FRM: Yield curves

Explanations, applications and practice for yield curves.

• Reprice three instruments after bootstrapping

  Solve two coupon-dependent factors and check every input price.

  Worked examples

• Bootstrap a discount curve from coupon prices

  Solve successive dated factors using earlier payments already valued.

  Lessons
• 

  Curve bootstrap reconciliation sheet

  Record coupon equations, solved factors and repriced residuals.

  Templates

• Coupon mix and a bond’s single yield

  Understand why one fitted yield depends on the cash-flow schedule.

  Concepts

• Match dated cash flows to a discount curve

  Build a maturity-matched present-value schedule before using a curve.

  Lessons

• Treating an implied forward as a certain future rate

  Separate a present curve relationship from a realized future rate.

  Common mistakes

• Forward-rate products and dated discounting

  Build a multi-period factor from successive forward intervals.

  Lessons

• Price one payment through successive forward factors

  Verify that a forward accumulation product reproduces a spot price.

  Worked examples
• 

  Curve input and payment-date log

  Keep payment dates, curve conventions and interpolation choices together.

  Templates

• Curve interpolation is a model choice

  Specify what is interpolated when a payment falls between curve nodes.

  Concepts

• Annual effective forward over an arbitrary interval

  Use start and end factors to annualize a forward accumulation.

  Formulas

• Averaging spot rates before discounting

  Diagnose a rate average that changes the payment-date valuation.

  Common mistakes

• Compare log-factor and zero-rate midpoint prices

  Show how two interpolation choices differ between the same endpoints.

  Worked examples

• Discount factors above one and negative rates

  Interpret a positive factor above one using its stated convention.

  Concepts

• Using a par coupon as every spot rate

  Diagnose a single par-coupon percentage applied to all maturities.

  Common mistakes

• Par coupon from dated discount factors

  Solve the coupon rate that makes a scheduled bond worth its face amount.

  Formulas

• Curve dates and rate-convention practice

  Check annualization, negative rates and interpolation with explained choices.

  Practice

• Curve valuation and reconciliation practice

  Check coupon extraction, relative factors and a local curve scenario.

  Practice

• Replicate a coupon schedule with zero payments

  Match the dates and quantities before comparing package prices.

  Worked examples

• Discount-curve reconciliation checklist

  Review payment dates, quotation bases and repricing before accepting a curve result.

  Revision

• Reprice cash flows under a curve scenario

  Change the specified maturities and recompute each payment factor.

  Lessons

• Yield-curve learning sequence

  Move from dated factors to curve extraction and scenario interpretation.

  Study plans

• Price a coupon bond with dated spot factors

  Discount a three-year schedule using a different rate at each date.

  Worked examples

• Spot-curve valuation and single YTM

  Compare maturity-specific pricing inputs with a rate fitted to one bond.

  Comparisons

• Value a forward-starting coupon schedule

  Convert later payments into a value at a specified future start date.

  Worked examples

• Adding forward rates instead of compounding

  Identify a percentage sum that omits cross-period accumulation.

  Common mistakes

• Forward start date and interval tenor

  Distinguish when a rate begins from how long it applies.

  Concepts

• Trace a curve twist to one payment node

  Reprice a coupon when only its dated spot rate changes.

  Worked examples

• Effective zero rate from a discount factor

  Recover an annual effective rate from one positive dated factor.

  Formulas