CFA level 1 Derivatives

What is the defining characteristic of a derivative security?

A derivative is a financial instrument that derives its value from the value or performance of another security, asset, or variable (referred to as the underlying) at a specific future date.

At contract initiation, what is the typical initial value of a standard forward contract to both counterparties?

Zero value. The forward price is set such that neither party pays an upfront fee at contract initiation.

Contrast deliverable forward contracts with cash-settled forward contracts.

In a deliverable contract, the underlying asset is physically exchanged for the agreed forward price at settlement. In a cash-settled contract, only the net financial gain or loss is exchanged between counterparties at maturity.

An investor enters a deliverable forward contract to buy 100 shares of Acme at USD 30 per share in 3 months. At settlement, the spot price of Acme is USD 40. Calculate the buyer's payoff and profit.

Payoff / Profit = (USD 40 − USD 30) × 100 = USD 1,000. The buyer purchases shares worth USD 4,000 for USD 3,000, realizing a profit of USD 1,000.

An investor enters a forward contract to buy 100 shares of Acme at USD 30. At maturity, the spot price of Acme is USD 25. Under cash settlement, what cash payment must be made?

The forward buyer must pay USD 500 to the seller. Net loss = (USD 25 − USD 30) × 100 = −USD 500.

Distinguish hard commodities from soft commodities and give an example of each.

Hard commodities are mined or extracted (e.g., crude oil, gold, aluminum). Soft commodities are agricultural products grown or raised (e.g., wheat, soybeans, cattle, cotton).

What is the common rule for hedging risk using futures or forwards?

"Do in the futures/forwards market today what you must do in the spot market in the future." (e.g., if you must buy wheat in 3 months, buy wheat futures today).

If a derivative contract is used to partially offset an existing risk rather than fully eliminate it, how is the strategy classified?

A partial hedge.

Define stand-alone derivative vs. embedded derivative and give an example of each.

Stand-alone derivative: A distinct, separate contract (e.g., exchange-traded stock option, OTC forward). Embedded derivative: A derivative provision contained within a host security (e.g., call provision in a callable bond, conversion feature in convertible debt).

What is the primary operational difference between Over-the-Counter (OTC) derivatives and Exchange-Traded Derivatives (ETDs)?

ETDs are standardized contracts traded on centralized exchanges backed by a central clearinghouse with low counterparty risk. OTC derivatives are customized contracts negotiated bilaterally between end users and dealers.

Define novation in the context of exchange-traded derivative markets and central clearinghouses (CCH).

Novation is the process where a central clearinghouse steps between the original buyer and seller, becoming the counterparty to both sides of the trade and guaranteeing payment performance.

What regulatory mechanism was instituted for many OTC derivative swap trades following the 2008 global financial crisis?

The central clearing mandate, which requires OTC swap trades executed on Swap Execution Facilities (SEFs) to be novated to a Central Counterparty (CCP).

What is the primary risk trade-off associated with implementing a central clearing mandate for OTC derivatives?

It reduces bilateral counterparty credit risk between dealers but concentrates systemic risk into Central Counterparties (CCPs).

How do market makers earn trading profits in exchange-traded derivative markets?

Market makers earn profits primarily from the bid-ask spread between buying and selling derivative contracts simultaneously, rather than holding speculative directional positions.

Define Swap Execution Facility (SEF).

A Swap Execution Facility (SEF) is a regulated electronic trading platform where multiple dealer quotes are aggregated to execute OTC derivative swap trades prior to central clearing novation.

How do futures contracts differ from OTC forward contracts with respect to daily cash flows?

Futures require daily mark-to-market cash settlement of gains and losses into margin accounts, whereas forwards settle cumulatively at contract maturity.

In a futures margin account, what occurs when the account balance falls below the maintenance margin level?

The investor receives a margin call and must deposit additional funds to restore the account balance back to the initial margin level (not just the maintenance margin level).

A long position is taken in a futures contract for 100 ounces of gold at USD 1,950/oz on Day 0. Initial margin is USD 5,000; maintenance margin is USD 4,700. Day 1 settlement price: USD 1,947.50. Day 2 settlement price: USD 1,945.00. Calculate the margin account balance at the end of Day 2 prior to any margin call.

Day 1 loss = (1,947.50 − 1,950) × 100 = −USD 250. Margin balance = USD 4,750. Day 2 loss = (1,945 − 1,947.50) × 100 = −USD 250. Margin balance = USD 4,500.

In the gold futures scenario where the margin balance drops to USD 4,500 on Day 2 against a USD 4,700 maintenance margin and USD 5,000 initial margin, how much cash must the long buyer deposit?

The buyer must deposit USD 500 to bring the balance back to the initial margin level of USD 5,000 (USD 5,000 − USD 4,500 = USD 500).

Define price limits and circuit breakers in exchange-traded futures markets.

