lecture 2 - forwards and futures ppt

LECTURE 2 - Forwards & Futures 26/02/14

Lecture 2 - Forwards and Futures

FINM2002 Derivatives

FINM7041 Applied Derivatives

1

Review of Previous Lecture

• In last week’s lecture we went through a

broad overview/revision of forward, futures

and options contracts.

• In particular we focussed on the

mechanics of forward and futures markets.

2

Lecture Overview

• In today’s lecture we will discuss forwards and

futures contracts in greater detail, and how they are

related to the spot price of the underlying asset. We

will focus on the following topics:

– Determination of interest rates;

– What is short selling?;

– Determination of forwards and futures prices; and,

– Hedging strategies using forwards and futures contracts.

3

1. Consumption vs Investment Assets

• When considering forward and futures contracts, it is important to

distinguish between investment assets and consumption

assets.

– Investment assets are assets held by significant numbers of people

purely for investment purposes.

• Examples of investment assets are stocks, bonds, gold and silver.

– Consumption assets are assets held primarily for consumption and

NOT usually for investment purposes.

• Examples of consumption assets are commodities such as copper,

oil and pork bellies.

– We can use arbitrage arguments to determine the forward and futures

price of an investment asset from its spot price and other observable

market variables. We CANNOT do this for consumption assets.

4


2. Short Selling • Short selling (also called shorting), involves selling an asset that is NOT

owned.

• It is NOT possible for all investment assets.

• Your broker borrows the securities from another client and sells them

in the market in the usual way.

• At some stage you must buy the securities back so they can be replaced

in the account of the client you originally borrowed them from.

• You MUST pay dividends and other benefits that would have accrued to

the client you borrowed from, if they had still held the shares. In other

words, the client you borrowed from should be no worse off as a result of

lending you their shares.

• Likewise, the client can be no better off. Therefore, if you borrow a

physical asset such as gold off a client, the client must pay you for the

storage costs of the gold.

• The investor (the person who has shorted the asset) benefits if prices fall,

as they sell the asset for a higher price than what they buy it back for.

5

2. Short Selling

• The investor is required to maintain a margin

account with the broker.

• The initial margin is required so that possible

adverse movements (increases) in the price of

the asset that is being shorted are covered.

• The margin account consists of cash or marketable

securities deposited by the investor with the broker

to guarantee that the investor will NOT walk away

from the short position if the share price

increases.

6

2. Short Selling

• Regulators in the United States currently allow a stock to be shorted ONLY on an uptick – that is, when the most recent movement in the price of the stock was an increase.

• In Australia, ONLY a limited number of stocks are allowed to be short sold, called the ASX Approved Securities List.

• Further, in 2008, we saw a ban on various forms of short selling in markets around the world.

7

3. Measuring Interest Rates

• The compounding frequency used for an interest

rate is the unit of measurement.

• From Foundations of Finance, you are familiar

with the need to calculate the effective rate of

interest.

• The difference between quarterly and annual

compounding is analogous to the difference

between miles and kilometres.

8



• In this course, we will mainly use continuous

compounding. As such you will need to ensure that your

interest rates are expressed as continually

compounded interest rates. We will detail any

exceptions to this rule as we progress through the course.

• In the limit, as we compound more and more frequently,

we obtain continuously compounded interest rates.

• $100 grows to $100eRT when invested at a continuously

compounded rate R for time T.

• $100 received at time T discounts to $100e-RT at time

zero when the continuously compounded discount

rate is R.

9


• If a nominal rate of 10% p.a. is

compounded continuously what is the

effective rate?

• eR-1

• e0.10-1

• 2.71828…0.10-1= 10.51%

10

4. Assumptions

• In this lecture we make the following

assumptions regarding market participants:

1. They are subject to no transaction costs when they

trade;

2. They are subject to the same tax rate on all net

trading profits;

3. They can borrow money at the same risk-free rate of

interest as they can lend money; and,

4. They take advantage of arbitrage opportunities as

they occur.

