Derivatives Flashcards
7.1 Pricing & Valuation of Forward Commitements
– Describe how equity forwards and futures are priced, and calculate and interpret their no-arbitrage value.
– Describe the carry arbitrage model without underlying cashflows and with underlying cashflows.
– Describe how interest rate forwards and futures are priced, and calculate and interpret their no-arbitrage value.
– Describe how fixed-income forwards and futures are priced, and calculate and interpret their no-arbitrage value.
– Describe how interest rate swaps are priced, and calculate and interpret their no-arbitrage value.
– Describe how currency swaps are priced, and calculate and interpret their no-arbitrage value.
– Describe how equity swaps are priced, and calculate and interpret their no-arbitrage value.
Forward Commitments
1- Definition of Forward Commitments: Agreements to transact at a specified future date under predetermined terms.
– Forward commitments include forwards, futures, and swaps.
– Forwards and futures involve a single transaction, whereas swaps consist of a series of transactions.
2- Key Characteristics:
– Terms such as price, transaction date, and quantity are agreed upon when the contract is established.
– Payoffs are linear, exposing parties to both gains and losses based on market movement.
3- Pricing and Valuation:
– Pricing is based on value additivity, arbitrage, and the law of one price.
– Example:
— An interest rate swap’s value can be determined using the prices of two bonds that replicate the swap.
4- Applications of Forward Commitments:
– 1- Diversification: Equity index swaps help investors diversify concentrated asset holdings.
– 2- Duration Adjustments: Pension funds use interest rate swaps to modify the duration of their portfolios.
– 3- Tactical Allocation: Swaps enable portfolio rebalancing without the need to liquidate positions.
– 4- Debt Nature Conversion: Borrowers use interest rate swaps to change their debt from fixed-rate to floating-rate, or vice versa.
– 5- Market Expectations: Forward contract prices, such as volatility swaps, can infer market expectations.
Key Takeaways
– Forward commitments include forwards, futures, and swaps, offering flexibility and risk management.
– Pricing relies on arbitrage and the law of one price.
– Applications include diversification, duration adjustments, and tactical portfolio rebalancing.
Principles of Arbitrage-Free Pricing and Valuation of Forward Commitments
1- Key Definitions:
– Forward Price: The fixed price agreed upon at the initiation of a forward commitment, which remains constant throughout the term of the contract. It is typically set so the initial value of the contract is zero.
– Value of a Forward Commitment: The fluctuating value of the forward contract over its term, which changes based on the underlying asset’s price movements.
2- Arbitrage-Free Pricing:
– Arbitrage-free pricing relies on the assumption that no arbitrage opportunities exist in the market, meaning no risk-free profit can be earned without using capital.
– Arbitrage is possible when identical cash flows for the same risk do not have the same price.
– The law of one price ensures identical investments have the same price, while the value additivity principle states that the total portfolio value equals the sum of its individual asset values.
3- Simplifying Assumptions for No-Arbitrage Pricing:
– Replicating instruments are available for the derivative.
– No market frictions exist (e.g., no taxes or transaction costs).
– Short selling is allowed.
– Borrowing and lending occur at the risk-free rate.
4- Carry Arbitrage Model:
– Definition: This model combines a position in the underlying asset with a forward contract. It demonstrates how to arbitrage the cost of carrying an asset over time.
– Example:
– Borrow money to purchase the underlying asset today.
– Simultaneously, enter a forward contract to sell the asset at a predetermined price in the future.
– This is called the cost-of-carry model or cash-and-carry arbitrage.
– Use: The carry arbitrage model is a foundational tool for deriving the observed prices of forward contracts.
Key Takeaways
– Forward price and value are distinct: the forward price is fixed, while the value changes with the underlying asset price.
– Arbitrage-free pricing assumes no arbitrage opportunities, enabled by the law of one price and value additivity.
– The carry arbitrage model is crucial for understanding forward contract pricing based on cost-of-carry.
Notation for Forward and Futures Contracts
1- S_t:
– Represents the price of the underlying instrument at a specific point in time, t.
2- F₀(T):
– Denotes the forward price agreed upon at time 0 for a contract that expires at time T.
3- f₀(T):
– Indicates the futures price agreed upon at time 0 for a contract that expires at time T.
4- V_t(T):
– The value of a forward contract at time t, which will expire at time T.
5- v_t(T):
– The value of a futures contract at time t, which will expire at time T.
Pricing and Valuing Generic Forward and Futures Contracts
1- Differences Between Forwards and Futures:
– Forwards: Over-the-counter (OTC) contracts with customizable terms and no assurance from a clearinghouse.
– Futures: Exchange-traded contracts with standardized terms, guaranteed by a clearinghouse requiring margin deposits and daily marking-to-market.
– While operational differences exist, pricing of both instruments can be analyzed using the same carry arbitrage models.
2- Key Definitions and Notations:
– S0: Spot price of the underlying asset at initiation (time 0).
– Ft: Forward price of a futures or forward contract at a specific point after initiation (time t).
– FT: Forward price when the contract expires at time T.
– ft: Current price of a futures contract after initiation but before expiration.
3- Price Movements During the Contract Term:
– The spot price of the underlying asset (St) will fluctuate after initiation, influencing forward and futures prices.
– Example: If an asset’s forward price is initially F0 = €100 and its spot price increases to St > S0, the forward price for a shorter contract (Ft) will likely rise above F0, all else equal.
4- Convergence to Spot Price:
– As the contract approaches expiration, forward and futures prices tend to converge with the spot price of the underlying asset.
– At expiration: FT = ft = ST.
– Forward price reflects the value for immediate delivery, ensuring it matches the spot price at expiration.
5- Cost and Benefits of Carrying the Underlying Asset:
– Carrying Costs: Borrowing costs to hold the asset increase the forward price. Higher risk-free rates result in a higher forward price.
– Carrying Benefits: Benefits such as dividends reduce the forward price.
– Basis: The difference between the spot price and the forward or futures price at any time, reflecting these costs and benefits.
Key Takeaways
– Forwards and futures pricing depends on changes in the spot price and cost-of-carry assumptions.
– Convergence ensures forward prices align with the spot price at expiration.
– The forward price incorporates both the costs and benefits of holding the underlying asset.
Valuing Forward and Futures Contracts
1- Valuing Forward Contracts:
– The value of a forward contract depends on the difference between the prevailing spot price (St) and the forward price agreed upon at initiation (F0).
– Formulas:
— Long position: Value at time T (VT) = St - F0.
— Short position: VT = F0 - St.
– Observations:
— The long party benefits if the spot price rises above the forward price, while the short party benefits if the spot price falls below the forward price.
— Forward commitments are typically structured as “at market” contracts, meaning the initial value (V0) is zero.
2- Valuing Futures Contracts:
– Futures contracts are marked-to-market daily, resetting the value to zero after settlement each day.
– Formula for the value of a futures contract (vt) for the long party (before daily settlement):
vt = Nf × (ft - ft-1).
