Determination of Forward
and Futures Prices
International Investments and Capital Markets Prof. Hyounggoo KANG
Agenda
• Investment asset vs. consumption asset • Short selling& examples
• Forward price under known yield& example • Forward price vs. futures price
• Delivery
• Index arbitrage • Cost of carry
Types of Assets
INVESTMENT ASSET CONSUMPTION ASSET
-Asset that is held for
investment purposes by a representative number of investors
- E.g. Stocks, bonds, gold
silver
- Investment assets do not
have to be held exclusively for investment
- Forward and futures prices
can be derived for
investment assets from its spot price or other
observable market variables
-Asset held primarily for
consumption purposes
- E.g. Copper, oil, pork
- Forward and futures prices
cannot be determined from their spot prices for
Short Selling
• A type of arbitrage strategy
• Involves the sale of an asset that is not owned • Possible only for some investment assets
• Suppose an investor instructs a broker to short 500 IBM shares
• Broker will borrow the shares from another client and sell them in the market as per normal
• Investor will have to close out the position by purchasing 500 IBM shares
Short Selling cont’d
• Investor takes a profit if the stock price has
declined and a loss if the stock price has risen
• If the broker is unable to borrow any shares at
any time while the contract is still open, investor is forced to close out the position
• Investor with a short position has to pay the
Short Selling - Table
5.1
•
Cash flows from the short sale
and purchase of shares
April Purchase 500 shares for $120
-$60,000
May Receive dividend +$500
July Sell 500 shares for $100 per share
+
$50,000
Short Selling - Table
5.1
•
Short sale of shares
April Borrow 500 shares and sell them for $120
+
$60,000 May Pay dividend -$500
July Buy 500 shares for
$100 per share
Replace borrowed
shares to close short position
-$50,000
Example: Table 5.2
• Arbitrage opportunities when forward price is out of line with spot price for asset providing no income:
• Asset price = $40; interest rate = 5%; maturity of forward contract = 3 months.
• When Forward Price = $43;
• Borrow $40 at 5% now for 3 months
• Enter into a forward contract to sell asset in 3 months for $43
• Sell asset at $43 in 3 months time
• Use $40.50 to repay loan with interest
Example: Table 5.2
cont’d
• When Forward Price = $39;
• Short 1 unit of asset to realized $40 now
• Invest $40 at 5% for 3 months
• Enter into a forward contract to buy asset in 3 months for $39
• Buy asset for $39 in 3 months time
• Close short position
• Receive $40.50 from investment
Example: Table 5.3
• Arbitrage opportunities when 9-month forward price is out of line with spot price for asset providing
known cash income:
• Asset price = $900; income of $40 occurs at 4 months; 4-month and 9-month rates are 3% and 4% per annum respectively.
• When Forward Price = $910;
• Borrow $900 now: $39.60 for 4 months and $860.40 for 9 months
• Buy 1 unit of asset
Example: Table 5.3
cont’d
• Receive $40 income on asset in 4 months
• Use $40 to repay first loan with interest
• Sell asset for $910 in 9 months
• Use $886.20 to repay second loan with interest
• Profit realized = $23.40
• When Forward Price = $870;
Example: Table 5.3
cont’d
• Invest $39.60 for 4 months and $860.40 for
9 months
• Enter into a forward contract to buy asset in 9 months for $870
• Receive $40 from 4-month investment
• Pay income of $40 on asset
• Receive $886.20 from 9-month investment
• Buy asset for $870, close out short position
Forward Price On A Security
That Provides A Known Yield
• Asset provides a known yield rather than a
known cash income
• Income is known when expressed as a
percentage of the asset’s price at the time of income payment
• Define ‘q’ as the average yield per annum
on an asset during the life of a forward contract with continuous compounding:
Example: Known Yield
• 6-month forward contract on an asset
• Expected to provide income equal to 2% of
asset price once during every 6-month period
• Risk-free rate of interest = 10% after
continuous compounding
• Asset price = $25
Forward Price vs. Futures
Price
• In theory, they are different because:
• Gains (losses) of a futures contract are invested (financed) at the short term interest rate, while the gains (losses) of a forward contract are not realize until it matures or is liquidated.
• A strong positive correlation between interest rates and the asset price implies the futures price is
slightly higher than the forward price
• A strong negative correlation implies the reverse. • When interest rate is uncorrelated with the price of
Futures Price and Value
• Unlike stocks or other assets, a futures price
and its value are different concepts
• The interest rate effect is usually ignored in
Delivery
• If a futures contract is not closed out before maturity, it is usually settled by delivering the assets underlying the contract
• When there are alternatives about what is delivered, where it is
delivered, and when it is delivered, the party with the short position chooses
• Closing out a futures position involves entering into an offsetting trade
• Most contracts are closed out before maturity
• Speculators always close out their positions before .
Index Arbitrage
• If F0 > S0e(r-q)T, profits can be made by buying the stocks underlying the index at the spot price and shorting futures contracts;
- Usually done by corporations holding short-term money market investments
• If F0 < S0e(r-q)T, profits can be made by shorting the stocks underlying the index and taking a long
position in futures contracts. of stock;
Cost of Carry (CoC)
• Relationship between futures prices and
spot prices
• Measures the storage cost plus the interest
that is paid to finance the asset less the income earned on the asset
• Defining the CoC as ‘c’:
• Investment asset futures price: F0 = S0ecT
• Consumption asset futures price: F0 = S0e(c-y)T
Futures Prices and
Future Spot Prices
• Refers to the market’s average opinion
about what the spot price of an asset will be at a certain future time as the expected
spot price of the asset
• Keynes and Hicks: If hedgers hold short
positions and speculators hold long
NormalBackwardation
andContango
• Normal Backwardation: Situation whereby
the futures price is below the expected future spot price
• Contango: Situation whereby the futures
