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Value a forward on a financial asset in two cases: when the asset generates income or yield and when it generates none.

FRM® Part I · Financial Markets and Products · Pricing Financial Forwards and Futures · FRM1.FM.10.f

The idea

Income from the asset is a benefit the forward holder misses, so it lowers the forward price. If the income over the contract's life has present value I, the forward price is F0 = (S0 − I) × e^(rT). A long forward with delivery price K and time T to delivery is worth f = S0 − I − K × e^(−rT). For an asset paying a continuous yield q, such as an index's dividend yield or a currency's foreign rate, the forward price is F0 = S0 × e^((r − q)T). With that yield, a long forward with delivery price K is worth f = S0 × e^(−qT) − K × e^(−rT). With no income, set I or q to zero, and the forward is spot grown at the risk-free rate: F0 = S0 × e^(rT). A long forward is then worth the spot price less the present value of K: f = S0 − K × e^(−rT). Discount each known dividend at r over its own time to payment. Common errors are subtracting undiscounted dividends, adding them to spot, and forgetting the income when valuing an existing contract.

Formulas

Forward price of an asset paying known income

F0 = (S0 − I) × e^(r × T)

S0
the spot price today
I
the present value of the income paid during the contract
r
the continuously compounded risk-free rate, per year
T
the time to delivery, in years

Discount each dividend at r over its own time to payment before subtracting. Adding the income to spot, or subtracting it undiscounted, are the usual errors.

Forward price of an asset paying a continuous yield

F0 = S0 × e^((r − q) × T)

S0
the spot price today
r
the continuously compounded risk-free rate, per year
q
the continuous income yield, such as an index's dividend yield or a currency's foreign rate
T
the time to delivery, in years

The net cost of carry is r − q, so the forward lies below spot when q exceeds r.

Value of a long forward on an asset paying a continuous yield

f = S0 × e^(−q × T) − K × e^(−r × T)

S0
the spot price now
q
the continuous income yield
K
the delivery price written into the existing contract
r
the continuously compounded risk-free rate, per year
T
the time left to delivery, in years

With known cash income instead, the value is S0 − I − K × e^(−r × T). Setting q to zero gives the non-income case.

A practice question

From the question bank, with the reason each wrong option is wrong.

A stock will pay dividends with present value I over the life of a forward contract on it. With spot price S, rate r and maturity T, the forward price is:

  1. S x e^(rT)Why not: Ignores the known income altogether.
  2. (S + I) x e^(rT)Why not: Adds the income instead of subtracting it; the forward holder does not receive the dividends.
  3. (S - I) x e^(-rT)Why not: Discounts instead of compounding the net spot price.
  4. (S - I) x e^(rT) - correct

The forward holder misses the dividends, so the income's present value is deducted from spot before compounding at r.

A harder one

A stock trades at USD 60.00. It will pay a dividend of USD 2.00 in five months and another of USD 2.00 in eleven months. The risk-free rate is 10% continuously compounded. The one-year forward price is closest to:

  1. USD 61.89Why not: Subtracts the two dividends at face value instead of their present value.
  2. USD 62.17 - correct
  3. USD 66.31Why not: Ignores both dividends and compounds the full spot price.
  4. USD 70.45Why not: Adds the present value of the dividends instead of subtracting it.

PV of dividends = 2 x e^(-0.10 x 5/12) + 2 x e^(-0.10 x 11/12) = 3.7432. F = (60 - 3.7432) x e^(0.10) = USD 62.17.

The hardest tier

A company holds a long forward to buy EUR 5 million at USD 1.0900 per EUR in one year. The spot rate is USD 1.1000 per EUR, the USD risk-free rate is 6% and the EUR risk-free rate is 2%, both continuously compounded. The currency pays the foreign rate as a yield. The value of the forward is closest to:

  1. USD -162,000Why not: Swaps the two rates: discounts spot at the USD rate and K at the EUR rate.
  2. USD 258,000 - correct
  3. USD 274,000Why not: Takes the forward price minus K without discounting the difference.
  4. USD 367,000Why not: Ignores the foreign interest rate, treating the currency as a non-yielding asset.

f = (S x e^(-r_f T) - K x e^(-r_d T)) x notional = (1.1000 x e^(-0.02) - 1.0900 x e^(-0.06)) x 5,000,000 = USD 258,476, about USD 258,000.

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