– Where:
— Nf = Contract multiplier.
— ft = Current futures price.
— ft-1 = Previous day’s settlement price.
3- Example (Futures Contract Valuation):
– If a contract for 1,000 barrels of oil rises from $80 to $82:
vt = 1,000 × ($82 - $80) = $2,000.
– This gain is credited to the long party’s margin account, while the short party records an equivalent loss. After daily settlement, the value returns to zero.
Key Takeaways
– Forward contract valuation is straightforward and depends on the difference between the spot price and the agreed forward price.
– Futures contracts require daily marking-to-market, with gains and losses settled daily, ensuring the contract’s value resets to zero at the close of each trading day.
Price vs. Value in Forward Contracts
1- Forward Price:
– The forward price is the rate agreed upon at the initiation of the forward contract for a future transaction.
– Key points:
— It is fixed and does not change over the life of the contract.
— This is the price scheduled to occur at the contract’s expiration.
2- Forward Value:
– The forward value measures the worth of the contract during its term.
– Key points:
— It fluctuates as the underlying asset’s spot price changes.
— The value reflects the financial gain or loss for one party, offset by the equivalent loss or gain for the other party.
Key Takeaways
– The forward price remains constant throughout the contract, whereas the forward value changes based on market conditions.
– Forward price determines the transaction terms, while forward value assesses the contract’s worth over time.
Marking to Market and Valuation Differences in Futures and Forward Contracts
1- Futures Contracts and Marking to Market:
– The value of a futures contract is adjusted daily through a process known as marking to market.
– The difference between the futures price of the current day (f_t(T)) and the futures price of the previous day (f_(t-)(T)) determines the daily gain or loss.
– Value before marking to market (v_t(T)):
— For a long futures position: v_t(T) = f_t(T) - f_(t-)(T).
— For a short futures position: v_t(T) = f_(t-)(T) - f_t(T).
2- Cash Flow Adjustments:
– Any gains or losses are immediately added to or subtracted from the investor’s margin account daily.
– Because of this daily settlement, the value of the futures contract is reset to 0 at the end of each trading day.
3- Impact of Marking to Market:
– The differences between f_t(T) and f_(t-)(T) are generally small due to the daily adjustment process.
– This ensures minimal carryover of risk compared to forward contracts.
4- Forward Contracts and Counterparty Risk:
– Forward contracts do not undergo daily marking to market, which means gains or losses are realized only at expiration.
– This lack of interim cash flow adjustments results in greater counterparty risk.
– No intermediate transactions occur prior to contract expiration, unlike futures.
Key Takeaways
– Futures contracts reduce counterparty risk due to daily settlements, while forward contracts carry greater risk as adjustments occur only at the end of the contract.
– The value of a futures contract fluctuates daily, whereas forward contracts maintain a fixed value until expiration.
Forward and Futures Contracts: Value at Inception and Expiration
1- Forward Contract:
– At inception (time 0):
— The forward contract is typically structured as an “at market” contract, meaning the initial value is set to zero.
— Formula: V₀(T) = 0.
– At expiration (time T):
— For a long forward position: The value is the difference between the spot price of the underlying asset (S_T) and the forward price agreed at initiation (F₀(T)).
— Formula: V_T(T) = S_T - F₀(T).
— For a short forward position: The value is the reverse, where the forward price is subtracted from the spot price.
— Formula: V_T(T) = F₀(T) - S_T.
2- Futures Contract:
– Futures contracts are marked to market daily, so their value resets to zero after each trading session.
– Before marking to market:
— For a long futures position: The value is the difference between the current futures price (f_t(T)) and the previous day’s futures price (f_(t-)(T)).
— Formula: v_t(T) = f_t(T) - f_(t-)(T).
— For a short futures position: The value is the difference between the previous day’s futures price and the current futures price.
— Formula: v_t(T) = f_(t-)(T) - f_t(T).
Carry Arbitrage Model Without Underlying Cash Flows
1- Concept of Carry Arbitrage:
– The carry arbitrage model is used to exploit mispricing in forward contracts by carrying the underlying asset over the life of the contract.
– The portfolio is designed to require no initial capital and involve no risk, provided the ending value is certain.
2- Steps to Establish a Carry Arbitrage Position:
– 1- Take a short position in a forward contract, with an initial value of zero.
– 2- Borrow funds equal to the current value of the underlying asset (S₀).
– 3- Use the borrowed funds to purchase the underlying asset.
3- Cash Flows:
– At Time 0:
— The investor borrows S₀ and uses it to purchase the underlying, resulting in zero net cash flow.
– At Time T:
— The investor repays the borrowed amount with interest, calculated as FV(S₀), and delivers the asset under the forward contract.
— Net payoff is determined by the difference between the forward price (F₀) and the future value of the borrowed funds [FV(S₀)].
4- Future Value of Borrowed Amount (FV(S₀)):
– Continuous compounding: FV(S₀) = S₀ * e^(r_c*T).
– Annual compounding: FV(S₀) = S₀ * (1 + r)^T.
— Where r_c = continuously compounded risk-free rate, and r = annually compounded risk-free rate.
5- Arbitrage Condition:
– If the forward contract is correctly priced, the overall position generates a return equivalent to the risk-free rate.
– If the forward price (F₀) deviates from its fair value, arbitrage opportunities arise.
Reverse Carry Arbitrage Strategy
1- Definition of Reverse Carry Arbitrage:
– This strategy is the inverse of the carry arbitrage model. It is used when the forward price of an asset is too low relative to its fair value.
– The objective is to exploit the mispricing by profiting from borrowing the underlying asset and selling it short while entering into a long forward contract to lock in a repurchase price at maturity.
2- Steps to Execute Reverse Carry Arbitrage:
– 1- Take a long position in a forward contract:
— Enter into a forward contract to buy the underlying asset at a future date (at price F₀).
– 2- Sell the underlying asset short:
— Borrow the underlying asset and sell it in the market at its current spot price (S₀).
— The proceeds of this short sale equal S₀.
– 3- Lend the proceeds from the short sale:
— Invest the proceeds from selling the underlying asset at the risk-free rate.
3- Cash Flows:
– At Time 0:
— Borrow the underlying asset and sell it at S₀, earning S₀ in cash.
— Lend the proceeds at the risk-free rate, resulting in a zero net cash flow at initiation.
– At Time T:
— Receive the repayment from the loan, including interest, calculated as FV(S₀).
— Use the forward contract to buy back the asset at F₀ and return it to the lender.
— The net payoff is the difference between the future value of the loan [FV(S₀)] and the forward price (F₀).
4- Arbitrage Condition:
– The strategy is profitable if the forward price (F₀) is lower than the fair forward price, calculated as FV(S₀).
— If F₀ < FV(S₀), the arbitrageur earns risk-free profits.
— Fair pricing ensures that the forward price accounts for the cost of carrying the underlying asset, preventing arbitrage opportunities.
