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FRM Part I · FRM Exam Part I

Bond Yields and Return Calculations for FRM Part I

This chapter covers how bonds are priced as the present value of cash flows, and how spot, forward, par and yield-to-maturity rates relate. To solve questions, identify the cash flows, apply the correct compounding and discount rate, then compute price, yield or return. Check the day count last.

What this chapter covers

This chapter is about turning a bond's promised cash flows into a price, a yield or a return. You start with present value, then learn the family of interest rates used in fixed income: spot rates, forward rates, par rates and yield to maturity. You then add compounding conventions, realized return, and the market details of Treasury securities, STRIPS and day counts.

The core idea is simple. A bond is a bundle of zero-coupon payments, and each payment is discounted at the rate for its own maturity. Almost every question is a variation on that: price from spot rates, a forward rate from two spot rates, a par rate from a discount curve, or a return over a holding period.

The chapter connects to the rest of Part I in several ways. It feeds into duration and convexity, which assume you can price a bond and shift its yield. It supports derivatives pricing, where forwards and discount factors reappear. It also supports valuation and risk models, where term structures, interest rate risk and yield curve moves are central. Fluency here makes later chapters easier.

The exam has 100 multiple-choice questions in 4 hours, and many are short calculations that reward speed and accuracy. Bond pricing and rate conversions are some of the most mechanical, so they are marks you can secure with practice. The same skills sit underneath duration, hedging, forward pricing and term structure questions. A candidate who is slow or error-prone here loses time and confidence across the Financial Markets and Products and Valuation and Risk Models topics. Learn the formulas, and learn to run them on a financial calculator.

Bond Yields and Return Calculations: topics in the order to study them

  1. 1Bond Pricing and Present ValueEverything else depends on discounting cash flows, so master price as the sum of discounted coupons and principal first.
  2. 2Spot, Forward and Par RatesOnce you can discount, you learn that each cash flow has its own rate, and how forwards and par rates are derived from the same discount curve.
  3. 3Yield Measures and CompoundingYield to maturity is a single rate that summarizes the curve, and you need compounding rules to convert and compare quoted yields correctly.
  4. 4Return Calculations and Realized YieldWith pricing and yield clear, you can separate promised yield from the return you actually earn when rates change or coupons are reinvested.
  5. 5Treasury Securities, STRIPS and Day CountsThese are the market conventions that adjust your calculations, so they come last, once the underlying math is solid.

How to prepare Bond Yields and Return Calculations

Treat this chapter as a set of repeatable calculations. Your aim is to set up each problem in under a minute and check it against a sensible answer.

  1. Write the core formulas from memory: P = Σ C ÷ (1 + y/m)^(mt) + F ÷ (1 + y/m)^(mn), and the forward rate link (1 + s₂)² = (1 + s₁)(1 + f₁,₂) for annual compounding.
  2. Learn your financial calculator keys for N, I/Y, PV, PMT and FV. Set payments per year correctly and clear memory between problems.
  3. Practice pricing bonds from a spot curve, then from a single yield, and see why the two usually give different setups but consistent answers.
  4. Drill compounding conversions: annual, semiannual and continuous. Use e^r for continuous compounding and convert before comparing yields.
  5. Work return problems with a clear timeline: purchase price, coupons received, reinvestment, sale price and holding period.
  6. Review day count conventions and accrued interest, and note the difference between clean and dirty price.
  7. Finish with timed mixed sets and a quick sanity check on every answer: a higher yield should mean a lower price, and a premium bond should have a coupon above its yield.

Common mistakes in Bond Yields and Return Calculations

  • Using the annual yield with semiannual periods

    Fix: Divide the yield by m and multiply the years by m, or set the calculator to the right payments per year.

  • Discounting every cash flow at the same rate when a spot curve is given

    Fix: Use the spot rate for each cash flow's own maturity, and only use a single yield when the question gives one.

  • Mixing compounding conventions when comparing rates

    Fix: Convert everything to the same basis, for example effective annual, before comparing or combining rates.

  • Confusing clean and dirty price

    Fix: Read the question carefully, and add accrued interest, computed with the stated day count, when cash paid is asked.

  • Treating yield to maturity as the return you will actually earn

    Fix: Remember it assumes coupons reinvested at the yield and hold to maturity. Compute realized return when the question gives different reinvestment rates or an early sale.

  • Skipping a sanity check on the answer

    Fix: Check direction: a discount bond must price below face, and a forward rate should be consistent with an upward or downward sloping curve.

Last-day revision: Bond Yields and Return Calculations

  • Bond price = present value of all coupons and principal, discounted at the appropriate rates.
  • Price and yield move in opposite directions.
  • A bond priced at par has coupon rate equal to yield; premium means coupon is above yield; discount means below.
  • A spot rate is the yield on a zero-coupon bond for a given maturity.
  • Forward rate link, annual compounding: (1 + s₂)² = (1 + s₁)(1 + f₁,₂).
  • The par rate is the coupon rate that prices a bond at par given the discount curve.
  • Yield to maturity is the single discount rate that equates price to cash flows.
  • Continuous compounding: FV = PV × e^(rt); convert before comparing with periodic rates.
  • Effective annual rate = (1 + y/m)^m − 1.
  • Dirty price = clean price + accrued interest.
  • A STRIP is a zero-coupon security created from separated Treasury coupon or principal payments.
  • Realized return depends on reinvestment rates and the sale price, so it can differ from the original yield.

Bond Yields and Return Calculations practice questions

Bond Yields and Return Calculations in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Bond Yields and Return Calculations: frequently asked questions

How much calculation is in this chapter?

Most questions are calculations, though some are conceptual, such as why forward rates exceed spot rates on an upward sloping curve. Practice both types, because the exam mixes them within the same set of 100 questions.

Can I use a financial calculator for bond questions?

Yes, a financial calculator is very useful for pricing and yield-to-maturity problems. Check GARP's current calculator policy and use only an approved model. Practice with it before exam day so you do not make keying errors.

What is the difference between spot, forward and par rates?

A spot rate is the yield on a zero-coupon bond to a given maturity. A forward rate is an implied rate for a future period, derived from spot rates. A par rate is the coupon rate that makes a bond price at par.

Do I need to memorize day count conventions?

You should know the common ones, such as actual/actual for Treasury bonds and 30/360 for many corporate bonds, and how they affect accrued interest. Always follow the convention stated in the question.