# How and Why Interest Rates Affect Futures

Interest rates are one of the most significant elements influencing futures prices; however, other factors such as the underlying price, interest (dividend) income, storage costs, the risk-free rate, and convenience yield are also essential.

### Key Takeaways

- Many variables influence futures prices, including interest rates, storage costs, and dividend income.
- A non-dividend-paying and non-storable asset’s futures price is determined by the risk-free rate, spot price, and time to maturity.
- Assets that are projected to pay an income will lower the price of futures contracts.
- Storage costs always raise futures prices since the seller integrates the cost into the contract.
- Convenience yields, which represent the advantage of holding another asset rather than futures, reduce the price of futures.

## Effect of the Risk-Free Rate

If a trader purchases a non-interest producing asset and immediately sells futures on it, the trader must discount the futures cash flow at a risk-free rate to determine the asset’s present value. The outcome must be equal to the asset’s current price under no-arbitrage circumstances. A trader may borrow and lend at risk-free rates, and the price of futures with a time to maturity of T will be equivalent to the following:

Where:

- S0 is the underlying’s spot price at time 0.
- F0,T is the underlying futures price at time 0 with a time horizon of T.
- R is the risk-free rate.

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

If the underlying price of a non-dividend (interest) paying and non-storable asset is S0 = $100 and the annual risk-free rate, r, is 5%, and the one-year futures price is $107, we can demonstrate that this situation creates an arbitrage opportunity, which the trader can exploit to earn a risk-free profit. The trader may carry out the following activities at the same time:

- Borrow $100 at a 5% risk-free rate.
- Purchase the item at the current market price using borrowed cash and keep it.
- Sell one-year futures at $107.

At maturity after one year, the trader will provide the underlying profits of $107, refund the loan and interest of $105, and net risk-free $2.

Assume that everything else remains same from the preceding example, but the one-year futures price is $102. This condition creates another arbitrage opportunity, when traders may benefit without jeopardizing their cash by carrying out the following simultaneous actions:

- Short sell the asset at $100.
- Invest the proceeds of the short sale in a risk-free asset to earn 5% compounded on a continual basis.
- Purchase one-year futures contracts on the asset at $102.

After a year, the trader will get $105.13 from their risk-free investment, pay $102 to accept delivery through futures contracts, and return the item to the owner from whom they borrowed for the short sale. From these concurrent positions, the trader makes a risk-free profit of $3.13.

These two instances demonstrate that in order to prevent the arbitrage opportunity, the potential futures price of a non-interest paying and non-storable asset must equal $105.13 (calculated using continuous compounded rates).

## Effect of Interest Income

If the asset is projected to generate revenue, the asset’s futures price will fall. If the present value of an asset’s anticipated interest (or dividend) income is indicated as I, then the potential futures price is as follows:

Alternatively, given the asset’s known yield, the futures pricing formula would be:

When there is a known interest income, the futures price falls since the long side purchasing the futures does not own the asset and hence loses the interest advantage. Otherwise, if the buyer held the item, they would earn interest. In the case of stocks, the long side loses out on dividends.

### Income Paying Assets

Because the purchasing side does not own the asset and hence does not get the interest income, any asset that provides an income reduces the price of a futures contract.

## Effect of Storage Costs

Certain goods, such as crude oil and gold, must be held in order to be traded or used later. The owner incurs storage expenses as a result of storing the asset, and these costs are added to the futures price if the item is sold via the futures market. Until it really possesses the asset, the long side incurs no storage expenses. As a result, the short side charges the long side to cover storage expenses and the futures price. This includes the storage cost, which has the following current value:

If the storage cost, c, is stated as a continuous compounding yield, the formula is:

The general method for calculating the futures price of an asset that generates interest income while simultaneously incurring storage costs is:

- F
_{0,T}=S_{0 }eor F_{0,T}=(S_{0 }– I + C)e

## Effect of Convenience Yield

A convenience yield has the same influence on futures prices as interest income. As a result, the futures prices fall.

A convenience yield denotes the advantage of holding another asset rather than purchasing futures. A convenience yield may be found, especially in commodity futures, since certain traders gain more from actual asset ownership. For example, with an oil refinery, there is greater profit to holding the asset in a warehouse rather than anticipating delivery via futures since the inventory may be put into production immediately and react to growing market demand. Consider the convenience yield, y:

The final formula demonstrates that three of the five components (spot price, risk-free interest rate, and storage cost) are positively connected with futures prices.

For example, if we look back in time to see the correlation between futures price changes and risk-free interest rates, we can estimate the correlation coefficient between the June 2015 S&P500Index futures price change and the 10-year U.S. Treasury bond yields based on a historical sample data set for the entire year of 2014.

As a consequence, the coefficient is 0.44. The connection is positive, but it may not seem to be as strong as it appears since the overall impact of the futures price shift is dispersed across multiple factors, including spot price, risk-free rate, and dividend income. (The S&P 500 should have no storage costs and a negligible convenience yield.)

## The Bottom Line

Changes in futures prices are influenced by many variables, including changes in the underlying’s spot price, the risk-free interest rate, interest income, the underlying asset’s storage cost, and the convenience yield.

The spot price, the risk-free rate, and storage costs all have a positive connection with futures prices, whereas the others have a negative association. The link between risk-free rates and futures prices is based on a no-arbitrage opportunity assumption, which should hold true in efficient markets.

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