5- Formula for Fair Forward Pricing:
– The forward price is considered fair if F₀ = FV(S₀).
— FV(S₀) = S₀ * e^(r_c*T) under continuous compounding, or S₀ * (1 + r)^T under annual compounding.
— If the forward price deviates from this condition, arbitrage opportunities exist.
6- Key Takeaways:
– Reverse carry arbitrage locks in profits by capitalizing on a forward price that is lower than its fair value.
– The profit results from lending proceeds of the short sale (FV(S₀)) and buying the asset back at a cheaper forward price (F₀).
– This strategy ensures risk-free profits if the mispricing persists.
7- Illustration of Arbitrage Profit:
– Suppose:
— S₀ = $100, r = 5%, T = 1 year, F₀ = $103.
— Fair forward price = FV(S₀) = $100 * (1 + 0.05) = $105.
— Arbitrage profit = FV(S₀) - F₀ = $105 - $103 = $2 per unit.
Carry Arbitrage and Reverse Carry Arbitrage: Key Explanation
1- Carry Arbitrage:
– A carry arbitrage position is established when the observed forward price is too high relative to the arbitrage-free forward price.
– This situation is represented mathematically as:
F₀ > S₀ × (1 + r)^T,
where:
— F₀ is the forward price.
— S₀ is the spot price today.
— r is the risk-free rate.
— T is the time to maturity.
In this case, the forward price exceeds its arbitrage-free level, creating an opportunity to sell the forward contract (short forward), borrow funds, and purchase the underlying asset.
2- Reverse Carry Arbitrage:
– A reverse carry arbitrage position is established when the observed forward price is too low relative to the arbitrage-free forward price.
– This situation is represented mathematically as:
F₀ < S₀ × (1 + r)^T.
Here, the forward price is below its fair value, presenting an opportunity to buy the forward contract (long forward), sell the underlying asset short, and lend the proceeds.
3- Key Implications:
– The arbitrage-free forward price is solely determined by the current price of the underlying (S₀) and the risk-free interest rate (r), compounded over time (T).
– Importantly, the forward price is not influenced by the expected future price of the underlying asset. This ensures that arbitrage opportunities are strictly a function of mispricing relative to the cost of carrying the underlying.
Linking Back to the Previous Example:
In the reverse carry arbitrage example:
– If the forward price F₀ = 78 is correct, there are no arbitrage profits because gains on one leg of the strategy are fully offset by losses on the other.
– If the forward price were observed to be too low, say F₀ = 76, a reverse carry arbitrage strategy could be executed to lock in a risk-free profit by:
— Going long the forward contract.
— Short-selling the stock at S₀ = 75.
— Lending the proceeds at the risk-free rate to earn FV(S₀) > F₀.
Similarly, if F₀ were too high, a carry arbitrage position could exploit the mispricing by shorting the forward contract and buying the underlying asset.
This ensures that prices align with the no-arbitrage condition, maintaining market efficiency.
Valuing a Forward Contract with No Cash Flows After Initiation
1- Initial Value of the Contract:
– At initiation, a forward contract is structured to have a value of zero by setting the forward price appropriately.
– This eliminates the need for any payment between the two parties at the beginning of the contract.
2- Value Over Time:
– As time progresses, the value of the forward contract changes due to fluctuations in the spot price St of the underlying asset and time remaining to expiration.
3- Formula for the Value of a Long Forward Contract:
– Name of Formula: Long forward value formula.
– Formula: “Vt = St - [F0 ÷ (1 + r)^(T-t)]”
Where:
– Vt: Value of the forward contract at time t.
– St: Current spot price of the underlying asset at time t.
– F0: Forward price agreed upon at initiation.
– r: Risk-free interest rate.
– T: Time to expiration (in years).
– t: Time elapsed since initiation (in years).
4- Explanation of the Formula:
– The formula calculates the forward contract’s value by finding the difference between:
– 1- The current spot price of the underlying asset (St).
– 2- The present value of the forward price (F0) discounted back to time t using the risk-free rate.
5- Key Assumptions:
– The underlying asset generates no interim cash flows, such as dividends or coupons, during the contract’s life.
– The risk-free rate is constant over the contract’s duration.
6- Numerical Example:
– Suppose the following:
– 1- St = 110 (current spot price of the underlying).
– 2- F0 = 105 (forward price agreed at initiation).
– 3- r = 5% (risk-free rate).
– 4- T = 1 year, and t = 0.5 years (6 months have passed since initiation).
Using the formula:
“Vt = St - [F0 ÷ (1 + r)^(T-t)]”
Substituting values:
“Vt = 110 - [105 ÷ (1 + 0.05)^(1-0.5)]”
Step-by-step:
“Vt = 110 - [105 ÷ (1.05)^0.5]”
“Vt = 110 - [105 ÷ 1.0247]”
“Vt = 110 - 102.45”
“Vt = 7.55”
– The value of the forward contract at time t = 0.5 is 7.55.
Key Takeaways
– Forward contracts are priced to have a zero value at initiation by setting an appropriate forward price.
– The contract’s value fluctuates over time as the spot price changes and time passes.
– The formula accounts for the current spot price and the discounted forward price, ensuring consistency with no-arbitrage principles.
Carry Arbitrage Model When Underlying Has Cash Flows
1- Introduction: The pricing of a forward contract must account for the presence of cash flows in the underlying asset. These cash flows can be positive (benefits) or negative (costs).
– Carry benefits (CB): Include cash flows like dividends and bond coupon payments.
– Carry costs (CC): Include costs such as storage, waste, and insurance.
2- Formula for Forward Price with Cash Flows:
– Name of Formula: Forward price with cash flows.
– Formula: “F0 = FV[S0 + CC0 − CB0]”
— Where:
— F0: Forward price.
— FV: Future value operator.
— S0: Spot price of the underlying asset.
— CC0: Present value of carry costs.
— CB0: Present value of carry benefits.
– Explanation:
— Carry benefits (CB) decrease the forward price as they add value to holding the underlying.
— Carry costs (CC) increase the forward price as they impose additional burdens.
3- Special Case for Dividend-Paying Stocks:
– Name of Formula: Forward price for dividend-paying stocks.
– Formula: “F0 = FV(S0) − FV(Dividends)”
— Where:
— FV(S0): Future value of the spot price at time T.
— FV(Dividends): Future value of dividends paid during the contract period.
– Explanation:
— For dividend-paying stocks, the forward price decreases by the future value of dividends, assuming there are no other carrying costs.
Key Takeaways:
– Cash flows in the underlying affect forward pricing, requiring adjustments for benefits and costs.
– Carry benefits (e.g., dividends) lower the forward price, while carry costs (e.g., storage) increase it.
Valuing a Forward Contract with Cash Flows after Initiation
1- Introduction: Forward contracts with cash flows follow similar valuation principles as those without cash flows. However, adjustments must account for carry costs (CC) and carry benefits (CB).
2- General Valuation Formula:
– Name of Formula: Value of a forward contract with cash flows.
