Lesson 2 of 4 6–9% of the exam

By Daniil Tarakanov Updated Oct 20, 20262027 curriculum

Valuing a forward when the asset pays income or has costs

Adjust today's spot price for the carry still to come: subtract the present value of income still to be paid, add the present value of costs still to be incurred, then subtract the present value of the forward price. V_t(T) = [S_t − PV_t(I) + PV_t(C)] − F₀(T)(1 + r)^−(T−t).

6 min cram, 28 min full read. Based on the 2027 CFA® Program Level I curriculum, Derivatives, Learning Module 5. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Forward price vs forward value: at initiation, during the life, at expiry, Cost of carry: income, storage and convenience yield

On the exam

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

Explain the value of a forward during its life when the asset has income or costs. Expect a dividend-paying stock forward, sometimes a breakeven spot price (value equals zero), and occasionally the direction of a currency forward's value.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

How does the cost of carry change a forward’s value?

The cost of carry is the net of what it costs to own the asset (storage, insurance) and what owning it pays (dividends, coupons). The no-arbitrage forward price already includes it. During the life of the contract it also changes the value, because the long does not own the asset and so does not collect the income or pay the costs.

Start from the pricing formula you met in the arbitrage and cost-of-carry module, with present values taken at t = 0.

Forward price with income and costs Use it

F_0(T) = \left[S_0 - PV_0(I) + PV_0(C)\right](1 + r)^T

Because F₀(T) includes the carry, the value at initiation is still zero, and at expiration the value is still S_T − F₀(T). Only the value during the life needs an adjustment.

What is the value during the life with income and costs?

Adjust the spot price for the carry still ahead. The present value of income still to be paid is subtracted, and the present value of costs still to be incurred is added. Then subtract the present value of the forward price over the remaining time, as before.

Value during the life with income and costs, long Memorize

V_t(T) = \left[S_t - PV_t(I) + PV_t(C)\right] - F_0(T)(1 + r)^{-(T-t)}

The short’s value is the negative. The subscript t matters: the present values are measured at time t and cover only the items between t and T.

How remaining carry changes the long's value

Carry items between t and Tonly what is still ahead of you counts

Income still to come Memorize dividends, coupons: PV_t(I) is subtracted from S_t
- Effect on the long value is lower
- Why the long will not receive the income; the owner of the asset will
Costs still to come Memorize storage, insurance: PV_t(C) is added to S_t
- Effect on the long value is higher
- Why the long avoids the costs; the owner of the asset pays them
Items already paid Memorize ignored; recompute the present values at t

Each remaining payment is discounted over the time until it is paid.

Present value of a remaining payment Use it

PV_t = D\,(1 + r)^{-m}

Rearrange itPresent value of a remaining dividend or cost: solve for any input

The question asks for

A payment of 1.80 is due in 0.166667 years. The risk-free rate is 6.00%. Its present value is closest to:

Payment still to come, DRisk-free rate, r (annual)%Time until the payment (years)

P V_{t} = D (1 + r)^{- m}

1.80 × (1 + 0.06)^−0.166667 = 1.7826

Present value, PV_t1.78

On your calculator

1.06y^x0.166667+|-=×1.8=1.7826

y^x power · +|- change sign

1.06ENTER0.166667CHSy^x1.8×1.7826

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Worked example: a dividend-paying stock forward

A dividend-paying stock forward is the standard case. Take a share at EUR 85 that pays EUR 1.80 in eight months, with r = 6% and a one-year forward.

Worked example Value of the forward six months in

At initiation. The dividend is 8 months away: PV₀(I) = 1.80 × 1.06^−(8/12) = 1.7314. The forward price is F₀(1) = (85 − 1.7314) × 1.06 = EUR 88.26.

Six months later. The spot price is EUR 89 and the dividend is two months away. Six months remain.

PV_t(I) = 1.80 × 1.06^−(2/12) = 1.7826.
Present value of the forward price: 88.26 × 1.06^−0.5 = 85.73.
V_t = (89 − 1.7826) − 85.73 = EUR 1.49 for the long (and −1.49 for the short).

Without the dividend adjustment you would get 89 − 85.73 = 3.27. That overstates the long’s value: the long will not receive the dividend.

The long is worth zero when S_t = PV_t(I) + PV of F₀ = 1.78 + 85.73 = EUR 87.51. That is the breakeven spot price.

