Lecture-1, Section 16, Reading 58, Forward Markets and Contracts

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Lecture 1Session 16, Reading 58

Forward Markets And Contracts

Dr. Stanley Gyoshev Xfi Centre for Finance and Investment

University of Exeter

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Dr.Stanley B. Gyoshev – Module Leader

Kingkan Ketsiri – Tutorial Instructor

Learning Outcomes Explain how the value of a forward contract is

determined at initiation, during the life of the contract, and at expiration;

Calculate and interpret the price and the value of an equity forward contract, assuming dividends are paid either discretely and continuously;

Calculate and interpret the price and the value of 1) a forward contract on a fixed income security, 2) a forward rate agreement (FRA), and 3) a forward contract on a currency;

Evaluate credit risk in a forward contract and explain how market value is a measure of the credit risk to a party in a forward contract.

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1. Introduction

Definition of a Forward Contract A forward contract is an agreement between two parties in which one party, the buyer, agrees to buy from the other party, the seller, an underlying asset or other derivative, at a future date at a price established at the start of the contract.

Long Position: buyer Short Position: seller 1.1 Delivery and Settlement 1.2 Default Risk 1.3 Termination of a Contract

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2. The Structure of the Global Forward Market

• For individual reading

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3. Types of Forward Contracts 3.1 Equity Forwards

Forward contracts on individual stocks Forward contracts on stock portfolios Forward contracts on stock indices The effect of dividends

3.2 Bond and Interest Rate Forward Contracts Forward contracts on individual bonds and bond

portfolios Forward contracts on interest rates: forward rate

agreements (FRA)• Eurodollar: the primary time deposit

instrument• London Interbank Offer Rate (LIBOR):

lending rate in derivative contracts.

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3. Types of Forward Contracts

3.3 Currency Forward Contracts

3.4 Other Types of Forward Contracts Commodity forwards: oil, a precious metal, et al.

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4. Pricing and Valuation of Forward Contracts Definition:

Value is what you can sell something for or what you must pay to acquire something. Valuation is the process of determining the value of an asset or service.

Definition:A contract price is the fixed price or rate at which

the transaction scheduled to occur at expiration will take place, which is commonly called the forward price or forward rate.

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4.1 Generic Pricing and Valuation of a Forward Contract

Time-line of the contract Today is identified as time 0 and is the date the

contract is created. The expiration date is time T. Time t is an arbitrary time between today and the expiration.

0 T

t(today)The day the contract is created

(expiration)

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4.1 Generic Pricing and Valuation of a Forward Contract

Variable Definitions S0: spot price at time 0; ST: spot price at time t; F(0, T): price of a forward contract initiated at

time 0 and expiring at time T; Vt(0, T): the value at time t of a forward contract

initiated at time 0 and expiring at time T; Value at expiration of a forward contract

established at time 0VT(0, T) = ST – F(0, T) (58-1)

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4.1 Generic Pricing and Valuation of a Forward Contract Determining Forward Price - an Example 1 of 2 Suppose the underlying asset is worth $100 and the forward

price is $108, the interest rate is 5%.VT(0, T) = ST – F(0, T)=108 – 105=$3

This is an arbitrage profit and the derivative price would have to come down to $105.

Consider if F = $103 with T = 1, the value of the contract at present would be

V0(0, T) = S0 – F(0, T) / (1+r)T = 100 – 103 / (1+0.05) = $1.9048

If a party is longing a contract, it must pay $1.9048 to the party shorting the contract.

NB! “Parties going long must pay positive values; parties going short pay negative values.”

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4.1 Generic Pricing and Valuation of a Forward Contract Determining Forward Price – Example 2 of 2 If the forward price were $108, the value would be

V0(0, T) = S0 – F(0, T) / (1+r) = 100 – 108 / (1+0.05) = -$2.8571

To eliminate this arbitrage profit, this value would have to be paid from the short to the long.

Determining Forward Price It is customary in the forward market for the initial value to be

set to zero.V0(0, T) = S0 – F(0, T)/(1+r) F(0, T) = S0(1+r)T (58-2)

So in the above example:F(0, T) = S0 * (1+r)T = 100 * (1+ 0.05) = $105

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4.1 Exhibit 3: Generic Pricing and Valuation of a Forward Contract

Determining Forward Price (2)

F(0, T) = S0(1+r)T

Off-market FRA: a contract in which the initial value is intentionally set at a nonzero value.

