FRM Part I · FRM Exam Part I
Pricing Conventions, Discounting, and Arbitrage for FRM Part I
This chapter teaches you to value cash flows by discounting them at the right rate and to price instruments so no risk-free profit exists. You solve questions by choosing the correct compounding and day count, converting rates to discount factors, then applying no-arbitrage logic to bonds, forwards and swaps.
What this chapter covers
This chapter is the pricing toolkit for the whole paper. It starts with the time value of money and compounding, then moves to discount factors, spot rates, forward rates and par yields. These are different ways of describing the same term structure. You learn to move between them quickly.
The second half adds market detail and logic. Bond pricing conventions and day counts tell you how quoted prices and accrued interest are calculated. The law of one price and arbitrage give you the principle behind fair value. You then use that principle to price forwards and swaps from spot prices, financing rates and discount factors.
The chapter connects to much of the rest of Part I. Bond duration and convexity, futures and forwards, options, swaps and valuation models all assume you can discount correctly and price without arbitrage. Quantitative questions in other topics often hide a discounting step. If this chapter is solid, many later chapters feel easier.
The exam has 100 multiple-choice questions in 4 hours, and many are short calculations where one wrong convention costs the whole question. Discounting, compounding and no-arbitrage pricing sit underneath bonds, forwards, swaps and risk measures, so mastering them helps you across several topics, not only one. The skills are also mechanical, which makes them a reliable place to gain marks through practice. You do not need new theory for each question. You need accurate conversions, a clear formula and careful reading of the conditions given.
Pricing Conventions, Discounting, and Arbitrage: topics in the order to study them
- 1Time Value of Money and CompoundingEverything else uses present and future values, so learn annual, periodic and continuous compounding first.
- 2Discount Factors and Spot RatesDiscount factors turn any cash flow into a present value, and spot rates are the standard way to quote them.
- 3Forward Rates and Par YieldsBoth are derived from discount factors or spot rates, so they come after you are comfortable with those.
- 4Bond Pricing Conventions and Day CountsOnce the rate concepts are clear, add the market rules for quoted price, accrued interest and day count fractions.
- 5Law of One Price and ArbitrageThis gives the principle that justifies the earlier relationships and prepares you for derivative pricing.
- 6Pricing Forwards and Swaps via No-ArbitrageIt combines discounting, forward rates and arbitrage logic, so it works best as the final topic.
How to prepare Pricing Conventions, Discounting, and Arbitrage
Treat this chapter as a set of conversions and one core principle. Build speed on the conversions, then apply the principle to products.
- Write a one-page sheet of formulas: FV = PV × (1 + r ÷ m)^(m×n), continuous FV = PV × e^(r×n), and discount factor d(T) = 1 ÷ (1 + z)^T for annual compounding.
- Practise converting between discount factors, spot rates and forward rates until you can do it without looking. Use the financial calculator for powers and roots, such as the y^x key.
- Solve par yield problems by setting the bond price to par and solving for the coupon, using the discount factors for each payment date.
- Learn each day count convention (such as actual/360, actual/365 and 30/360). Compute accrued interest as the periodic coupon payment × (days since the last coupon ÷ days in the coupon period), or as the annual coupon × the day count fraction under the stated convention.
- Explain the arbitrage argument for a forward in your own words: if the forward price differs from the no-arbitrage value, you can buy low and sell high with no risk. Then price forwards and swaps with it.
- Finish with timed mixed sets. Allow about two to three minutes per question and review every error by naming the convention or step you missed.
Common mistakes in Pricing Conventions, Discounting, and Arbitrage
Mixing compounding bases when comparing or combining rates
Fix: Convert every rate to one basis before any calculation, and check the compounding stated in the question.
Treating forward rates as simple differences of spot rates
Fix: Use the growth-factor relationship over the full period, then solve for the forward rate with the correct exponent.
Using the wrong day count or days in the period for accrued interest
Fix: Read which convention is stated and count days as that convention requires. Then compute accrued interest as the periodic coupon × (days accrued ÷ days in the coupon period), or as the annual coupon × the day count fraction.
