While interest rates are not the only factors that affect the futures prices (other factors are underlying price, interest (dividend) income, storage costs, and convenience yield), in a no-arbitrage environment, risk-free interest rates should explain futures prices.

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:

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; and r is the risk-free rate. Thus, the futures price of a non-dividend paying and 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 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 risk-free profit. The trader can implement 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 earning of $107, will repay the debt and interest of $105 and will net risk-free of $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 a rise in 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 annually.
- Buy one-year futures on the asset at $102.

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

These two examples show that theoretical futures prices of a non-interest paying and 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 will 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 get interest if he or she owned the asset. In the case of stock, the long side loses the opportunity to get dividends.

### Effect Storage Costs

Certain assets such as crude oil and gold must be stored in order to trade or to use in the future. Therefore, the owner holding the asset incurs storage costs, and these costs are added to the futures price if the asset is sold through the futures. 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 *is 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 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.

To see the correlation between the futures price change and risk-free interest rates demonstrated, one can estimate the correlation coefficient between June 2015 S&P 500 Index futures price change and the 10-year US 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 very small convenience yield.)

### The Bottom Line

There are at least four factors that affect change in futures prices (excluding any transaction costs of trading): a change in the spot price of the underlying, the risk-free interest rate, 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 influence 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.