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Explain how forward prices relate to futures prices.

FRM® Part I · Financial Markets and Products · Pricing Financial Forwards and Futures · FRM1.FM.10.g

The idea

Forward and futures prices for the same asset and maturity are equal when interest rates are constant and the same for all maturities. The difference in practice arises because futures are settled daily, so gains and losses are reinvested or financed along the way. If the asset price is positively correlated with interest rates, a long futures holder tends to gain when rates are high, reinvesting at high rates, and to lose when rates are low, funding at low rates. Futures are then more attractive and the futures price is above the forward price. With negative correlation the futures price is below the forward price. For contracts of a few months the gap is usually too small to matter next to transaction costs, margin requirements and taxes, but it grows with maturity. In exam calculations, treat the two as equal unless told otherwise.

A practice question

From the question bank, with the reason each wrong option is wrong.

When interest rates are constant and the same for all maturities, the forward price and the futures price for the same asset and maturity are:

  1. such that the forward price is higher, because of counterparty credit riskWhy not: Credit risk is not part of the standard pricing relationship.
  2. such that the futures price is higher, because of daily settlementWhy not: Daily settlement has no effect on price when rates are constant, since reinvestment rates are known.
  3. equal - correct
  4. such that the futures price is lower, because the clearinghouse reduces riskWhy not: Risk reduction by a clearinghouse is not priced in the standard relationship.

With known, constant rates the daily cash flows of a futures contract can be reinvested at a known rate, so the futures price equals the forward price.

A harder one

A long-dated bond futures contract is considered where the underlying price is negatively correlated with interest rates. Compared with the forward price, the futures price should be:

  1. equal, because bond prices are determined only by the yield curveWhy not: Assumes that bond pricing removes the daily-settlement effect, which stays whenever rates move with the price.
  2. lower, because the long's daily gains arrive when rates are low and its losses when rates are high - correct
  3. higher, because the short's daily losses arrive when rates are highWhy not: Mixes up the effect on the short with that on the long.
  4. higher, because the long's daily gains arrive when rates are lowWhy not: Takes low-rate gains as an advantage; gains reinvested at low rates are worth less.

With negative correlation, the long futures holder gains when rates are low and loses when rates are high, which is unattractive. The futures price must be lower than the forward price.

The hardest tier

A risk manager asks how much the difference between forward and futures prices matters in practice. Which statement is most accurate?

  1. The difference is largest for short-dated contracts because daily settlement occurs most oftenWhy not: Reverses the maturity effect; the gap accumulates over longer horizons.
  2. The difference is always zero because arbitrage between forwards and futures is costlessWhy not: Ignores margin, taxes, credit risk and the correlation effect, which can leave small differences.
  3. The difference depends only on contract size and not on interest ratesWhy not: Ignores that the difference comes from the interaction of price moves and interest rates.
  4. For contracts of a few months the difference is usually small enough to ignore, while it grows with maturity and with the volatility of rates - correct

The correlation effect is small over short horizons and grows with the maturity and with rate volatility; in practice forwards and futures are often treated as equal.

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