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CFA Level I Exam · Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Forward Price with Dividends, Coupons, Storage Costs and Convenience Yield

Updated 7 October 2026 · Fact-checked

A forward price on an asset with income or carry costs equals the spot price grown at the risk-free rate, minus the future value of benefits (dividends, coupons, convenience yield) and plus the future value of costs (storage). It is the no-arbitrage price: F0 = (S0 − PV benefits + PV costs)(1 + r)^T.

Understand Forwards on Assets with Income or Carry Costs

A forward price is set so that nobody can make a risk-free profit. Imagine you want the asset at time T. You can sign a forward, or you can buy the asset today with borrowed money and hold it. Both routes must cost the same at T. This is the cost of carry idea.

Holding the asset has costs and benefits. Interest on the money you tie up is a cost. Storage costs (warehousing, insurance) are also costs. Benefits include dividends on a share, coupons on a bond, and convenience yield on a commodity. A forward holder gets none of these, so the forward price must be lower for benefits and higher for costs.

Convenience yield is the non-monetary benefit of holding a physical commodity, such as keeping a factory running during a shortage. It is not a cash flow you can see. It acts like a dividend: it lowers the forward price. Storage cost acts the other way: it raises the forward price. A high convenience yield can push the forward price below spot, which is backwardation.

For an equity or bond, you subtract the present value of the cash flows paid before expiry. For a commodity, you add the present value of storage costs and subtract the present value of convenience yield. For a continuous dividend or yield, use the exponential form. Accrued interest matters for bonds: the forward is calculated from the full (dirty) spot price, which exams often give directly. A quoted forward price may be flat, meaning it is net of the interest accrued at expiry, so check which price the question asks for.

Key formulas to remember

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.

How to solve Forwards on Assets with Income or Carry Costs questions

Use the same sequence for equities, bonds and commodities. The only thing that changes is what you subtract and what you add.

  1. 1Identify the asset and list every cash flow or benefit between today and expiry: dividends, coupons, storage costs, convenience yield.
  2. 2Note the time of each flow in years and the risk-free rate. Check whether rates are annual or continuous.
  3. 3Discount each benefit and each cost to today at the risk-free rate: PV = CF ÷ (1 + r)^t.
  4. 4Compute the adjusted spot: S0 − PV of benefits + PV of costs.
  5. 5Compound the adjusted spot to expiry: multiply by (1 + r)^T, or use e^(rT) for continuous rates.
  6. 6Ignore any flow that occurs after expiry, and for bonds start from the full price.
  7. 7Sanity check: more benefits mean a lower forward, more costs mean a higher one.

Quickest way: Compound the spot, then net the compounded flows

When to use it: Use when there are one or two cash flows and the answer options are far apart.

  1. Compute S0(1 + r)^T first and note it as the no-carry forward.
  2. Compound each dividend or coupon forward to expiry: CF × (1 + r)^(T − t). Subtract these.
  3. Add compounded storage costs the same way.
  4. Compare with the three options. If benefits exceed costs, the forward is below S0(1 + r)^T, so discard any option above it. If costs exceed benefits (net carry is positive), the forward is above S0(1 + r)^T, so discard any option below it. The check only works in the direction that matches the sign of net carry.
  5. On the BA II Plus use 1.05 ^ 0.5 via the yx key (enter 1.05, press yx, enter 0.5, press =).

Common mistakes in Forwards on Assets with Income or Carry Costs

  • Subtracting benefits without discounting them

    The formula looks like simple subtraction from spot.

    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

    Students reuse the exponent T from the final step.

    Fix: Use the time from today to that payment. A dividend due in 3 months is discounted for 0.25 years.

  • Adding convenience yield instead of subtracting it

    It is called a yield, and students confuse it with a cost.

    Fix: Treat convenience yield like a dividend: it lowers the forward price. Storage cost raises it.

  • Including coupons paid after the forward expires

    Students list every coupon of the bond.

    Fix: Only cash flows between today and expiry belong in the calculation.

  • Using the discount rate of the asset instead of the risk-free rate

    Equity and bond questions mention required returns or yields.

    Fix: Forward pricing is no-arbitrage, so always use the risk-free rate for the period.

  • Mixing discrete and continuous rates

    Both forms appear in the curriculum.

    Fix: If the question gives a continuously compounded rate, use e^(rT) throughout and do not mix with (1 + r)^T.

