FRM Exam Part I · Pricing Financial Forwards and Futures

Forward Price on Assets with Known Income or Yield

Q: What is the forward price formula with a dividend yield?

F0 = S0 × e^((r − q)T), where r is the continuously compounded risk-free rate and q is the continuous dividend yield. It is used for stock indices and other assets paying a yield proportional to price.

Q: How do I price an index futures contract with dividends?

Treat the index as an asset paying a continuous yield q. Apply F0 = S0 × e^((r − q)T). Use the average dividend yield over the life of the contract.

Q: How is covered interest rate parity linked to forward pricing?

A foreign currency earns the foreign risk-free rate, so it acts like an asset with yield rf. The forward rate is S0 × e^((r − rf)T), with the rate quoted as domestic currency per foreign unit. This stops riskless profit from borrowing in one currency and investing in the other.

Q: Why does known income lower the forward price?

The forward holder does not receive the income paid before maturity, but the spot holder does. To keep the two routes equal in value, the forward price must be lower by the compounded value of that income.

Updated 11 October 2026 · Fact-checked

A forward on an asset with known income is priced by no-arbitrage. With discrete cash income, F0 = (S0 − I) × e^(rT), where I is the present value of the income. With a continuous yield q, F0 = S0 × e^((r − q)T). For currencies, q is the foreign risk-free rate.

Understand Forwards on Assets with Known Income or Yield

A forward price is the delivery price that gives the contract zero value today. You find it by asking what it costs to own the asset until maturity, using a buy-and-hold replication.

Without income, you borrow S0, buy the asset and hold it. At maturity you owe S0 × e^(rT). That is the forward price. Now suppose the asset pays income while you hold it. The holder of the forward does not receive that income, so the forward must be cheaper than in the no-income case. Income lowers the forward price.

There are two cases. If the income is a known cash amount, such as a bond coupon or a fixed stock dividend, you subtract the present value of the income from the spot price, then compound what is left. If the income is a continuous yield, such as a stock index dividend yield, you subtract q from the interest rate in the exponent.

Currencies fit the yield case. If you hold foreign currency, you can deposit it and earn the foreign risk-free rate rf. So rf plays the role of q, and the domestic rate r is the cost of funding. This gives covered interest rate parity. The currency with the higher interest rate trades at a forward discount.

If the market forward price differs from the formula, you can lock in a riskless profit. Forward too high: sell the forward, buy the asset with borrowed money. Forward too low: buy the forward, short the asset and invest the proceeds.

Key formulas to remember

Forward price, no income: F0 = S0 × e^(rT); Continuous compounding. Use as the base case.
Forward price, known cash income: F0 = (S0 − I) × e^(rT); I = present value of all income paid before maturity, discounted at r for the time to each payment.
Present value of income: I = Σ Di × e^(−r × ti); Di is the cash amount paid at time ti.
Forward price, continuous yield: F0 = S0 × e^((r − q)T); q is the dividend yield, annualised, continuously compounded.
Currency forward (covered interest parity): F0 = S0 × e^((r − rf)T); S0 and F0 are domestic currency per one unit of foreign currency. r is domestic, rf is foreign.
Discrete compounding version of parity: F0 = S0 × (1 + r × T) ÷ (1 + rf × T); Use for simple interest periods, such as money market rates under one year.

How to solve Forwards on Assets with Known Income or Yield questions

Use this routine for any forward on an asset with income, an index or a currency.

1Identify the asset type: known cash income, continuous yield, or currency.
2Write the quote direction. For currencies, confirm which currency is the unit and which is the price.
3List S0, T in years, the rate r, and q or rf or the income amounts and dates.
4Check the compounding. Convert to continuous if the formula needs it, or use the discrete parity form if the question gives simple rates.
5For cash income, compute the PV of each payment as D × e^(−r × t) and sum to get I.
6Apply the right formula: (S0 − I) × e^(rT), or S0 × e^((r − q)T), or S0 × e^((r − rf)T).
7Sanity check: income or a higher foreign rate should push F0 below the no-income price.
8If asked about arbitrage, compare F0 with the market quote and state the trade.

Quickest way: Exponent shortcut

When to use it: Use when the question gives yields or rates and asks for a forward price or a quick comparison.

