fi8000 valuation of financial assets spring semester 2010 dr. isabel tkatch assistant professor of...

Fi8000Fi8000Valuation ofValuation of

Financial AssetsFinancial Assets

Spring Semester 2010Spring Semester 2010

Dr. Isabel TkatchDr. Isabel TkatchAssistant Professor of FinanceAssistant Professor of Finance

Forward and Futures ContractsForward and Futures Contracts

☺ The spot marketThe spot market

☺ Forward ContractsForward Contracts☺ DefinitionDefinition☺ PricingPricing

☺ Futures ContractsFutures Contracts☺ DefinitionDefinition☺ The role of the clearinghouseThe role of the clearinghouse☺ Margin accountsMargin accounts☺ Futures price versus Forward priceFutures price versus Forward price

The Spot MarketThe Spot Market

The spot market is the market for immediate The spot market is the market for immediate delivery (or trade) of a financial asset or a delivery (or trade) of a financial asset or a commoditycommodity

The trade occurs immediately between the The trade occurs immediately between the buyer and the seller:buyer and the seller:

The buyer pays cash to the seller;The buyer pays cash to the seller;

The seller hands over the asset.The seller hands over the asset.

Forward ContractForward Contract

DefinitionDefinition::A A forward contractforward contract is an agreement between a is an agreement between a buyer and a seller, to trade a specific quantity and buyer and a seller, to trade a specific quantity and quality of an asset (or commodity), at a specific quality of an asset (or commodity), at a specific time in the future (maturity), at a specific price.time in the future (maturity), at a specific price.

Buyer / longBuyer / long position in the forward contract – the position in the forward contract – the trader who agrees to buy the asset in the future;trader who agrees to buy the asset in the future;

Seller / shortSeller / short position in the forward contract – the position in the forward contract – the trader who agrees to sell the asset in the future;trader who agrees to sell the asset in the future;

Forward priceForward price – the agreed upon asset price. – the agreed upon asset price.

Forward Contract TimelineForward Contract Timeline

Time t = 0Time t = 0::Contract is negotiated and the following terms are Contract is negotiated and the following terms are determined:determined:

Specification of the asset;Specification of the asset;Quantity and quality;Quantity and quality;Delivery method and date;Delivery method and date;Forward price.Forward price.

Time t = TTime t = T::The buyer pays cash to the seller;The buyer pays cash to the seller;The seller hands over the asset.The seller hands over the asset.

Pricing Gold ForwardsPricing Gold Forwards

Example of a forward contract on gold:Example of a forward contract on gold:

- The spot price of gold is $432/oz;- The spot price of gold is $432/oz;

- The one year forward price of gold is $468/oz;- The one year forward price of gold is $468/oz;

- The annual risk-free rate is 8%.- The annual risk-free rate is 8%.

Is there an opportunity to make arbitrage Is there an opportunity to make arbitrage profits?profits?

Arbitrage PricingArbitrage Pricing

Perfect market assumptionsPerfect market assumptions::

No transaction costs; No taxes; No risk of No transaction costs; No taxes; No risk of default; no storage costs.default; no storage costs.

Arbitrage pricingArbitrage pricing::

Replicate the present and future payoffs of the Replicate the present and future payoffs of the forward contract;forward contract;

Use the cash-flows of the replicating portfolio to Use the cash-flows of the replicating portfolio to determine the no-arbitrage forward price.determine the no-arbitrage forward price.

The Cost of Carry ModelThe Cost of Carry Model

☺ There are two ways to pay cash and end of with There are two ways to pay cash and end of with gold on date t = 1:gold on date t = 1:

☺ Buy a forward contractBuy a forward contract☺ Buy a replicating strategy:Buy a replicating strategy:

☺ Buy gold on the spot market for cash, and carry it into the Buy gold on the spot market for cash, and carry it into the future (store it);future (store it);

☺ Borrow the cash needed to fund the purchase, and repay Borrow the cash needed to fund the purchase, and repay the loan on date t = 1.the loan on date t = 1.