11

5. Forward and Futures Contract Prices

• Remember from Foundations of Finance that:

– It is well known in practice that if interest rates are

constant, a futures contract has the same value

as an otherwise identical forward contract. That is,

although a futures contract has a complicated cash

flow pattern (via the marking to market feature) it can

be valued as though it were a forward contract. Since

a forward contract has only a single cash flow, it is

easy to value. Consequently, it is industry practice to

value futures contracts as though they were forward

contracts.

12



• Remember that:

– Forward and futures contracts can be valued

by recognizing that, in many cases, forward

and futures markets are redundant. This

occurs when the payoff from a forward or

futures contract can be replicated by a

position in:

1. The underlying asset, and

2. Riskless bonds.

13


• Before illustrating this concept, we define the

cost of carry (q) of the underlying commodity.

This is the cost of holding a physical quantity

of the commodity. For wheat the cost of carry

is the storage cost; for live hogs it consists of

storage and feed costs; and for gold it consists

of storage and security costs. Some

commodities have a negative cost of carry. For

example, holding a stock index provides the

benefit of receiving dividends.

14


• Consider the strategy of:

– Borrowing enough money to buy one unit of an

investment asset that provides the holder with no

income, and has no holding costs. Non-dividend paying

stocks and zero-coupon bonds are examples of such

investment assets. The borrower incurs the obligation to

pay for the associated interest through time T; and,

– Entering into a forward or futures contract to sell the

commodity at time T.

The value of this position in terms of the initial (time 0)

and terminal (time T) cash flows is tabulated in the

following table. 15


• Arbitrage Relationship Between Spot and

Forward Contracts

Position Initial Cash Flow Terminal Cash Flow

Borrow and incur cost of carry

S0 -S0erT

Buy one unit of commodity

-S0 ST

Enter 6-month forward sale

0 F-ST

Net portfolio value

0 F- S0erT

16



• Example:

– Consider a four-month forward contract to

buy a zero-coupon bond that will mature one

year from today. (This means that the bond

will have eight months to go when the forward

contract matures.) The current price of the

bond is $930. We assume that the four-month

risk-free rate of interest (continuously

compounded) is 6% per annum. The forward

price, F0, is given by:

40.06

120 930 $948.79F e

17

6. Arbitrage

• In general if:

– Arbitrageurs can make a riskless profit from buying the asset and

entering into a short forward contract on the asset. This strategy

is financed by borrowing funds at the risk free-rate if interest.

– Arbitrageurs can make a riskless profit by shorting the asset and

entering into a long forward contract. The excess funds are

invested at the risk-free rate of interest until they are needed to

buy back the asset.

0 0

rTF S e

0 0

rTF S e

18


on Assets with Known Income

• We now consider a forward contract on an investment

asset that will provide a perfectly predictable cash

income to the holder.

Examples are stocks paying known dividend yields and

coupon-bearing bonds.

• Where D is the present value of the income.

0 0( ) rTF S D e

19

7. Forward and Futures Contract Prices on

Assets with Known Income

• Consider a long forward contract to purchase a

coupon-bearing bond whose current price is $900.

We will assume that the forward contract matures in 9

months. We also assume that a coupon payment of

$40 is expected after 4 months. The 4 - month and 9 -

month risk-free interest rates continuously

compounded are 3% and 4% per annum, respectively.

20



Assets with Known Income

40.03

12

0.04 0.75

0

40 39.60

(900 39.60) $886.60

I e

F e

21


Assets with Known Yield

• How do we deal with a situation where the asset

underlying a forward contract provides a known yield

rather than known cash income?

• A yield implies that the known income is expressed

as a percentage of the asset’s price at the time the

income is paid.

• We defined as the average yield per annum on an

asset during the life of a forward contract with

continuous compounding.

• The formula we use is:

( )

0 0

r d TF S e

22


Assets with Known Yield

• Consider a six-month forward contract on an asset that is expected to provide an income equal to 2% of the asset price once during a six-month period. The risk-free rate of interest with continuous compounding is 10% per annum. The asset price is $25.

• The yield is 4% per annum with semi-annual compounding. Converting this to continuous compounding we get:

• This formula is one of the many located on page 84 to convert nominal rates to continuously compounded rates.