Step 3 on your calculator

1.06y^x(2÷12)=1/x×1.8=STO1 1.7826
1.06y^x0.5+|-=×88.26= 85.73
+|-+89−RCL1= 1.49

y^x power · 1/x reciprocal · STO store · +|- change sign · RCL recall a stored number

1.8ENTER1.06ENTER2ENTER12÷y^x÷ 1.7826
CHS89+ 87.2174
1.06ENTER0.5CHSy^x88.26×− 1.49

ENTER enter (on the HP, puts the number on the stack) · y^x power · CHS change sign

Rearrange itForward value with income and costs: solve for any input

The question asks for

A forward was agreed at 88.26. With 0.5 years left the spot price is 89.00, the present value of the income still to come is 1.78, the present value of the costs still to come is 0.00 and the risk-free rate is 6.00%. The value of the long position is closest to:

Spot price now, S_tPV of income still to come, PV_t(I)PV of costs still to come, PV_t(C)Forward price, F₀(T)Risk-free rate, r (annual)%Time remaining, T − t (years)

V_{t} (T) = [S_{t} - P V_{t} (I) + P V_{t} (C)] - F_{0} (T) (1 + r)^{- (T - t)}

(89.00 − 1.78 + 0.00) − 88.26 × (1 + 0.06)^−0.5 = 87.22 − 85.73 = 1.49

Value of the long position, V_t(T)1.49

On your calculator

1.06y^x0.5+|-=×88.26=85.7257
+|-+89−1.7826+0=1.4917

y^x power · +|- change sign

1.06ENTER0.5CHSy^x88.26×85.7257
CHS89+1.7826−0+1.4917

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

InteractiveValuing a forward with a dividend or a cost along the way

Try this 0 of 3

Drag today past the dividend date
Find the spot price today at which the long is worth zero
Switch the carry item to a storage cost, with today before it

Todaymonth 6

Spot today S_t (EUR)

Contract terms (fixed at month 0)

Spot at month 0, S₀Dividend amount

Dividend paid atmonth 8

Risk-free rate r6%

F₀(T) set at month 088.26

PV of income still ahead1.7826

PV of F₀(T), 6 months left85.73

Long value V_t+1.49

Long value, step by step
Spot today S_t	89.00
− PV of income still ahead	−1.7826
− PV of F₀(T)	−85.73
= Long value	+1.49

The dividend is still ahead, so its PV comes off spot. The long will not receive the dividend; only the holder of the share does. Breakeven spot today: 87.51.

Show working

At month 0: PV₀ of the dividend = 1.80 × 1.06^−8/12 = 1.7314; F₀(T) = (85.00 − 1.7314) × 1.06 = 88.26

Today (month 6): PV_t = 1.80 × 1.06^−2/12 = 1.7826; PV of F₀(T) = 88.26 × 1.06^−6/12 = 85.73

Long V_t = [S_t − PV_t(I)] − F₀(T)(1 + r)^{−(T − t)} = (89.00 − 1.7826) − 85.73 = +1.49; short = −1.49.

Traps: ignoring the dividend gives +3.27; using 1.80 instead of its PV gives +1.47.

The long's value on your calculator

1.06y^x(2÷12)=1/x×1.8=STO11.7826, PV of the dividend still ahead
1.06y^x(6÷12)=1/x×88.2647=STO285.73, PV of F₀(T)
89−RCL1−RCL2=+1.49, long value

y^x power · 1/x reciprocal · STO store · RCL recall a stored number

1.8ENTER1.06ENTER2ENTER12÷y^x÷STO11.7826, PV of the dividend still ahead
88.2647ENTER1.06ENTER6ENTER12÷y^x÷STO285.73, PV of F₀(T)
89ENTERRCL1−RCL2−+1.49, long value

ENTER enter (on the HP, puts the number on the stack) · y^x power · STO store · RCL recall a stored number

Only carry still ahead of you counts; once the dividend is paid, it drops out.

The “today” marker is the whole story. Drag it past the dividend date and PV_t(I) drops to zero: income already paid no longer counts.

Breakeven spot price Use it

S_t^{*} = PV_t(I) - PV_t(C) + F_0(T)(1 + r)^{-(T-t)}

Exam tip Same signs as the pricing formula: income comes off, costs go on. Only the clock changes, from 0 to t.

How is a currency forward valued?

The currency forward value follows the same logic with two interest rates. In a quote f/d, the long buys the base currency d. Its value is today’s spot rate minus the forward rate, discounted at the interest rate differential over the remaining time. Level I uses continuous compounding here.