☺ Note that the interest rate is your “cost of carry”:Note that the interest rate is your “cost of carry”:☺ Borrowing cash on date t = 0 allows you to carry the Borrowing cash on date t = 0 allows you to carry the

gold forward to date t = 1;gold forward to date t = 1;☺ The interest rate is the cost that you pay for carrying The interest rate is the cost that you pay for carrying

gold forward.gold forward.

Detecting the Arbitrage OpportunityDetecting the Arbitrage Opportunity

Buy a forward contractBuy a forward contract::CFCF00 = 0 = 0 CFCF11 = -$468 = -$468

Replicating strategyReplicating strategy::1. Buy gold in the spot market:1. Buy gold in the spot market:

CFCF00 = -$432 = -$4322. Borrow cash at 8%:2. Borrow cash at 8%:

CFCF00 = +$432 = +$432 CFCF11 = -$466.56 = -$466.56

Total CF:Total CF:CFCF00 = 0 = 0 CFCF11 = -$466.56 = -$466.56

Cash and Carry ArbitrageCash and Carry Arbitrage

Date t = 0Date t = 0 Date t = 1Date t = 1

Sell a forward Sell a forward contractcontract

00 Deliver gold Deliver gold against forwardagainst forward

+$468.00+$468.00

Buy gold on the Buy gold on the spot marketspot market

-$432-$432

Borrow cashBorrow cash

at 8%at 8%

+$432+$432 Repay loanRepay loan -$466.56-$466.56

TotalTotal 00 +$1.44+$1.44

Pricing Gold ForwardsPricing Gold Forwards

Another example of a forward contract on gold:Another example of a forward contract on gold:

- The spot price of gold is $432/oz;- The spot price of gold is $432/oz;

-The one year forward price of gold is $464/oz;-The one year forward price of gold is $464/oz;

-The annual risk-free rate is 8%.-The annual risk-free rate is 8%.

Is there an opportunity to make arbitrage Is there an opportunity to make arbitrage profits?profits?

Detecting the Arbitrage OpportunityDetecting the Arbitrage Opportunity




CFCF00 = +$432 = +$432 CFCF11 = -$466.56 = -$466.56


Reverse Cash and Carry ArbitrageReverse Cash and Carry Arbitrage


Buy a forward Buy a forward contractcontract

00 Get gold Get gold against forwardagainst forward

-$464.00-$464.00

Sell gold on the Sell gold on the spot marketspot market

+$432+$432

Lend cashLend cash

at 8%at 8%

-$432-$432 Collect loanCollect loan +$466.56+$466.56

TotalTotal 00 +$2.56+$2.56

Forward PricingForward Pricing

The cost-of-carry arbitrage restriction:The cost-of-carry arbitrage restriction:

FF00 = S = S00(1+rf)(1+rf)TT

WhereWhere

FF00 = forward price = forward price

SS00 = spot price = spot price

rf = risk-free rate for one periodrf = risk-free rate for one period

T = number of periods to maturityT = number of periods to maturity


One derivationOne derivation::Use the cost-of-carry modelUse the cost-of-carry model

Another derivationAnother derivation::A long position in the underlying asset and a short position A long position in the underlying asset and a short position in the forward contract on that asset is a perfect hedge, and in the forward contract on that asset is a perfect hedge, and therefore should pay the risk-free rate of return. If the therefore should pay the risk-free rate of return. If the forward contract matures one period from nowforward contract matures one period from now

((FF00 - S - S00)/S)/S00 = rf or F = rf or F00 = S = S00(1+rf )(1+rf )

If the forward contract matures T periods from nowIf the forward contract matures T periods from now

FF00 = S = S00(1+rf )(1+rf )TT


Cost-of-carry arbitrage restriction:Cost-of-carry arbitrage restriction:

FF00 = S = S00(1+rf)(1+rf)TT

If FIf F00 > S > S00(1+rf)(1+rf)T T then the forward price is too high:then the forward price is too high:

Buy one unit of the asset in the spot market, sell a Buy one unit of the asset in the spot market, sell a forward contract on one unit of the asset and sell a risk-forward contract on one unit of the asset and sell a risk-free bond.free bond.

