# How and why interest rates affect futures

Interest rate is one of the most important factors affecting futures prices; however, other factors, such as underlying prices, interest (dividend) income, storage costs, risk-free interest rates, and convenience rates of return, also play an important role in determining futures prices.

### Key points

• Many factors affect the price of futures, such as interest rates, storage costs, and dividend income.
• The futures prices of non-distributing and non-storable assets are a function of the risk-free interest rate, the spot price, and the expiry time.
• Assets that are expected to pay income will lower futures prices.
• Storage costs always increase the futures price because the futures seller incorporates the cost into the contract.
• The convenience rate of return indicates the benefit of owning another asset instead of futures and reduces the price of futures.

## The impact of risk-free interest rates

If a trader buys a non-interest-bearing asset and immediately sells the futures of that asset, because the futures cash flow is certain, the trader will have to discount it at a risk-free interest rate to find the present value of the asset. The no-arbitrage condition stipulates that the result must be equal to the spot price of the asset. Traders can borrow and lend at risk-free interest rates, and under no-arbitrage conditions, the futures price with expiration time T will be equal to the following:

Where:

• second It is the spot price of the subject matter at time 0.
• F0,T It is the futures price of the subject matter in the time range T at time 0.
• R is the risk-free interest rate.

Therefore, the futures prices of non-dividends and non-storable assets (assets that do not need to be stored in a warehouse) are a function of the risk-free interest rate, the spot price, and the expiry time.

If the underlying price of non-dividend (interest) payment and non-storable assets is S = 100 US dollars, and the annual risk-free interest rate r is 5%. Assuming that the one-year futures price is 107 US dollars, we can prove that this situation creates arbitrage opportunities and traders can earn risk-free profits. Traders can perform the following operations at the same time:

1. Borrow \$100 at a risk-free rate of 5%.
2. Purchase and hold assets at spot market prices by paying borrowed funds.
3. Sell ​​one-year futures at a price of \$107.

One year later, when it expires, the trader will deliver a basic income of \$107, will repay \$105 of debt and interest, and will receive a risk-free net income of \$2.

Suppose everything else is the same as the previous example, but the one-year futures price is \$102. This situation once again created arbitrage opportunities, traders can make profits by simultaneously implementing the following operations without having to bear capital risks:

1. Sell ​​the asset short for \$100.
2. Investing the proceeds from short selling in risk-free assets can earn a 5% return and continue to compound interest on a continuous basis.
3. Purchase one-year futures of the asset at a price of \$102.

One year later, traders will receive \$105.13 from their risk-free investment, pay \$102 to accept delivery through a futures contract, and return the assets to the assets they borrowed for short selling. Traders realized a risk-free profit of US\$3.13 from these simultaneous positions.

These two examples show that in order to avoid arbitrage opportunities, the theoretical futures price of non-interest payments and non-storable assets must be equal to \$105.13 (based on continuous compound interest calculations).

## The impact of interest income

If the asset is expected to provide income, this will lower the asset’s futures price.Suppose the present value of the expected interest (or dividend) income of an asset is expressed as A generation, Then the theoretical futures prices are as follows:

Or, given the known rate of return on the asset q, The futures price formula is:

When the interest income is known, the futures price drops because the long side buying the futures does not own the asset and therefore loses the interest income. Otherwise, if the buyer owns the asset, they will receive interest. In the case of stocks, the bulls have lost the opportunity to receive dividends.

### Income payment assets

Any asset that pays income will lower the price of the futures contract because the buyer does not own the asset and therefore will suffer a loss when it receives interest income.

## The impact of storage costs

Certain assets (such as crude oil and gold) must be stored before they can be traded or used in the future. Therefore, the owner of the asset will incur storage costs, and if the asset is sold through the futures market, these costs will be added to the futures price. The bulls will not incur any storage costs before they actually own the assets. Therefore, shorts charge multiple parties to compensate for storage costs and futures prices.This includes storage costs, the present value of which is C as follows:

If the storage cost is expressed as a continuous compound interest rate of return, C, Then the formula is:

For assets that provide interest income and carry storage costs, the general formula for futures prices is:

• F0,T=S0 electronic(r-q+c)T Or F0,T=(S0 -I + C)etime

## Convenience benefits

The effect of the convenience rate of return in futures prices is similar to the effect of interest income. Therefore, it reduces the futures price.

The convenience rate of return indicates the benefit of owning other assets instead of buying futures. In particular, convenience gains can be observed in commodity futures, as some traders find that they benefit more from the ownership of physical assets. For example, for an oil refinery, owning assets in a warehouse is more beneficial than expecting delivery through futures because the inventory can be put into production immediately and can respond to increased market demand. In general, considering the convenience rate of return, y:

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

For example, if we look at the correlation between futures price changes and risk-free interest rates from a historical perspective, we can estimate the correlation coefficient between the changes in S&P 500 index futures prices in June 2015 and the 10-year US Treasury bond for the entire year of 2014 The rate of return of historical sample data.

The result is a coefficient of 0.44. The correlation is positive, but the reason it does not seem so strong may be because the total impact of futures price changes is distributed in many variables, including spot prices, risk-free interest rates, and dividend income. (The S&P 500 should exclude storage costs and very small convenience gains.)

## Bottom line

The factors affecting futures price changes include the following (excluding the transaction costs of any transaction): changes in the underlying spot price, risk-free interest rates, interest income, storage costs of the underlying assets, and convenience benefits.

Spot prices, risk-free interest rates and storage costs are positively correlated with futures prices, and the rest are negatively correlated with futures. The relationship between risk-free interest rates and futures prices is based on the assumption of no arbitrage opportunities, which will prevail in efficient markets.

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