Chapt 17

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Forwards and Futures

A spot contract is an agreement between two parties ( bi-lateral) to transact, involving the immediate (time - 0) exchange of assets (securities) and funds (cash). A spot bond quote of 97% for a 20-year maturity bond is the cash price that the buyer pays the seller for the delivery of $100,000 face value of the security; (definition - bi-lateral transaction between two parties may introduce counterparty risk.

A forward contract is a nonstandardized (contractual) agreement between a buyer and a seller at time - 0, to transact involving the future exchange of a set amount of assets at a set price ( OTC and bi-lateral) and a set date (the "expiration date"); forwards allow the market participant (the "buyer/seller") to hedge/speculate on the risk that the future spot prices on an asset will move adversely (increase/decrease in cost); forward prices are fixed over the life of the contract; cash is only exchanged between the parties at the expiry date of the forward agreement - there is counterparty risk

A forward contract agreement between a buyer and a seller at time - 0, to transact involving the future exchange of a $100,000 face value 20-year bond from the seller, at a set price of 97% paid by the buyer, at set date in three months (the "expiration date").

A futures contract is a standardized futures agreement between two parties, whose payment is guaranteed by the exchange on which the futures are listed and trade (no-counterparty risk), involving the exchange's clearinghouse as an credit intermediators, for an exchange of a set amount of an asset for a price that is settled daily; The price of the futures contract changes daily as the market value of the asset underlying the futures contract fluctuates (mark-to-market).

Hedging Strategies Using Futures

Investor chooses to use futures markets in order to hedge a risk, the objective is to take a position that neutralizes the risk as far as possible

Short Hedge Futures: If the price of the underlying goes down, the gain on the futures position offsets the loss on the rest of the company's business in the cash market

Short Hedge Futures: If the price of the underlying goes up, the loss on the futures position offsets the gain on the rest of the company's business in the cash market

A short hedge is appropriate when hedger already owns an asset and expects to sell it in the future

A copper fabricator knows it will require 100,00 pounds of copper on May 15th to meet a certain contract. The spot price of copper is 140 cents per pound and the May 15 futures price is 120 cents per pound

A Long Hedge StartegyJanuary 15: Take a long position in 4 May futures contracts (20,000 contract size)

May 15: Close out the position

Long Hedge Futures: A long hedge is appropriate when a company knows it will have to purchase a certain asset in the future and wants to lock in a price now

ExampleA Long Hedge - January 15th

Interest rates and future prices move in opposite directions; when interest rates go up, the price of the asset underlying the futures contract goes down, this, in turn causes the futures price itself to go down; an investor who is exposed to higher interest rates, should hedge by taking a short position

When interest rates go down, the reverse happens, and the future price goes up; the hedger exposed to lower interest rates should take the long-position

Result: The company ensures that its cost will be close to 120 cents per pound.

Example 1: Cost of copper on May 15 is 125 cents per pound; the compnay gains: 100,000 X (1.25 - 1.20) = 5,000 on the future contract; total cost 125,000 - 5,000 = 120,000

Example 2: Cost of copper on May 15 is 105 cents per pound; the compnay losses: 100,000 X (1.05 - 1.20) = -15,000 on the future contract; total cost 105,000 + 15,000= 120,000

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Example: $ 100,000.00 90-05 0.15625 90.00 0.9016 90,156.25$ $ 100,000.00 95.16 0.50 96.00 0.9650 96,500.00$

► Cash Price = Quoted Price +

132.3438% 134.0313%

Futures Price

Payoff Profit

Profit or Loss on a Futures Position in Treasury Bonds taken on November 15, 2007

Buy (long) or sell (short) December treasury bond futures contract traded on the CBT at a price of 133.1875 ; percent of the face value of the T-bond futures contract at $100,000

Long position in one bond can be taken at 133.1875% of $100,000 or $133,187.50

Long Position

Payoff Loss

Futures Price Rise

0

Rates ( r )

$ 843.75

$ (843.75)

133.1875%( r )Incr.

Futures Price Falls

( r ) Decr.