Price limits are exchange-mandated bounds on daily price movements relative to the prior day's settlement. If market equilibrium falls outside these limits, trades cannot occur outside the bounds. Circuit breakers temporarily suspend trading when limit prices are hit.

Define an interest rate swap and identify the roles of the counterparties.

An interest rate swap is an agreement to exchange periodic cash flows based on a specified notional principal. The fixed-rate payer pays a fixed rate and receives floating interest; the floating-rate payer pays floating interest based on a market reference rate (MRR) and receives the fixed rate.

A 2-year interest rate swap has a notional principal of USD 10 million with quarterly payments. The fixed rate is 2.0% per annum. At inception, 90-day SOFR is 1.6%. Calculate the net settlement payment at the end of the second quarter if 90-day SOFR at the end of Q1 was 1.6%.

Fixed quarterly payment = USD 10,000,000 × 0.02 / 4 = USD 50,000. Floating payment = USD 10,000,000 × 0.016 / 4 = USD 40,000. Net payment: Fixed-rate payer pays USD 10,000 to the floating-rate payer (USD 50,000 − USD 40,000).

Describe the mechanics of a Credit Default Swap (CDS) and the obligations of both protection buyer and seller.

The protection buyer pays fixed periodic premiums (credit spread) to the protection seller. The protection seller makes a payout only if a credit event (default, bankruptcy) occurs on the reference credit instrument to compensate for the loss in value.

An oil producer enters a forward contract to sell 1,000 barrels of oil in 2 months at forward price USD 64/barrel. At maturity, spot price is USD 58.50/barrel. Calculate buyer payoff and net cash settlement.

Buyer payoff per barrel = ST − F0(T) = USD 58.50 − USD 64.00 = −USD 5.50. Net settlement paid by buyer to seller = 1,000 × USD 5.50 = USD 5,500.

Why does an interest rate swap equal a portfolio of forward contracts?

A swap involves multiple settlement dates where net payments are exchanged based on a fixed rate vs. market reference rate, making each payment equivalent to a individual Forward Rate Agreement (FRA) maturing on that settlement date.

In a forward rate agreement (FRA), who realizes a gain when the market reference rate (MRR) at settlement is higher than the agreed FRA fixed rate?

The FRA buyer (long position), who is the fixed-rate payer and floating-rate receiver. Net payment to the long = (MRR − FRA fixed rate) × notional × period.

What is the formula for the payoff of a long call option at expiration T?

cT = max(0, ST − X), where ST is the spot price of the underlying at expiration and X is the exercise price.

What is the formula for the profit Π of a long put option at expiration T?

Π = max(0, X − ST) − p0, where X is the exercise price, ST is the spot price at expiration, and p0 is the initial put premium paid.

Compare the maximum profit and maximum loss profiles for a short call option position vs. a short put option position.

Short call: Maximum profit is limited to the call premium c0; maximum loss is unlimited as ST → ∞. Short put: Maximum profit is limited to the put premium p0; maximum loss is limited to X − p0 (occurring if ST = 0).

A call option with exercise price X = USD 50 trades at premium c0 = USD 5. Calculate the breakeven underlying price at expiration for the call buyer.

Breakeven spot price = X + c0 = USD 50 + USD 5 = USD 55.

A put option has an exercise price X = USD 50 and premium p0 = USD 5. At expiration, ST = USD 35. Calculate the profit for the short put position.

Put payoff to buyer = max(0, 50 − 35) = USD 15. Short put profit = p0 − payoff = USD 5 − USD 15 = −USD 10 (a loss of USD 10).

A stock trades at USD 42. A call option with X = USD 40 costs USD 3, and a put option with X = USD 40 costs USD 0.75. Calculate the long call profit and long put profit if ST = USD 35.

Long call profit = max(0, 35 − 40) − 3 = −USD 3. Long put profit = max(0, 40 − 35) − 0.75 = USD 5 − USD 0.75 = USD 4.25.

An investor purchases a 6-month call option on an index with exercise price X = USD 1,240 for premium c0 = USD 24.85. Calculate the index spot price at maturity for breakeven.

Breakeven index price = X + c0 = USD 1,240 + USD 24.85 = USD 1,264.85.

A put option seller receives a premium p0 = USD 5 for an option with strike X = USD 30. Calculate maximum profit and maximum loss.

Maximum profit = p0 = USD 5 (if option expires unexercised, ST ≥ 30). Maximum loss = X − p0 = USD 30 − USD 5 = USD 25 (if spot price drops to ST = 0).

TRAP: Can an option buyer ever face a negative payoff at expiration?

NO. The payoff to an option buyer at expiration is max(0, ST − X) for calls and max(0, X − ST) for puts, which is always ≥ 0. (Net profit can be negative due to the premium paid, but payoff cannot).

TRAP: Is trading options a positive-sum game, negative-sum game, or zero-sum game before transaction costs?

Zero-sum game. Every dollar gained by the option buyer is equal to a dollar lost by the option writer.

Contrast a forward commitment with a contingent claim based on rights and obligations.

