Derivative Products

Futures Contract Specification

A FUTURES CONTRACT is a legal contract in the US under the Commodities Exchange Act of 1850. The futures contract allow market participants to take or make delivery of a commodity on some date in the future at a price agreed to today. In many cases businesses plan their raw supply needs and/or their production ahead of time.  Often businesses can make contracts between each other, but the benefits offered by the futures markets centralized trading standard contract terms and the risk mitigation of the Clearinghouse made futures well in demand as our country grew. By having a central marketplace (electronic bulletin board) where buyer and seller can meet and transact also begets liquidity. Then, the investment Managers will tend to get interested in a products along with the speculator & arbitrageur. Futures are OBLIGATIONS. Both buyer and seller are obligated to perform to the terms of their contract. Futures contracts are standardized with respect to all terms and conditions:

  • The quantity of the commodity covered by each contract
  • The quality of the commodity to be delivered
  • The Geographical location to which the commodity will be delivered
  • The date on which each contract matures/expires
    • N.B. : At or before maturity the contract can be canceled by reversing the position.
    • If at maturity the contracts no longer wish to make delivery (SHORT) of or take (LONG) delivery of the underlying commodity, the position can be closed out as above, otherwise, all remaining open contracts go through the delivery process.

West Texas Intermediate Crude Oil Contract Specifications

Symbol: CL_7 <comdty> P [go] (for the price of the __month* (7=2017) on Bloomberg
Quantity: 1,000 barrels of WTI Crude Oil
Quality: WTI describes the area in West Texas where the crude oil a very light & very sweet
Delivery Point: Cushing, Oklahoma

Contract Value & Value of One Unit of Currency per $1 Contract Price Change
Contract Value WTI trading @ $50 per barrel * 1,000 barrels = $50,000 CONTRACT VALUE WTI trading @ $51 per barrel * 1,000 barrels = $51,000 CONTRACT VALUE CHANGE IN CONTRACT VALUE FROM $50 TO $51 = $1,000
Execution of Futures Contracts
Execution Futures contracts trade on an “exchange”, electronic or telephone. Futures Options still trade on exchange floors due to the complexity of order types.
Clearing for Futures Contracts
Clearing All Futures contracts are cleared at a Clearinghouse In the US futures exchanges had their own clearinghouses. Each futures contract traded and cleared on the same exchange. As trading moved to electronic market and a wider range of types of crude oil’s trading, Clearinghouses for each product are not as connected to the execution location.

Futures Clearinghouses – Organizational Structure

Clearinghouses are Organization of Mutuality that is the risk belongs to the owners of the Clearinghouse. If a client defaults on a trade, i.e., cannot post more margin, the introducing broker has a choice: they can cover the loss themselves, or if they’re unable to afford the loss they can bring it to the clearinghouse committee. Each member will share in the loss on a pro-rata basis.

Initial Margin

As mentioned above, futures contracts clear at a clearinghouse. Long & Short both post INTIAL MARGIN or a good faith deposit. Thereafter the positions are marked to the market: the counterparty making money receives more margin. The counterparty losing money pays more margin. This additional margin is called variation margin. Initial Margin acts as a cushion and variation margin allows for the settlement of P&L daily. This system allows clearinghouses to mitigate the counterparty risk. Here is a list of futures contracts with specifications and margin

Example of Futures Margin

Initial Margin = $3,190 Both long and short post $3,190 for each contract they want to keep open overnight. While they may choose to get out of the trade when Asia or Europe are trading, they will have to post $3,190 to their US-based clearinghouse. Variation or Maintenance Margin = $2,900 Maintenance margin is a trigger. The client will be called for additional margin if the account falls below the maintenance level. During the credit crisis, Maintenance Margin wasn’t used and some firms have chosen to maintain that policy. Herefore a client must always have the initial margin for the contract set aside in a segregated account as a good-faith or performance bond. Check with your broker prior to trading so you know their specific guidelines.