Quoted price in dollars and in thirty seconds of a dollar; with face value of $100

Accrued Interest

132.3438% 134.0313%

See Swaps FM&I Chapter 10

TSoKfF

r

time to maturityprice of asset underlying forward contract todaydelivery price in forward contractvalue of long forward contract todayforward price today

risk-free rate of interest per annum today, continuous compounding thru maturity in T years.

At contract initiation: F = K; and f = zero

0 Futures Price

$ 843.75

Short Position

( r ) Incr. Payoff Profit $ 843.75 Futures Price

Falls

Rates ( r )

133.1875%

( r )decr. Payoff LossFutures

Price Rise

So 40T 3 monthsr 0.05 p.a.D 40.5

12

Initial CF0 T = 3 mos.1 -40 0.02 0 40.5

SUM -40 40.5

Return: 1.26%

A Arbitrage: F 43B Arbitrage: F 39

Borrow:

Sell Forward Contract

Initial CF0 T = 3 mos.1 40 -40.52 0 43.0

SUM 40 2.5

Enter into a forward contract to sell the shares in 3 months

40 at .05; buy one share

short forward contract in three mos.

A

FORWARD CONTRACTS ON A SECURITY THAT PROVIDES NO INCOME

Combined Strategies: F = K; and f = zero

Buy one stock

Sell:

But Forward Contract

Initial CF0 T = 3 mos.1 40 40.52 0 -39.0

SUM 40 1.5

f

S =

F =

f =

Sell So at 40; invest proceeds at .05; buy one share

Long forward contract at 39 in three mos.

B

Current Price B = 900T = 1.0r = 6 mos. = 0.09r = 12 mos. = 0.1Coupon s.a = 40.00Fo = 1 year = 912.39

f = 0 -900 + 40e^-0.09*0.5+40e^-0.1*1 + Fe-.10*1f = 0 -900 + 74.43 + 825.57

Fo = 912.39

Buy Bond -900Borrow Proceeds 6 mos at 0.0938.23 40.0Borrow Proceeds 1 year at 0.10861.77 952.40Total Proceeds: 900Sell Fo 930At T Recv coupon 2nd. Coupon 40At T Recv.Fo 930Profit = 17.60

Sell Bond Short (owe coupons)900Invest PV of 40 Proceeds 6 mos at 0.09-38.23 40.0Invest Proceeds 1 year at 0.10-861.77 952.40Total Proceeds: 0Buy Fo -905At T Pay 2nd. Coupon -40At T Buy Fo -905Profit = 7.40

Example

FORWARD CONTRACTS ON A SECURITY THAT PROVIDES A KNOWN CASH INCOME

Arbitrage Opportunity when Forward Price on Coupon Paying Bond Is Too High

Arbitrage Opportunity when Forward Price on Coupon Paying Bond Is Too Low

(S - I)^erT

Portfolio A

825.57

f =

Fo =

PPP Therom:

Foreign currency exchange rates between two countries adjust to reflect changes in each countries price levels (inflation rates) as consumers and importers switch their demands for goods from relatively high inflation rate countries to low inflation countries.

the change in the exchsnge rate between two countries' currencies is proportional to the difference in the inflation ratres in the two countries; that is:

IPus - IPS = ∆S/us/s/Sus/s

According to PPP, the most important factor determining exchange rates is the fact that in open economies, the differences in prices (and by implication price level changes with inflation drive trade flows and thus demand for and supplies of currencies

Example

Purchasing Power Parity (PPP)Impact of difference between relative inflation rates between two countries on the currency exchange rate between them:Directly reflected in relative interest rates in these countries:

ius = IPus + RIRus and is = IPs + RIRsAssuming: RIRus = RIRs; thenius - is = IPus - IPS; the nominal interest rate spread between the United States and Switzerland reflects the difference in inflation rates

Interaction of Interest Rates, Inflation, and Exchange Rates

Higher domestic interest rates may attract foreign financial investment and impact the value of the domestic currencyFisher Effect:: the relationship between nominal interest rates (I), real interest rates (RIR), and expected inflation (IP)(1) compensates investors for any reduced purchasing power due to inflationary price (IP) changes, and

