Investment vs Consumption Assets, Short Selling

• Investment assets are held for investment purposes by at least some traders, examples include stocks, bonds, gold, and silver.
• Consumption assets are held primarily for consumption and not normally for investment, such as copper, crude oil, corn, and pork bellies.
• Arbitrage arguments can be utilized to determine the forward and futures prices of an investment asset from its spot price and other market variables.
• This approach cannot be used for consumption assets.

Short Selling

• Short selling (or "shorting") involves selling an asset an investor does not own.
• It is possible for some investment assets.
• To short sell, a broker borrows shares from an owner and sells them in the market.
• Later, the investor buys the same number of shares to replace the borrowed ones, closing the position.
• Profit is made if the stock price declines, while a loss occurs if it rises.
• If the borrowed shares must be returned and no others can be borrowed, the investor must close the position.
• A fee may be charged for lending the shares.
• Short sellers must pay the broker any income (dividends, interest) that would normally be received on the shorted securities, and the broker transfers this income to the lender.
• Example: Shorting 500 shares at $120 in April, dividend of $1 per share in May, and buying back at $100 in July results in a net gain of $9,500, without borrowing fees.
• Cash flows from a short sale mirror those from purchasing shares and selling them later.
• Investors must maintain a margin account with the broker to cover potential losses if the share price increases.
• Initial margin is required; additional margin may be needed for adverse price movements.
• Failure to provide additional margin leads to the short position being closed out.
• Interest is usually paid on margin account balances, or Treasury bills can be used to meet margin requirements.
• Regulations on short selling have evolved, including the uptick rule (allowing shorting only on an "uptick").
• The SEC abolished the uptick rule in 2007, but introduced an "alternative uptick" rule in 2010, restricting short selling when a stock's price drops over 10% in one day.
• Temporary bans on short selling have occurred to curb market volatility.

Assumptions and Notation

• Common assumptions for market participants include no transaction costs, the same tax rate on net trading profits, and borrowing/lending money at the same risk-free interest rate.
• Market participants take advantage of arbitrage opportunities when they arise.
• These assumptions need to be true for only a few key market participants, such as large derivatives dealers.
• Notation:
  • T: Time until delivery date (in years)
  • S0: Current price of the underlying asset
  • F0: Forward or futures price today
  • r: Zero-coupon risk-free interest rate per annum with continuous compounding for an investment maturing at the delivery date.
• The risk-free rate (r) is the rate with no credit risk.

Forward Price for an Investment Asset

• The easiest forward contract to value is one on an investment asset that provides no income.
• Non-dividend-paying stocks and zero-coupon bonds are examples.

Generalization

• For a forward contract on an investment asset with price S0 that provides no income, the relationship between F0 and S0 is F0 = S0 * e^(rT).
• If F0 > S0 * e^(rT), arbitrageurs can buy the asset and short forward contracts on the asset.
• If F0 < S0 * e^(rT), arbitrageurs can short the asset and enter into long forward contracts on it.
• A long forward contract and a spot purchase both result in asset ownership at time T.
• The forward price exceeds the spot price because of the cost of financing the spot purchase during the contract's life.

Example 5.1

• For a 4-month forward contract to buy a zero-coupon bond maturing in 1 year, with a current price of $930 and a 4-month risk-free rate of 6% per annum, the forward price F0 is $930 * e^(0.06 * 4/12) = $948.79.

What If Short Sales Are Not Possible?

• Short selling isn't always possible, and fees may apply.
• Equation (5.1) can still be derived without short selling.
• It only requires market participants who hold the asset purely for investment.
• If the forward price is too low, these participants will sell and take a long position in a forward contract.
• If F0 > S0 * e^(rT), investors can borrow S0 dollars at rate r for T years, buy the asset, and short a forward contract, profiting F0 - S0 * e^(rT) at time T.
• If F0 < S0 * e^(rT), asset owners can sell the asset for S0, invest at rate r, and take a long forward position, profiting S0 * e^(rT) - F0 relative to keeping the asset.
• The forward price will adjust to eliminate these arbitrage opportunities, ensuring the relationship in equation (5.1) holds.

Known Income

• Forward contracts on investment assets offer predictable cash income such as stocks paying known dividends and coupon-bearing bonds.