Futures Volume & Open Interest

Using the Futures Margin Example above, FOCUS ON THE TWO RIGHT MOST COLUMNS TO SEE HOW Volume & Open Interest Work. Volume and open interest are specific to contract markets. But by measuring the volume relative to the open interest a trader can glean a lot of clues as to:

  • Knowing the relationship between the amount of volume and open interest, you will know who is involved in your market

Volume & Open Interest — Summary & Uses

Volume: Contracts traded during the trading day. Volume is additive throughout the day. Open Interest: Open Contracts held in position. Open Interest measures the net OPEN contracts. If a market has very high daily volume, but open interest remains stable, one could surmise that market has more speculators involved in that market. By contrast, a futures market with high volume sporadically but where open interest grows on those days and progressively over the life of the contract, one could surmise that market has more hedgers involved than speculators. FUTURES _OPEN INTERESTFUTURES _OPEN INTEREST

Market Participants Uses & Applications

Market Participants

Any market we look at, we need to understand who uses the product and why. It’s worth saying at the outset that derivatives contracts, in general, have four participants. Viewing the diagram below, you will see two participants on top labeled HEDGERS. These are your participants needing futures contracts to hedge their actual business risk. A scan to the bottom of the diagram will show you the speculator and arbitrageur. These participants are our leveraged players. They use the derivatives market for the leverage it offers and provides liquidity to the hedgers. Going forward you may want to look at the type of participants in a specific product you’re viewing to see if all four participants are in the market. Without all four participants actively trading the product, the contract is unlikely to provide significant liquidity and transparency.

Uses & Applications

Hedgers a.k.a “Commercials”

Unleveraged players, a.k.a. “commercials” are corporations with market risk of crude or corn or any future contract. We differentiate between their risk and other types of market participants because these corporations are hedging their business risk to the end product they use or produce:

End Users

An end user of grains and sugar such as Kellogg & Co. doesn’t buy & store their corn, wheat & sugar all at once. Rather they make their purchases periodically over the course of the year. This way they don’t bear the risk of spoilage or the cost of storing the raw products. If the price of their commodity goes up, they will have to pay a higher price to make their product. Companies can choose to take this risk or they can choose to hedge it. Companies can hedge their price risk directly with the producers of the products or they can choose to hedge using futures. Futures, cleared through a central clearinghouse provides anonymity, price transparency and the mitigation of counterparty risk. The trade-off of using futures & central clearing is the company doesn’t get the exact product they use, delivered to the exact location they need. They take “basis risk”, which we’ll discuss in a moment.

End Producers

End producers create a product or make a raw product usable. For example, refiners buy crude oil and refine it into gasoline & heating oil. Kellogg’s purchases grains from farm consortiums. The farmers are the End Producers of corn. End producers are at risk to the price going down. To hedge their market risk they could contract directly with the end user when it’s ready for sale, or they can use futures to hedge their price risk. The end producer, such as the farm consortium choosing to sell futures makes the same trade-off as the End User. By using a central clearinghouse, their advantages include anonymity, price transparency and the mitigation of counterparty risk. The trade-off is that the futures contract may not call for delivery of the exact product they create or delivered to the same location where they are located. The producer takes “basis risk”, which we’ll discuss in a moment.

Leveraged Players

“Leveraged players” in any derivative market describe those people who will borrow money to express their trade. Futures contracts are leveraged due to the low margin cost relative to the contract value. Recall our crude oil contract, with crude @ $50 per barrel = $50,000.  Yet our initial margin is only $3190.

  • The leverage of the contract =Initial margin $3,190 divided by the contract value $50,000.
  • The buyer & seller are posting 6.38% of the contract’s value as initial margin
  • They have a leverage factor of  15.67 x margin (capital used).

Margin levels can and will change as prices rise or the commodity becomes more volatile. Generally speaking, clearinghouses want initial margin to cover 3 times the average daily move. Or the average volatility of the most recent 3 days, whichever is higher.


Speculators love leverage and with good reason. For a relatively small investment, they can reap great returns. Of course, the ever optimist living within us rarely chooses to look at the double edge sword. In a futures contract, you can LOSE MORE THAN YOU INVEST.

