Welcome to derivatives.  This is the first of two modules on derivatives, covering forward and futures contracts.  Candidates will need to know:
<ul>
	<li>The similarities (such as pricing) between forwards and futures;</li>
	<li>The differences (such as value) between forwards and futures;</li>
	<li>How to price and value forwards and futures; and</li>
	<li>Basic the characteristics of forwards and futures.</li>
</ul>
When approaching derivatives, it is important that the candidate understand both the formulas as well as the concepts behind them.

Forwards and futures are a good way to wade into derivatives, as the material gets more complex in session two.  Pricing any type of forward or future is done with the same base formula, but cash flows may vary by underlying asset.  Never forget that forwards accumulate positive or negative value, but futures are reset to a value of zero every day. Lastly, candidates must understand the qualitative differences between futures and forwards.

Candidates can reasonably expect two item sets (twelve questions), drawn from both derivatives sessions.

This module aligns with Study Session 16 material in the Level II CFA Program Curriculum ©.

CFA Level 2: Derivatives Part 1 – Introduction

A forward contract is an agreement to buy or sell an underlying asset, such as gold or an index basket of stocks, at a specified date in the future.

A forward contract is a private, over-the-counter contract between two parties; it is not exchange traded. Alternatively a futures contract is exchange traded and the parties post margin upon contract initiation.  More on the differences between forwards and futures will be discussed throughout the module.

A forward contract can be customized to meet the specific needs of the two parties.

Since a forward is over the counter, it can be difficult to exit the position prior to expiration.  The party with negative contract value can attempt to buy out the other contract party or one party can attempt to sell his/her position to a third party.

Forward Contract Price and Forward Contract Value

Forward Contract Price: The set price or interest rate agreed to at contract initiation that will be paid by the forward buyer upon expiration, at the agreed settlement date.
<ul>
	<li>The buyer of the contract is in the long position and will purchase the underlying asset at the contract price and the seller is in the short position and must deliver the underlying asset to the contract buyer.</li>
	<li>The forward contract price set at initiation is based upon an arbitrage relationship between the contract and the underlying asset; this arbitrage relationship causes the forward contract’s value to equal zero at initiation.</li>
	<li>Arbitrage Relationship: based on the sequence where the underlying asset is purchased with borrowed capital at the risk free rate and the forward contract is sold (note that adjustments will be made for cash inflows and outflows associated with the underlying during the contract term). The arbitrage transaction is made with funds borrowed at the risk free rate, so the accurate forward price assumes that no profit will be made and the value at contract initiation is zero.</li>
	<li>A forward contract is priced at initiation and the price is set at an amount that is theoretically fair to both the buyer and seller.</li>
</ul>
Forward Contract Value: This is the amount required to assume ownership of an existing forward contract.
<ul>
	<li>The value of a forward contract upon initiation is zero as a result of the arbitrage relationship between the contract and the underlying asset.</li>
	<li>A forward contract can be valued at any time during the life of the contract.</li>
	<li>A forward contract price is set at initiation and will not change regardless of market movements; alternatively, the forward contract’s value will most likely change from its initial value of zero between initiation and settlement as market conditions change.</li>
	<li>Forward Contract Value at Expiration = Price of the Underlying Asset at Expiration less the Forward Contract’s Price at Initiation</li>
	<li>For example, if a gold speculator agreed to a forward contract price to buy one ounce of gold at $1,300 in one month; if at expiration, the price of one ounce of gold is $1,350 then the contract will have a value of $50 to the gold speculator.</li>
</ul>
Off Market Forward: A forward contract where the initial contract value is not set equal to zero and this value is exchanged between the buyer and seller at initiation.

What are Forward Contracts?

An equity <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contract</a> is an agreement between two parties to buy a pre-specified number of an equity stock (or stock index) at a given price at a given date.

<ul>
	<li>Notation</li>
		<ul>
			<li>F(0,T) = forward price for a contract initiated at time 0 and expiring in time T</li>
			<li>S0 = spot price of the underlying equity at time 0</li>
			<li>r = risk-free rate (rc indicates continuous compounding)</li>
			<li>&delta; = dividend yield (&delta;c indicates continuous compounding)</li>
			<li>T = total time of the contract, where T of 1 year = 1 and T of six months = 0.5</li>
		</ul>
	<li>Equity Forward Contracts for Discrete Dividends</li>
		<div class="formula1">
		Price at initiation: F(0,T) = (S0 - PV(&delta;))(1+r)T
		 