– Formula: “Vt = PV[FT − F0]”
— Where:
— Vt: Value of the forward contract at time t.
— PV: Present value operator.
— FT: Forward price at time T.
— F0: Forward price agreed upon at initiation.
3- Adjusted Valuation Formula:
– Name of Formula: Adjusted value of forward contracts considering cash flows.
– Formula: “Vt = St − [F0 ÷ (1 + r)^(T−t)] + PV(CC) − PV(CB)”
— Where:
— Vt: Value of the forward contract at time t.
— St: Spot price of the underlying asset at time t.
— F0: Forward price at initiation.
— r: Risk-free rate.
— T: Time of forward contract expiration.
— PV(CC): Present value of carry costs.
— PV(CB): Present value of carry benefits.
4- Explanation:
– 1- The first component, “St − [F0 ÷ (1 + r)^(T−t)],” represents the intrinsic value of the contract based on the difference between the current spot price and the present value of the agreed forward price.
– 2- The additional terms, “PV(CC) − PV(CB),” adjust for the effects of carry costs and benefits:
— PV(CC): Increases the forward value, reflecting the costs incurred.
— PV(CB): Decreases the forward value, reflecting the benefits received.
Key Takeaways:
– Forward valuation after initiation must account for adjustments due to carry costs and benefits.
– The formula integrates the intrinsic value of the contract and the cash flow effects of the underlying asset.
Equity Forward and Futures Contracts
1- Forward Pricing Formula with Cash Flows:
The formula for forward price is adjusted to account for continuous compounding, carry costs, and carry benefits when the timing and amount of interim cash flows (e.g., dividends) are uncertain.
– Name of Formula: Equity forward price formula (adjusted for interim cash flows).
– Formula: “F0 = S0 × e^[(rc + CC - CB) × T]”
– Where:
— F0: Forward price at time 0.
— S0: Current spot price of the equity or index.
— rc: Continuous compounding rate (risk-free rate).
— CC: Carry costs (e.g., storage, borrowing costs).
— CB: Carry benefits (e.g., dividends, coupon payments).
— T: Time to contract maturity, expressed in years.
2- Explanation:
– The formula works best when interim cash flows (dividends, etc.) are limited and known.
– If dividend payments or other cash flows have uncertain timing and amounts, the formula adjusts for these variables through carry benefits and costs, alongside continuous compounding rates.
Key Takeaways
– This formula incorporates both costs and benefits of carrying equity over the life of the contract.
– Continuous compounding ensures precision in calculating forward prices.
– Adjustments for uncertain cash flows make the model applicable to stock indexes with complex dividend structures.
Interest Rate Forward and Futures Contracts
1- Libor as Reference Rate:
The London Interbank Offered Rate (Libor) has traditionally been the reference interest rate for many derivatives. It represents the rate at which London banks borrow from each other. Despite its phasing out due to rate-rigging scandals, Libor is still used in this context.
2- Notation for Interest Rate Forward Pricing:
– Lm: Libor on an m-day deposit on day i.
– NA: Notional amount, or funds deposited.
– NTD: Number of days in a year for interest calculations (360 days by convention).
– tm: Fraction of year for an m-day deposit, calculated as m ÷ NTD.
– TA: Terminal amount or funds repaid, calculated as follows:
— “TA = NA × [1 + Lm × tm]”
3- Example of Terminal Amount Calculation:
– Suppose a 180-day Libor deposit has a notional amount of $100 and a 4% annualized rate.
– Using “TA = NA × [1 + Lm × tm]”, the terminal amount is:
— “TA = 100 × [1 + 0.04 × (180 ÷ 360)] = 102”
– Here, $2 represents the interest paid over 180 days.
4- Formula for Interest Paid:
Interest can be calculated using either of the following methods:
– Formula 1: “Interest = TA - NA”
– Formula 2: “Interest = NA × (Lm × tm)”
Key Takeaways
– The pricing of Libor-based forwards uses simple, linear calculations based on the actual/360 convention.
– Terminal amounts and interest paid depend on the notional amount, the Libor rate, and the fraction of the year.
– These formulas are foundational for understanding Libor-based derivatives.
Forward Rate Agreements (FRAs)
1- Definition of FRAs:
Forward Rate Agreements are forward contracts with an interest rate as the underlying.
– The fixed-rate payer (short) and floating-rate receiver (long) are the two counterparties.
– The long party benefits if interest rates rise, while the short party benefits if rates fall.
– The FRA price is the fixed rate that ensures the FRA’s initial value is zero.
2- FRA Notation and Format:
FRAs are written as “X × Y”.
– X: Time to FRA expiration in months.
– Y: Time to the maturity of the underlying deposit in months from the contract’s initiation.
– Example: A 3 × 9 FRA expires in 3 months and is based on a 6-month deposit maturing in 9 months.
3- Timeline for FRAs:
– Time 0: Initiation date, when the FRA is created and priced.
– Time h: FRA expiration, when the floating rate is determined and compared to the fixed rate.
– Time h + m: The maturity of the underlying deposit, also denoted as T (where T = h + m).
– The FRA’s value depends on the time remaining until h and the rates observed during this period.
4- FRA Pricing and Settlement:
– The fixed forward rate, “FRA0”, is the interest rate ensuring that the FRA’s initial value is zero.
– FRAs are typically advanced set (rate set at time h) and advanced settled (settled at time h).
– Settlement in arrears means the payment would occur at h + m, but FRAs usually settle early.
5- Settlement Formula:
For the floating rate receiver (long position), the settlement amount is calculated as:
– Formula: “Settlement = [NA × (Lh(m) - FRA0) × tm] ÷ [1 + Dm × tm]”
— NA: Notional amount.
— Lh(m): Floating rate for the m-day deposit determined at time h.
— FRA0: Fixed forward rate agreed at time 0.
— tm: Fraction of the year for the m-day deposit.
— Dm: Discount rate, typically assumed equal to Lm (the m-day Libor rate).
Key Takeaways
– FRAs are useful for hedging interest rate exposure by locking in future borrowing or lending rates.
– They are structured for flexibility with “advanced set, advanced settled” being the standard for payment.
– Pricing and settlement involve comparing the fixed and floating rates, with settlement discounted to the FRA expiration date.
Simple Explanation of Forward Rate Agreements (FRAs)
A Forward Rate Agreement (FRA) is like a bet on future interest rates. Two parties agree to a specific interest rate for a loan or deposit that will start in the future. Here’s how it works:
Two Roles in an FRA:
Fixed-rate payer: This party “locks in” an interest rate and will pay it no matter what happens in the market.
Floating-rate payer: This party agrees to pay whatever the actual interest rate is at the future time.
Why Do FRAs Exist?
FRAs are used to manage the risk of future interest rate changes. For example, a company that knows it will borrow money in six months can use an FRA to “lock in” the borrowing cost today.
How the Settlement Works:
On the agreement’s end date (called the expiration date), the actual interest rate in the market is compared to the fixed rate that was agreed upon.