Currency forward value, long Use it

V_t(T) = S_{t,f/d} - F_{0,f/d}(T)\,e^{-(r_f - r_d)(T-t)}

Worked example A CAD per EUR forward three months in

A bank agreed to buy EUR 5,000,000 against CAD at F₀ = 1.4848 CAD per EUR, one year out. At initiation the spot rate was 1.4700, r_CAD = 3.5% and r_EUR = 2.5% (continuous), which gives 1.4700 × e^0.01 = 1.4848. Three months later the spot rate is 1.4760, rates are unchanged and 0.75 years remain.

Discount the forward rate: 1.4848 × e^−(0.035 − 0.025) × 0.75 = 1.4848 × e^−0.0075 = 1.4737.
Value per EUR = 1.4760 − 1.4737 = 0.0023 CAD.
On EUR 5,000,000 that is CAD 11,472, a gain for the bank.

If the CAD rate then rose to 4.5% (spot unchanged), the differential doubles to 2%. The present value of the forward rate falls to 1.4848 × e^−0.015 = 1.4627 and the long’s gain grows to about CAD 66,529. A wider r_f − r_d helps the buyer of the base currency.

Steps 1 to 3 on your calculator

0.035−0.025=×0.75=+|-2NDe^x×1.4848= 1.4737
+|-+1.476= 0.0023
×5000000= 11472

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.035ENTER0.025−0.75×CHSge^x1.4848× 1.4737
CHS1.476+ 0.0023
5000000× 11472

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Rearrange itCurrency forward value: solve for any input

The question asks for

A long forward to buy the base currency was agreed at 1.4848 units of the price currency per unit. With 0.75 years left the spot rate is 1.476. The continuously compounded rate is 3.50% in the price currency and 2.50% in the base currency. The value of the long position per unit of base currency is closest to:

Spot rate now, S_t (price currency per base)Forward rate agreed, F₀(T)Price-currency rate, r_f (continuous)%Base-currency rate, r_d (continuous)%Time remaining, T − t (years)

V_{t} (T) = S_{t, f / d} - F_{0, f / d} (T) e^{- (r_{f} - r_{d}) (T - t)}

1.476 − 1.4848 × e^(−(0.035 − 0.025) × 0.75) = 0.00229

Value of the long position per unit of base currency0.0023

On your calculator

0.035−0.025=×0.75=+|--0.0075
2NDe^x×1.4848=1.473706
+|-+1.476=0.002294

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.035ENTER0.025−0.75×CHS-0.0075
ge^x1.4848×1.473706
CHS1.476+0.002294

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

The exam usually asks only for the direction. If the differential r_f − r_d narrows, the present value of F₀ rises and the long loses. If it widens, the long gains.

Go deeper Coupon bonds as the underlying

For a forward on a coupon bond, the coupons still to be paid before expiration are the income I. Discount each from its payment date to time t. Coupons already paid drop out, exactly as a paid dividend does.

Go deeper What if the dividend changes after initiation?

The forward price F₀(T) was set net of the dividends expected at initiation, so a dividend that arrives as expected creates no gain or loss for the long. It appears in the value formula only because the spot price still includes a dividend the long will not receive. If the dividend is later cut, PV_t(I) falls and the long’s value rises by the same amount. If it is raised, the long’s value falls.

Memorize this

MemorizeCarry and the long forward value

Item	In the formula	Effect on long's value
Income still to come	$-\,PV_t(I)$	lower
Costs still to come	$+\,PV_t(C)$	higher
Income or costs already paid	not counted	none

Memory hook Same sign pattern as the price formula, but with t instead of 0: income comes off, costs go on, and only items between t and T count.

Know it cold $V_t(T) = [S_t - PV_t(I) + PV_t(C)] - F_0(T)(1+r)^{-(T-t)}$ . Short is the negative. Breakeven: set $V_t = 0$ .

How remaining carry changes the long's value

Carry items between t and Tonly what is still ahead of you counts

Income still to come Memorize dividends, coupons: PV_t(I) is subtracted from S_t
- Effect on the long value is lower
- Why the long will not receive the income; the owner of the asset will
Costs still to come Memorize storage, insurance: PV_t(C) is added to S_t
- Effect on the long value is higher
- Why the long avoids the costs; the owner of the asset pays them
Items already paid Memorize ignored; recompute the present values at t

Forward price with income and costs

F_0(T) = \left[S_0 - PV_0(I) + PV_0(C)\right](1 + r)^T

Quick example

Share at 64, PV of dividends before expiry 1.1922, r = 4%, six months. (64 − 1.1922) × 1.04^0.5 = 64.05.