		Value at time "t": Vt(0,T) = St - PV(&delta;) - PV[F(0,T)]

		</div>
		 
		Remember, the price will represent the amount that the contract buyer will pay at contract expiration to acquire the underlying asset; alternatively, the value reflects a change to the intrinsic value of the contract (the spot price at time "t" minus the present value of the dividends, minus the present value of the price of the forward contract). 
	<li>Equity Forward Continuously Compounded Dividends</li>
		<div class="formula1">
		Price at initiation: F(0,T) = (S0e-&delta;cT)ercT
		 
		Value at time "t": Vt(0,T) = Ste-&delta;c(T-t) - F(0,T)e-&delta;rc(T-t)

		</div>
	<li>The formulas for discrete and continuously compounding dividends, when pricing and valuing equity forward contracts, should lead to similar answers.</li>

</ul>

Equity Forward Contracts

A fixed income <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contract</a> is an agreement between two parties to buy a pre-specified number of a fixed income security at a given price at a given date.


The formulas for pricing and valuing fixed income forward contracts are similar to that of equity forward contracts, simply replace stock price with bond price (including accrued interest) and dividend payment with interest coupon payment.
<div class="formula1">
Price at initiation: F(0,T) = (S0 - PV(coupon pmts over contract life))(1+r)T 
 
Value at time "t": Vt(0,T) = St - PV(coupon pmts remaining in contract life) - PV[F(0,T)]

</div>

Fixed Income Forward Contracts

<ul>
	<li>A currency <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contract</a> involves two currencies and two interest rates. A currency forward contract lets you lock-in a pre-defined price at which you can buy/sell a currency on a future date.</li>
	<li>Contract parties commonly enter into currency forwards with the objective of hedging exchange rate risk exposure.</li>
	<li>The interest rates can be interpreted as a net cash flow yield of investing one currency in a risk free asset and borrowing the other currency.</li>
	<li>Price of a Currency Forward with Simple Compounding:</li>
		<div class="formula1">
			F(0,T)X/Y = S0,X/Y &times; ((1+rX)/(1+rY))T
			 
			Where:
			<ul>
				<li>S0 = Spot exchange rate for X/Y (#X units domestic per 1Y foreign) at initiation</li>
				<li>rX = Risk free rate in country X (domestic currency)</li>
				<li>rY = Risk free rate in country Y (foreign currency)</li>
			</ul>	
		</div>
	<li>Price of a Currency Forward with Continuous Compounding</li>
		<div class="formula1">
		F(0,T)X/Y = (S0,X/Y e-ryT ) erxT 
		</div>
	<li>Value of a Currency Forward with Simple Compounding</li>
		<div class="formula1">
		Vt(0,T)X/Y = (St,X/Y / (1+rY)(T-t)) - (F(0,T)X/Y / (1+rX)(T-t))
		</div>	
	<li>Value of a Currency Forward with Continuous Compounding</li>
		<div class="formula1">
		Vt(0,T)X/Y = (St,X/Y e-ry(T-t)) - (F(0,T)X/Y e-rx(T-t))
		</div>
</ul>

Currency Forward Contracts

<ul>
	<li>The <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contracts</a> discussed thus far in the module have focused on situations where a buyer and seller want to lock in a future transaction price to buy/sell an asset such as a stock, a bond, or a foreign currency.</li>
	<li>A <a href="https://financetrain.com/introduction-to-forward-rate-agreements/" title="Introduction to Forward Rate Agreements">forward rate agreement (FRA)</a> enables a borrower to lock a fixed interest rate for borrowing and a lender to lock a fixed interest rate for lending.</li>
	<li>The borrower (buyer, long the contract) in an FRA is seeking protection against rising interest rates and the lender (seller, short the contract) is seeking protection against falling interest rates.</li>
	<li>FRAs are commonly tied to LIBOR.</li>
	<li>FRA Price: An FRA's "price" is the fixed interest rate set at contract initiation; this is the rate that the contract buyer pays.</li>
	<li>FRA Value: At contract initiation, an FRA's value is zero (just like other forwards). As interest rates move over the life of the contract, the fixed rate stays the same; the floating rate that will be set upon contract expiration is changing - thus, the one of the contract parties is gaining value and the other party is losing value.</li>
		<ul>
			<li>Rising Rates: Fixed rate buyer is gaining value; alternatively, the seller is losing value.</li>
			<li>Falling Rates: Fixed rate buyer is losing value; alternatively, the seller is gaining value.</li>
		</ul>
	<li>Like <a href="https://financetrain.com/equity-forward-contracts/" title="Equity Forward Contracts">equity forwards</a>, <a href="https://financetrain.com/fixed-income-forward-contracts/" title="Fixed Income Forward Contracts">fixed income forwards</a>, and <a href="https://financetrain.com/currency-forward-contracts/" title="Currency Forward Contracts">currency forwards</a>, FRAs are priced at initiation and can be valued at any time during the contract's life and of course there are formulas associated with these.</li>
		<ul>
			<li>The formulas associated with FRAs are not shown. Candidates are advised to perform practice conceptual problems for FRAs and then move on to understand the formulas.</li>
		</ul>		