If the market rate is higher than the agreed fixed rate, the fixed-rate payer gains money because they locked in a lower rate.
If the market rate is lower, the floating-rate payer benefits because they pay less interest than expected.
Key Points to Remember:
FRAs are settled in cash, not by actually borrowing or lending money.
Settlement is based on the difference between the agreed fixed rate and the actual floating rate.
Difference Between FRAs and Swaps
While FRAs and swaps both involve agreements about future interest rates, they work differently:
Timeframe:
FRA: A one-time agreement about a single period in the future. For example, you agree on the interest rate for a six-month loan starting three months from now.
Swap: A longer-term agreement that involves multiple periods. For example, you might exchange interest rate payments every six months for five years.
Structure:
FRA: Only one settlement happens (at the expiration date of the agreement).
Swap: Many settlements happen, one for each payment period over the life of the swap.
Use:
FRA: Typically used to manage risk for a specific short-term future loan or deposit.
Swap: Used for longer-term risk management or to change the structure of ongoing debt (e.g., from fixed to floating interest payments or vice versa).
Determining the FRA Rate (FRA0)
1- Setting the FRA Rate:
To make the FRA’s initial value zero, the fixed forward rate (FRA0) is set using no-arbitrage principles.
– Assumptions:
— Borrowing and lending occur at the Libor rate (Lm).
— Notional amount (NA) is standardized as 1.
— Discount rate is equal to Lm at time h.
2- Cash Flow Table Summary:
The table summarizes cash flows for the following:
– Depositing funds for T days.
– Borrowing for h days.
– Borrowing for m days starting at time h.
– Receiving floating-rate payments based on FRA.
Key Rows from the Table:
At time 0: Deposits and borrowings are initiated.
At time h: Borrowing m-day funds begins; floating payments are calculated.
At time T (h + m): The total payoff of deposits and borrowings is settled.
The net cash flow at time T is adjusted to ensure the ending value is zero, eliminating any arbitrage opportunities.
3- FRA Pricing Formula:
The no-arbitrage condition results in the following equation:
“1 + LrT * tT ÷ (1 + Lth * th) - FRA0 * tm ÷ (1 + Lth * th) = 0”
Rearranging this equation, the FRA0 rate is calculated as:
“FRA0 = [ (1 + LrT * tT) ÷ (1 + Lth * th) - 1 ] ÷ tm”
– Where:
— LrT: Libor rate for T days.
— Lth: Libor rate for h days.
— tT and th: Time fractions for T and h days.
— tm: Fraction of the year for m days.
4- Simplifying the Equation:
To better illustrate equivalence, the cash flows from:
– Two successive deposits (one for h days and one for m days).
– A single deposit for T days.
The equivalence can be expressed as:
“[1 + Lth * th] * [1 + FRA0 * tm] = 1 + LrT * tT”
Example: Calculating the FRA Fixed Rate
1- Problem Setup:
The task is to calculate the rate of a 3 × 9 forward rate agreement (FRA).
– Given:
— h = 90 days.
— m = 180 days.
— T = 270 days.
— Euribor for 90 days (Lth) = 5.6% (or 0.056).
— Euribor for 270 days (LrT) = 6.1% (or 0.061).
2- Formula for FRA0:
The FRA fixed rate (FRA0) can be calculated using:
“FRA0 = [ (1 + LrT * tT) ÷ (1 + Lth * th) - 1 ] ÷ tm”
– Where:
— LrT: Rate for T days.
— Lth: Rate for h days.
— tT = T ÷ 360: Fraction of a year for T days.
— th = h ÷ 360: Fraction of a year for h days.
— tm = m ÷ 360: Fraction of a year for m days.
3- Substituting the Values:
Step 1: Calculate tT, th, and tm:
– tT = 270 ÷ 360 = 0.75.
– th = 90 ÷ 360 = 0.25.
– tm = 180 ÷ 360 = 0.5.
Step 2: Substitute into the formula:
“FRA0 = [ (1 + 0.061 * 0.75) ÷ (1 + 0.056 * 0.25) - 1 ] ÷ 0.5”
Step 3: Simplify the terms:
– Numerator: (1 + 0.061 * 0.75) = 1.04575.
– Denominator: (1 + 0.056 * 0.25) = 1.014.
– Fraction: (1.04575 ÷ 1.014) - 1 = 0.0313.
– Divide by tm: 0.0313 ÷ 0.5 = 0.0626 (or 6.26%).
Final Result:
The FRA fixed rate (FRA0) is 6.26%.
Valuing an Existing Long FRA (Floating Receiver)
1- Formula for Valuing a Long FRA:
The value of an existing long FRA, where the investor receives floating payments, can be calculated using:
“Vg = [(FRAg - FRA0) * tm] ÷ [1 + D(T-g) * t(T-g)]”
– Where:
— Vg: Value of the FRA at time g.
— FRAg: Current forward rate for the underlying period at time g.
— FRA0: Fixed forward rate established at initiation.
— tm: Fraction of the year for the maturity period (m days ÷ 360).
— D(T-g): Discount rate for the remaining time (T-g days).
— t(T-g): Fraction of the year for the remaining time (T-g days ÷ 360).
2- Explanation of Components:
– The numerator, “(FRAg - FRA0) * tm”, calculates the difference in cash flows based on the current FRA rate and the original rate for the maturity period.
– The denominator discounts this difference back to the current time (g), reflecting the time value of money.
3- Key Points:
– A positive Vg indicates a gain for the floating-rate receiver.
– A negative Vg indicates a loss for the floating-rate receiver.
– The formula accounts for changes in forward rates and adjusts for the time remaining until maturity.
Key Takeaways
– The value of an FRA is determined by the difference between the current forward rate (FRAg) and the original rate (FRA0), adjusted for the maturity period and discounted for time.
– This approach ensures that the FRA’s value reflects the prevailing market conditions and no-arbitrage principles.
Valuation of a Long FRA
1- Problem Summary:
The goal is to calculate the value of a long FRA, given that the FRA is 25 days into the contract. The following data is provided:
– h = 65 days (time until the FRA expires).
– m = 180 days (maturity of the underlying rate).
– T = 245 days (total time until the maturity of the underlying rate).
– The 65-day Euribor = 5.9%.
– The 245-day Euribor = 6.5%.
– The original FRA rate (FRA0) at time 0 = 6.26%.
2- Steps to Solve:
Step 1: Calculate the current FRA rate (FRAg) at day 25.
The formula for FRAg is:
“FRAg = [ (1 + LrT * tT) ÷ (1 + Lh * th) - 1 ] ÷ tm”
– LrT = 6.5% (245-day Euribor).
– Lh = 5.9% (65-day Euribor).
– tT = 245 ÷ 360 = 0.6806.
– th = 65 ÷ 360 = 0.1806.
– tm = 180 ÷ 360 = 0.5.