(2) provide an additional premium above the expected rate of inflation for forgoing present consumption due to the time value of money (which reflects the real interest rate); such that: I = IP + RIR

Current spot rate:us/s 0.9691$ IPs 0.1Ipus 0.04

IPus - IPS = ∆S/us/s/Sus/s= -0.06 = ∆Sus/s/.9691 IPus - IPS = ∆S/us/s/Sus/s= (0.0581) = ∆Sus/s

0.9110

Current spot rate: us/C 0.9622$ 6 - mos Forward: us/C 0.9586$

IRPT: =Return on U.S. investment =

The hedged dollar return on foreign investments just equals the return on domestic investments

1 + iust = (1/St) X (1 + iukt) X FtHedged return on foreign (U.K.) investment

Thus; the Swiss Franc now purchases 5.81 cents less against the dollar:Application of PPP: over the course of one year, the Swiss franc will depreciate in value by 5.81% against the U.S. dollar.

Law of one Price: in the long-run exchange rates should move toward rates that would equalize the price of an identical basket of goods and services in any two countries.

Interest Rate Parity Theorem (IRPT): The relationship that links spot exchange rates, interest rates and forward exchange rates is described as the interest rate parity theorem

The forward exchange rate is determined by the spot exchange rate and the interest rate differential between the two countries

Investors have an opportunity to invest in domestic or foreign markets, the IRPT implies that, by hedging in the forward exchange rate market, an investor should realize the same returns, whether investing domestically or in a foreign country

Application of PPP

T = 1 monthiuk 0.0050 ius 0.0048 = 0.641272284 X 1.005 X 1.5591St: $1.5594/₤ 1.5594Ft: 1.5591/₤ 1.5591 -0.000192 -0.000192 100,480.67

Portfolio A

Portfolio B

Portfolio A =

Discount spreads betw: dom.and fr. interest rates are equal in perct. terms to the spread bet. Forw. & Spot Exchange

Example

=

(iust - iukt)/(1+iukt) = (Ft - St)/St

One long forward contract plus an of cash equal to Ke^-rT

An amount e^-rfT

Portfolio B

Both portfolios will become worth the same as one unit of the foreign currency at time T; they therefore must be equally valuable at time t.

Forward and Futures Contracts on Currencies

Variable, S, is the current price of dollars of one unit of the foreign currency

Variable, K, is the delivery price agreed to in the forward contract

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; invest in a foreign denominated bond

Define rf as the value of this foreign risk-free interest rate with continuous compounding.

The two portfolios that enable us to price a forward contract on a foreign currency are:

Fo = Se^(r - rf)T

$ Per Fr. Crry. Fr. Crry per $Futures Buys In $ $ Buys

Canadian Dollar 0.8378$ 1.19 British Pound 1.6822$ 0.59 Japanese Yen 0.0076$ 131.49 Euro area euro 1.2819$ 0.78 China Yuan 0.1473$ 6.79 Swiss Franc 0.9691$ 1.03

Current Price Spot = 0.5243T = 0.500

r Domestic = 0.072

rf Foreign = 0.036Fo = 0.5338

F = 0.5338 0.5243 0.27175933

Canadian DollarBritish PoundJapanese Yen

If rf > r; F < S and F is always less than S, and that F decreases as the maturity of the contract, T, increases

If rf < r; F > S and F is always greater than S, and that F increases as the maturity of the contract, T, increases

Example

Euro area euroChina YuanSwiss Franc

3 month future contract to buy a S&P 500.

f + Ke^-rT = Se^-rfT

f = Se^-rfT - Ke^-rT

The forward price (or forward exchange rate), F, is the value of K that makes f = 0.

This is the well known interest rate parity relationship from the field of international finance

Forwards

Current Price S = 400T = 0.250r = 0.08q = 0.03Fo =

F = 405.03

Are Forward Prices and Future Prices Equal:

Ceterus paribus, a long futures contract will be more attractive than a long forward contract; when S is strongly positively correlated with interest rates, futures prices will tend to be higher than forward prices.