</ul>

Forward Rate Agreements (FRA)

There is <a href="https://financetrain.com/overview-of-credit-risk/">credit risk</a> involved in <a title="What are Forward Contracts?" href="https://financetrain.com/what-are-forward-contracts/">forward contract</a> because the counterpart may not deliver the asset to you at the time of delivery.

Since a forward contract is not exchange traded, a buyer or seller cannot lock in gains/losses on the contract’s value prior to the agreed settlement date. The contract must be held till its maturity.

Credit Risk Amount: Given this characteristic of forwards, the contract’s value serves as a quantification of credit risk for the counterparties involved in the contract.

Expected Loss: The expected loss on a forward contract is equal to the notional amount of the contract multiplied by the contract’s value multiplied by the probability of default.

If the contract has a positive value, then the buyer is at risk that the seller will not deliver the underlying at settlement.

If the contract has a negative value, then the seller is at risk that the buyer will not purchase the underlying at settlement.

Credit Risk and Forward Contracts

Futures contacts can be viewed as modifications on forward contracts, but is imperative to understand the differences.

Futures cover either commodities or financial instruments.

Like forward contracts, futures contracts are priced based on an arbitrage relationship.

NOTE: Futures contracts for commodities can incorporate additional variables related to storing the underlying asset.

<h3>Futures Contract Characteristics</h3>
<ul>
	<li>Futures are traded on regulated exchanges (not over the counter).</li>
</ul>
<ul>
	<li>Contract terms are standardized with respect to the: deliverable, contract size, settlement date, and method of settlement (alternatively, forwards are customized).</li>
</ul>
<ul>
	<li>A clearinghouse guarantees contract performance, which eliminates counter-party credit risk; alternatively forward contract participants face the risk that the counter-party will not perform on the contract.</li>
</ul>
<ul>
	<li>Counterparties post margin with the clearinghouse in order to help ensure contract performance.</li>
</ul>
<ul>
	<li>The value of a futures contract is reset to zero daily, as the daily gain/loss on the contract is netted against the margin posted by the counter parties.</li>
</ul>
<ul>
	<li>The mechanics of a futures trade is standardized.</li>
</ul>
<h2>Closing a Futures Contract – 3 ways</h3>
<ol>
	<li>Enter an offsetting position before contract expiration.</li>
	<li>Deliver the underlying asset at expiration.</li>
	<li>Settle in cash at expiration.</li>
</ol>
For some contracts on financial assets, physical delivery is not allowed and cash settlement is required.

<h3>Value of a Futures Contract</h3>

Forward contracts can accrue significant positive or negative value during their life.

Because the daily gain or loss on a futures contract is netted against the margin posted by the counterparties, the value of a futures contract remains approximately zero during the contract term.

Introduction to Futures Contracts

Given that <a href="https://financetrain.com/introduction-to-futures-contracts/" title="Introduction to Futures Contracts">futures contracts</a> are standardized modifications on <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contracts</a>, the same pricing formula and arbitrage relationship applied.

<div class="formula1">
Futures Price: f0(T) = [S0 - PV(CF)](1+r)T
 
Futures Price (alternative formula): f0(T) = S0(1+r)T - FV(CF)

 
A lower case "f" notation distinguishes futures from forwards ("F").
</div>

<ul>
	<li>At the expiration of a futures contract, the futures contract price must equal the spot price of the underlying; if this is not the case then an arbitrage opportunity exists.</li>
	<li>If the futures contract is less than the spot price of the underlying at expiration, then a trader can buy the future and sell the spot.</li>
	<li>If the futures contract is greater than the spot price of the underlying at expiration, then a trader can buy the spot and sell the futures contract.</li>

