1. Chapter 5 Determination of Forward and Futures Prices
Geng Niu

2. Consumption vs Investment Assets
Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver)
Consumption assets are assets held primarily for consumption (Examples: copper, oil)

3. Short Selling (Page )
Short selling involves selling securities you do not own
Your broker borrows the securities from another client and sells them in the market in the usual way

4. Short Selling (continued)
At some stage you must buy the securities so they can be replaced in the account of the client
You must pay dividends and other benefits the owner of the securities receives
There may be a small fee for borrowing the securities

5. Example
You short 100 shares when the price is $100 and close out the short position three months later when the price is $90
During the three months a dividend of $3 per share is paid
What is your profit?
What would be your loss if you had bought 100 shares?

6. Purchase of shares
Purchase 100 shares for $ ,000
Receive dividend
Sell 100 shares for $
Net profit=-700
Short sale of shares
Borrow 100 shares and sell them for $100
+10,000
Pay dividend
Buy 100 shares for
Net profit =700

7. Assumptions 1. No transaction costs
2.Same tax rate on all net trading profits
3. Investors can borrow money at the same risk-free rate of interest as they can lend money
4. Investors take advantage of arbitrage opportunities as they occur

8. Notation for Valuing Futures and Forward Contracts
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T

9. Current stock price is $40
Consider:
A long forward contract to purchase a non-dividend-paying stock in 3 months
Current stock price is $40
3-month risk-free interest rate is 5% per annum.
What should be the forward price?
Wulin Suo

10. An Arbitrage Opportunity?
Suppose that:
The spot price of a non-dividend-paying stock is $40
The 3-month forward price is $43
The 3-month US$ interest rate is 5% per annum
Is there an arbitrage opportunity?

11. Today:
Borrow $40 at the risk-free rate of 5%
Buy one share
Short a forward contract
After 3 month
Pay 40e0.05*3/12 =40.5 for the 40 borrowed
Delivers the share and receives 43
Lock in a profit: =2.5

12. Another Arbitrage Opportunity?
Suppose that:
The spot price of nondividend-paying stock is $40
The 3-month forward price is US$39
The 1-year US$ interest rate is 5% per annum
Is there an arbitrage opportunity?

13. Today:
Short one share and get $40
Invest the $40 at 5% per annum for 3 months
Long the forward contract
After 3 month
Get 40e0.05*3/12 =40.5
Pay 39 to get the share according to the forward contract
Return the share to close out short selling position
Lock in a profit: =1.5

14. The Forward Price
If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then
F0 = S0erT
where r is the T-year risk-free rate of interest.
In our examples, S0 =40, T=0.25, and r=0.05 so that
F0 = 40e0.05×0.25 = 40.50

15. If Short Sales Are Not Possible..
Formula still works for an investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low

16. When an Investment Asset Provides a Known Income
A 10-month forward contract on a stock
Stock price is $50
The risk-free rate is 8% per annum
Dividends of $0.75 per share are expected after 3, 6 and 9 months.
What should be the forward price?
Wulin Suo

17. When an Investment Asset Provides a Known Income
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract

18. When an Investment Asset Provides a Known Income
Buy one unit of the asset
Enter into a short forward contract to sell it for F0 at time T
This costs S0 and leads to a cash inflow of F0 at time T and an income with a present value of I
The present value of the inflow is I +F0e-rt
Hence, S0 = I +F0e-rt
Or, F0 = (S0 – I ) ert

19. Arbitrage opportunity if F0 < (S0 – I )erT ?

20. Example The present value of the dividends income, I, is
I =0.75*e-0.08*3/ *e-0.08*6/12
+ 0.75*e-0.08*9/12 =2.162
T is 10 months, so
F0 = ( )e0.08*10/12 =51.14

21. When an Investment Asset Provides a Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)

22. Example
A 6-month forward contract on an asset is expected to provide income equal to 2% of the asset price once during a 6-month period
Risk free rate is 10% per annum
Asset price is 25
4% per annum with seminannual compounding is 3.96% per annum with continuous compounding, then q=0.0396
F0 = 25e( )* =25.77

23. Valuing a Forward Contract
A forward contract is worth zero (except for bid-offer spread effects) when it is first negotiated
Later it may have a positive or negative value
Suppose that K is the delivery price and F0 is the forward price for a contract that would be negotiated today

24. Valuing a Forward Contract
By considering the difference between a contract with delivery price K and a contract with delivery price F0 we can deduce that:
the value of a long forward contract, ƒ, is
(F0 – K )e–rT
the value of a short forward contract is
(K – F0 )e–rT

25. Forward vs Futures Prices
When the maturity and asset price are the same, forward and futures prices are usually assumed to be equal.
When interest rates are uncertain they are, in theory, slightly different:
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

26. Forward vs Futures Prices
Suppose that the price of the underlying asset, S, is strongly positively correlated with interest rates, R
If S increases, a long position trader makes an immediate gain because of the daily settlement
It is also likely that interest rate have also increased, thus the gain can be invested at a higher rate.