Types of Speculators

  • Short-Term Speculators or Day Traders generally make their decision using technical analysis.
  • Medium to longer-term speculators generally make their decisions based on the fundamental supply-demand of the underlying commodity. They may use technical points once they’ve decided to buy or sell in order to get the best price on the trade.
    • Commodity Trading Advisors, structured product desks and commodity hedge funds would fall into this user type as well.


Arbitrageurs take advantage of price anomalies between similar products when the price between the products reaches a level outside of normal range. The arbitrageur enters a trade, looking to reverse the trade when the prices between the products normalize. In effect, a market maker is also an arbitrageur. A market maker is a liquidity provider to the market. Their mandate is to provide a bid and an offer, prices at which market participants can trade. There is a misconception that arbitrage is exploiting an obvious mispricing in the market. There Ain’t No Such Thing As A Free Lunch (TANSTAAFL). Typically arbitrageurs are simultaneously buying and selling similar contracts. For example, an arbitrageur buying US crude oil and selling UK Brent crude oil will profit if US crude rises more or falls less than Brent Crude.

This arbitrageur is still assuming risk on their trade and will lose money if Brent rises more or falls less than US Crude.

Arbitrageurs look for anomalies in relationships between two. assets.

Hedging a Futures Position – Revisited

In the typical futures market, it’s the End User & End Producer that create the need for the futures product. Consider crude oil:

  • The end user of crude oil is the refiner that needs to purchase oil at different points in time to refine the crude oil into gasoline and heating oil.
  • The end producer of crude oil is the oil & exploration company.

In the last section we covered market risk: we learned that end users of a commodity are at risk to the price of the commodity going up if they choose not to hedge. The end producer of a commodity has price risk of the commodity going down.

Why do corporations hedge using futures?

Because they don’t need to purchase or sell the crude oil today, but rather on specific dates in the future. They can hedge directly with the company who buys (sells) the underlying commodity. But the futures contract offers several advantages. Since all futures contracts are cleared at a central clearinghouse the advantages are:

  1. Privacy: no one other than the clearinghouse knows whose position is it
  2. Counterparty risk mitigation: is limited to the clearinghouse
  3. Transparency & Liquidity: provided by the central trading location of futures (electronic bulletin board)

If they hedge directly with the buyer (seller) of their commodity, they will receive (deliver) the exact commodity at the exact location they need. But they lose the advantages of the clearinghouse and the price discovery/liquidity of active futures contracts.

What is Basis Risk?

The contract specifications for crude oil called for delivery of a certain quality of crude oil. Specifically, the crude must have a Sulphur content of .42% of less; and the API gravity between 37-42 degrees. The crude oil must be delivered into Cushing Oklahoma. Basis risk is the differences between the WTI crude oil (“THE DELIVERABLE GRADE”) and the EXACT crude oil our client is hedging. The exact crude oil the client is hedging can be quite different. The hedger will adjust the number of contracts to account for this risk. But they still use the futures contract as their choice for hedging because of the reasons mentioned above:

  1. A clearinghouse offers more anonymity.
  2. A clearinghouse mitigates counterparty risk.
  3. A clearinghouse provides more liquidity and price discovery so the hedger doesn’t need to give up price advantage due to a need to hedge.
    • The hedger may have to get out of their hedge for a variety of reasons. Hedging with another commercial company will limit their ability to do so at a good price.

Volume & open interest statistics show that many hedgers prefer to use futures. For example in crude oil only ~ 3% of contracts are still open & go through the delivery process. Further, as the open interest of the maturing futures contract goes down, the open interest of the next available futures contract rises. This shows the hedgers are “rolling” their positions into the next delivery month to maintain their price hedge.

Pricing Forwards & Futures

Future Value of a Cash Flow (FV)

Pricing a Futures Contract

To price a futures contract, we start by calculating the Future Value of the spot price of the commodity and adjust from there. The Deliverable grade of crude oil futures “WTI” has Sulfur: 0.42% or less by weight, and Gravity: Not less than 37 degrees nor more than 42 degrees (measured by API)

STEP ONE – pricing a futures contract

UNDERLYING WTI Crude Oil (deliverable grade )
MATURITY one month Crude Oil Futures Contract
SPOT PRICE OF CL $50 per barrel

STEP TWO transportation & storage costs

Other costs or income associated with the underlying asset adjust the Future Value number above.