Compute the net carry rate: r − q, or r − rf.
Multiply by T to get the exponent.
Estimate e^(x) ≈ 1 + x for small x and multiply by S0.
Check the sign: positive net carry gives F0 > S0, negative gives F0 < S0.
Pick the closest answer, then confirm with the calculator's e^x key only if two options are close.

Common mistakes in Forwards on Assets with Known Income or Yield

Subtracting the undiscounted income instead of its present value.
The cash amounts look simple, so students subtract them directly.
Fix: Always discount each payment to today at r before subtracting from S0. Then compound the remainder to maturity.
Discounting income for the full time T instead of the time to each payment.
Students reuse T for everything.
Fix: Use each payment's own time ti in e^(−r × ti).
Flipping the currency quote, so r and rf swap places.
Quotes such as EUR/USD versus USD/EUR are confusing.
Fix: Write the quote as domestic per one foreign unit first. The numerator currency's rate is r, the other is rf.
Adding q instead of subtracting it.
Students confuse a yield with a cost of carry such as storage.
Fix: Income lowers the forward price, so use r − q. Storage costs would be added.
Mixing continuous and annual compounding in one formula.
Questions give rates in different conventions.
Fix: Convert everything to one convention before computing, or use the matching parity form.
Ignoring a dividend that falls after the forward matures.
Students include every dividend listed.
Fix: Include only income paid between today and maturity.

Worked examples

Example 1

A stock trades at $50. It will pay a dividend of $1.00 in 3 months and $1.00 in 6 months. The risk-free rate is 6% continuously compounded. Find the 9-month forward price.

Show the solution

Present value of the first dividend: 1.00 × e^(−0.06 × 0.25) = e^(−0.015) = 0.98511.
Present value of the second dividend: 1.00 × e^(−0.06 × 0.5) = e^(−0.03) = 0.97045.
I = 0.98511 + 0.97045 = 1.95556.
S0 − I = 50 − 1.95556 = 48.04444.
Compound for 9 months: e^(0.06 × 0.75) = e^(0.045) = 1.04603.
F0 = 48.04444 × 1.04603 = 50.256.

Answer: The 9-month forward price is about $50.26.

Example 2

The spot EUR/USD rate is 1.0800 USD per EUR. The USD risk-free rate is 5% and the EUR risk-free rate is 3%, both continuously compounded. Find the 1-year forward rate and say whether the EUR trades at a forward premium or discount.

Show the solution

USD is the domestic currency, so r = 5% and rf = 3%.
Use F0 = S0 × e^((r − rf)T).
Exponent: (0.05 − 0.03) × 1 = 0.02.
e^(0.02) = 1.020201.
F0 = 1.0800 × 1.020201 = 1.10182.
F0 is above S0, so the EUR trades at a forward premium.

Answer: The 1-year forward rate is about 1.1018 USD per EUR. The EUR is at a forward premium because its interest rate is lower than the USD rate.

Exam tips

Check whether the question gives a dollar dividend or a yield. Choose the cash-income form or the yield form accordingly.
For currency questions, spend a few seconds fixing which currency is the price. Most lost marks come from swapping r and rf.
Expect arbitrage wording. If the market forward exceeds the formula value, sell the forward and buy the asset funded by borrowing.
Use the sign check. A higher foreign rate must give a forward below spot, and a higher domestic rate must give a forward above spot.
On a financial calculator, use the e^x function and keep at least five decimals until the final step.

Practice questions from Pricing Financial Forwards and Futures

Forwards on Assets with Known Income or Yield in other exams

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

CFA Level IForwards on Assets with Income or Carry Costs

Forwards on Assets with Known Income or Yield: frequently asked questions

What is the forward price formula with a dividend yield?

F0 = S0 × e^((r − q)T), where r is the continuously compounded risk-free rate and q is the continuous dividend yield. It is used for stock indices and other assets paying a yield proportional to price.

How do I price an index futures contract with dividends?

Treat the index as an asset paying a continuous yield q. Apply F0 = S0 × e^((r − q)T). Use the average dividend yield over the life of the contract.

How is covered interest rate parity linked to forward pricing?

A foreign currency earns the foreign risk-free rate, so it acts like an asset with yield rf. The forward rate is S0 × e^((r − rf)T), with the rate quoted as domestic currency per foreign unit. This stops riskless profit from borrowing in one currency and investing in the other.

Why does known income lower the forward price?

The forward holder does not receive the income paid before maturity, but the spot holder does. To keep the two routes equal in value, the forward price must be lower by the compounded value of that income.