	</ul>

Futures: Convergence of Spot and Futures Prices at Expiration

While a futures contract is priced in the same general manner as a forward contract, there are some small differences between futures and forwards.
<ul>
	<li>Because the daily gain/loss is settled daily on outstanding <a title="Introduction to Futures Contracts" href="https://financetrain.com/introduction-to-futures-contracts/">futures contracts</a> via margin account transfers credit risk is eliminated.  Alternatively, forward contracts can accumulate significant <a title="Overview of Credit Risk" href="https://financetrain.com/overview-of-credit-risk/">credit risk</a>, as the value of the forward contract will change with changes to the price of the underlying asset.</li>
	<li>The margin accounts for futures contracts are invested in short term interest securities.  This difference from forward contracts adds an element to the returns from futures contracts, affecting the pricing relationship.</li>
	<li>The pricing of futures contracts is affected by the correlation between interest rates and futures prices.</li>
	<li>When there is positive correlation the futures contract buyer benefits as he/she is gaining more from the margin account interest and the contract price is rising.</li>
</ul>
<h2>Holding the Underlying Asset</h2>
Futures prices are based on the same arbitrage relationship applied when pricing <a title="Equity Forward Contracts" href="https://financetrain.com/equity-forward-contracts/">forward contracts</a> – the price of the future should equal the cost of buying the underlying asset at the spot price with borrowed funds.

When this relationship is not reflected in a futures contract price, an opportunity for arbitrage exists for traders and in an efficient market, the mispricing will on exist for a very short period of time.

Benefits of holding the underlying asset, as opposed to holding the futures contract:
<ul>
	<li>Storage and carrying costs: The futures contract does not give the contract holder physical ownership; therefore the contract holder avoids storage and carrying costs. This fact increases the price of the futures contract.</li>
	<li>Cash flows on the underlying: Physical ownership gives entitles the asset owner to the asset’s cash flows (such as dividends, interest coupons, etc.); the contract holder does not receive these cash flows. This fact reduces the price of the futures contract.</li>
	<li><a href="https://financetrain.com/storage-convenience-yield-and-commodity-prices/">Convenience yield</a>: Physical ownership of the underlying gives the asset owner the benefit of actually being able to consume the asset; alternatively, the holder of a futures contract cannot immediately consume the underlying. This fact reduces the price of the futures contract.</li>
	<li>Daily margin cash flows: Technically there should be futures price adjustments based on the margin cash flows and interest, but this is not expected to be on the exam.</li>
</ul>
<div class="formula1">Theoretical Futures Price = Future Value of the Spot Price + Future Value of Storage Costs – Future Value of Asset Cash Flows – Future Value of Asset’s Convenience Yield</div>

Futures Prices vs. Forward Prices

<ul>
	<li>When an asset does not involve any storage costs, cash flows, and/or convenience yield, then one could expect the price of a future contract to equal the spot multiplied by one plus the periodic risk free interest rate.</li>
	<li>However, if there are other cost/benefit variables associated with the underlying asset, then the futures price could equal, exceed, or be below the spot price of the underlying asset.</li>
	<li><a href="https://financetrain.com/what-is-contango/">Contango</a>: Futures price exceeds current spot price (f0 > S0); interest and storage costs to finance asset purchase exceed physical ownership benefits.</li>
	<li><a href="https://financetrain.com/what-is-backwardation/">Backwardation</a>: Futures price is below current spot price (f0 < S0); benefits of physical ownership exceed costs of storage and interest to finance asset purchase.</li>
	<li>"Normal" contango and backwardation describes the relationship between futures prices and the expected spot price.</li>
	<li>Normal Contango: The futures price exceeds the expected spot price (f0 > E(ST)); indicates that the futures price is biased high.</li>
	<li>Normal Backwardation: The futures price is below the expected spot price (f0 < E(ST)); indicates that the futures price is biased low.</li>
	<li>When thinking about contango and backwardation versus normal contango and normal backwardation, keep in mind that the futures price and the expected spot price reflect different degrees of risk. The futures price is based on risk free rate arbitrage pricing, while the movement from the current spot price to the expected future spot price is the price change on a risky asset.</li>
	<li>A futures price can be a biased estimate for the expected spot price and this bias is commonly low.</li>

</ul>

Contago and Backwardation

Equity Index Futures are a type of futures contracts that try to replicate the performance of an equity index such as S&P, FTSE, or ay other index. Buyers can invest in a basket of equities without trading the individual stocks. 

These contracts are used to hedge against an existing equity position for for speculation.

