. New lessons await subject-matter review. No CFA designation or official PSM credit is awarded.

Learning goals

  • Explain cash-flow discounting using mechanisms and limitations.
  • Explain curve conversion using mechanisms and limitations.
  • Explain interpretation limits using mechanisms and limitations.

Draft · approximately 35 minutes, excluding further revision.

Cash-flow discounting

A spot rate discounts a single cash flow from its date to today; a coupon bond requires the relevant rate for each date rather than one arbitrary spot rate.

A par coupon makes the present value of promised coupons and principal equal face value using the spot discount factors.

Curve conversion

No-arbitrage forwards are implied by ratios of discount factors; they make adjacent investment paths equivalent under the specified compounding convention.

A forward rate is an implied break-even rate, not a guaranteed future spot outcome; risk premia can separate the two.

Interpretation limits

Spot, par and forward curves represent different transformations of the same discount system; their shapes need not coincide when rates vary by maturity.

Bootstrap sequentially from reliable cash-flow prices and consistent conventions; inconsistent day counts or negative discount factors signal a model problem.

Bootstrap par and spot rates from two prices

One-year spot4% implies DF1=1/1.04=0.961538. A two-year par bond pays5 annually and100 principal, so100=5DF1+105DF2. Therefore DF2=(100−5DF1)/105=0.906593 and z2=DF2^(−1/2)−1=5.0252%. This sequence removes the known first coupon before solving the final discount factor.

Conversely use spot4% and5%: DF1=0.961538 and DF2=0.907029. The two-year par coupon per100 is100(1−DF2)/(DF1+DF2)=4.9755. Forwards reproduce the same discount factors by multiplying successive one-period gross rates; rate conventions must remain identical in every transformation.

Formula and assumptions

(1+z2)²=(1+z1)(1+f1,1); par coupon=(1−DFn)/ΣDFt

Original applied case: Spot, par and forward curves

  1. One-year spot is 4%, two-year spot 5%, both annual effective.
  2. One-year forward starting in year one=(1.05²/1.04)−1=6.0096%.
  3. This is a no-arbitrage implication of today’s curve, not a forecast certainty.

Errors to catch

  • Do not assume: One spot rate must discount all maturity dates.
  • Do not assume: Par coupon means every spot rate is identical.
  • Do not assume: Forward rates are chosen independently of discount factors.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.