If FIf F00 < S < S00(1+rf)(1+rf)T T then the forward price is too low:then the forward price is too low:Sell one unit of the asset in the spot market, buy a Sell one unit of the asset in the spot market, buy a forward contract on one unit of the asset and buy a risk-forward contract on one unit of the asset and buy a risk-free bond.free bond.


Cost-of-carry arbitrage restriction:Cost-of-carry arbitrage restriction:

FF00 = S = S00(1+rf)(1+rf)T T = $432·1.08 = $466.56= $432·1.08 = $466.56

If $468 = FIf $468 = F00 > S > S00(1+rf)(1+rf)T T = $466.56 = $466.56 then:then:Buy 1oz of gold, sell a forward contract on 1oz Buy 1oz of gold, sell a forward contract on 1oz of gold and sell a risk-free bond.of gold and sell a risk-free bond.

If $464 = FIf $464 = F00 < S < S00(1+rf)(1+rf)T T = $466.56 = $466.56 then:then:Sell 1oz of gold, buy a forward contract on 1oz Sell 1oz of gold, buy a forward contract on 1oz of gold and buy a risk-free bond.of gold and buy a risk-free bond.

Cost of Carry ModelCost of Carry Model Imperfect Markets Imperfect Markets

Usually the borrowing rate is higher than the Usually the borrowing rate is higher than the lending rate.lending rate.

Example:Example:The spot price of gold is $432/oz;The spot price of gold is $432/oz;

The borrowing rate is 9% (annual);The borrowing rate is 9% (annual);The lending rate is 8%.The lending rate is 8%.

What is the no-arbitrage What is the no-arbitrage price rangeprice range (lower and (lower and upper bounds) of the one year forward price of upper bounds) of the one year forward price of gold?gold?

Cash and Carry ArbitrageCash and Carry Arbitrage


Sell forward Sell forward contractcontract

00 Deliver gold Deliver gold against forwardagainst forward

+F(9%)+F(9%)

Buy gold on the Buy gold on the spot marketspot market

-$432-$432

Borrow cashBorrow cash

at 9%at 9%

+$432+$432 Repay loanRepay loan -$470.88-$470.88

TotalTotal 00 00

Reverse Cash and Carry ArbitrageReverse Cash and Carry Arbitrage


Buy forward Buy forward contractcontract

00 Get gold Get gold against forwardagainst forward

-F(8%)-F(8%)

Sell gold on the Sell gold on the spot marketspot market

+$432+$432

Lend cashLend cash

at 8%at 8%

-$432-$432 Collect loanCollect loan +$466.56+$466.56

TotalTotal 00 00

Cost of Carry ModelCost of Carry Model – Imperfect Markets – Imperfect Markets

Cash and carry: Cash and carry: forward price ≤ $470.88forward price ≤ $470.88

Reverse cash and carry:Reverse cash and carry:forward price ≥ $466.56forward price ≥ $466.56

The no-arbitrage price rage:The no-arbitrage price rage:

LB = SLB = S00(1+rf(1+rfLendLend))TT ≤ F ≤ F00 ≤ S ≤ S00(1+rf(1+rfBorrowBorrow))T T = UB= UB

LB = $466.56 ≤ forward price ≤ $470.88 = UBLB = $466.56 ≤ forward price ≤ $470.88 = UB

Show that if the forward price is $470 you can not make Show that if the forward price is $470 you can not make arbitrage profits.arbitrage profits.

Cash and CarryCash and Carry




CFCF00 = +$432 = +$432 CFCF11 = -$470.88 = -$470.88


Reverse Cash and CarryReverse Cash and Carry



CFCF00 = -$432 = -$4322. Borrow cash at 8% ( not 9%! ):2. Borrow cash at 8% ( not 9%! ):

CFCF00 = +$432 = +$432 CFCF11 = -$466.56 = -$466.56


Cost of Carry ModelCost of Carry Model

Arbitrage pricingArbitrage pricing::We have solved for the We have solved for the forward priceforward price taking the spot taking the spot price and interest rate as given.price and interest rate as given.We can use the same framework to solve for the We can use the same framework to solve for the spot spot priceprice or the or the interest rateinterest rate..