27. Forward vs Futures Prices
If S decreases, a long position trader will incur an immediate loss.
It is likely that interest rates have also decreased.
Thus, the loss could be financed at a lower rate.
A forward contract holder is not affected in this way.
Hence, in this case future contracts are more attractive and futures prices tend to be slightly higher.

28. Stock Index
Can be viewed as an investment asset paying a dividend yield
The futures price and spot price relationship is therefore
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract

29. Stock Index (continued)
For the formula to be true it is important that the index represent an investment asset
In other words, changes in the index must correspond to changes in the value of a tradable portfolio
The Nikkei index viewed as a dollar number does not represent an investment asset (See Business Snapshot 5.3, page 113)

30. Example A 3-month futures contract on an index
Dividend yield of 1% per annum
Current value of the index is 1300
Risk-free rate is 5%
F0 =1300e( )*0.25 =1,313.07
Note: dividend yield=dividend/price

31. Index Arbitrage
When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures
When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

32. Index Arbitrage (continued)
Index arbitrage involves simultaneous trades in futures and many different stocks
Very often a computer is used to generate the trades
Occasionally simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold (see Business Snapshot 5.4 on page 114)

33. Futures and Forwards on Currencies
S0 : the current 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.
r : US risk-free rate
rf : foreign risk-free rate

34. Futures and Forwards on Currencies
A foreign currency is analogous to a security providing a yield
The yield is the foreign risk-free interest rate
It follows that if rf is the foreign risk-free interest rate
Interest rate parity in international finance.

35. Explanation of the Relationship Between Spot and Forward
1000 units of foreign currency (time zero)
1000S0 dollars at time zero
1000S0erT
dollars at time T

36. Futures on commodities
Some assets provide income to the holders, e.g. gold and silver
Commodities also have storage costs.

37. Income and storage costs
If U is the present value of all the storage costs, net of income, then F0 = ( S0 + U) erT If u denotes the storage costs per annum as a proportion of the spot price net of any yield earned on the asset, we have F0 = S0 e(r+u)T

38. Arbitrage strategies for consumption asset
Suppose that,
F0 > (S0 + U)erT
an arbitrageur can:
Borrow S0 + U at the risk-free rate and use it to purchase one unit of the commodity and to pay storage costs,
Short a futures contract on one unit of the commodity

39. Arbitrage strategies for consumption asset
Suppose next that,
F0 < (S0 + U)erT
an arbitrageur can:
Sell the commodity, save the storage costs, and invest the proceeds at the risk-free rate
Long futures contract.

40. Arbitrage strategies for consumption asset
However, companies who own a consumption commodity usually plan to use it in some way and are reluctant to sell the commodity in the spot market and buy forward.

41. Arbitrage strategies for consumption asset
Thus, the inequality can hold and we have:
F0  S0 e(r+u )T
where u is the storage cost per unit time as a percent of the asset value.
Alternatively,
F0  (S0+U )erT
where U is the present value of the storage costs.

42. Convenience Yields
Convenience yields, y, refer to the benefits from holding the physical asset.
y is defined such that
F0eyT = (S0 + U)erT Or
F0eyT = S0 e(r+u)T
which gives: F0 = S0 e(r+u-y)T

43. The Cost of Carry
The cost of carry, c (=r+u), is the storage cost plus the interest costs less the income earned
For an investment asset F0 = S0ecT
For a consumption asset F0  S0ecT
The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T

44. Futures Prices & Expected Future Spot Prices
Consider a speculator who takes a long position in a futures contract
The speculator puts the present value of the futures price into a risk-free investment today
And simultaneously takes a long futures position
Today’s cash flow: -F e-rT
Cash flow in the end of the contract: ST

45. Futures Prices & Expected Future Spot Prices
Suppose k is the expected return required by investors in an asset
The present value of the investment is
-F0 e-rT + E (ST )e-kT
We can assume that the present value is zero, so
-F0 e-rT + E (ST )e-kT =0

46. Futures Prices & Future Spot Prices (continued)
No Systematic Risk
k = r
F0 = E(ST)
Positive Systematic Risk
k > r
F0 < E(ST)
Negative Systematic Risk
k < r
F0 > E(ST)