• So the forward price is given by:

0.04ln(1 ) 2ln(1 ) 0.0396

2

mc

RR m

m

(0.10 0.0396) 0.5

0 25 $25.77F e

23


Stock Indices

• A stock index can be viewed as an investment

asset paying a dividend yield.

• The futures price and spot price relationship is

therefore:

– Where d is the dividend yield on the portfolio represented

by the index.

– Remember, with indices, they are stated as points.

Therefore, the number of points must be multiplied by

a factor to end up with a dollar value for the contract.

In Australia, the convention is $25 per point.

( )

0 0

r d TF S e

24



Stock Indices

• Consider a 1-year futures contract on the ASX S&P200. Suppose

that the stocks underlying the index provide a dividend yield of 5%

per annum continuously compounded, that the current value of the

index is 3529, and that the continuously compounded risk-free

interest rate is 10% per annum.

• If we multiply this value by $25 per point, each futures contract will

hedge a dollar value of $92,750.

( )

0 0

r d TF S e

(0.10 0.05)

0 3529 3710F e

25


on Stock Indices

• In general if:

– An arbitrageur can make a riskless profit by buying the stocks

underlying the index and shorting index futures contracts.

This strategy will be financed by borrowing funds at the risk-free

interest rate.

– An arbitrageur can make a riskless profit by shorting the stocks

underlying the index and taking a long position in index

futures contracts. The excess funds will be invested at the risk-

free interest rate until needed to buy back the stocks.

( )

0 0

r d TF S e

( )

0 0

r d TF S e

26


Stock Indices

• Index arbitrage involves simultaneous trades in

futures and many different stocks.

• Very often a computer is used to generate the

trades.

• Occasionally (e.g., on Black Monday) simultaneous

trades are NOT possible and the theoretical no-

arbitrage relationship between F0 and S0 does NOT

hold.

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10. Futures and Forwards on Currencies

• The underlying asset in a forward or futures currency contract is a

certain number of units of a foreign currency.

• A foreign currency is analogous to a security providing a dividend

yield. 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. For example, the holder can invest the foreign

currency in a foreign-denominated bond.

• Thus, the continuous dividend yield is the foreign risk-free

interest rate.

• It follows that if rf is the foreign risk-free interest rate, S0 as the

current spot price in dollars of one unit of the foreign currency and

F0 as the forward or futures price in dollars of one unit of the

foreign currency,

28

F S er r Tf

0 0( )


11. Forwards and Futures on Commodities

• First, let us consider the futures prices of commodities that are investment assets such as gold and silver.

• In the absence of storage costs and income the forward price of a commodity that is an investment asset is given by:

• If there are storage costs, Q is the present value of all of the storage costs less all income during the life of the forward contract, and the forward price is given by:

0 0

rTF S e

0 0( ) rTF S Q e

29


• If storage costs and income are given as a

percentage, then q is the percentage storage

costs less the percentage income during the

life of the forward contract, and the forward price

is given by:

( )

0 0

r q TF S e

30


• Now let us consider consumption commodities.

Commodities that are consumption assets rather than

investment assets usually provide no income, but

can be subject to significant storage costs.

• Individuals and companies who keep such a commodity

in inventory do so because of its consumption value,

not because of its value as an investment. They are

reluctant to sell the commodity and buy forward

contracts because forward contracts cannot be

consumed. There is therefore nothing to keep the

previous equations holding (ie arbitrage).

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• As a result:

– Due to the high storage costs of consumption

commodities, Q is the present value of all of the

storage costs, and the forward price is given by:

– If storage costs are expressed as a proportion q of

the spot price, the equivalent formula is:

0 0( ) rTF S Q e

32

( )

0 0

r q TF S e


11. Forwards and Futures on Commodities • The reason we do not have equality in the formula’s on the previous slide is because

users of a consumption commodity may feel that ownership of the physical

commodity provides benefits that are not obtained by holders of futures

contracts.

• For example, an oil refiner is unlikely to regard a futures contract on crude oil in the

same way as crude oil held in inventory.