Substituting values:
“FRA25 = [ (1 + 0.065 * 0.6806) ÷ (1 + 0.059 * 0.1806) - 1 ] ÷ 0.5”
“FRA25 = 6.65%”.
Step 2: Calculate the value of the FRA (V25).
The value formula for a long FRA is:
“Vg = [ (FRAg - FRA0) * tm ] ÷ [ 1 + D(T-g) * t(T-g) ]”
– FRAg = 6.
Valuation of a Long FRA
1- Problem Summary:
The objective is to calculate the value of a long FRA at 25 days into the contract. The following information is provided:
– h = 65 days (time to FRA expiration).
– m = 180 days (maturity of the underlying rate).
– T = 245 days (total time to the maturity of the underlying rate).
– The 65-day Euribor = 5.9%.
– The 245-day Euribor = 6.5%.
– The original FRA rate (FRA0) at time 0 = 6.26%.
2- Steps to Solve:
Step 1: Calculate the Current FRA Rate (FRAg) at Day 25.
The formula for FRAg is:
“FRAg = [ (1 + LrT * tT) ÷ (1 + Lh * th) - 1 ] ÷ tm”
– LrT = 6.5% (245-day Euribor).
– Lh = 5.9% (65-day Euribor).
– tT = 245 ÷ 360 = 0.6806.
– th = 65 ÷ 360 = 0.1806.
– tm = 180 ÷ 360 = 0.5.
Substitute the values into the formula:
“FRA25 = [ (1 + 0.065 * 0.6806) ÷ (1 + 0.059 * 0.1806) - 1 ] ÷ 0.5”
“FRA25 = 6.65%”.
Step 2: Calculate the Value of the FRA (V25).
The formula for valuing the FRA is:
“V25 = [ (FRAg - FRA0) * tm ] ÷ [ 1 + Lr(T-g) * t(T-g) ]”
– FRAg = 6.65%.
– FRA0 = 6.26%.
– tm = 180 ÷ 360 = 0.5.
– Lr(T-g) = 6.5%.
– t(T-g) = 245 ÷ 360 = 0.6806.
Substitute the values into the formula:
“V25 = [ (0.0665 - 0.0626) * 0.5 ] ÷ [ 1 + 0.065 * 0.6806 ]”
“V25 = 0.0019”.
3- Key Insights:
– The value is positive, indicating that the long position benefits because interest rates have increased.
– The FRAg being greater than FRA0 signifies a favorable shift for the floating receiver.
Key Takeaways
– The value of a long FRA depends on the difference between the current and original FRA rates, adjusted for time to maturity.
– Discounting ensures that the value reflects the time value of money for the remaining term.
Pricing a Two-Year Forward Contract
1- Problem Summary:
The S&P 500 Index is currently valued at 4,000. Key details include:
– A continuous dividend yield of 1.90%.
– A continuously compounded annual interest rate of 0.80%.
– The goal is to calculate the price of a two-year forward contract.
2- Observations:
– The forward price is calculated based on the current index value, the risk-free rate, and the dividend yield.
– The forward price reflects no arbitrage and is unaffected by an investor’s expectations about the future price of the index.
3- Key Insights:
– The forward price is less than the current index value because the dividend yield (1.90%) is greater than the risk-free rate (0.80%).
– When the dividend yield exceeds the risk-free rate, dividends reduce the forward price since they represent benefits received before contract maturity, reducing the cost of holding the index.
The expected future prices of the S&P is based on investors opinion, and not relevant in the calculation
Understanding a 3 × 9 Forward Rate Agreement (FRA)
1- Overview of a 3 × 9 FRA:
– A 3 × 9 FRA refers to a forward rate agreement that expires in 3 months, with a payoff based on the 6-month Libor rate that starts at that time.
– The “3 × 9” format means:
— “3” is the number of months until the FRA expires (deferral period).
— “9” is the total number of months until the underlying deposit matures (3 months deferral + 6 months deposit).
2- Structure of a Short Position in a 3 × 9 FRA:
– A short position involves:
– 1- Going short on a 9-month Libor deposit to avoid locking in a lower rate in the future.
– 2- Going long on a 3-month Libor deposit to hedge against rate increases in the short term.
3- Key Payment Mechanics:
– When the FRA expires in 3 months:
— The floating interest rate for a 6-month period (Libor) will determine the payoff.
— The FRA’s settlement is made based on the difference between the agreed fixed rate and the floating rate.
— Payments are netted at this point.
Key Takeaways
– The payoff of the FRA is calculated based on the 6-month Libor rate at the expiration of the FRA.
– The settlement allows participants to hedge interest rate risk for the 6-month deposit that begins in 3 months.
– The agreement can be “advanced set, advanced settled,” meaning rates are fixed at expiration and settlement happens immediately.
Understanding FRA Settlements and Payments
1- Structure of FRA Payments:
– The fixed interest rate is established at the FRA initiation date (time 0).
– The floating interest rate is determined at the FRA expiration date, based on the underlying rate (e.g., Libor) for the maturity period specified.
2- Settlement Process:
– The difference between the fixed and floating interest rates is settled in cash at the FRA expiration date.
– This settlement represents the payoff of the FRA, which is based on the agreed notional amount.
3- Example: A 3 × 9 FRA:
– This FRA expires in 3 months, with the payoff determined by the 6-month Libor rate.
– At expiration (3 months):
— The fixed rate was agreed upon at initiation.
— The floating rate is based on the actual 6-month Libor rate observed at that time.
– The cash settlement occurs immediately after expiration.
4- Key Takeaways:
– The floating rate is observed at the FRA expiration and determines the payoff.
– The FRA allows for hedging or speculation on interest rate movements over the specified time frame.
– Payments are settled in cash, netting the difference between fixed and floating interest obligations.
Settlement Amount for FRA (Floating Receiver/Long Position)
1- Overview of the Settlement Process:
– The floating receiver (long position in an FRA) benefits if the floating rate exceeds the fixed rate agreed upon at initiation.
– The settlement amount reflects the difference between the actual floating rate (Lh) and the fixed FRA rate (FRA0).
2- Advanced Settlement:
– FRA settlements occur at the expiration date, but the underlying interest period typically starts later (maturity of the underlying rate).
– The settlement payment is discounted back to the FRA expiration date using the discount rate for the period.
3- Formula for Settlement Amount:
– Settlementamount = [NA × (Lh - FRA0) × tm] ÷ [1 + D × tm]
– Where:
— NA: Notional amount.
— Lh: Floating rate determined at FRA expiration.
— FRA0: Fixed FRA rate agreed upon at initiation.
— tm: Fraction of the year for the maturity of the underlying rate.
— D: Discount rate (typically the floating rate).
4- Key Steps in the Timeline:
– At contract initiation (time 0), the fixed rate (FRA0) is agreed upon, ensuring the initial value of the FRA is zero.
– At FRA expiration (time h), the floating rate (Lh) is observed, and the settlement amount is determined.
