CFA Level I · CFA Level I Exam

Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities: formula sheet

Full chapter guide

Key formulas

Forward price, no cash flows (discrete compounding): F0(T) = S0 × (1 + r)^T; r is the annual risk-free rate, T is in years. This is the future value of the spot price.
Forward price, continuous compounding: F0(T) = S0 × e^(rT); Use only if the question gives a continuously compounded rate.
Forward price with cash flows and costs: F0(T) = (S0 − PV of benefits + PV of costs) × (1 + r)^T; Benefits such as dividends or coupons lower the price. Storage costs raise it. Discount them to today first.
Forward price with a continuous yield: F0(T) = S0 × e^((r − δ)T); δ is the continuous dividend or benefit yield. Subtract a convenience yield in the same way, add a storage cost rate.
Value of a long forward during its life, no cash flows: Vt(T) = St − [F0(T) ÷ (1 + r)^(T − t)]; Only F0 is discounted, over the time remaining (T − t). The short's value is the negative of this.
Value of a long forward at expiration: VT(T) = ST − F0(T); The long gains when the spot price at expiry exceeds the agreed price.
General value of a long forward: Vt = PV of (new forward price − original forward price); Discount over the remaining life at the risk-free rate.
Forward price, discrete compounding: F0(T) = S0 × (1 + r)^T; r is the annual risk-free rate and T is in years. Use this when the question gives an effective annual rate.
Forward price, continuous compounding: F0(T) = S0 × e^(rT); Use when the rate is stated as continuously compounded. On the BA II Plus, e^x is 2ND then LN.
Spot price from forward price: S0 = F0(T) ÷ (1 + r)^T; Rearrange to find the spot or the implied rate. This is the present value of the forward price.
Implied risk-free rate: r = (F0 ÷ S0)^(1/T) − 1; Discrete form. For continuous compounding, r = ln(F0 ÷ S0) ÷ T.
Forward price with discrete benefits and costs: F0 = (S0 − PVB + PVC)(1 + r)^T; PVB = present value of dividends, coupons or convenience yield; PVC = present value of storage costs; all discounted at the risk-free rate.
Equivalent future-value form: F0 = S0(1 + r)^T − FVB + FVC; FVB and FVC are values compounded to time T. Use whichever form is simpler.
Continuous dividend yield or carry: F0 = S0 × e^((r − q + c)T); r, q and c are continuously compounded risk-free rate, yield (dividend or convenience) and storage cost rate. Check the rate is continuous.
Present value of a benefit: PV = CF ÷ (1 + r)^t; t is the time of the cash flow in years from today, not the time to expiry.
Bond forward (coupons): F0 = (B0 − PVCoupons)(1 + r)^T; B0 is the full (dirty) spot price including accrued interest; only coupons paid before expiry are removed.
Value at initiation: V0(long) = V0(short) = 0; The forward price F(0) is set to make the contract worth zero to both parties.
Value of a long forward at time t (using forward prices): Vt(long) = [F(t) − F(0)] ÷ (1 + r)^(T − t); F(t) is the new forward price for the same expiry T. r is the annual rate for the remaining time T − t. Use continuous compounding as e^(−r(T − t)) if the question says so.
Value of a short forward at time t: Vt(short) = −Vt(long) = [F(0) − F(t)] ÷ (1 + r)^(T − t); Same size as the long, opposite sign.
Value at expiration: VT(long) = ST − F(0); VT(short) = F(0) − ST; No discounting, because T − t = 0 and F(T) = ST.
Value of a long forward on an asset with no cash flows: Vt(long) = St − F(0) ÷ (1 + r)^(T − t); Equivalent to the first formula when F(t) = St × (1 + r)^(T − t). Use it when the question gives the spot price.
Value with benefits or costs of holding the asset: Vt(long) = [St − PVt(benefits) + PVt(costs)] − F(0) ÷ (1 + r)^(T − t); Adjust the spot price for the present value of income (subtract) and storage costs (add) over the remaining life.
Implied forward rate (add-on rates): F(A,B) = [(1 + R_B × t_B) ÷ (1 + R_A × t_A) − 1] ÷ (t_B − t_A); A is the start, B is the end, in years. For a 3x6 FRA, A = 0.25 and B = 0.5. R_A and R_B are the spot rates for those terms.
No-arbitrage link: (1 + R_B × t_B) = (1 + R_A × t_A) × (1 + F × (t_B − t_A)); Rearrange this to get the forward rate. It is the idea behind the formula, so you can rebuild it under pressure.
FRA settlement at expiry (paid at start of loan period): Payoff to long = Notional × (L − K) × (m ÷ 12) ÷ (1 + L × m ÷ 12); L is the floating rate at expiry. K is the contract (FRA) rate. m is the number of months in the loan period. Positive means the long receives.
FRA value before expiry (long position): V = Notional × (F_new − K) × (m ÷ 12) ÷ [(1 + F_new × m ÷ 12) × (1 + R_A × t_A)]; F_new is the current forward rate for the same loan period. The (1 + F_new × m ÷ 12) term moves the amount from the loan end to expiry. The (1 + R_A × t_A) term discounts from expiry to today, where R_A is the current spot rate for the remaining time t_A (in years) to expiry. Short value is the negative.
Long and short value: Value of short = − Value of long; An FRA is a zero-sum contract. The two sides have equal and opposite values.
Covered interest rate parity (annual periods): F = S × (1 + r_price)^T ÷ (1 + r_base)^T; S and F are price/base. Rates must match the currency and the term. T is in years.
Short-term (simple interest) form: F = S × [1 + r_price × (days ÷ 360)] ÷ [1 + r_base × (days ÷ 360)]; Use the day-count given in the question (360 or 365). Money market quotes are usually simple interest.
Forward points: Forward points = (F − S) × 10,000; Use ×100 for yen quotes. Positive means base currency at forward premium.
Forward premium or discount: Base currency at premium if r_price > r_base; at discount if r_price < r_base; The currency with the higher interest rate trades at a forward discount relative to the other currency. When r_price > r_base, the price currency is at a discount and the base currency is at a premium.
Value of an FX forward at time t: V_t (long base) = [F_t − F_0] ÷ (1 + r_price)^(remaining T) × notional; Ft is the new forward rate for the remaining term. Result is in the price currency.
Uncovered interest rate parity: Expected % change in spot ≈ r_price − r_base; Based on expectations, no hedge. Not an arbitrage relation.
Forward price, asset with no cash flows: F0(T) = S0 × (1 + r_T)^T; Use the spot rate r_T for the forward's expiry T, with annual compounding unless told otherwise.
Implied forward rate: (1 + z_(A+B))^(A+B) = (1 + z_A)^A × (1 + f(A,B))^B; f(A,B) is the rate for a period of B years starting A years from now. Solve: f(A,B) = [(1 + z_(A+B))^(A+B) ÷ (1 + z_A)^A]^(1/B) − 1.
Forward price of a zero-coupon bond: F0 = P0(A+B) ÷ P0(A) = P0(A+B) × (1 + z_A)^A; P0(n) is today's price of a zero maturing in n years. The contract expires at A and the bond matures at A+B.
Forward price of a coupon bond: F0(T) = (B0 + AI0 − PVCI) × (1 + r_T)^T − AI_T; B0 is today's flat price, AI is accrued interest, PVCI is the present value of coupons paid before T. Discount each coupon at the spot rate for its own date. If the quoted price is flat, remove AI_T from the result.
Value of a long forward during its life: V_t(long) = [F_t(T) − F_0(T)] ÷ (1 + r_(T−t))^(T−t); F_t(T) is the new forward price for the same delivery date. The short position has the opposite value.