Ceterus paribus, a long futures contract will be less attractive than a long forward contract; when S is strongly negatively correlated with interest rates, forward prices will tend to be higher than future prices.

Futures Prices of Stock IndicesTo a reasonable approximation, the stock underlying the index can be assumed to provide a continuous dividend yield. If q is the dividend yield F = Se^(r-q)T

Example

Stock Index Futures:A stock index tracks the changes in the value of a hypothetical portfolio of stocks

The percentage increases in the value of a stock index over a small interval of time is usually defined so that it is equal to the percentage increase in the total value of the stocks comprising the portfolio at that time

A stock index is usually not adjusted for cash dividends; dividends are ignored when percentage changes in most indices being calculated.

3 month future contract to buy a S&P 500.

We observe that future prices appear to increase with maturity by an amount by which the risk-free rate exceeds the dividend yield.

Futures on Commodities

Current Price Gold (oz) = 450T = 0.250r = 0.07U (storage per oz) = 2Fo =

U (2e^-0.07) 1.86

F = 484.63

However, it turns out that they can only be used to give an upper bound to the futures price in the case of consumption commodities.

Gold and SilverStorage costs can be regarded as negative income. If U is the present value of all storage costs that will be incurred during the life of a futures contract, it follows:

F = (S + U)e^rT

Arbitrage arguments can be used to obtain exact futures prices in the case of investment commodities

F <= (S + U)e^rT; condition can hold due to users unwilling to sell needed commodity (inventory input to production)

For the consumption commodity:

Convenience Yield: yFe^yT = (S + U)e^rt

If storage costs incurred at any time are proportional to the price of the commodity, they can be regarded as providing a negative dividend yield:

F = Se^(r+u)T

Example1- year futures contract on gold.

The convenience yield simply measures the extent to which the left-hand side is less than the right-hand side

Non-dividend paying stock= rStock Index = r - qCurrency = r - rfCommodity; w/ storage cost= r + u

Asset

Provides no income

Provides known income with PV I

Provides known dividend yield, q

no storage Cts.& no incomeincome earned at rate q on the asset

proportional finance & storage cost

The future prices of commodities such as coffee, copper, and crude oil decrease as the maturity of the contract increases; this suggest that the convenience yield, y, is greater than r + u

F = Se^(r + u - y)

The Cost of Carry

Se^ -qT - Ke^ -rT Se^(r -q)T

F = Se^cT; for an investment assetF = Se^(c-y)T; for a consumption asset

Summary of Results for a Contract with Maturity T on an Asset with Price S

S - Ke^ -rT Se^rT

Value of Long forward Contract w/ Delivery Price K Forward/Futures Price

S - I - Ke^ -rT (S - I)^ erT

The relationship between future prices and spot prices

This cost measures the storage cost plus the interest that is paid to finance the asset less the income earned on the asset.

The Cost of Carry

Define The Cost of Carry as C

Forwards and Futures

A spot contract is an agreement between two parties ( bi-lateral) to transact, involving the immediate (time - 0) exchange of assets (securities) and funds (cash). A spot bond quote of 97% for a 20-year maturity bond is the cash price that the buyer pays the seller for the delivery of $100,000 face value of the security; (definition - bi-lateral transaction between two parties may introduce counterparty risk.

A forward contract is a nonstandardized (contractual) agreement between a buyer and a seller at time - 0, to transact involving the future exchange of a set amount of assets at a set price ( OTC and bi-lateral) and a set date (the "expiration date"); forwards allow the market participant (the "buyer/seller") to hedge/speculate on the risk that the future spot prices on an asset will move adversely (increase/decrease in cost); forward prices are fixed over the life of the contract; cash is only exchanged between the parties at the expiry date of the forward agreement - there is counterparty risk

A forward contract agreement between a buyer and a seller at time - 0, to transact involving the future exchange of a $100,000 face value 20-year bond from the seller, at a set price of 97% paid by the buyer, at set date in three months (the "expiration date").