– The maturity of the underlying interest rate (h + m) marks the end of the interest period, but the settlement occurs earlier, at time h, on an advanced-settled basis.
5- Key Takeaways:
– The difference between the floating and fixed rates determines the FRA payoff.
– The settlement is discounted to account for the time value of money, as the underlying period extends beyond the FRA expiration.
Unique Issues Affecting Fixed-Income Forward and Futures Contracts
1- Accrued Interest in Bond Pricing
– Bonds can be quoted in two ways:
— 1- Clean price: Excludes accrued interest.
— 2- Dirty price: Includes accrued interest.
– For forward and futures contracts, adjustments may need to be made to account for accrued interest, especially when determining the actual settlement amount.
2- Multiple Bonds Deliverable by the Seller
– Certain contracts allow sellers to choose from a set of eligible bonds for delivery.
– This flexibility can introduce pricing variations and potential uncertainty for the buyer.
3- Cheapest-to-Deliver Bonds
– When multiple bonds are eligible for delivery, the seller often selects the cheapest-to-deliver bond, minimizing their cost of fulfilling the contract.
– The cheapest-to-deliver bond is determined by comparing the cost of the bond to the delivery price specified in the contract.
Accrued Interest in Fixed-Income Pricing
1- Definition of Accrued Interest (AI)
– Accrued interest refers to the interest that has been earned on a bond since the last coupon payment but has not yet been paid.
– Bonds are quoted in two ways:
— 1- Clean price: Excludes accrued interest and reflects only the bond’s market value.
— 2- Dirty price: Includes accrued interest, representing the total amount a buyer pays for the bond.
2- Relevance in Derivative Pricing
– The distinction between clean and dirty prices is essential for derivative pricing, as different markets and countries may use either method, potentially impacting pricing conventions.
3- Calculation of Accrued Interest
– Accrued interest is calculated using linear interpolation, based on:
— 1- NAD: Number of days since the last coupon payment.
— 2- NTD: Total number of days in the coupon payment period.
— 3- n: Number of coupon payments per year.
— 4- C: Annual coupon amount.
Bond Forward Pricing Adjustments
1- Cheapest-to-Deliver (CTD) Bonds
– Sellers in bond forward contracts often have the flexibility to deliver multiple bonds. A conversion factor is applied to standardize values across bonds.
– However, differences remain, leading sellers to choose the cheapest-to-deliver bond for maximizing profits or minimizing costs.
2- Forward Price Calculation with Accrued Interest
– If the full bond price, including accrued interest, is quoted, the forward price is determined using the general formula:
— Forward price adjusts for carry costs (CC) and carry benefits (CB).
3- Customizing the Formula for Bonds
– Key adjustments for bonds include:
— 1- Carry costs (CC) are assumed to be zero.
— 2- Carry benefits (CB) consist of coupon payments, represented by the present value of coupon interest (PVCI).
– The resulting adjusted formula accounts for coupon benefits:
— Forward price reflects the clean price (B0) and accrued interest (AI0).
4- Accrued Interest and Settlement Price
– The accrued interest at contract expiry (AIT) is excluded from the settlement because contracts settle based on the clean bond price.
– This exclusion ensures the calculation of an arbitrage-free forward price.
Key Takeaways
– Bond forward pricing relies on clean bond price (excluding accrued interest) and coupon cash flow adjustments.
– Cheapest-to-deliver bonds and accrued interest considerations are critical for ensuring fair valuation in bond derivatives.
F0=[B0 + AI0] - AI_T - FVCI
F0=[B0 + AI0] - AI_T - FVCI –> Formula must be true to avoid arbitrage
Quoted Futures Price and Conversion Factor for Bond Forwards
1- Use of Conversion Factor (CF)
– When multiple bonds can be delivered, the forward price is calculated as the product of the quoted futures price (Q0) and the conversion factor (CF):
— Forward price formula: Forward price = Q0 × CF.
2- Purpose of Conversion Factor
– The conversion factor adjusts the quoted futures price to account for the seller’s choice of the cheapest-to-deliver (CTD) bond.
– This adjustment is necessary because the bond used to calculate the forward price may differ from the bond ultimately delivered.
3- Formula for Quoted Forward Price
– The quoted forward price (Q0) is derived as the forward price adjusted for the conversion factor. The formula is:
— Quoted forward price accounts for:
— Clean bond price (B0),
— Accrued interest (AI0),
— Accrued interest at expiry (AIT),
— Present value of coupon interest (PVCI).
Q0 = [1/CF] {FV [B0 + AI0] -AI_T - FVCI}
Q0 = [1/CF] {FV [B0 + AI0] -AI_T - FVCI}
Key Takeaways
– Conversion factors ensure fair pricing by standardizing futures prices when multiple bonds can be delivered.
– The quoted price reflects the actual forward price adjusted for the chosen bond’s conversion factor.
Quoted Futures Price and Conversion Factor for Bond Forwards
1- Use of Conversion Factor (CF)
– When multiple bonds can be delivered, the forward price is calculated as the product of the quoted futures price (Q0) and the conversion factor (CF).
— Formula: F0 = Q0 × CF.
2- Purpose of Conversion Factor
– The conversion factor adjusts the quoted futures price to account for the seller’s choice of the cheapest-to-deliver (CTD) bond.
– This adjustment is necessary because the bond used to calculate the forward price may differ from the bond ultimately delivered.
3- Forward Price and Adjusted Formulas
– The general forward price formula is: F0 = FV(S0 + CC0 - CB0).
– For bond forward contracts, since there are no carry costs (CC0 = 0) and the carry benefit is the present value of coupon interest (PVCI), the adjusted formula is:
— F0 = FV(S0 + 0 - PVCI).
— Additionally, the bond price can be separated into clean price (B0) and accrued interest (AI0), giving:
— S0 = B0 + AI0.
4- Final Forward Price Formulas
– Substituting into the adjusted formula, we get:
— F0 = FV(B0 + AI0 - PVCI).
– If the accrued interest at expiration (AIT) and future value of coupon interest (FVCI) need to be deducted at settlement, the formula becomes:
— F0 = [B0 + AI0] - AIT - FVCI.
5- Quoted Forward Price Formula with Conversion Factor
– When using the conversion factor (CF), the quoted forward price (Q0) is calculated as:
— Q0 = [1/CF] {FV [B0 + AI0] - AIT - FVCI}.
Comparing Forward and Futures Contracts
1- Overview of the Concept
– Forward and futures prices both reflect the future value of the underlying asset, adjusted for carry costs and carry benefits.
– The value of a forward contract after initiation is calculated as the present value of the difference between the current forward price and the original forward price.
– Futures positions are marked to market at the end of each trading day, resetting their value to zero after settlement.
2- Formula for the Value of a Forward Contract
– Formula: Vt = PV[Ft - F0].
3- Explanation of Variables
– Vt: Value of the forward contract at time t.
– Ft: Current forward price of the contract.
– F0: Forward price agreed upon at the initiation of the contract.
