Level I · Derivatives

Learning Module 5
Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Key Outcomes Summary & Practice Problems

Learning Outcomes

What you must be able to do

Curriculum Year: 2026

LOS 1

Explain how the value and price of a forward contract are determined at initiation, during the life of the contract, and at expiration.

LOS 2

Explain how forward rates are determined for interest rate forward contracts and describe the uses of these forward rates.

1 · Pricing vs. Valuation — Key Distinction

    • Price (F0(T)) is fixed at inception and never changes over the contract's life.

    • Value (Vt(T)) is the mark-to-market (MTM) amount — what the contract is worth at any point in time — and changes as the underlying spot price, time, and rates change.

    • At inception: V0(T) = 0 for both parties (no-arbitrage). At maturity: VT(T) = ST − F0(T) (long) — the settlement amount.

AT T = 0

V0(T) = 0

AT T = T

VT(T) = ST − F0(T)  (long)  |  F0(T) − ST  (short)

2 · Valuation During the Life of the Contract (No Costs/Benefits)

MTM VALUE

Vt(T) = St − F0(T)(1 + r)−(T−t)  (long position)

    • The long position's value = current spot price minus the present value of the (fixed) forward price; the short position's value is the negative of this.

    • An instantaneous rise in St (other things equal) → MTM gain to the long / loss to the short.

    • An increase in the risk-free rate r (other things equal) → present value of F0(T) falls → MTM gain to the long / loss to the short.

    • At any time t, if St exactly equals the PV of F0(T), then Vt(T) = 0 (the contract's "breakeven" spot price).

3 · Valuation With Costs and Benefits

PRICE (T=0)

F0(T) = [S0 − PV0(I) + PV0(C)](1 + r)T

MTM (T)

Vt(T) = [St − PVt(I) + PVt(C)] − F0(T)(1 + r)−(T−t)

    • If remaining benefits (PVt(I)) exceed remaining costs (PVt(C)) → MTM is lower than the no-cost-benefit case.

    • If remaining costs exceed remaining benefits → MTM is higher than the no-cost-benefit case.

    • If remaining costs and benefits exactly offset → MTM = St − PV of F0(T) (the simple case).

4 · FX Forward Valuation

FX PRICE

F0,f/d(T) = S0,f/d × e(rf − rd)T

FX MTM

Vt(T) = St,f/d − F0,f/d(T) × e−(rf − rd)(T−t)

    • An increase in the interest-rate differential (rf − rd) increases the discount rate applied to F0,f/d(T) — this generally produces an MTM gain to the long position (buyer of the base/domestic currency), all else equal, since the PV of the forward price falls.

    • A rate change affects MTM value, but the original contract price, F0(T), never changes over the contract's life.

5 · Zero Rates, Discount Factors, and Bootstrapping

    • Zero (spot) rate — the yield on a single cash flow received at a future date; for a one-period bond, the YTM equals the zero rate.

    • Bootstrapping — sequentially deriving zero rates from coupon-bond prices, using already-solved shorter zero rates to back out the next maturity's zero rate.

    • Discount factor (DFi) — the price today of $1 received at time i; also the no-arbitrage price of a zero-coupon bond per unit face value.

DF

DFi = 1 / (1 + zi)i

6 · Implied Forward Rates (IFR)

    • The implied forward rate is the breakeven reinvestment rate that equates (1) investing short-term and rolling over at the forward rate, to (2) investing directly to the longer maturity.

    • Notation: IFRA,B−A = forward rate starting in period A, with tenor (B − A); e.g., "2y3y" = 2-year forward starting point, 3-year tenor.

IFR

(1 + zA)A × (1 + IFRA,B−A)B−A = (1 + zB)B

7 · Forward Rate Agreements (FRAs)

    • An FRA is an OTC derivative in which counterparties agree to apply a fixed rate to a hypothetical notional deposit over a future period; the underlying is a market reference rate (MRR).

    • FRA buyer / fixed-rate payer pays the fixed rate (set at t=0, equal to the no-arbitrage IFR) and receives the floating MRR set at the start of the forward period.

    • Like a swap, no notional is exchanged — only net interest; and like other forwards, V0(T) = 0 if priced at the no-arbitrage IFR.

    • FRAs settle at the start of the rate period (not the end) — the net settlement amount must be discounted back from the end of the period to the settlement date using the realized MRR.

    • A fixed-rate payer FRA benefits if MRR rises above the contracted fixed rate — useful for hedging exposure to rising floating-rate borrowing costs.

NET CF (END)

Net payment = (MRRB−A − IFRA,B−A) × Notional × Period
Settled at start of period → discount back using MRRB−A