Generalization

• When an investment asset provides income with a present value of I during the life of a forward contract, we have F0 = (S0 - I) * e^(rT).
• If F0 > (S0 - I) * e^(rT), an arbitrageur can buy the asset and short a forward contract.
• If F0 < (S0 - I) * e^(rT), an arbitrageur can short the asset and take a long position in a forward contract.
• If short sales are not possible, asset owners will sell and enter into long forward contracts.

Example 5.2

• For a 10-month forward contract on a stock at $50, with a risk-free rate of 8% and dividends of $0.75 expected after 3, 6, and 9 months, the present value of dividends I is $2.162. The forward price is F0 = (50 - 2.162) * e^(0.08 * 10/12) = $51.14.

Known Yield

• This considers the scenario where the asset underlying a forward contract provides a known yield.
• The income is known as a percentage of the asset’s price at the time of payment.
• Define q as the average yield per annum with continuous compounding.

Generalization

• The formula is F0 = S0 * e^((r - q) * T).

Example 5.3

• With a six-month forward contract on an asset, income equals 2% of the $25 asset price once during the period.
• Given 10% risk-free rate, S0 = 25, r = 0.1, T = 0.5, q = 0.0396, the forward price is F0 = 25 * e^((0.10 - 0.0396) * 0.5) = $25.77.

Valuing Forward Contracts

• The starting value of a forward contract is around zero.
• Later, it can gain or lose value.
• Institutions need to value the contract daily.
• K is the delivery price, T is the delivery date (in years), r is the risk-free interest rate, and F0 is the present forward price.
• It is important to understand the meaning of F0, K, and f.
• f is the value of forward contract today.

General Result

• f = (F0 - K) * e^(-rT) is applicable to all long forward contracts
• Portfolio: (a) forward contract to buy at K, and (b) forward contract to sell at F0. Payoff is known today as (F0 - K) * e^(-rT) at time T.
• Because the forward contract is to sell at F0 is worth zero, the value of a long forward contract must be (F0 - K) * e^(-rT).
• Similarly, value of a short forward contract is (K - F0) * e^(-rT).

Example 5.4

• A long forward contract on a non-dividend-paying stock with 6 months to maturity.
• With a risk-free rate of 10%, a $25 stock price, and a $24 delivery price, S0 = 25, r = 0.10, T = 0.5, K = 24.
• F0 = 25 * e^(0.1 * 0.5) = $26.28.

General Result

• By assuming the price of the asset at maturity equals the forward price F0 you can value forward contracts.
• Using f = (F0 - K) * e^(-rT) and F0 = S0 * e^(rT), the value of a forward contract on a no income bearing investment is f = S0 * e^(rT) - K * e^(-rT)
• Using formula f = (F0 - K) * e^(-rT) and F0 = (S0-I) * e^(rT), the value of a forward contract on a known income investment is f = S0 - I - Ke^(-rT)
• Using formula f = (F0 - K) * e^(-rT) and Fo = Soe^((r-q) * T) is f = S0e^(-qT) - Ke^(-rT)
• When a futures trades, the gain or loss on a futures contract is calculated as the change in the futures price times the size of the position.
• From Equation (5.4) the gain or loss is the present value of the change in the forward price times the size of the position

Are Forward Prices and Futures Prices Equal?

• Theoretical argument suggests that forward and futures prices should be the same when the short-term risk-free interest rate is constant, or is a known function of time.
• When the price of the underlying asset, S, is positively correlated with interest rates, futures prices tend to be slightly higher than forward prices.
• Conversely if interest rates are negatively correlated with the futures price, forward prices will tend to be slightly higher than futures prices
• In practice forward prices and future prices are different for reasons not stated in the theoretical models like taxes, transaction costs, and margin requirements
• However, for most practical purposes it is reasonable to assume they forward and futures prices are the same

Futures Prices of Stock Indices

• Stock indices can be seen as prices of investment assets while paying dividends
• q = dividend yield rate
• Fo = So * e^((r-q)*T) this shows that the future price increases at r - q with the maturity of the futures contract

Index Arbitrage

• If F0 > S0e!r qÞT , profits can be made by buying the stocks underlying the index at the spot price (i.e., for immediate delivery) and shorting futures contracts.
• If F0 < S0e!r qÞT , profits can be made by doing the reverse—that is, shorting or selling the stocks underlying the index and taking a long position in futures contracts.
• These profits are known as index arbitrage

