Skip to content

FRM Part I · FRM Exam Part I

Pricing Financial Forwards and Futures for FRM Part I

Forward pricing uses no-arbitrage: the forward price equals the spot price grown at the cost of carry. Without income, F = S × e^(rT). With income or yield, subtract it. For commodities, add storage and subtract convenience yield. Hedging then uses h* = ρ × σS ÷ σF to size the futures position.

What this chapter covers

This chapter shows how forwards and futures are priced and used. You start with contract mechanics: delivery, margining, daily settlement and the difference between a forward and a future. Then you build the core idea, cost of carry. If you can buy the asset today, finance it and store it, the forward price must match that cost, or an arbitrageur earns a risk-free profit.

You then extend the base formula step by step. Known cash income (such as coupons or dividends) and known yields reduce the forward price. Storage costs raise it. Convenience yield lowers it. Then you learn how the forward price relates to the expected future spot price, and how the value of an existing forward differs from its price. The chapter ends with interest rate contracts, Treasury bond futures and Eurodollar futures, and with hedging using the minimum-variance hedge ratio.

This chapter links to the rest of Part I in several ways. It uses continuous and discrete compounding from quantitative analysis, correlation and regression for the hedge ratio, and the interest rate and bond material from valuation and risk models. It also feeds into options and swaps, where the same no-arbitrage logic returns. Part II builds on it for market risk and hedging in practice.

Questions here are numerical and usually have one clean path to the answer, so careful preparation turns directly into marks. Cost of carry is a pattern you apply repeatedly, and the same few formulas cover many question types. The chapter also sits in the Financial Markets and Products topic, and its ideas support later options and swap pricing. Since the exam is 100 questions in 4 hours, speed matters. If you know which formula fits each asset type, you save time for harder questions elsewhere. GARP publishes no pass mark, so aim for solid command of every formula rather than guessing which areas matter most.

Pricing Financial Forwards and Futures: topics in the order to study them

  1. 1Forward and Futures Contracts BasicsYou need the vocabulary, payoffs, margining and daily settlement before any pricing formula makes sense.
  2. 2Cost of Carry and Forward Pricing Without IncomeThis is the base case, F = S × e^(rT), and every later formula is a modification of it.
  3. 3Forwards on Assets with Known Income or YieldIt is the first extension: subtract the present value of income, or the yield q, from the carry.
  4. 4Storage Costs, Convenience Yield and Commodity ForwardsIt adds two more carry terms, so study it once income and yield are clear.
  5. 5Futures Prices, Expected Spot Prices and Valuation of ForwardsWith pricing done, you separate the forward price from the value of a contract already in place, and link forwards to expected spot prices.
  6. 6Treasury Bond Futures and Eurodollar FuturesThese apply the pricing ideas to interest rates and bring in conventions such as conversion factors and the cheapest-to-deliver bond.
  7. 7Hedging with Futures and Optimal Hedge RatioIt comes last because it uses futures knowledge plus statistics, and it ties the whole chapter to risk management.

How to prepare Pricing Financial Forwards and Futures

Treat this chapter as one formula family. Learn the logic first, then drill the variants until you can pick the right one in seconds.

  1. Read the contract basics once and write down how a forward differs from a future in settlement, credit risk and standardisation.
  2. Derive F = S × e^(rT) from the arbitrage argument. If you can explain why a higher or lower price creates a profit, you will remember the formula.
  3. Build a one-page formula sheet: no income, known cash income, known yield, storage cost, convenience yield. Note which terms add and which subtract.
  4. Practise with a financial calculator. Use the e^x function, store S and r, and check whether the question gives continuous or annual compounding before you compute.
  5. Work forward valuation separately: value of a long forward = S − PV of income − K × e^(−rT) when there is no storage or other carry. Practise with new and old contracts.
  6. Solve hedge ratio problems: h* = ρ × σS ÷ σF, then number of contracts = h* × position size ÷ contract size. Finish with timed mixed sets of 15 to 20 questions.

Common mistakes in Pricing Financial Forwards and Futures

  • Mixing compounding conventions, using an annual rate in an exponential formula without conversion

    Fix: Read the compounding statement first. If the rate is annual, either use (1 + r)^T or convert it before using e^(rT).

  • Adding income instead of subtracting it

    Fix: Ask who gets the cash. The spot holder gets it, so the forward buyer does not, and the forward price falls.

  • Confusing forward price with forward value

    Fix: The forward price is the delivery price that makes a new contract worth zero. The value of an existing contract changes with spot and time.

  • Using the wrong sign for convenience yield

    Fix: Treat it like a dividend yield: it reduces the forward price in the carry formula.

  • Reading Eurodollar quotes as rates

    Fix: Convert first: a quote of 97.50 means an implied rate of 2.50%, and a lower quote means a higher rate.

  • Applying the hedge ratio formula with the wrong standard deviations or inverted ratio

    Fix: Write σS on top and σF on the bottom. Then scale by position size over contract size to get the contract count.

Last-day revision: Pricing Financial Forwards and Futures

  • No-income forward: F = S × e^(rT), with r the continuously compounded rate.
  • Known yield q: F = S × e^((r − q)T).
  • Known cash income with present value I: F = (S − I) × e^(rT).
  • Storage cost rate u adds to carry: F = S × e^((r + u − y)T), where y is convenience yield.
  • Convenience yield lowers the forward price and reflects the benefit of holding the physical asset.
  • Value of a long forward with delivery price K and no income: f = S − K × e^(−rT).
  • A new forward has value zero at inception, since K is set equal to F.
  • Futures are marked to market daily; forwards usually settle at maturity.
  • Futures and forward prices can differ when interest rates are correlated with the asset price.
  • Eurodollar futures quotes are 100 minus the rate, so a price rise means a fall in the implied rate.
  • Bond futures: the short chooses the cheapest-to-deliver bond, and conversion factors adjust for coupon and maturity.
  • Minimum-variance hedge ratio: h* = ρ × σS ÷ σF; contracts = h* × QA ÷ QF.

Pricing Financial Forwards and Futures practice questions

Pricing Financial Forwards and Futures in other exams

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

Pricing Financial Forwards and Futures: frequently asked questions

Which formulas must I memorise for this chapter?

Learn the cost of carry family: F = S × e^(rT), with adjustments for yield, income, storage and convenience yield. Also learn the forward value formula, the Eurodollar quote convention and the minimum-variance hedge ratio. These cover most numerical questions.

Are forward and futures prices the same?

Not always. They are close when interest rates are steady or uncorrelated with the asset price. Daily settlement of futures creates differences when rates and the asset price move together.

How is the forward price different from the value of a forward?

The forward price is the delivery price that gives a new contract zero value. Once a contract is in place, its value changes as spot prices and time to maturity change. A long forward is worth S minus the present value of K, adjusted for any income.

Can a financial calculator help here?

Yes. The e^x and square root functions speed up the carry and hedge calculations. Practise entering the full expression in one pass so that you avoid rounding errors in intermediate steps.