Quick revision

Forward price comes from no arbitrage, not from expected spot price.
No cash flows: F₀ = S₀ × (1 + r)^T.
Income or benefits lower the forward price; costs raise it.
With discrete income and costs: F₀ = S₀(1 + r)^T − FV(benefits) + FV(costs), which equals (S₀ − PV of benefits + PV of costs) × (1 + r)^T. Here FV and PV are taken to the delivery date and to today at rate r.
A forward has zero value at inception when the price is set fairly.
Long forward value at time t = PV of (new forward price − original contract price).
The short position value is the negative of the long position value, ignoring credit risk.
An FRA locks in a rate for a future borrowing or lending period.
The FRA forward rate F is implied by spot rates: (1 + S_B × t_B) = (1 + S_A × t_A) × (1 + F × (t_B − t_A)), where A is the earlier date and B the later one, with times in years under the stated day-count.
Long FRA (the party that pays fixed and gains when rates rise) value at time t has two steps. Step 1: value at the FRA settlement date = [(new forward rate − contract rate) × notional × (days ÷ 360)] ÷ [1 + new forward rate × (days ÷ 360)], where days is the length of the underlying period from settlement to the end of the period. Step 2: discount that amount back from the settlement date to the valuation date at the spot rate for the time remaining to settlement. The short FRA has the opposite sign.
Covered interest rate parity, for a quote of price currency per 1 unit of base currency: F = S × (1 + r_price)^T ÷ (1 + r_base)^T, with the price currency rate in the numerator.
The currency with the higher interest rate trades at a forward discount.
Check that your answer fits the smallest-to-largest option order and the sign logic.

Common mistakes

Treating the forward price and the forward value as the same thing. Fix: Remember the price is a fixed number in the contract. The value starts at zero and changes. Ask yourself which one the question wants.
Forgetting to subtract the present value of dividends or coupons before compounding. Fix: Read the stem for any income on the asset. Subtract its PV from the spot first, then compound.
Using the expected future spot price instead of the spot price today. Fix: Always start from today's S0. The forward price depends only on S0, r and T.
Using (1 + r)^T when the rate is continuously compounded, or e^(rT) when it is annual. Fix: Read the rate description before choosing the formula. 'Continuously compounded' means e^(rT).
Subtracting benefits without discounting them Fix: Discount the dividend or coupon to today at the risk-free rate before subtracting, or compound it to expiry if using the future-value form.
Discounting a cash flow over the full life T instead of its own time t Fix: Use the time from today to that payment. A dividend due in 3 months is discounted for 0.25 years.
Forgetting to discount the price difference. Fix: The gain is received at T. Always divide by (1 + r)^(T − t) when time remains.
Using the original time to expiry instead of the time remaining. Fix: Use T − t. If 3 months have passed on a 1-year contract, discount for 0.75 years.
Reading 3x6 as a 6-month loan or as a 3-year contract. Fix: Both numbers are months from today. The loan starts at 3 and ends at 6, so it lasts 3 months.
Forgetting to divide the forward rate by the forward period. Fix: Always divide by (t_B − t_A) in years. For 3 months divide by 0.25, or multiply by 4.

Exam tips

Read for the word 'value' versus 'price'. At initiation the value is zero, and exam writers use this as a trap option.
Underline any dividend, coupon or storage cost in the stem before you start calculating.
Because options run smallest to largest, use rough compounding to decide whether the answer is just above or just below the spot.
For arbitrage questions, decide first whether the market forward is above or below the fair price, then pick the matching trade.
Convert months to years carefully: 3 months is 0.25, 9 months is 0.75.
Look for the words 'continuously compounded' in the stem. They decide which formula you use.
Convert months to years first. This is the most frequent error in time-pressured questions.
With three options, a sense check (F0 above S0 by roughly rT) often removes two of them.