Forward and Futures Contracts on Currencies

• S0 : the spot price in US dollars of one unit of the foreign currency
• F0 : the forward or futures price in US dollars of one unit of the foreign currency
• A foreign currency has the property that the holder of the currency can earn interest at the risk-free interest rate prevailing in the foreign country

Generalization

• rf : foreign risk-free interest rate when money is invested for time T
• r: risk-free rate in US dollar.
• F0 = S0 * e^(r-rf) * T
• Two ways to convert 1,000 units of foreign currency into dollars at time T,
  • Invest in 1,000 units for "T" years at rf and convert at F0 . This generates 1,000 e^(rfT)F0 dollars.
  • Exchange for dollars at spot rate and invest at "r". Results in, 1,000 S0e^(rT)
  • 1,000 e^(rf)F0 = 1,000 S0e^(rT) because in absence of arbitrage opportunities both strategies must have the same results.
• Currency can be regarded as an investment asset that pays you a dividend
• The value of interest depends on the value of the currency

Futures on Commodities

• First look at future prices of commodities that are investment assets such as gold and silver

Income and storage costs:

• Gold owners such as bank charge interest when they land gold "gold lease rate"
• Therefore, gold can provide income to the holder. The same is true for silver.
• Commodities generally include storage costs

Generalization

• in the absence of storage costs and income, the forward price of a commodity that is an investment asset is F0 = Soe^(rT)
• If you treat storage costs as negative income, and U: the present value of storage costs, that in all equations leads to Fo = ( S0+U ) e^(rT)
• Alternatively, if storage costs are proportional to the price of the commodity Fo = Soe^((r+u)*T) where u denotes proportional storage cost
• Commodities that are consumption assets, and in significant storage cost
• Assume same delivery location for spot and futures

Commodity Arbitrage

• Fo > ( So + U ) *e^(rT) implementation of the trading strategies lead to profit of Fo - ( So + U )e^(rT) at time "T"
• As arbitrageurs execute this there is tendency for "So" to increase and "Fo" to decrease until conditions no longer there -> conclusion cannot hold.

Generalization

• Fo < ( So = U )*e^(rT) which cannot be used on commodities
• (1) Investors decide to sell the commodity and invest the proceeds at the risk free interest
• (2) Buy a long position in a futures contract
• Because this cant be guaranteed F0 < ( S0 + U)e^(rT), and all we can assert for a consumption commodity is F0 cannot equal ( S0 + U )e^(rT)

Convenience Yields

• Manufacturers view spot prices in terms of storage because a future cannot be used as a resource, the benefits form holding the physical assets are the convenience yield of a commodity
• For the dollar amount if storage costs y that is know with a present value U, then the convenience yield is y = F0e^(yT)
• Then from then Fo^(yT) = ( So+U )e^(rT)
• = So * e^(r+u)- Soe^(r+u)*T

The cost of carry

• It measures storage cost + the interest - the investments, (dividends for stocks, rates in foreign currency contracts)
• C: defines the cost of carry

Generalization

• Future prices that lead Soe^(C*T)
• *Future prices that lead Soe^(C-y)*T
• y: Convenience yield

Delivery Options

• Futures allow for shorting positions to choose a time during a period allowing some complication
• Increasing price = benefits of holding asset are less than risk ( deliver early )
• Declining price = the opposite is true and deliver late
• Contract should match maturity

Futures Prices and Expected Future Spot Prices

• Futures price is below the spot price at maturity expecting the September futures price to the client, so those with short positions again while those long lose

Risk and Return

• Higher risk, higher expected return
• The capital asset pricing model has
  • Systematic - cant be diversified away
  • Non-systematic - can be eliminated

The Risk in a Futures Position

• • ST: asset price at time T, end fo futures contract
• Expected returns on an investment depend on its systematic risk

Generalization

• Returns on an investment depend on It's systematic risk
• With returns as a set if K= R
• With S0 expected returns on an investment depend on It's systematic risk
• Results are summarized in Table 5.5

Normal Backwardation and Contango

• Normal backwardation - Expected spot price exceeds a future
• Contango - expected spot price is less than future