  • If our underlying asset were a bond, then we would include the income of the coupon.
    • Storage of crude oil for one month
    • Transportation of crude oil for one month
  • Same with Stock Index futures where dividends are paid. But since we are pricing oil, we only have costs to add:

Transportation by Sea requires a bit of background, so this example will keep us on dry land. Due to larger tankers, investment in canals and other issues, historical numbers skew the cost of transporting oil. To get our futures Price and presume our oil will travel by pipeline from Calgary to Cushing, Oklahoma and stored for the remaining days. The total cost of storage and transportation for 30- days will be $3.78 per barrel. To make our calculation easier, convert it to an annual percentage figure: $3.78/$50.00  = 7.56%.

Hedging Crude Oil Risk

Hedging using futures is done using the SIMPLE HEDGE RATIO or the OPTIMAL HEDGE RATIO. The simple hedge ratio is more focused on the timing of the hedge. The Optimal Hedge Ratio also wants to adjust for the difference in volatility & correlation between the futures & the exact grade/location of crude oil the hedger buys (sells). Example: Let’s assume a refiner is looking to hedge his crude oil purchases beginning in April 2017. Recall the refiner buys crude oil to convert it into gasoline and heating oil. Our refiner needs to purchase a total of 150 million barrels of crude oil over four months from May to August 2017. REFINER’S SIMPLE HEDGE = 150,000,000/1000 = 37,500 contracts each month from May, June July & August. REFINER’S SIMPLE HEDGE: BUY 37,500 May, June, July & August Crude Oil Futures.

Optimal Hedge Ratio

Let’s continue with the same example used to calculate the simple hedge ratio and introduce the concept of optimal hedge ratio. Our refiner still needs to purchase 150 million barrels over four months beginning in May 2017. But in reality, the crude he will purchase is Gulf Coast crude oil. His refinery is along the gulf coast and it’s easier for him to take delivery at that port. To hedge using the futures contracts we need to adjust for two major variables.

  1. The difference in volatility between the two crude oils and
  2. The correlation between the two types of crude oils

We need to adjust for both of these variables because even though we expect all oils to move in the same direction, if the two oils are less than +.985 correlated, the futures are not likely to be the most efficient hedge. Volatility measures something very different. Volatility measures the magnitude of price returns. If we buy too many contracts we will be over-hedged. If the price of WTI goes down more than we expected we will lose money using futures as our hedge. Adjusting for both of these variables the futures will mimic the performance of Calgary Crude.

Relative Volatility
WTI 29%
Relative Volatility = 89.7%
Positive Correlation = +.99

If you’d like to learn more about Commodities forward curves, physical versus futures arbitrage and others topics, they will be covered in Our Learning Center

Contract Terms & Conditions

An interest Rate Swap is a contract whereby each counterparty gets to transform the type of risk management they run. Specifically, it’s a contract where the counterparties exchange a fixed cash flow, which will remain unchanged for the duration of the swap for a floating rate cash flow which will change at pre-defined times and using a pre-defined rate at periodic dates over the duration of the swap.

Interest Rate Swap Diagrams

There are three ways to illustrate an interest rate swap. The first is called a box & arrow diagram.

The timeline Diagram. The timeline diagram show the payment of cash flows over the passage of time:

The third way is called the cash flow diagram. We will discuss this diagram in greater detail in the SWAPS PRICING section.

Interest Rate Swaps

To An interest rate swap is a contract where one type of interest rate is applied to one leg of the swap while the other leg’s cash flow is calculated using a different type of interest rate. For example: Fixed Pay for 5 yrs & Received 3 Month LIBOR The fixed leg will be paid for 5 years at the same rate The floating leg (LIBOR) will change every 3 months, for 5 years. If interest rates go down, this swap will lose money as the floating rate heir receiving is going down. When we speak about swaps, we speak in terms of the fixed leg. “I’ll pay fixed on $100 million for 5 years” It’s already implied I’m receiving the floating leg. PAY FIXED = make money if rates go higher RECEIVE FIXED = makes money if rates go lower.