A futures contract is a standardized futures agreement between two parties, whose payment is guaranteed by the exchange on which the futures are listed and trade (no-counterparty risk), involving the exchange's clearinghouse as an credit intermediators, for an exchange of a set amount of an asset for a price that is settled daily; The price of the futures contract changes daily as the market value of the asset underlying the futures contract fluctuates (mark-to-market).

Hedging Strategies Using Futures

Investor chooses to use futures markets in order to hedge a risk, the objective is to take a position that neutralizes the risk as far as possible

Short Hedge Futures: If the price of the underlying goes down, the gain on the futures position offsets the loss on the rest of the company's business in the cash market

Short Hedge Futures: If the price of the underlying goes up, the loss on the futures position offsets the gain on the rest of the company's business in the cash market

A short hedge is appropriate when hedger already owns an asset and expects to sell it in the future

A copper fabricator knows it will require 100,00 pounds of copper on May 15th to meet a certain contract. The spot price of copper is 140 cents per pound and the May 15 futures price is 120 cents per pound

A Long Hedge StartegyJanuary 15: Take a long position in 4 May futures contracts (20,000 contract size)

May 15: Close out the position

Long Hedge Futures: A long hedge is appropriate when a company knows it will have to purchase a certain asset in the future and wants to lock in a price now

ExampleA Long Hedge - January 15th

Interest rates and future prices move in opposite directions; when interest rates go up, the price of the asset underlying the futures contract goes down, this, in turn causes the futures price itself to go down; an investor who is exposed to higher interest rates, should hedge by taking a short position

When interest rates go down, the reverse happens, and the future price goes up; the hedger exposed to lower interest rates should take the long-position

Result: The company ensures that its cost will be close to 120 cents per pound.

Example 1: Cost of copper on May 15 is 125 cents per pound; the compnay gains: 100,000 X (1.25 - 1.20) = 5,000 on the future contract; total cost 125,000 - 5,000 = 120,000

Example 2: Cost of copper on May 15 is 105 cents per pound; the compnay losses: 100,000 X (1.05 - 1.20) = -15,000 on the future contract; total cost 105,000 + 15,000= 120,000

1.1541

1,154.06 1,177.8125 23.75 133,187.5000

0.84375%

117.2500

Profit or Loss on a Futures Position in Treasury Bonds taken on November 15, 2007

Buy (long) or sell (short) December treasury bond futures contract traded on the CBT at a price of 133.1875 ; percent of the face value of the T-bond futures contract at $100,000

Long position in one bond can be taken at 133.1875% of $100,000 or $133,187.50

Traders in futures market can be either speculators or hedgers.

Speculator in interest rates buys T-bond future at 133.1875 betting interest rates will go down; so as to sell at a higher price at the expiry date of the futures contract 134.0313

Hedger (pension fund) buys (long position) in the T-bond futures as protection against a decrease in fixed rates related to large cash inflow near-term that it plans to invest in long-

term bonds. It is concerned that interest rates may fall by the time it can make the investment and wants to lock in the

yield

Quoted price in dollars and in thirty seconds of a dollar; with face value of $100

Value of a Forward Contract: F ≠ K; and f ≠ zero

Hedgers (Bond Issuer) take a short position in the T-bond futures contract as protection against a decrease in prices or increase in rates (will sell fixed rate debt in one month at the

market); lock-in price (rate).

Hedger (pension fund) buys (long position) in the T-bond futures as protection against a decrease in fixed rates related to large cash inflow near-term that it plans to invest in long-

term bonds. It is concerned that interest rates may fall by the time it can make the investment and wants to lock in the

yield

Speculator in interest rates sells (shorts) T-bond future at 133.1875 betting interest rates will go up; so as to buy at a

lower price at the expiry date of the futures contract 132.3238

So 40T 3 monthsr 0.05 p.a.D 38

12

Initial CF0 T = 3 mos.1 -37.53 38.02 -40.5

SUM -37.53 -2.50

f = 2.47

Arbitrage: f 2.60Arbitrage: f 2.40

Borrow:

Sell Forward Contract

Initial CF0 T = 3 mos.1 40.00 -40.52 -403 40.60

SUM 0.00 0.10

(40.50 - 38)e^-.05*.25

A

Combined Strategies:

Invest PV of Delivery Price 38 at .05 for 3 mos.Enter into a forward contract (value = f) to buy the shares in 3 months

40 at .05; buy one share

short forward contract in three mos.

