# Futures Prices: Known Income, Cost of Carry, Convenience Yield

How the prices of forward and futures contracts are affected when the underlying asset pays a known income, has a cost of carry, such as storage costs, or offers any convenience yield, which is the additional benefit of holding the asset rather than holding a forward or futures contract on the asset, such as being able to take advantage of shortages.

As shown in the previous article, the prices of futures and forward contracts are determined by the spot price, the risk-free interest rate, and any income earned by holding the asset. This article discusses in greater detail how the price of such contracts may be modified by the cost of carry and other benefits that may be gained by holding the asset, such as being able to take advantage of shortages.

When calculating present or future values of contracts, it is conventional to use the continuously compounded risk-free interest rate:

Continuously Compounded Interest Rate = e^{rT}

- where r = interest rate and T = years. Hereafter, e
^{rT}will be referred to as the**growth factor**.

The risk free rate is often assumed to be provided by LIBOR rather than Treasury rates; many investors considered Treasury rates to be lower than the actual risk free rate because:

- regulations require banks and other institutions to hold government bonds;
- Treasuries are not taxed by the state or local government;
- the banks' requirement for capital is less for government bonds, since they are considered free of credit risk.

Because a short party holds the asset until the delivery date, the forward price must be adjusted by adding any costs incurred by the short party for holding the asset minus any income earned from the same property. Income is usually earned in the form of interest or dividends. Cost of carry includes the interest that is forfeited by holding the asset and costs of storage.

Because the spot price of the underlying on the delivery date is unknowable, the futures price of an underlying asset that has no storage costs nor pays any income is equal to the spot price multiplied by the growth factor that can be earned on the value of the underlying during the time until delivery:

F_{0 }= S_{0}e^{rT}

Interest payments are generally calculated by multiplying the interest rate by the value of the underlying asset while dividend payments are a specific amount independent of the value of the underlying.

## Known Income

If an asset provides a known **income** (**I**), then the value of a forward contract on the asset is modified by subtracting the present value of the income from the spot price, then finding the future value of that result:

- F
_{0}= (S_{0}– I)e^{rT}- I
- = PV(I
_{1}) + PV(I_{2}) … - = I
_{1}e^{-rt1}+ I_{2}e^{-rt2}…

- = PV(I

- I

Current Stock Price | $100 | ||

Continuously Compounded Risk-Free Interest Rate per Year | 0.06 | ||

Contract Duration | 10 months | ||

Quarterly Dividends | $0.50 | ||

Present Value of Dividend Payment 1 | $0.49 | ||

Present Value of Dividend Payment 2 | $0.49 | ||

Present Value of Dividend Payment 3 | $0.48 | ||

Present Value of Dividend Payment 4 | $0.47 | ||

Total PV of Dividends (Income) | $1.93 | ||

Forward Price | $103.10 | = (100 – 1.93) × e^{(0.06 × 10/12)} |

## Forward Contract with a Known Yield

When the amount of income earned by holding an asset is a fixed percentage of the asset's value rather than a specified sum, then the yield is subtracted from the interest rate used to calculate the future value of the forward contract:

F_{0}=S_{0}e ^{(r–y)T}

S&P 500 Index | 800.00 | |

Time Remaining until Delivery (fraction of year = 3 months) | 0.25 | |

Risk-Free Interest Rate | 6% | |

Dividend Yield | 1% | |

Forward Contract Value | $810.06 | = 800 × e^{((.06 – .01) × 0.25)} |

Note that if the yield of the asset is greater than the risk-free interest rate, then the forward price will be less than the spot price, since the holder of the asset is earning a greater yield than the risk-free interest rate. Similarly, for forward contracts on currencies, the interest rate that can be earned on the foreign currency is subtracted from the domestic interest rate:

F_{0}=S_{0}e ^{(rd–rf)T}

Foreign Exchange Rate | $0.95 | |

Foreign Interest Rate | 5% | |

Domestic Interest Rate | 3% | |

Time (in years) | 10.0 | |

Forward Price | $1.16 | = $0.95 × e^{(.03 – .05) × 10} |

## Mark to Market Value of Forward and Futures Contracts

When a forward or futures contract is 1^{st} agreed upon, the value of the contract must be 0; otherwise, there would be no agreement. As time passes, the spot price of the underlying will vary, but the delivery price will remain fixed, in which case, the value of the contract will vary with the value of the underlying. Oftentimes, banks or other financial institutions will need to know the value of forward contracts, so their value must be marked to market, to determine their value before the delivery date. Futures contracts are always marked to market daily.

The present value of a forward or futures contract can be calculated by subtracting the present value of the spot price minus the present value of the delivery price for the long party. The value of the contract to the short party is the present value of the delivery price minus the present value of the spot price:

- Value of Contract to Long Party
- = PV(Spot Price) – PV(Delivery Price)
- = (F
_{0 }– K)e^{– rT }

- Value of Contract to Short Party
- = PV(Delivery Price) – PV(Spot Price)
- = (K– F
_{0})e^{–rT}

(Noting that since the value of the futures or forward contract is equal and opposite for the long and short party, the remaining equations will apply to the long party.) If the asset pays a set income, then the value of the contract is equal to the following:

- Contract Value
- = PV (Spot Price) – PV (Income) – PV(Delivery Price)
- = S
_{0}– I – Ke^{–rT}

By combining the above equations, the value of a forward contract with a given yield, *y*, is:

Contract Value of Asset with Known Yield = S_{0}e^{–yT}– Ke^{–rT}

Note that in the above equations, the present value of the spot price is, of course, just the spot price.

## The Relationship of Forward and Futures Prices

Especially with short-term maturities, forward and future prices are generally equal, but they diverge for longer maturities. The primary difference arises because futures contracts are marked to market daily, so if the value of the contract increases, then the additional equity can be reinvested for greater returns; if the value of the contract decreases, then additional margin may have to be posted. Since forward contracts are usually not marked to market, their varying value does not present reinvestment opportunities nor are there any margin requirements unless it is specified in the contract.

## Commodities

Commodities generally have storage costs, which can be treated as negative income or a negative yield. If the storage cost is a fixed cost, independent of the value of the underlying asset, then the present value of the storage cost is added to the spot price, then multiplied by the growth factor:

F_{0}=(S_{0}+U)e^{rT}

If the storage cost is a percentage of the underlying value, then it is treated as a negative yield:

F_{0 }= S_{0 }e^{(r + u)T }

Note that since the cost of carry is a positive cost, it is added to the interest rate, which is also a positive opportunity cost, with both factors increasing the initial price of the futures contract with respect to the spot price.

## Convenience Yield

For some contracts, the value of the contract will depend on whether it is an investment asset or a consumption asset. An **investment asset** is an asset held primarily for investment by a significant number of investors, while a **consumption asset** is one held primarily for consumption. Even if an asset is consumable, like silver, the asset can still be considered an investment asset if held by a significant number of investors as an investment. This distinction allows the use of arbitrage to determine the forward and future prices of an investment asset based on the spot price, but the same cannot be done for a consumption asset, because a consumption asset may have what is called a convenience yield. **Convenience yield** is the additional value gained by holding the asset rather than having a long forward or futures contract on the asset, such as the ability to take advantage of shortages. Oil, for instance, is primarily a consumption asset and has a convenience yield because the holder of the asset can sell at higher prices during shortages.

However, the convenience yield cannot be calculated directly; like implied volatility, it can only be inferred by the deviation of the actual contract value from the value calculated without considering the convenience yield, using one of the above applicable equations. And like implied volatility, convenience yield changes continually.