Memory Tip: Interest Rate Swaps

If you’re long a bond, you receive a coupon and make money if rates go lower. If you’re receiving fixed on a swap you make money if rates go lower. If you’re short a bond you pay the coupon and make money if rates go higher. If you’re paying fixed on a swap, you make money as rates go higher.

Interest Rate Swap Market Participants

To convert the language from futures to Rates, recall with crude oil futures we spoke of End Users & End Producers as participants who would use Crude Oil Futures as Hedges against price risk.

Market participants by Uses & Applications

  1. Corporations
    • Borrowers of money by issuing bonds;
    • Issuing stock allows investors to participate in the company’s Performance
  2. Unleveraged Buyers of Assets: Investors, Pension Funds, Mutual Funds
    • Hedge a fixed income portfolio by decreasing or increasing the modified duration
    • Create synthetic assets yielding a higher rate than available in the market
  3. Speculators (leveraged)
    • Interest rate swaps can be used as synthetic bonds (long or short) where the credit risk is pure bank credit
  4. Arbitrageurs (leveraged)
    • Swaps Dealers: will assume the risk of a client’s swap and hedge the swap which limits any residual risk. Best case scenario is the Swap dealers makes a BP or 2 BPs
    • Hedge fund & other leveraged players

Market Terminology: Rates & Spreads

When speaking about other debt instrument relative to interest rate swaps, we speak in terms of the spread between the two products. Below is a corporation bond trading at 4.50%. Today, since interest rate swaps are so liquid, we would say the bond is trade +50 bps over swap rates. The 150 bps is partially the credit rating spread. But relative liquidity between the bond and the swap, the relative duration etc would also be taken into account. Another spread tracked closely is the swap spread. In the illustration below, the swap spread is 100 bps, the swap rate is 4.00% (US Treasuries + Swap Rate).

Comparative Advantage

With respect to interest rate hedgers, it may be worthwhile to discuss how corporations raise capital. Corporations can either borrow money from a bank, or if their credit rating is high enough and they are well known in the market can issue bonds in the public bond market. The determination will be which market they receive a lower rate.

Examples of Applications

In the Interest Rate Swap Market the concept of Comparative Market Advantage is useful to understanding why each market participant uses swaps. As we go through the hedger examples, we’ll speak to this comparative advantage with respect to each player.

Higher Credit Rated Corporates

A AA-rated Corporation typically can borrow in the public bond market (fixed rate) at a lower rate than they would pay to a bank on a bank loan. They will issue a bond and pay a fixed coupon to bondholders. If the Higher Rated Corporation wants to pay fixed and it fits their asset-liability management, their fixed-floating mix and cash flow then they will remain in a fixed rate liability. If not, they will enter into a swap, swapping from a fixed rate liability to a floating rate liability. Let’s take a look.

Let’s assume the following rates
5 yr BOND ISSUE 2.25%
5 YR SWAP RATE 2.50%

This AA corporate could have remained a fixed rate borrower for 5 yrs. There are several reasons why corporations choose to swap.

  1. In an upper sloping yield curve, 3 month LIBOR will be lower than 5 yr rates. The company’s interest expense will be lower.
  2. Most companies want a mix of fixed – floating liabilities. Depending on the type of business they’re in they will prefer 50% – 80% fixed rate liabilities. Conversely, they may prefer 50 – 80% floating rate liabilities. This is to create efficient cash flow.
    • Example: an internet subscription corporation receives payments for their services every month. They may prefer to pay floating interest, even though they want to lock in the time for which they have use of the money.

Lower Credit Rating Corporation

A BBB Corporation borrows from a bank (floating rate loan) at a lower rate than the public would lend them money, they will take out a bank loan and pay LIBOR plus a spread. If a Lower Rated Corporation wants to pay LIBOR and the bank loan fits their asset-liability, fixed-floating and cash flow mix, they will remain in a bank loan. If a Lower Rated Corporation would prefer to pay a fixed rate, they will enter into an interest rate swap to PAY FIXED AND RECEIVE LIBOR.