Buy asset at S0

Sell forward contract at F(0, T)Outlay: S0

Hold asset and lose interest on outlay

Deliver assetReceive F(0, T)

0 T

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4.1 Generic Pricing and Valuation of a Forward Contract The Value of a Forward Contract at Time t

Vt(0, T) = St – F(0, T)/(1+r)(T – t) (58-3) Example: St = $102, F(0, T) = $105, t = 0.25, T = 1

Vt(0, T) = V0.25(0, 1) = 102 – 105 / (1 + 0.05)^(1 – 0.25)

= $0.7728 If the market value is positive, the value of the asset

exceeds the present value of what the long promises to pay. Thus it makes sense that the short must pay the long and vice versa if the market value is negative.

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4.1 Exhibit 4: Generic Pricing and Valuation of a Forward Contract

The Value of a Forward Contract at Time t (2)

Vt(0, T) = St – F(0, T)/(1+r)(T – t) Example: St = $71.19, F(0, T) = $62.25, t = 1.5, T = 2, r = 7%

Vt(0, T) = V1.5(0, 2) = 71.19 – 62.25/(1 + 0.07)0.5 = $11.01

Went long forward contract at price F(0, T)Outlay = 0

Hold a claim on asset currently worth ST

Obliged to pay F(0, T) at T

Receive asset worth ST

Pay F(0, T)

0 T

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4.1 Exhibit 5: Pricing and Valuation for a forward Contract

Value of a forward contract at any timeVt(0, T) = St – F(0, T)/(1+r)(T – t)

Value of a forward contract at expiration (t=T)VT(0, T) = ST – F(0, T)

Value of a forward contract at initiation (t=0)V0(0, T) = S0 – F(0, T)/(1+r)T

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Example 1A An investor holds title to an asset worth $125.72.

To raise money for an unrelated purpose, the investor plans to sell the asset in nine months. The investor is concerned about uncertainty in the price of the asset at that time thus he enters a forward contract to sell the asset in 9 months. The risk-free interest rate is 5.625%.

A. Determine the appropriate price for the forward. Solution S0 = 125.72 T = 9/12 = 0.75 r = 0.05625 F(0, T) = 125.72(1.05625)0.75 = 130.99

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Example 1BB. Suppose the counterparty to the forward contract is

willing to engage in such a contract at a forward price of $140, Explain what type of transaction the investor could execute to take advantage of the situation. Calculate the rate of return (annualized), and explain why the transaction is attractive.

Solution The rate of return: (140/125.72) - 1 = 0.1136 in 9 months,which can be annualized to (1.1136)12/9 – 1 = 0.1543

The return is obviously larger than the risk-free rate of 5.625%. The position is not only hedged but also earns an arbitrage profit.

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Example 1CC. Two months later, the price of the asset is

$118.875. Determine the market value of the forward contract at this point in time from the perspective of the investor in Part A.

Solution t = 2/12 T – t = 9/12 – 2/12 = 7/12 St = 118.875 F(0, T) = 130.99Vt(0, T) = V2/12(0, 9/12)

= 118.875 – 130.99/(1.05625)7/12 = – 8.0 The contract has a negative value. This investor is

short thus the value to the investor in this problem is gain of 8.0.

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Example 1DD. Determine the value of the forward contract at

expiration assuming the contract is entered into at the price you computed in A and the price of the underlying asset is $123.50 at expiration. Explain how the investor did on the overall position of both the asset and the forward contract.Solution ST = 123.50VT(0, T) = V9/12(0, 9/12) = 123.50 – 130.99 = – 7.49

The investor is short so she gains 7.49 on the forward contract.

She incurred a loss on the asset of 125.72 – 123.50 = 2.22.

Therefore the net gain is 7.49 – 2.22 = 5.27, which represents a return of 5.27/125.72 = 4.19%.

When annualized the return equals (1.0419)12/9 = 0.05625, the same as the risk-free rate.

