Interest rates are one of the most important factors that affect futures prices; however, other factors, such as the underlying price, interest (dividend) income, storage costs, the risk-free rate, and convenience yield, play an important role in determining futures prices as well.

### Key Takeaways

- Many factors affect the price of futures, such as interest rates, storage costs, and dividend income.
- The futures price of a non-dividend-paying and non-storable asset is the function of the risk-free rate, spot price, and time to maturity.
- Assets that are expected to pay an income will decrease the price of the futures.
- Storage costs always increase the futures price as the seller of the futures incorporates the cost into the contract.
- Convenience yields, which indicate the benefit of owning another asset rather than the futures, decrease the futures price.

## Effect of the Risk-Free Rate

If a trader buys a non-interest earning asset and immediately sells futures on it, because the futures cash flow is certain, the trader will have to discount it at a risk-free rate to find the present value of the asset. No-arbitrage conditions dictate that the result must be equal to the spot price of the asset. A trader can borrow and lend at the risk-free rate, and with no-arbitrage conditions, the price of futures with time to maturity of T will be equal to the following:

- F
_{0,T}=S_{0}*e^{r*T}

Where:

- S
_{0}is the spot price of the underlying at time 0. - F
_{0,T}is the futures price of the underlying for a time horizon of T at time 0. - R is the risk-free rate.

Thus, the futures price of a non-dividend-paying and a non-storable asset (an asset that does not need to be stored at a warehouse) is the function of the risk-free rate, spot price, and time to maturity.

If the underlying price of a non-dividend (interest) paying and a non-storable asset is S_{0} = $100, and the annual risk-free rate, r, is 5%, assuming that the one-year futures price is $107, we can show that this situation creates an arbitrage opportunity and the trader can use this to earn a risk-free profit. The trader can implement the following actions simultaneously:

- Borrow $100 at a risk-free rate of 5%.
- Buy the asset at spot market price by paying borrowed funds and hold.
- Sell one-year futures at $107.

After one year, at maturity, the trader will deliver the underlying earnings of $107, will repay the debt and interest of $105, and will net risk-free $2.

Suppose that everything else is the same as in the previous example, but the one-year futures price is $102. This situation again gives rise to an arbitrage opportunity, where traders can earn a profit without risking their capital, by implementing the following simultaneous actions:

- Short sell the asset at $100.
- Invest the proceeds of the short sell in the risk-free asset to earn 5%, which continues to be compounded on a continuous basis.
- Buy one-year futures on the asset at $102.

After one year the trader will receive $105.13 from their risk-free investment, pay $102 to accept the delivery through the futures contracts, and return the asset to the owner from which they borrowed for the short sell. The trader realizes a risk-free profit of $3.13 from these simultaneous positions.

These two examples show that the theoretical futures price of a non-interest paying and a non-storable asset must be equal to $105.13 (calculated based on continued compounded rates) in order to avoid the arbitrage opportunity.

## Effect of Interest Income

If the asset is expected to provide an income, this will decrease the futures price of the asset. Suppose that the present value of the expected interest (or dividend) income of an asset is denoted as *I, *then the theoretical futures price is found as follows:

- F
_{0,T}=(S_{0}- I) e^{rT}

Or, given the known yield of the asset *q, *the futures price formula would be:

- F
_{0,T}=S_{0 }e^{(r-q)T}

The futures price decreases when there is a known interest income because the long side buying the futures does not own the asset and, thus, loses the interest benefit. Otherwise, the buyer would receive interest if they owned the asset. In the case of stock, the long side loses the opportunity to get dividends.

### Income Paying Assets

Any asset that pays an income will reduce the price of a futures contract because the buying side does not own the asset and, therefore, loses out on receiving the interest income.

## Effect of Storage Costs

Certain assets such as crude oil and gold must be stored in order to trade or to use in the future. The owner holding the asset thus incurs storage costs, and these costs are added to the futures price if the asset is sold through the futures market. The long side does not incur any storage costs until it actually owns the asset. Therefore, the short side charges the long side for the compensation of storage costs and the futures price. This includes the storage cost, which has a present value of *C *as follows:

- F
_{0,T}=(S_{0}+ C) e^{rT}

If the storage cost is expressed as a continuous compounding yield, *c*, then the formula would be:

- F
_{0,T}=S_{0 }e^{(r+c)T}

For an asset that provides interest income and also carries a storage cost, the general formula of the futures price would be:

- F
_{0,T}=S_{0 }e^{(r-q+c)T}or F_{0,T}=(S_{0 }- I + C)e^{rT}

## Effect of Convenience Yield

The effect of a convenience yield in futures prices is similar to that of interest income. Therefore, it decreases the futures prices.

A convenience yield indicates the benefit of owning some other asset rather than buying futures. A convenience yield can be observed particularly in futures on commodities because some traders find more benefit from ownership of the physical asset. For example, with an oil refinery, there is more benefit from owning the asset in a warehouse than in expecting the delivery through the futures because the inventory can be put immediately into production and can respond to the increased demand in the markets. Overall, consider convenience yield, *y:*

- F
_{0,T}=S_{0 }e^{(r-q+c-y)T}

The last formula shows that three components (spot price, risk-free interest rate, and storage cost) out of five are positively correlated with futures prices.

For example, if we take a historical look to see the correlation between the futures price change and risk-free interest rates demonstrated, one can estimate the correlation coefficient between the June 2015 S&P 500 Index futures price change and the 10-year U.S. Treasury bond yields on a historical sample data for the whole year of 2014.

The result is a coefficient of 0.44. The correlation is positive but the reason why it may not seem that strong could be because the total effect of the futures price change is distributed among many variables, which include spot price, risk-free rate, and dividend income. (The S&P 500 should include no storage cost and a very small convenience yield.)

## The Bottom Line

There are a few factors that affect changes in futures prices (excluding any transaction costs of trading): a change in the spot price of the underlying, the risk-free interest rate, interest income, the storage cost of the underlying asset, and the convenience yield.

Spot price, the risk-free rate, and storage costs have a positive correlation with futures prices, whereas the rest have a negative correlation on futures. The relationship of risk-free rates and futures prices is based on a no-arbitrage opportunity assumption, which shall prevail in markets that are efficient.