Let’s assume the following rates
5 yr BOND ISSUE 4.00%
5 YR SWAP RATE 2.50%

It’s worth noting that the interest rate swap provides flexibility for corporations to separate their source of funds from the way they manage their interest rate risk. As we saw with our AA borrower, they borrowed in the market which provided them the lowest rate. But due to the speed of their asset conversion cycle, as a subscription company, they can better manage their interest rate risk by borrowing LIBOR.

Asset Managers

Asset managers can also be seen as Lenders of money by buying bonds from corporations. The type of asset buyer we’re speaking about in the upper right hand portion of our participants circle is an unleveraged buyers of assets. Asset managers use Swaps to create an asset swap. In an asset swap, the asset buys a fixed rate note or bond and enters into a swap to PAY FIXED AND RECEIVE LIBOR. The end result is the asset manager receives LIBOR plus a spread. Effectively they own a synthetic floating rate note. The asset manager can purchase the bond and enter into the swap on their own, as in the diagram below.

Asset Managers also use interest rate swaps as a hedge. Asset managers are paid by client to outperform some pre-agreed bond index. That index has an average maturity or duration of the bonds included in the index. An asset manager can outperform their index in several ways. They can buy more long dated bonds if they think the market will go up or purchase more bonds of the corporations they like. Conversely, an Asset Manager may choose not to buy some of the bonds in the index because they don’t like the credit risk. However once a portfolio of bonds has been put in place, if the manager thinks the market will trade lower, managers don’t want to pay the slippage (bid-ask spread) of selling the bonds and rebuy them later. So they take an alternate route to hedging their rate risk. Asset managers will overlay an interest rate swap to shorten the duration of their portfolio. For those unfamiliar with DV01, it’s the amount the bond will change for a 1 basis point change in rates.

Down the second column of the matrix, you’ll note an increasing column of numbers. Taking a look at the 5 yr row, the present value for each number of years at the rate in force when the matrix was built shows a 5 yr bond portfolio has a DV01 = $481.96. To illustrate the most extreme hedge, we’ll use the following illustration:

By putting on the hedge 1:1, we’ve turned the portfolio into one similar to our asset swap. But for an asset manager whose index has a 5 yr average maturity, we need to look at the difference in risk to see the hedge. Earlier we saw the average risk was $481.96. With the hedge, the portfolio’s DV01 is now only$25!!  if rates go up as expected, this portfolio manager will only lose $25 per basis point (for each $ 1 million invested).  The unhedged portfolio manager will lose $481.96 per basis point (for each $1 million invested). This is NOT A LONG TERM HEDGE. This hedge will be in place for two weeks but not much longer. Once the market goes down, the hedge will be taken off. This type of brute force hedging, without refinement for the credit spread risk, etc. is not elegant, but it works. Particularly in fast markets.

Hedge Funds

Leveraged funds use swaps as synthetic financed bond positions. To a hedge fund, they view swaps as synthetic bonds which they’ve repo’d to purchase and have the credit quality of swap dealers. To understand this in greater detail let’s look at how a long bond position would look like the following diagram: A client calls and buys a bond, Calls the repo desk to lend the bond and borrow money, Uses the money to pay for the bond. (N.B.: leverage) Hedge funds use swaps to create synthetic bond positions. The following diagrams show the logic behind this:

Looking just at the coupon flows, the position is receiving the fixed coupon on the bond and paying the repo rate on the financing of the bond position.

But looking at just the cash flows, perhaps we don’t need to use bonds or “real assets” unless they better fit our goal. When ee view this diagram

It doesn’t have anything different than this diagram

On the short side, where the trader is also taking on delivery risk, the bond gets tight in repo, the trade takes on a different angle:

Again focusing on the cash flows:

The above doesn’t look much different with respect to the cash flows below.

The differences between the bond position and the swap position are:

  1. Credit rating of the bond position would be subject to whatever the credit rating of the issuing corporation (or sovereign entity) is.
  2. Credit rating of the swap is counterparty credit risk, which is mitigated by posting initial collateral on the swap as well as paying losses and receiving profits daily.
  3. The floating leg on the swap is 3 Month LIBOR; the bond is financed at the repo rate, which is typically an overnight rate. The spread between 3 ML and Repo is fairly stable unless the bond is hard to borrow (“special”).