• The crude oil in inventory can be used in the refining process whereas a futures

contract cannot.

• The benefits from holding the physical asset is referred to as the convenience

yield.

• As such we can re-write the equations on the previous slide, where y, the

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

the right hand side in those previous equations:

0 0

( ) ( )

0 0 0 0

( )

and if the storage costs are a percentage then,

yT rT

yT r q T r q y T

F e S Q e

F e S e or F S e

33

12. Valuing Forward Contracts • The value of a forward contract at the time it is first entered into is

zero. At a later stage it may prove to have a positive or negative value.

• Suppose that – K is delivery price in a forward contract (the initial forward price when the

contract was negotiated some time ago);

– F is the current forward price for the contract that was negotiated some time ago;

– The delivery date is T years from today;

– r is the T-year risk-free interest rate; and,

– f is the value of the forward contract today.

• The value of a long forward contract (on both investment and consumption assets, ƒ, is:

• Similarly, the value of a short forward contract is

( ) rTf F K e

( ) rTf K F e

34

12. Valuing Forward Contracts

• A long forward contract on a non-

dividend paying stock was entered into

some time ago. It currently has six

months to maturity. The risk-free rate of

interest (with continuous compounding) is

10% per annum, the stock price is $25,

and the delivery price is $24. What is the

value of the forward contract?

35

12. Valuing Forward Contracts

0.1 0.5

0

0.1 0.5

25 $26.28

(26.28 24) $2.17

F e

f e

36

• The value of the forward contract is:


13. Forward vs Futures Prices

• Forward and futures prices are usually assumed to be

the same. 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. This is due to the person in the long position in

a futures contract receiving an immediate gain because of

daily settlement. The positive correlation indicates that interest

rates are also likely to have risen, therefore the gain will be

invested at a higher than average interest rate.

– A strong negative correlation implies the reverse.

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14. Delivery

• In a futures contract, the party in the

short position has the right to choose to

deliver the asset at any time during a

certain period (called the delivery

period).

• The person in the short position has to

give at least a few days notice of their

intention to deliver.

38

Hedging Strategies

Using Futures

39

15. Hedging

• Hedgers aim to use futures markets or forward

contracts to reduce a particular risk they may

face. This risk may relate to the price of an

asset such as gold, a move in the foreign

exchange rate, or the level of the stock market.

• A perfect hedge is one that completely

eliminates the risk. However, a perfect hedge

is very rare.

40


15. Hedging

• A short hedge is a hedge which involves a short

position in futures contracts. It is appropriate

when the hedger already owns an asset (or will

own it at some definite date) and expects to sell

it at some time in the future. This allows them

to lock in the price they will receive.

• A long hedge involves taking a long position in

futures contracts. It is appropriate when a

company knows it will have to purchase a

certain asset in the future and wants to lock in

the price now.

41

15. Hedging

• Arguments in FAVOUR of hedging include:

– Companies should focus on the main business

they are in and take steps to minimize risks

arising from interest rates, exchange rates, and

other market variables; and,

– By hedging, they avoid adverse movements such

as sharp rises in the price of a commodity.

42

15. Hedging

o Arguments AGAINST hedging include:

• Shareholders are usually well diversified

and can make their own hedging

decisions;

• It may increase risk to hedge when

competitors do not; and,

• Explaining a situation where there is a loss

on the hedge and a gain on the underlying

can be difficult

43

16. Basis Risk

• Hedges are NOT always perfect and

straightforward. Some of the reasons for

this are:

– The asset whose price is to be hedged may

NOT be exactly the same as the asset underlying

the futures contract;

– The hedger may NOT be certain of the exact

date the asset will be bought or sold; and,

– The hedge may require the futures contract to

be closed out before its delivery month. 44


16. Basis Risk

• What is basis risk:

– If the asset to be hedged and the asset underlying the futures

contract are the same, the basis risk should be zero at the

expiration of the futures contract.

– Prior to expiration, the basis may be positive or negative.

– When the spot price increases by more than the futures price,

the basis increases. We call this strengthening of the basis.