LimitationsLimitations::This model is useful only for contracts on assets that have This model is useful only for contracts on assets that have certain characteristics:certain characteristics:

StorabilityStorabilityEase of short sellingEase of short sellingUnderlying asset must be tradedUnderlying asset must be traded

Futures ContractsFutures Contracts

☺ The futures contract is a standardized form of The futures contract is a standardized form of the forward contract, which is traded on the the forward contract, which is traded on the exchange:exchange:

☺ The selection of The selection of underlying assetsunderlying assets is limited, but is limited, but there is a liquid secondary market for the contracts;there is a liquid secondary market for the contracts;

☺ All contract parameters are standardized, except for All contract parameters are standardized, except for the the futures pricefutures price which is determined by supply and which is determined by supply and demand;demand;

☺ In order to eliminate counter-party risk, the In order to eliminate counter-party risk, the clearinghouseclearinghouse is always the seller if the trader takes is always the seller if the trader takes the long position, and the buyer if he takes the short the long position, and the buyer if he takes the short position.position.

Why Trade Futures?Why Trade Futures?

SpeculationSpeculation: futures offer the opportunity to place : futures offer the opportunity to place “bets on the future spot price of a commodity.“bets on the future spot price of a commodity.

HedgingHedging: futures offer the opportunity to hedge : futures offer the opportunity to hedge against the risk of the future spot price.against the risk of the future spot price.

Futures provide leverageFutures provide leverage: allow for big bets on : allow for big bets on the value of the underlying asset with little cash the value of the underlying asset with little cash (the (the initial margininitial margin).).

Creating a Liquid MarketCreating a Liquid Market

☺Ex ante:Ex ante:☺ The buyer and the seller managed to The buyer and the seller managed to

eliminate the risk of the future spot priceeliminate the risk of the future spot price☺ Both buyer and seller are “winners”Both buyer and seller are “winners”

☺Ex post:Ex post:☺ The future price is either higher or lower than The future price is either higher or lower than

the contracted pricethe contracted price☺ One side is a “loser” with an incentive to One side is a “loser” with an incentive to

renege on the futures contract.renege on the futures contract.

Creating a Liquid MarketCreating a Liquid Market

☺Eliminating counter-party risk:Eliminating counter-party risk:☺ If all traders realize that the “losers” will If all traders realize that the “losers” will

renege, then no one will traderenege, then no one will trade☺ To promote a liquid market, we must To promote a liquid market, we must removeremove

the incentive to renege (the the incentive to renege (the counter-party counter-party riskrisk))

☺ The The individual verification individual verification of the “credit of the “credit worthiness” of every counter-party is worthiness” of every counter-party is costlycostly

☺ The solution is to establish the The solution is to establish the clearinghouseclearinghouse

The ClearinghouseThe Clearinghouse

☺ The clearinghouse is the The clearinghouse is the counter-party to counter-party to every tradeevery trade

☺ As long as the clearinghouse is “credit-worthy” As long as the clearinghouse is “credit-worthy” no one faces counter-party riskno one faces counter-party risk

☺ The The clearinghouse takes no position clearinghouse takes no position – for – for every buyer there is a sellerevery buyer there is a seller

☺ Traders still have an incentive to renege Traders still have an incentive to renege – to – to promote liquidity, the clearinghouse should promote liquidity, the clearinghouse should make sure that they are credit-worthy.make sure that they are credit-worthy.