Swaps Pricing

Time Value of Money

Before we delve into swaps pricing, this section reviews the Time Value of Money, introduces compounding & continuous compounding. Then we’ll have all of the math necessary for this section and your further studies at MH.

The Variables of Pricing a Cash Flow

  1. Dollar amount at the start of the calculation
  2. Rate: fixed or floating
    • Fixed rate: will be in force for the entire length of the calculation
    • Floating rate: will change periodically on a pre-agreed basis
  3. How often the rate will be paid (daily, weekly, quarterly, etc. )
  4. Day count convention of the cash flow (30/360, ACT/365, etc)

Present Value & Future Value

1. FUTURE VALUE calculates the value of a cash flow in the future for an investment made today. 2. PRESENT VALUE calculates the value of a cash flow today, presumed to be paid in the future.

Compounded Values

Compounding assume reinvestment of a cash flow received during the year.  If I own a bond paying 10% and interest is paid semi-annually, the annual rate at year end will be more than 10%.  The cash flow received at 6 months is reinvested for the remainder of the year.

Compounded Present Value

To calculate the compounded present value, we use the following calculation where R = the annual rate, C = compounding periods per year & t = time to maturity.


Compounded Future Value

To calculate the compounded future value, we use the following calculation where R = the annual rate, C = compounding periods per year & t = time to maturity.



Continuously Compounded Values

We use continuous compounding to convert all rates to a single compounding period so we can compare the rates apples to apples.

Continuously Compounded Present Values

To calculate a continuously compounded cash flow, we use the Exponent e notated (and used in Excel) as EXP where r = rate per year & t = time to maturity. The only difference between continuously compounded present value and future value is the use of a negative rate -r*t, to calculate a present value, continuously compounded.       The right hand column in the table below are Present Values continuously compounded using the same assumptions asa above: $100 invested for one year at 10%


Continuously Compounded Future Values

To calculate a continuously compounded cash flow, we use the Exponent e notated (and used in Excel) as EXP where r = rate per year & t = time to maturity. Since we’re calculating the continuously compounded FUTURE cash flow, we leave the RATE ( r) as a positive number.




Interest Rate Swaps Pricing

The “First Commandment” of Derivatives pricing is using the most liquid instrument whose terms and conditions most closely resemble what you’re trying to price. The logic is:

  1. if the contract is liquid enough to hedge the swap it will be liquid enough to price the swap.

The Timeline Diagram With Dates

The Timeline diagram shows the cash flows of a swap.  on the right-hand side you can see the floating rates we’ll use to price the following swap:

$1 million; MATURITY: 3 years; PAY fixed; RECEIVE floating (both legs netted & paid annually)

in order to price a new At-Market swap, the Net Present Value of all cash flows must equal zero.  In order to find the fixed rate of the swap, we use the floating rates.  After bringing the floating forward rates to present value, we iterate and solve for the FIXED rate which results in an NPV of zero.


Forward Rate Agreements (FRA’s)Are A Cornerstone Product for Swaps Pricing

Forward Rate Agreements (FRA’s) are agreements to pay (receive) a fixed rate of interest and receive (pay) a floating rate at some time in the future (maturity).

FRA’s are really single period interest rate swaps.

  • In a payer FRA the payer agrees to pay a fixed rate at some date in the future, in exchange for receiving whatever rate is in force for that term to maturity. In a payer FRA you make money as rates go higher (prices fall). 


  • In a Receiver FRA the receiver agrees to receive a fixed rate and pay the floating rate in force for that term at maturity. You make money on a receiver FRA as rates go DOWN (price goes up)


Pricing An Interest Rate Swap using Forward Rate Agreements

Notional $1,000,000
Maturity Three Years
Fixed Pay Annual
Floating Rec Annual
Using the following 12 ML%
1 yr 6.25%
2 yr 7.00%
3 yr 7.50%

We have our LIBOR rates so we can calculate the Forward Rate Agreements:

The first FRA is the 12 x 24 (12 month libor = 24-12; forward start date: 12 months)

Then we need to price the 24 x 36 FRA (12 month libor = 36-24; forward start = 24 months forward)

Now we have all of our rates & forward rates to price the swap. Here’s the resulting timeline showing the Rate Agreement (rate reset today) and the 2 forward rates 12 x 24 & 24 x 36.


