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# Pricing Financial Forwards and Futures

After completing this reading, you should be able to:

• Define and describe financial assets.
• Define short-selling and calculate the net profit of a short sale of a dividend-paying stock.
• Describe the differences between forward and futures contracts and explain the relationship between forward and spot prices.
• Calculate the forward price given the underlying asset’s spot price and describe an arbitrage argument between spot and forward prices.
• Distinguish between the forward price and the value of a forward
• Calculate the value of a forward contract on a financial asset that does or does not provide income or yield.
• Explain the relationship between forward and futures prices.
• Calculate the value of a stock index futures contract and explain the concept of index arbitrage.

## Financial Assets

A financial asset is an asset that derives its value from a particular claim.

Assets held for the purposes of investing are referred to as investment assets. Examples of such assets include stocks and bonds issued by various financial institutions. On the other hand, assets primarily held for the purpose of consumption and not for investment or resale are referred to as non-investment or consumption assets. Examples of such assets include oil, coffee, tea, corn, e.t.c.

In this chapter, we consider three types of assets:

1. Assets providing no income,
2. Assets providing a known income that is a fixed amount, and
3. Assets providing a known income that is a percentage of their value.

## Short-Selling

Short selling involves the sale of a security not owned by an investor. The investor sells the security but purposes to buy it later. The investor will then realize profits if the price of the security goes down and losses if the price of the security goes up.

Short sales are transacted through a broker. The short seller must deposit some collateral to guarantee the eventual return of the security to the owner. In addition, the short seller is required to pay all accrued dividends, if any, to the lender. Thus, the net profit is equal to:

$$\text{Net profit}=\text{Sale price}–\text{Borrowing price}–\text{Dividend paid}$$

For example, if a trader shorts a stock today at $100, a dividend of$4 is paid next month, and the trader closes the short position the following month at $90, the net profit will be$100 – $90 –$4 = $6. His return would normally have been $$\frac {10}{100} = 10\%$$, but the dividend that he had to pay to the long position decreased his return to only $$\frac {(10 – 4)}{100} = 6\%.$$ Ignoring brokerage costs and assuming no borrowing fee, the cash flows obtained from a long position should mirror the cash flows from a short position, i.e., a profit of$200 to a short position trader should mean a loss of $200 to a long position trader. This reading examines the relationship between spot and forward prices of assets that provide no income, provide a known income amount, and provide an income that is a percentage of the asset. ### Assets that Provide No Income These assets include treasury bills, stocks that do not provide dividends, and zero-coupon bonds. #### Illustration Consider a financial asset that costs$50 as of now. The borrowing/lending rate of a financial institute is 5% per year. How can a trader maximize profits if the one-year forward price is:

1. $70 2.$40

#### Example 1: Forward Price of $70 To make a profit, a trader will have to buy the asset today at USD 50 and then sell it a year later at USD 70. For that one year, the cost of funding the asset will equal to $$(0.05 × \text{USD } 50)=\text{USD } 2.50$$ The profit made will therefore be equal to $$\text{USD } 70-\text{USD }50-\text{USD }2.50=\text{USD } 17.50$$ A forward price greater than USD 52.25 (spot price of the asset today plus the cost of funding the asset in one year) guarantees the trader a profit with zero risks. #### Example 2: Forward Price of$40

To make a profit, a trader will have to sell the asset (at USD 50) and enter into a contract to buy it back a year later (at USD 40).

## Question 3

The price of a six-month futures contract on an equity index is currently at USD 1,215. The underlying index stocks are valued at USD 1,200. The stocks also pay dividends at a rate of 3%. Given that the risk-free rate is 5%, determine the potential arbitrage profit per contract.

A. $3 B.$12

C. $15 D.$0

The fair value of the futures contract, $$F$$, is given by:

$$F=S\times \left(\frac{ (1+r)}{(1+r^{*})} \right)^{T}$$

Where:

$$S$$ = current value of the underlying

$${ r }^{ \ast }$$=rate of dividends

$$r$$=risk-free rate

$$T$$=time to maturity=$$\frac { 6 }{ 12 }$$

\begin{align*}F&=S\times \left(\frac{ (1.05)}{(1.03)} \right)^{0.5}\\ &=1,211.59\end{align*}

Thus, the actual futures price is too high by $$3\left( =1,215–1,212 \right)$$.

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2021-03-24
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2021-02-18
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