– When the futures price increases by more than the spot price,

the basis declines. We call this weakening of the basis.

– The formula for working out the basis in a hedging situation is:

Basis = Spot price of asset to be hedged - Futures price of contract used

45

Convergence of Futures to Spot

46

Time Time

(a) (b)

Futures

Price Futures

Price Spot Price

Spot Price

16. Basis Risk

• Basis risk with a long hedge:

– Suppose that

F1 : Initial Futures Price

F2 : Final Futures Price

S2 : Final Asset Price

– You hedge the future purchase of an

asset by entering into a long futures

contract

– Cost of Asset = S2 –(F2 – F1) = F1 + Basis

47

16. Basis Risk

• Basis risk with a short hedge:

– Suppose that

F1 : Initial Futures Price

F2 : Final Futures Price

S2 : Final Asset Price

– You hedge the future sale of an asset by

entering into a short futures contract

– Price Realized = S2+ (F1 –F2) = F1 + Basis

48


16. Basis Risk

• One key factor affecting basis risk is the choice of

the futures contract to be used for hedging. This

choice has two components:

– Choose a delivery month that is as close as possible

to, but later than, the end of the life of the hedge; and,

– When there is no futures contract on the asset being

hedged, choose the contract whose futures price is

most highly correlated with the asset price.

49

17. Cross Hedging

• Cross hedging occurs when the asset

underlying the futures contract is different to

the asset whose price is being hedged.

– For example an airline company may be concerned

about the future price of aviation fuel. However, as

there are no futures contracts on aviation fuel, the

company choose to use heating oil futures contracts

to hedge its exposure.

50

18. Optimal Hedge Ratio • The hedge ratio is the ratio of the size of the

position taken in futures contracts to the size of

the exposure.

• The optimal hedge ration is calculated by:

where

sS is the standard deviation of dS, the change in the

spot price during the hedging period;

sF is the standard deviation of dF, the change in the

futures price during the hedging period; and,

r is the coefficient of correlation between dS and dF.

51

h S

F

rs

s

19. Hedging Using Index Futures • Stock index futures can be used to hedge an

equity portfolio.

• To hedge the risk in a portfolio the number of

contracts that should be shorted is:

• where

P is the value of the portfolio,

b is its beta, and

A is the value of the assets underlying one futures

contract. 52

bP

A


19. Hedging Using Index Futures

• Reasons for using index futures to

hedge an equity portfolio include:

– Desire to be out of the market for a short

period of time. (Hedging may be cheaper

than selling the portfolio and buying it back.)

– Desire to hedge systematic risk

(Appropriate when you feel that you have

picked stocks that will outperform the market.)

53


• Imagine that the value of SPI200 is 3500

• The value of the portfolio to be hedged is $5 million

• The beta of the portfolio is 1.5

What position in SPI200 futures contracts is necessary to hedge the portfolio?

54


bP

A

55

= 1.5 x 5m/3500x25=86 contracts (SHORT)

What position is necessary to reduce the beta of the portfolio to 0.75?


• What position is necessary to reduce the beta of the

portfolio to 0.75 (b*)?

• Given we were hedging 100% with 86 contracts to reduce

our risk to zero, we can take 43 to hedge 50% to give us

half of our previous risk of 1.5.

• Generally:

5( *) (1.5 .75) 43

87500

P m

Ab b

56

What position is necessary to increase the beta of the portfolio to 2.00?



• What position is necessary to increase the

beta of the portfolio to 2.00?

( *)P

Ab b

5(1.5 2.0) 29

3500 25

M

x

57

Since this is negative we must go LONG


• We can use a series of futures contracts

to increase the life of a hedge.

• Each time we switch from 1 futures

contract to another we incur a type of

basis risk.

58

20. Conclusion

• In today’s lecture we have discussed futures

and forward contracts in detail.

• In particular we focussed on determining

forward/futures prices, valuing forward/futures

contracts, basis risk and hedging.

• In next week’s lecture we will discuss interest

rate contracts and swaps.

59

lecture 2 - forwards and futures ppt

Documents

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examples of consumption

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