On your calculator

1.04y^x0.5=×(64−1.1922)= 64.0516

y^x power

64ENTER1.1922−1.04ENTER0.5y^x× 64.0516

ENTER enter (on the HP, puts the number on the stack) · y^x power

Present value of a remaining payment

PV_t = D\,(1 + r)^{-m}

Quick example

A dividend of 1.50 is due in three months and r = 6%. PV = 1.50 × 1.06^−0.25 = 1.48.

On your calculator

1.06y^x0.25+|-=×1.5= 1.48

y^x power · +|- change sign

1.06ENTER0.25CHSy^x1.5× 1.48

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Value during the life with income and costs, long

V_t(T) = \left[S_t - PV_t(I) + PV_t(C)\right] - F_0(T)(1 + r)^{-(T-t)}

Quick example

Spot 30.00, PV of remaining dividends 0.80, forward agreed at 29.00, r = 4%, 0.5 years left. 30.00 − 0.80 − 29 × 1.04^−0.5 = 0.76.

On your calculator

1.04y^x0.5+|-=×29= 28.44
+|-+30−0.8= 0.76

y^x power · +|- change sign

1.04ENTER0.5CHSy^x29× 28.44
CHS30+0.8− 0.76

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Breakeven spot price

S_t^{*} = PV_t(I) - PV_t(C) + F_0(T)(1 + r)^{-(T-t)}

Quick example

Same forward: S = 0.80 + 29 × 1.04^−0.5 = 29.24. The long breaks even when the spot price is 29.24

On your calculator

1.04y^x0.5+|-=×29+0.8= 29.24

y^x power · +|- change sign

1.04ENTER0.5CHSy^x29×0.8+ 29.24

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Currency forward value, long

V_t(T) = S_{t,f/d} - F_{0,f/d}(T)\,e^{-(r_f - r_d)(T-t)}

Quick example

Forward agreed at 1.2500, spot now 1.2400, r_f = 1.5%, r_d = 2.5%, 0.25 years left. 1.2400 − 1.2500 × e^0.0025 = −0.0131 per unit of base currency.

On your calculator

0.015−0.025=×0.25=+|-2NDe^x×1.25= 1.2531
+|-+1.24= -0.0131

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.015ENTER0.025−0.25×CHSge^x1.25× 1.2531
CHS1.24+ -0.0131

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

How it is asked

Value with a dividend still to come

F₀ = 88.26, six months left, r = 6%, spot 89, a 1.80 dividend due in two months.

(89 − 1.80 × 1.06^−(2/12)) − 88.26 × 1.06^−0.5 = 1.49. Traps: 3.27 (ignores the dividend), 5.06 (adds it).
Breakeven spot price

At what spot price is the long worth zero?

Set V = 0: S* = PV_t(I) − PV_t(C) + PV of F₀. Trap: forgetting to add the PV of the dividend back.
Direction: income or costs

Effect of storage costs on the long's value?

Costs are added to S_t, so the value is higher. Trap: calling costs a deduction.
Currency forward direction

Interest rate differential r_f − r_d falls after the forward is agreed. Long (buys base currency)?

PV of F₀ rises, so the long loses.

Exam traps

Subtracting the dividend amount instead of its present value.
Including a dividend that has already been paid. Only income between t and T counts.
Using the PV of income from initiation at time t. Recompute it with the time left.
Adding income to S_t (or subtracting costs). Income comes off, costs go on.

Checkpoint

6 exam-style questions. About 90 seconds each.

1Two long forwards have the same forward price, the same expiration date and, today, the same spot price. The underlying of the first will pay a dividend before expiration; the underlying of the second pays no income. Compared with the second, the value of the first long position today is most likely:
2A forward agreed at USD 92.43 has nine months left. The risk-free rate is 5%, the spot price is USD 95 and a USD 2.00 dividend is due in one month. The value of the long position is closest to:
3For a forward on a stored commodity, the present value of the remaining storage costs is:
4A forward on a share was agreed at F₀(T) = 54.00. Six months remain, the risk-free rate is 4% and a dividend of 0.60 is due in three months. The spot price at which the long position is worth zero is closest to:
5Some months ago a trader went long a forward to buy EUR 1,000,000 against SEK at 11.40 (SEK per EUR). The remaining time is now 0.5 years, the spot rate is 11.30 and the continuously compounded rates are 3.0% in SEK and 2.0% in EUR. The value of the long position is closest to:
6A trader is long a currency forward (buys the base currency). After the contract is agreed, the price-currency risk-free rate falls, narrowing the difference r_f − r_d, and other parameters are unchanged. The mark-to-market value of the long position:

Key takeaways

V_t(T) = [S_t − PV_t(I) + PV_t(C)] − F₀(T)(1 + r)^−(T−t).
Count only income and costs still to come between now and expiration, each as a present value at t.
For a given spot price and forward price, income still to come lowers the long's value and costs raise it. The short is the mirror image.
A currency forward's value is S_t minus F₀ discounted at the rate differential over the remaining time.

Forward price with income and costs

F_0(T) = \left[S_0 - PV_0(I) + PV_0(C)\right](1 + r)^T

Quick example

Share at 64, PV of dividends before expiry 1.1922, r = 4%, six months. (64 − 1.1922) × 1.04^0.5 = 64.05.

On your calculator

1.04y^x0.5=×(64−1.1922)= 64.0516

y^x power

64ENTER1.1922−1.04ENTER0.5y^x× 64.0516

ENTER enter (on the HP, puts the number on the stack) · y^x power

Present value of a remaining payment

PV_t = D\,(1 + r)^{-m}

Quick example

A dividend of 1.50 is due in three months and r = 6%. PV = 1.50 × 1.06^−0.25 = 1.48.

On your calculator

1.06y^x0.25+|-=×1.5= 1.48

y^x power · +|- change sign

1.06ENTER0.25CHSy^x1.5× 1.48

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Value during the life with income and costs, long

V_t(T) = \left[S_t - PV_t(I) + PV_t(C)\right] - F_0(T)(1 + r)^{-(T-t)}

Quick example

Spot 30.00, PV of remaining dividends 0.80, forward agreed at 29.00, r = 4%, 0.5 years left. 30.00 − 0.80 − 29 × 1.04^−0.5 = 0.76.

On your calculator

1.04y^x0.5+|-=×29= 28.44
+|-+30−0.8= 0.76

y^x power · +|- change sign

1.04ENTER0.5CHSy^x29× 28.44
CHS30+0.8− 0.76

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Breakeven spot price

S_t^{*} = PV_t(I) - PV_t(C) + F_0(T)(1 + r)^{-(T-t)}

Quick example

Same forward: S = 0.80 + 29 × 1.04^−0.5 = 29.24. The long breaks even when the spot price is 29.24

On your calculator

1.04y^x0.5+|-=×29+0.8= 29.24

y^x power · +|- change sign

1.04ENTER0.5CHSy^x29×0.8+ 29.24

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Currency forward value, long

V_t(T) = S_{t,f/d} - F_{0,f/d}(T)\,e^{-(r_f - r_d)(T-t)}

Quick example

Forward agreed at 1.2500, spot now 1.2400, r_f = 1.5%, r_d = 2.5%, 0.25 years left. 1.2400 − 1.2500 × e^0.0025 = −0.0131 per unit of base currency.

On your calculator

0.015−0.025=×0.25=+|-2NDe^x×1.25= 1.2531
+|-+1.24= -0.0131

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.015ENTER0.025−0.25×CHSge^x1.25× 1.2531
CHS1.24+ -0.0131

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

All Derivatives formulas, every lesson

Practice exam-style questions on Pricing and Valuing Forward Contracts and FRAs

Next lesson Implied forward rates from spot rates

Questions about this lesson?

Where this comes back

Implied forward rates from spot rates Pricing and Valuing Forward Contracts and FRAs
Pricing and Valuing Futures Contracts Derivatives

Questions candidates ask

How do dividends affect the value of a forward contract?

For a given spot price and forward price, dividends still to be paid before expiration lower the value of the long position, because the long will not receive them. Subtract the present value of those dividends from the spot price, then subtract the present value of the forward price. Dividends already paid no longer count.

How do you value a forward on a dividend-paying stock?

V_t(T) = [S_t − PV_t(I)] − F₀(T)(1 + r)^−(T−t). Discount each remaining dividend from its payment date back to time t, subtract their sum from the spot price, and subtract the forward price discounted over the remaining time.

Do storage costs increase forward value?

Yes, for the long. The present value of storage costs still to be incurred is added to the spot price, because the long avoids paying them. Costs already paid no longer count, and the short's value moves the opposite way.

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