Worked examples

Example 1

A share trades at €80. It will pay a €2 dividend in 3 months. The risk-free rate is 4% per year, compounded annually. What is the price of a 6-month forward? Options: A) €79.54 B) €79.56 C) €81.58

Show the solution
  1. Discount the dividend: PV = 2 ÷ (1.04)^0.25 = 2 ÷ 1.009853 = 1.9805.
  2. Adjusted spot = 80 − 1.9805 = 78.0195.
  3. Compound for 0.5 years: (1.04)^0.5 = 1.019804.
  4. F0 = 78.0195 × 1.019804 = 79.5646, about €79.56.
  5. Check with the future-value form: 80 × 1.019804 = 81.5843; dividend compounded for 0.25 years = 2 × 1.009853 = 2.0197; 81.5843 − 2.0197 = 79.5646. Same result.
  6. Option C (€81.58) is the spot compounded with no dividend removed, which is a trap. Option A (€79.54) comes from subtracting the full €2 from spot without discounting it: (80 − 2) × 1.019804 = 79.5447. The dividend must be discounted for 0.25 years, so B is the answer.

Answer: F0 ≈ €79.56, option B.

Example 2

A commodity has spot price $500. Storage costs are $12 per year, paid at the end of the year. The convenience benefit of holding the physical commodity is worth $5 per year, stated as a monetary equivalent at the end of the year; it is not an actual cash receipt. The risk-free rate is 3% per year. What is the 1-year forward price? Options: A) $508.00 B) $515.00 C) $522.00

Show the solution
  1. Net carry in monetary terms = storage 12 − convenience benefit 5 = 7 at the end of the year. The $5 is a value placed on the convenience benefit, not cash you receive, but it still lowers the forward price like a dividend would.
  2. Compound spot: 500 × 1.03 = 515.
  3. Both amounts fall at expiry, so no further compounding: FV of costs = 12, FV of benefits = 5.
  4. F0 = 515 + 12 − 5 = 522.
  5. Check with the PV form: PVC = 12 ÷ 1.03 = 11.650; PVB = 5 ÷ 1.03 = 4.854; adjusted spot = 500 + 11.650 − 4.854 = 506.796; × 1.03 = 522.00.
  6. Net carry is positive here (costs exceed the benefit), so the forward must be above S0(1 + r)^T = 515. Option B (515) ignores carry. Option A (508) comes from 515 − 12 + 5, which reverses the signs of storage cost and convenience benefit. C is correct.

Answer: F0 = $522.00, option C. Storage cost raises the price and convenience yield lowers it; the net carry of $7 is added to the spot compounded at 3%. The convenience benefit is a non-cash value, not a payment received.

Exam tips

  • Read each cash flow's timing carefully. Many questions give dividends at 3 and 9 months for a 1-year forward, so discount each separately.
  • Questions often ask which factor raises or lowers the forward price. Remember: dividends, coupons and convenience yield lower it; storage costs and a higher risk-free rate raise it.
  • If the answer options are close together, avoid rounding the discount factors early. Keep at least four decimals.
  • Backwardation or contango questions: a convenience yield larger than interest plus storage gives a forward price below spot.
  • Check whether the question gives a quoted bond price or a full price. Accrued interest changes the answer.

Practice questions from Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Forwards on Assets with Income or Carry Costs in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Forwards on Assets with Income or Carry Costs: frequently asked questions

How do you calculate a forward price when the stock pays dividends?

Subtract the present value of the dividends paid before expiry from the spot price. Then compound the result at the risk-free rate over the life of the forward. Equivalently, compound the spot and subtract the compounded dividends.

What is the difference between convenience yield and storage cost?

Storage cost is a real cash cost of holding a physical commodity, and it raises the forward price. Convenience yield is a non-cash benefit of holding the physical good, and it lowers the forward price. They push the forward price in opposite directions.

How do you find the forward price of a bond with coupons?

Start with the full spot price including accrued interest. Subtract the present value of coupons paid before the forward expires. Compound the result to expiry at the risk-free rate. Coupons after expiry are ignored.

Can a forward price be lower than the spot price?

Yes. If the benefits of holding the asset, such as dividends or a high convenience yield, exceed the interest and storage costs, the forward price falls below spot. For commodities this is called backwardation.