Confusing clean and dirty prices
Fix: Remember the buyer pays the dirty price. Add accrued interest to the quoted clean price to get it.
Applying the forward formula without adjusting for income or costs
Fix: Ask whether the asset pays income or has costs. Subtract the present value of income from S (or use r − q for a yield q), and add the present value of storage costs (or use r + u for a storage cost rate u), as the question specifies.
Stating an arbitrage without checking it is risk-free and costless
Fix: List every cash flow today and at maturity. A true arbitrage needs no net investment today (or gives a cash inflow) and has a non-negative payoff in all outcomes, with a positive probability of profit.
Last-day revision: Pricing Conventions, Discounting, and Arbitrage
- Compounding more often at the same stated rate gives a higher future value; continuous compounding is the limit.
- Convert rates between compounding bases by matching growth over the same period: e^rc = (1 + r ÷ m)^m.
- A discount factor is the present value of 1 paid at time T; bond price = Σ cash flow × discount factor.
- A spot rate is the yield on a zero-coupon investment to a given maturity.
- With annual compounding, the forward rate between T1 and T2 satisfies (1 + z2)^T2 = (1 + z1)^T1 × (1 + f)^(T2 − T1).
- With continuous compounding, the forward rate = (z2 × T2 − z1 × T1) ÷ (T2 − T1).
- The par yield is the coupon rate that makes a bond price equal to its face value.
- Accrued interest depends on the day count convention; clean price = dirty price − accrued interest.
- The law of one price says identical cash flows must have the same price in a frictionless market.
- An arbitrage requires no net investment (or yields a cash inflow today) and has no risk of loss, with a positive probability of profit.
- A forward on an asset with no income, priced with continuous compounding: F = S × e^(r×T).
- A plain vanilla swap has a fixed rate set so the swap's value is zero at initiation, which is the par swap rate.
Pricing Conventions, Discounting, and Arbitrage practice questions
- Which day count convention is normally used to calculate accrued interest on U.S. corporate bonds?
- A 1-year zero-coupon Treasury with face value 100 trades at 96.00, and a 2-year zero-coupon Treasury with face value 100 trades at 91.00. A …
- Two default-free zero-coupon bonds each pay exactly 100 in one year. Bond A trades at 95.00 and Bond B trades at 96.00. Assuming no transact…
- Discount factors are d(1) = 0.9600 and d(2) = 0.9100. What is the implied one-year forward rate starting one year from now, with annual comp…
- A bond pays 50 in one year and 1,050 in two years. The continuously compounded rate is 4% per annum for both maturities. What is the bond's …
- Annual-pay zero-coupon bonds trade at 95.00 (1-year) and 88.00 (2-year) per 100 face. A 2-year bond with a 5% annual coupon and face value 1…
- The discount factors are d(1) = 0.9700, d(2) = 0.9300 and d(3) = 0.8900. A 3-year bond with face value 100 pays an annual coupon of 4%. What…
- Annual-pay discount factors are 0.9700 for 1 year, 0.9300 for 2 years and 0.8900 for 3 years. What is the 3-year par yield with annual coupo…
Pricing Conventions, Discounting, and Arbitrage in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Pricing Conventions, Discounting, and Arbitrage: frequently asked questions
Is this chapter calculation heavy?
Yes, most questions need a short calculation. The steps are mechanical, such as converting a rate or discounting cash flows. With practice and a financial calculator, they become quick.
Which calculator functions should I practise for this chapter?
Practise the power and exponential keys, the y^x and e^x functions, and the natural log. These handle compounding, discount factors and forward rates. Use the calculator GARP permits and learn its keys well before exam day.
How do I tell forward rates and par yields apart?
A forward rate is the rate for borrowing or lending between two future dates, implied by spot rates. A par yield is the coupon that prices a bond at face value today. Both come from discount factors, but they answer different questions.
What is the best way to understand arbitrage questions?
Build the trade step by step and write down the cash flows now and at maturity. If you can create a zero-cost position with a non-negative payoff, you have found an arbitrage and the price is not fair. This is more reliable than memorising formulas.