Viability of the ClearinghouseViability of the Clearinghouse

☺Margin accountMargin account☺ Initial marginInitial margin☺ Variation marginVariation margin

☺Marking-to-marketMarking-to-market☺ Daily settlementDaily settlement

☺Daily price limitsDaily price limits

Margin AccountMargin Account

☺ The The marginmargin is a security account consisting of is a security account consisting of cash or near-cash securitiescash or near-cash securities

☺ Note that both the seller and the buyer of the futures Note that both the seller and the buyer of the futures contract are exposed to lossescontract are exposed to losses

☺ Initial marginInitial margin – the initial deposit in the margin – the initial deposit in the margin account (usually 5%-15% of the value of the account (usually 5%-15% of the value of the contract)contract)

☺ Marking-to-marketMarking-to-market – the gains and loses from – the gains and loses from the daily changes in the futures price are the daily changes in the futures price are realized on daily basisrealized on daily basis


☺ Maintenance (variation) marginMaintenance (variation) margin – once the – once the value of the account falls below this value the value of the account falls below this value the trader receives a trader receives a margin callmargin call

☺ If the trader does not deposit the required funds in the If the trader does not deposit the required funds in the margin account the broker may close out his positionmargin account the broker may close out his position

☺ Convergence propertyConvergence property – the futures price – the futures price converges to the spot price at maturityconverges to the spot price at maturity

☺ The sum of all daily settlements is the total profit or The sum of all daily settlements is the total profit or loss from the position in the futures contractloss from the position in the futures contract


ExampleExample::

Assume that the current futures price of silver, for Assume that the current futures price of silver, for delivery in 5 days, is $5.20 per ounce.delivery in 5 days, is $5.20 per ounce.

Each silver contract on the Commodity Exchange Each silver contract on the Commodity Exchange (CMX) calls for purchase and delivery of 5,000 (CMX) calls for purchase and delivery of 5,000 ounces.ounces.

Assume that the initial margin is 10% of the value Assume that the initial margin is 10% of the value of the contract and the maintenance margin is 8% of the contract and the maintenance margin is 8% of that value. of that value.

Margin Account (Buyer)Margin Account (Buyer)

Initial margin = 10%; Maintenance margin = 8%Initial margin = 10%; Maintenance margin = 8%

Day PriceDay Price Market Market ValueValue

Profit Profit (loss)(loss)

Acct Acct BalanceBalance

Cash inCash in

1 $5.201 $5.20 $26,000$26,000 $2,600$2,600 $2,600$2,600

2 $5.152 $5.15

3 $5.093 $5.09

4 $5.104 $5.10

5 $5.125 $5.12


Initial margin=$2,600; Maintenance margin=$2,080Initial margin=$2,600; Maintenance margin=$2,080




Cash inCash in

1 $5.201 $5.20 $26,000$26,000 $2,600$2,600 $2,600$2,600

2 $5.152 $5.15 $25,750$25,750 -$250-$250 $2,350$2,350

3 $5.093 $5.09

4 $5.104 $5.10

5 $5.125 $5.12






Cash inCash in

1 $5.201 $5.20 $26,000$26,000 $2,600$2,600 $2,600$2,600

2 $5.152 $5.15 $25,750$25,750 -$250-$250 $2,350$2,350

3 $5.093 $5.09 $25,450$25,450 -$300-$300 (!!!$2,050)(!!!$2,050)

$2,600$2,600$550$550

4 $5.104 $5.10

5 $5.125 $5.12






Cash inCash in

1 $5.201 $5.20 $26,000$26,000 $2,600$2,600 $2,600$2,600

2 $5.152 $5.15 $25,750$25,750 -$250-$250 $2,350$2,350

3 $5.093 $5.09 $25,450$25,450 -$300-$300 (!!!$2,050)(!!!$2,050)

$2,600$2,600$550$550

4 $5.104 $5.10 $25,500$25,500 +$50+$50 $2,650$2,650

5 $5.125 $5.12 $25,600$25,600 +$100+$100 $2,750$2,750

Daily Price LimitsDaily Price Limits

☺ The maximum amount that a futures price may The maximum amount that a futures price may change in one day is restrictedchange in one day is restricted

☺ If there is a large change in the value of the If there is a large change in the value of the underlying asset’s spot price there should also be a underlying asset’s spot price there should also be a large change in the futures pricelarge change in the futures price