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4.2 Pricing and Valuation of Equity Forward Contracts The Present Value of Dividends

Price of Equity Forward Contracts Paying DividendsF(0, T) = [S0 – PV(D,0,T)](1+r)T (58-4)

Example S0 = $40, dividend of $3 in 50 days, r = 6%, T = 0.5 F(0, T) = F(0, 0.5) = [40-3/(1.06)50/365](1.06)0.5 = $38.12 NB! Forward price should not be interpreted as a

forecast of the future price of the underlying.

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4.2 Pricing and Valuation of Equity Forward Contracts

Price of Equity Forward Contracts Paying Dividends using Future value

F(0, T) = S0*(1+r)T– FV(D,0,T) (58-5)

Price of Equity Forward Contracts Paying Dividends using Continues Compounding

(58-6) TrT cc

eeSTF 0,0

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4.2 Pricing and Valuation of Equity Forward Contracts

The Value of Equity Forward Contracts Paying Dividends

Vt(0, T) = St – PV(D,t,T) – F(0, T)/(1+r)(T – t) (58-7)

The Value of Equity Forward Contracts Paying Dividends with continuous compounding

(58-8)

tTrtTtt

cc

eTFeSTV ,0,0

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4.2 Exhibit 6: Pricing and Valuation Formulas for Equity Forward Contracts

Price of Equity Forward Contract Discrete Dividends

F(0, T) = [S0 – PV(D,0,T)](1+r)T or S0 (1+r)T – FV(D,0,T)

Continuous Dividends

Value of Equity Forward ContractDiscrete Dividends

Vt(0, T) = St – PV(D,t,T) – F(0,T)/(1 + r)(T – t)

Continuous Dividends

TrT cc

eeSTF *,0 0

tTrtTtt

cc

eTFeSTV ,0,0

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• 4.2 Pricing and Valuation of Equity Forward Contracts

• Example 2For individual review

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4.3 Exhibit 7: Pricing and Valuation Formulas for Fixed Income Forward Contracts

Price of Forward Contract on Bond with Coupons CI

F(0, T) = [B0

c(T + Y) – PV(CI,0,T)](1 + r)T

Or [B0c(T + Y)] (1 + r)T– FV(CI,0,T)]

Price of Forward Contract on Bond with Coupons CI

Vt(0, T) = Btc(T + Y) – PV(CI,t,T) – F(0,T)/(1 + r)(T-t)

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• 4.3 Pricing and Valuation of Fixed Income Forward Contracts


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4.3 Pricing and Valuation Formulas for Interest Rate Forward Contracts (FRAs)

Forward Price (Rate)

(58-13)

Value of FRA on Day g (58-14)

mhhL

mhmhLmh 3601

3601

3601

,,0FRA0

0

3601

360,,0FRA1

3601

1,,0gmhgmhL

mmh

ghghLmhV

gg

g

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4.3 Exhibit 8: Pricing and Valuation Formulas for Interest Rate Forward Contracts (FRAs)

Forward Price (Rate)

Value of FRA on Day g

mhhL

mhmhLmh 3601

3601

3601

,,0FRA0

0

3601

360,,0FRA1

3601

1,,0gmhgmhL

mmh

ghghLmhV

gg

g

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4.4 Pricing and Valuation Formulas for Currency Forward Contracts

Price of Forward Contract on Foreign Currency Interest rate Parity Discrete Interest:

(58-15)

Continuous Interest

(58-16)

TTfr

rSTF

1

1,0 0

TrTr cfc

eeSTF 0,0

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4.4 Exhibit 9: Pricing and Valuation Formulas for Currency Forward Contracts (1 of 2)

Price of Forward Contract on Foreign Currency Interest rate Parity Discrete Interest:

Continuous Interest

TTfr

rSTF

1

1,0 0

TrTr cfc

eeSTF 0,0

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4.4 Exhibit 9: Pricing and Valuation Formulas for Currency Forward Contracts (2 of 2)

Value of Forward Contract on Foreign CurrencyDiscrete Interest Rate

Continuous Interest Rate

tTtTfT

t rTF

rSTV

1,0

1,0

tTrtTrtt

cfc

eTFeSTV ,0,0

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Lecture-1, Section 16, Reading 58, Forward Markets and Contracts

Documents

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