Next we’ll use a cash flow diagram and continue on with pricing each cash flow and then solving for the fixed rate on the swap. We use the FRA’s to get the correct forward rates for the swap, but for the swap to be priced fairly, we need all of the cash flow to come back to a NET PRESENT VALUE OF ZERO.

Pricing Interest Rate Swaps using A Cash Flow Diagram

The steps to calculate a new At-Market Interest Rate Swap

  1. Collect the necessary LIBOR rates
  2. LIBOR rates converted to forward rates (for floating leg)
  3. Forum your cash flow diagram in pieces
    1. Columns for spot & Forward rates
    2. Columns Present value of floating leg
    3. Columns Present value of fixed leg
    4. Last column of Net present values
    5. Bottom of last column = NPV
      • Goal seek to solve for NPV -0-

The Cash Flow Diagram in one Piece

Here’s a snapshot of the whole swap. Now we’ll go through the columns piece by piece.

The Cash Flow Diagram Step by Step

The first several columns are for Forward Rate Calculations. The forward rates can be seen in the right hand column with their title to the left. i.e. , 7.755% = 1 yr. LIBOR, 1 year forward. The next four columns are used to calculate the present value of the floating leg of the swap.

The future Value is the notional &1,000,000 * the forward rates. The present value brings these cash flows back using the LIBOR rate for 1, 2 & 3 years. Not the sub-diagram Discount Factors. A Discount factor is the zero coupon rate or present value of one unit of currency. The discount rate used is the same as above 1, 2 & 3 year LIBOR deposit rates.

The next two columns calculate the Present Value of the fixed leg of the swap, followed by the last column showing the NPV (Floating PV – Fixed PV).

Notice, the YELLOW highlight around the cell atop the FIXED FV Column. Also note the column NET PV. To solve for a Par swap, use Goal Seek (Data > what if analysis) the second choice down should be Goal Seek, if you don’t see this choice you will need to install your analysis toolkit. Using the goal seek functions as shown above, will give you the fixed rate at which all of the cash flow solve for an NPV of zero.

The Cash Flow Spreadsheet for Pricing Interest Rate Swaps

Here is the spreadsheet in one piece. Note the forward rates are on the row above the cash flow. This tells us the floating rate is set in advance and paid in arrears.

Repricing Interest Rate Swaps – Two Curves Necessary

Both cleared and un-cleared derivatives are repriced the same way, using the OIS curve & the LIBOR Curve.

  • The LIBOR curve is used to calculate the market-to-market, but
  • The OIS curve is used to calculate collateral calls.

During the Great Credit Crisis and the reporting dealers forced the LIBOR reset rate lower, Since the LIBOR reset rate is used to reset all swaps & loans resetting that day, the rate impacts a large dollar amount. Market Participants decided, given the volatility of LIBOR a more stable rate should be used. The OIS Swap (LIBOR minus Effective Fed Fund) was chosen as the rate for COLLATERAL POSTING. Please note the difference: LIBOR curve is used to calculate Profit & Loss, but the OIS Spread to calculate the collateral one counterparty needs to post.

What is the OIS Spread?

The OIS Spread or OIS Swaps is the spread between LIBOR and the effective fed funds rate (EFF).

Why don’t we just use Fed Funds Then?

First, recall the Fed Funds are loans made between banking institutions CAN ONLY LEND THEIR FREE RESERVES IN THE FED FUNDS MARKET, and the market is short dated, it’s only liquid out 270 days.

Where do we find OIS Spreads to Reprice Long-Dated Swaps?

We use the OIs Swaps Market rates. Why? Because they’re so liquid. However there are other sources in the links section is you prefer to do your own work on the curve.

What are OIS Swaps?

The OIS swap is a floating for floating swap, LIBOR versus Effective Fed Funds. As with most floating – floating swaps, the maturity of the swap is long (i.e. , 5 yr swap) but the two floating rates are observed very frequently, with a net cash flow periodically.

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