☺ In such cases, the restriction will result in daily In such cases, the restriction will result in daily changes in the futures price without tradechanges in the futures price without trade

☺ The restriction gives the clearinghouse a chance to The restriction gives the clearinghouse a chance to weed out traders who may renegeweed out traders who may renege

☺ On the last month of the futures contract this On the last month of the futures contract this restriction is usually removed (convergence property)restriction is usually removed (convergence property)

Pricing Futures ContractsPricing Futures Contracts

☺We have priced forward contractsWe have priced forward contracts☺Futures contracts haveFutures contracts have

☺ Margin accountsMargin accounts☺ Marking-to-marketMarking-to-market☺ Price limitsPrice limits

☺The no-arbitrage approach to pricing The no-arbitrage approach to pricing forward contracts carries directly to pricing forward contracts carries directly to pricing futures contracts, but it involves more futures contracts, but it involves more tedious calculations.tedious calculations.

Synthetic Positions with Futures

Synthetic stock purchaseSynthetic stock purchase

Purchase of the stock index instead of actual shares Purchase of the stock index instead of actual shares of stockof stock

Lower transactions costsLower transactions costs

Can buy and sell all stocks in the market index quickly Can buy and sell all stocks in the market index quickly and relatively cheaplyand relatively cheaply

Synthetic Positions with Futures Example:Example:

Institution wants to hold a long position of $130 M in S&P index for Institution wants to hold a long position of $130 M in S&P index for one month. Current index level = 1,300; current 1-month delivery one month. Current index level = 1,300; current 1-month delivery futures price is 1,313.futures price is 1,313.

Convention: each contract calls for $250 * (level of the index) = Convention: each contract calls for $250 * (level of the index) = 250*1300 = $325,000 250*1300 = $325,000

Thus, the institution can do the following:Thus, the institution can do the following:

• 1. buy 130M/325,000 = 400 contracts1. buy 130M/325,000 = 400 contracts

• 2. Invest enough in T-bills to ensure that the payment is covered 2. Invest enough in T-bills to ensure that the payment is covered (need to buy $130 million in bills today, assume rf=10%)(need to buy $130 million in bills today, assume rf=10%)

Check: Check:

Theoretical Payoff Payoff

Profits from contract ST-F0 400*250*(ST – 1,313)

Face value of T-bills F0 131,300,000

Total payoff ST 100,000ST

We created a synthetic equity position

Hedging Systematic RiskHedging Systematic Risk

How would a portfolio manager hedge market How would a portfolio manager hedge market exposure?exposure?

To protect against a decline in level stock prices, To protect against a decline in level stock prices, short the appropriate number of futures index short the appropriate number of futures index contractscontractsLess costly and quicker to use the index Less costly and quicker to use the index contractscontractsUse the portfolio beta to determine the hedge Use the portfolio beta to determine the hedge ratioratio

Hedging Systematic Risk: Text Example

Suppose you manage $30M and want to hedge.

How many contracts should you go short?

Portfolio Beta = .8 S&P 500 = 1,000

Decrease = 2.5% S&P falls to 975

Project loss if market declines by 2.5% = (.8) (2.5) = 2%

2% of $30 million = $600,000

How much does each S&P500 index contract change for a 2.5% change in the index?

$250 * 25 point swing = $6,250

Hedge Ratio: Text Example

H =

=

Change in the portfolio value

Profit on one futures contract

$600,000

$6,250= 96 contracts short

Practice ProblemsPractice Problems

BKM Ch. 22BKM Ch. 22

7th Ed.: 3-4, 6-8,10-12, 16, 20.7th Ed.: 3-4, 6-8,10-12, 16, 20.

8th Ed.: 2-3, 8-9,11-12, 158th Ed.: 2-3, 8-9,11-12, 15

CFA: 1, 5.CFA: 1, 5.

Practice problems:Practice problems:

Forward and futures contracts 1-5.Forward and futures contracts 1-5.

fi8000 valuation of financial assets spring semester 2010 dr. isabel tkatch assistant professor of...

Documents

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