CFA® Level II · Derivatives · free lesson

Pricing and Valuation of Forward Commitments

Forwards, futures and swaps are priced by the same idea: a position that can be replicated with the underlying and borrowing must cost the same as that replication. This module applies carry arbitrage to equity, interest rate and fixed-income contracts, then extends it to interest rate, currency and equity swaps. Level II item sets rarely ask only for a price at inception; they ask what an existing contract is worth after markets have moved, so the emphasis is on mid-life valuation and on getting the sign right for each side.

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What you will learn

1. Price equity forwards and futures with the carry arbitrage model, with and without dividends, and value an existing equity forward before expiry.

Forward price = (spot - PV of dividends before expiry) compounded at the risk-free rate.
With a continuous dividend yield: F0 = S0 x exp[(rc - delta)T]; higher dividends lower the forward price.
A forward is worth zero at inception; later the long is worth St - PV(remaining dividends) - PV(F0).
Futures values reset to zero at each daily settlement; the futures price, not the value, carries the gain or loss.
Positive correlation between the underlying and interest rates makes futures prices exceed forward prices.

2. Price and value forward rate agreements, and price fixed-income forwards and futures using accrued interest and conversion factors.

The long FRA pays fixed and receives floating, so it gains when rates rise; borrowers hedge by going long.
The FRA rate is the implied forward rate from the h-day and (h + m)-day MRRs using simple interest.
Mid-life value = notional x (new FRA rate - old FRA rate) x m/360, discounted to today from the end of the underlying period.
Bond futures price = [(clean price + AI0)(1 + r)^T - AI_T - FV of coupons] / conversion factor.
The futures price tracks the cheapest-to-deliver bond because the short chooses which bond to deliver.

3. Determine the fixed rate on an interest rate swap from discount factors and value a swap during its life.

Swap fixed rate (periodic) = (1 - last discount factor) / sum of discount factors; annualise by multiplying by periods per year.
A pay-fixed swap = long floating-rate note + short fixed-rate bond, or a portfolio of FRAs.
Value to the pay-fixed party = NA x (new swap rate - old swap rate) x period fraction x sum of remaining discount factors.
Rising rates increase the value of pay-fixed swaps and reduce the value of receive-fixed swaps.
Use only the remaining payment dates and current discount factors when valuing mid-life.

4. Determine the fixed rates on currency swaps and value currency and equity swaps during their lives.

Currency swap fixed rates come from each currency's own discount factors; notionals are set at the inception spot rate.
Mid-life currency swap value = PV of leg received - PV of leg paid, with the foreign leg converted at the current spot rate.
Include the final notional exchange in both legs when valuing a currency swap.
Equity swap: equity leg = NA x (St / S at last reset); fixed leg = bond with the remaining coupons and notional.
The equity payer receives the absolute value of a negative equity return plus the rate leg.

Module at a glance

Carry arbitrage: forward price = (spot - PV of income + PV of costs) compounded at the risk-free rate.
Forward value is zero at inception; later the long is worth the PV of the difference between the new and original forward prices.
Futures are marked to market daily, so their value resets to zero; correlation between the underlying and rates separates futures and forward prices.
FRA rates are implied forward MRRs; value an FRA mid-life by discounting the rate difference from the end of the underlying period.
Bond futures price = carry-adjusted clean forward price of the deliverable bond divided by its conversion factor.
Swap fixed rate = (1 - last discount factor) / sum of discount factors; value = notional x rate difference x sum of remaining discount factors.
Value currency swaps leg by leg in each currency, converting at the current spot rate; value equity swaps as an equity leg against a bond or floating leg.

This free module has 4 lessons, 15 practice questions with full explanations.

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