# Value and Price of Futures Contracts

Recall that during the initiation of a forward commitment, no cash changes hands. Further, the forward commitment is neither a liability nor an asset to a buyer or the seller. As such, the value of both the forward contract and futures contract is zero: $$V_0(T)=0$$ Consider an underlying with no associated costs or benefits. Like forward contracts, the futures price is calculated by compounding the spot price of the underlying using the risk-free rate: $$f_T(0)=S_0(1+r)^T$$ Where: $$f_T(0)=$$ Futures forward price. $$S_0=$$ Spot price of the underlying at time $$t=0$$. $$r=$$ Risk-free rate. $$T=$$ Time to maturity. Note that, like forward contracts, we have used discrete compounding. However, continuous compounding is also preferred in futures contracts if the underlying assets comprise a portfolio, such as commodities, fixed income, and equity. Also, continuous compounding is preferred when the underlying is foreign exchange denominated in two currencies. Using continuous compounding, the future price is given by: $$f_T(0)=S_0e^{rT}$$

## Cost of Carry and Futures Price

Like forward contracts, the price of futures whose underlying has income (I) and costs (C) is adjusted as follows: $$f_T(0)=[S_0-PV_0(I)+PV_0(C)](1+r)^{T}$$ Where: $$PV_0(I)=$$ Present value of income or benefit associated with the underlying at time $$t=0$$. $$PV_0(C)=$$ Present value of costs associated with the underlying at time $$t=0$$

### Example: Futures Price Valuation

#### Solution

We need to start by calculating the contract’s BPV: \begin{align*}\text{Futures contract BPV}&=\text{Notional principal}\times0.01\%\times\text{Period}\\&=10,000,000\times0.01\%\times\frac{2}{12}\\&=\166.67\end{align*} We need to calculate corresponding market reference rates (MRRs) for both prices.  Note that $$f_{A, B-A}=100-(100\times\text{MRR}_{A, B-A})$$ Therefore; \begin{align*}96.75&=100-(100\times\text{MRR}_{A, B-A})\rightarrow MRR_{1,2}=3.25\%\\ 97.75&= 100-(100\times\text{MRR}_{A, B-A})\rightarrow\text{MRR}_{1,2}=2.25\%\end{align*} The bank has made a 100bps gain $$(=3.25\%-2.25\%)$$. In monetary terms, the bank has made a cumulative gain of $$\16,667 (= \text{Futures contract BPV} \times \text{100bps})$$ on the contract.

## Question

Which of the following best describes the difference between the price of a futures contract and its value? A. The price determines the profit to the buyer, and the value determines the profit to the seller. B. The futures price is fixed at the start, and the value starts at zero and changes throughout the contract’s life. C. The futures contract value is a benchmark against which the price is compared to determine whether a trade is advisable.

### Solution

The correct answer is B. The futures price is fixed at the start, whereas the value starts at zero and then changes, either positively or negatively, throughout the contract’s life.
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