Lecture-1, Section 16, Reading 58, Forward Markets and Contracts
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Transcript of 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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
2. The Structure of the Global Forward Market
• For individual reading
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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
• 4.2 Pricing and Valuation of Equity Forward Contracts
• Example 2For individual review
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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
• 4.3 Pricing and Valuation of Fixed Income Forward Contracts
• Example 3For individual review
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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
• 4.3 Pricing and Valuation of Equity Forward Contracts
• Example 4For individual review
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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
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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Dr.Stanley B. Gyoshev – Module Leader
Kingkan Ketsiri – Tutorial Instructor
• 4.4 Pricing and Valuation of Equity Forward Contracts
• Example 5For individual review