The pricing of stock index futures is performed in the same formulaic manner as presented earlier in the futures section.
	<div class="formula1">
	Equity Index Futures Price: f0(T) = [S0 - PV(CF)](1+r)T
	 
	Equity Index Futures Price (alternative formula): f0(T) = S0(1+r)T - FV(CF)
	
	<ul>
		<li>CF = Dividend expected to be paid during the remaining life of the contract term</li>
		<li>S0 = Spot price of the equity index</li>
	</ul>
	</div>

Pricing Stock Index Futures

<ul>
<li>The pricing of Treasury bond futures is performed in the same formulaic manner as presented earlier in the futures section.</li>
	<ul>
		<li>Note that the spot price includes any accrued interest for the bond.</li>
		<li>The Treasury bond future price must be divided by the conversion factor.</li>
			<ul>
				<li>Because the futures contract seller is allowed to deliver from a range of bonds at expiration to fulfill the contract, a conversion factor must be applied to the futures price.</li>
				<li>Treasury bond pricing is based on the "cheapest to deliver" (CTD) bond as this would be the most rational decision for the futures contract seller.</li>
			</ul>
	</ul>
<li>Step 1: Price Treasury Bond Future</li>
	<div class="formula1">
		Treasury Bond Futures Price: f0(T) = [S0 - PV(CF)](1+r)T
		 
		Treasury Bond Futures Price (alternative formula): f0(T) = S0(1+r)T - FV(CF)
		<ul>
			<li>CF = Coupon payment during the remaining life of the contract term</li>
			<li>S0 = Full bond price, including accrued interest</li>
		</ul>
	</div>
<li>Step 2: Apply the Conversion Factor</li>
	<div class="formula1">
		Treasury Bond Price = Futures Price of the CTD/Conversion factor
		 
		Note: expect the exam to provide the CTD bond and the conversion factor. The test taker may be required to price a futures contract, given that data.
		 
		Either of the formulas from step 1 could be divided by the conversion factor; either would yield the same result.
		
	</div>
</ul>

Pricing Interest Rate/Treasury Bond Futures

<ul>
	<li>Currency futures are priced in the same manner as currency <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contracts</a>.</li>
	<li>Price of a Currency Future with Simple Compounding:</li>
		<div class="formula1">
		f(0,T)X/Y = S0,X/Y ((1+rX)/(1+rY))T 
		
		<ul>
		<li>T = total contract length</li>
		<li>S0 = Spot exchange rate for X/Y (#X units domestic per 1Y foreign) at initiation</li>
		<li>rX = Risk free rate in country X (domestic currency)</li>
		<li>rY = Risk free rate in country Y (foreign currency)</li>
		</ul>
		</div>
	<li>Price of a Currency Future with Continuous Compounding</li>
		<div class="formula1">f(0,T)X/Y = (S0,X/Y e-ryT ) erxT </div>
</ul>

		<ul>
			<li>If the currency future is not "fairly" priced (i.e. not priced based on an arbitrage relationship based on risk free rates), then it is possible for a trader to earn a risk-free arbitrage profit.</li>
			<li>Futures Market Overvalues a Currency:</li>
				<ol>
					<li>Sell the overvalued futures contract</li>
					<li>Buy the underlying currency in the spot market</li>
					<li>Borrow and invest all cash flows</li>
				</ol>

		</ul>

Pricing Currency Futures

Eurodollar: Eurodollar refers to a U.S. dollar denominated deposit in a bank outside the U.S. and thus outside the jurisdiction of the Federal Reserve System.

Eurodollar Future: A Eurodollar Future is a <a title="Introduction to Futures Contracts" href="https://financetrain.com/introduction-to-futures-contracts/">future contract</a> for a notional Eurodollar deposit amount, whose value at expiration is based upon the term relevant LIBOR rate on the expiration date.

A Eurodollar future is comparable to a forward rate agreement.

Unlike other futures and forwards, Eurodollar futures face a pricing challenge because pure risk-free arbitrage is not possible.
<ul>
	<li>An arbitrageur cannot simultaneously take a position in the asset and the Eurodollar future because the net value at expiration is not certain.</li>
	<li>Both the value of the Eurodollar deposit and the Eurodollar future are dependent upon the relevant LIBOR rate and these cannot be offset.</li>
	<li>As an underlying asset, a Eurodollar pays incremental interest on top of a deposit; compare that to a Treasury bill, which is a discount instrument.</li>
	<li>Despite these pricing challenges, Eurodollar futures can still be an effective instrument for hedging exposure to floating interest rates.</li>
</ul>

Eurodollar Futures

Derivatives Part 1

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Derivatives Part 1

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