– PV: Present value operator that discounts the difference between Ft and F0 to the current value.
4- Comparison to Futures Contracts
– Futures contracts differ from forward contracts in that they are marked to market daily.
– The value of a futures contract is determined by the difference between the current price and the previous day’s settlement price.
– After marking to market, the futures contract value is reset to zero.
Accrued Interest (AI)
1- Overview of the Concept
– Accrued interest is the portion of the bond’s coupon payment that has been earned but not yet paid since the last coupon payment date.
– Bonds can be quoted with accrued interest (dirty price) or without it (clean price).
2- Formula for Accrued Interest
– Formula: AI = (NAD ÷ NTD) × (C ÷ n).
3- Explanation of Variables
– AI: Accrued interest since the last coupon payment.
– NAD: Number of accrued days since the last coupon payment.
– NTD: Total number of days in the coupon payment period.
– C: Stated annual coupon amount.
– n: Number of coupon payments per year.
4- Calculation Example
– If an investor earns $10 per month on a bond with an annual coupon of $120, the accrued interest after two months would be:
— NAD = 2 months (out of 12 total months).
— C = $120, and n = 12 (monthly payments).
— AI = (2 ÷ 12) × (120 ÷ 12) = $20.
Fixed-Income Forward and Futures Contracts
1- Unique Issues
– Bond prices may be quoted with or without accrued interest:
— This depends on the country, as some use clean price (excluding accrued interest) while others use dirty price (including accrued interest).
– Contracts often allow more than one bond to be delivered by the seller:
— Sellers use adjustment factors, such as a conversion factor, to make bonds roughly equal in price.
– Sellers typically choose the cheapest-to-deliver bond:
— If multiple bonds can be delivered under a contract, the seller will deliver the bond that minimizes their cost, referred to as the “cheapest-to-deliver” bond.
Example: Calculating the Futures Price for a Bond Contract
1- Scenario
A 4% coupon bond with a par value of $1,000 pays semi-annual coupons. The current price, including accrued interest, is $990. The next coupon payment of $20 occurs in 80 days, and the futures contract expires in 210 days. The current risk-free rate is 5.10%.
2- Steps for Calculation
Step 1: Determine the Forward Price Formula
The futures price for the bond is calculated using the formula:
F0 = FV(B0 + AI0) - AIT - FVIC
Where:
– F0: Futures price.
– B0: Clean bond price (excluding accrued interest).
– AI0: Accrued interest at initiation (current accrued interest).
– AIT: Accrued interest at contract expiry.
– FVIC: Future value of the coupon interest payments.
Step 2: Calculate Each Component
A. Present Value of Bond Price Including AI (FV[B0 + AI0])
FV[B0 + AI0] = 990 × (1 + 0.051 × 210 ÷ 360) = 1,019.15
B. Accrued Interest at Contract Expiry (AIT)
AIT = 20 × (130 ÷ 180) = 14.44
C. Future Value of Coupon Interest (FVIC)
FVIC = 20 × (1 + 0.051 × 130 ÷ 360) = 20.36
Step 3: Combine the Results
Substituting the values into the formula:
F0 = 1,019.15 - 14.44 - 20.36 = 984.35
Comparing Forward and Futures Contracts
1- Bond Futures Contracts
– Bond futures contracts are marked-to-market daily. This involves:
— Daily settlement of gains or losses based on the price change since the previous day’s settlement.
— The value of the bond futures contract reverts to zero immediately after settlement.
2- Bond Forward Contracts
– Bond forward contracts do not involve daily settlements. Instead, the value is determined by the present value (PV) of the difference between the forward price at time t and the initial forward price:
— Formula: Vt = PV(Ft - F0).
A receive-floating, pay-fixed swap is equivalent to being long a floating-rate bond and short a fixed-rate bond. The investor is borrowing at a fixed rate and investing in a floating rate. For the swap to have zero initial value, the price of the floating-rate bond and fixed-rate bond should be the same. This is usually done by assuming both are selling at par.
Swaps for Converting Fixed-Rate to Floating-Rate Assets
1- Overview of Swaps
– Swaps are widely used by market participants to transform the nature of assets or liabilities from fixed-rate to floating-rate or vice versa, depending on financial objectives.
2- Example: Converting a Fixed-Rate Asset to a Floating-Rate Asset
– A company holding a fixed-rate asset can convert it into a floating-rate asset by entering into a receive-floating, pay-fixed swap.
3- Mechanics of the Swap
– The company continues to earn the fixed rate on its asset.
– It pays the fixed rate on the swap while receiving the floating rate from the swap.
– The fixed-rate receipt from the asset cancels with the fixed-rate payment on the swap, leaving the company with the floating-rate receipt from the swap as its net outcome.
Interest Rate Swap Contracts
1- Overview of Interest Rate Swaps
– Interest rate swaps are agreements between counterparties to exchange cash flows, typically one based on a fixed rate and the other on a floating rate.
– Swaps can vary significantly in terms of notional amounts, settlement frequencies, and day count conventions.
2- Formula for Fixed-Rate Receiver’s Cash Flow (FS)
– The fixed-rate cash flow is calculated as:
— Formula: “FS = AP * rFIX”.
— Where:
—- FS: Fixed-rate cash flow.
—- Cash flows on a monthly basis using a 30/360 day count convention - AP: Accrual period (e.g., 30/360 or ACT/ACT).
—- rFIX: Fixed swap rate.
3- Formula for Floating-Rate Receiver’s Cash Flow (Si)
– The floating-rate cash flow is calculated as:
— Formula: “Si = AP * rFLT,i”.
— Where:
—- Si: Floating-rate cash flow.
—- rFLT,i: Floating rate at the ith reset date.
4- Net Cash Flow to the Fixed-Rate Receiver (FS - Si)
– The net cash flow to the fixed-rate receiver is:
— Formula: “FS - Si = AP * (rFIX - rFLT,i)”.
– Example Calculation:
— Given:
—- AP = 30/360, rFIX = 4%, rFLT,i = 5.4%.
— Net cash flow: “FS - Si = (30/360) * (0.054 - 0.04) = 0.0012”.
5- Valuation of Fixed and Floating Legs
– The value of the fixed leg (VFIX) and the floating leg (VFLT) must be equal when the swap is initiated to ensure a zero value:
— Formula for VFIX:
“VFIX = FS * T∑_i=1 PVi(1) + PVn(1)”.
— Formula for rFIX:
“rFIX = [1 - PVn] / [T∑_i=1 PVi] * (1 / AP)”.
— Where:
—- T∑_i=1 PVi: Present value of cash flows at each payment period.
—- PVn: Present value of the notional amount at maturity.
Key Takeaways
– Interest rate swaps are flexible instruments that allow market participants to hedge interest rate risk or transform fixed and floating exposures.
– The net cash flow for each party depends on the accrual period and the difference between the fixed and floating rates.
– Valuation of the fixed and floating legs ensures fair pricing at the swap’s initiation.