Portfolio A = Portfolio B

f = 2.47

S = 40.00 r = 0.05 T = 0.25

F = 40.00 * e^.05*.25 40.50

f = = 2.47 2.47

This means that: F = Se^rT

It follows that: f + Ke^ -rT = S

(40.50 - 38)e^-.05*.25

This equation corresponds to the observation that the value of a long forward contract is equal to the present value of the amount by which the Forward Price exceeds the delivery price.

OR

f = S - Ke^ -rT

(40.50 - 38)e^-.05*.25

The Forward Price, F, is therefore that value of K which makes f = 0 in the above equation

A more Formal Argument (determining f and F)

One long forward contract on the security plus an amount of cash equal to Ke^-rTPortfolio A

Portfolio B One security

Current Price S = 930T = 0.333r = 0.06

F = 948.79

If F (Fo) < 948.79 If F (Fo) > 948.79Buy Fo Buy (S)Sell S Sell FoInvest S at r 930 Borrow S at r for 4 mos.At T Recv. 948.79 At T Repay Ln.

At T Pay < 948.79 Profit = Fo - LoanProfit = Recv. Amt - Fo

Current Price B 900T 1r = 6 mos. 0.09r = 12 mos. 0.1Coupon s.a 40Delivery 890Fo 912.4

Value of a Forward Contract: F ≠ K; and f ≠ zero

4 month forward contract to buy discount bond that will mature in one year.

Example

PV of Coupons = 74.43PV of Delivery P = 805.31

f + 805.31 + 74.43 = 900f = 20.26 805.31 + 74.43 = 900

f = (912.40 - 890)*e^-10 20.27

Portfolio A = Portfolio B

825.57 = 825.57

f = 20.26

Fo = (S - I)e^-rTFo = 912.39

f = (F - K )e^-rT 20.26

f + Ke^-rT = S - I

f = S - I - Ke^-rT

Example

The difference between owning one bond $900 today and owning $805.31 (PV of delivery price) plus a forward contract is the present value of the coupon payments.

A more Formal Argument: analyzing a forward contract on a security paying known income (I)

Portfolio A One long forward contract on the security plus an amount of cash equal to Ke^-rT

The value of a long forward contract can again be shown to be the present value of the extent to which the current forward price exceeds the delivery price.

Portfolio B One security, plus borrowings of amount I at the risk-free rate

Current Price S 50

T 0.83

r = 3,6,9 mos. 0.08

Div. quarterly 0.75

I 2.162

Fo 51.14

I = 2.162

Fo = 51.14

25.683194

10 month forward contract to buy stock with a price of $50

A more Formal Argument: analyzing a forward contract on a security paying known dividend income (q)

Portfolio A One long forward contract on the security plus an amount of cash equal to Ke^-rT

0.980198726.465364

Portfolio A = Portfolio B

Fo = Se^(r-q)T

If F < Se^(r-q) buy forward; short the stock

If F > Se^(r-q) sell forward; buy stock

f = (F - K )e^-rT

Current Price S = 25

T = 0.500r = 0.1q = 0.04Delivery Price = 27.00

Example

6 month forward contract to buy a stock that is expected to provide a continuous dividend yield of 4%

Portfolio B e^-qT of the security with all income being reinvested in the security

f + Ke^-rT = Se^-qT

f = Se^-qT - Ke^-rT

f= -1.18f = (F - K )e^-rTf= -1.18

Fo = Se^(r-q)TFo = 25.76

f = Se^-qT - Ke^-rT

Delivery Price > Fo