A forward commitment legally obligates both counterparties to perform in the future. A contingent claim (e.g., an option) gives the buyer the right, but not the obligation, to transact depending on a future event or price level.

Why are forward commitments described as linear derivatives, whereas contingent claims are non-linear derivatives?

Forward commitments have symmetric payoff profiles that change linearly dollar-for-dollar with the underlying price. Options have asymmetric payoff profiles where derivative sensitivity changes depending on moneyness.

Identify four operational advantages of derivatives over cash market transactions.

1. Lower transaction costs. 2. Greater liquidity. 3. Ease of short selling without borrowing physical assets. 4. Higher implicit leverage.

Define basis risk in derivative hedging strategies.

Basis risk is the risk that arises when the underlying asset of a hedging derivative contract differs from the asset/position being hedged, or when settlement dates do not match the holding period.

How can daily mark-to-market margin calls create liquidity risk for a market participant holding a fully hedged position?

Even if offsetting gains will be realized in the underlying asset at harvest/maturity, interim price moves require cash deposits to satisfy daily margin calls. If the hedger lacks liquid cash, the position will be liquidated at a loss.

Describe how price discovery functions as a key economic benefit of derivative markets.

Futures and forward prices reveal information about the likely direction of future spot prices and rates (e.g., index futures before the open), and option prices reveal implied volatility. Trap: futures prices are NOT unbiased forecasts of future spot prices.

Define systematic risk vs. systemic risk in derivative markets.

Systematic risk is non-diversifiable market-wide price risk. Systemic risk is the risk of widespread collapse of financial institutions and entire financial markets resulting from interconnected default exposures or excessive leverage.

Compare the counterparty credit risk exposure of an option buyer versus an option writer.

The option buyer faces counterparty credit risk because the seller owes performance if the option is in-the-money. The option writer faces NO counterparty credit risk after collecting the initial premium because the buyer has no remaining performance obligations.

Under hedge accounting, distinguish between a cash flow hedge and a fair value hedge.

Cash flow hedge: Offsets variability in future cash flows (e.g., foreign currency receipts or floating debt interest payments). Fair value hedge: Offsets changes in the fair value of recognized balance sheet assets or liabilities (e.g., inventory or fixed-rate debt).

What hedge accounting category applies to foreign currency forward contracts used to reduce volatility in the reported equity of a foreign subsidiary?

Net investment hedge.

An investor holds a bond portfolio and wants to temporarily increase portfolio duration at low cost without buying physical bonds. What swap position should be taken?

Enter an interest rate swap as the fixed-rate receiver / floating-rate payer. Receiving fixed interest adds positive bond duration.

A corporation issues floating-rate debt and enters a fixed-for-floating interest rate swap as the fixed-rate payer. What is the net liability risk profile created?

A fixed-rate debt obligation. Floating interest received from the swap cancels out floating interest paid on the debt, leaving a synthetic fixed interest obligation.

Define arbitrage and explain the principle of no-arbitrage pricing.

Arbitrage is a risk-free profit trade executed with zero initial net investment by simultaneously buying a low-priced asset/portfolio and selling an identical higher-priced asset/portfolio. No-arbitrage pricing sets derivative prices equal to the cost of a replicating portfolio.

Define replication in derivatives valuation.

Replication is the creation of a synthetic portfolio of cash market instruments (e.g., underlying asset and risk-free bond) that yields identical payoffs to a derivative contract across all future market states.

How is a long position in a forward contract replicated using cash market instruments?

By borrowing funds at the risk-free rate to purchase one unit of the underlying asset in the spot market today and holding it until the forward maturity date.

Calculate the 1-year no-arbitrage forward price F0(1) for a non-dividend-paying stock currently trading at S0 = USD 30 if the annual risk-free rate is r = 5% (compounded annually).

F0(1) = S0 (1 + r)^1 = USD 30 × 1.05 = USD 31.50.

Suppose a stock trades at S0 = USD 30, r = 5%, and a 1-year forward contract is priced at F0(1) = USD 32.00. Describe the arbitrage strategy and calculate the profit.

Because the forward is overpriced (USD 32.00 > USD 31.50): 1. Sell the forward contract at USD 32.00. 2. Borrow USD 30.00 at 5% to buy the stock. At t = 1, deliver the stock for USD 32.00, repay loan of USD 30 × 1.05 = USD 31.50. Arbitrage profit = USD 32.00 − USD 31.50 = USD 0.50 per share.

Suppose a stock trades at S0 = USD 30, r = 5%, and a 1-year forward contract is priced at F0(1) = USD 31.00. Describe the arbitrage strategy.

Because the forward is underpriced (USD 31.00 < USD 31.50): 1. Buy the forward contract at USD 31.00. 2. Short sell the stock for USD 30.00 and lend the proceeds at 5%. At t = 1, investment yields USD 30 × 1.05 = USD 31.50. Buy stock via forward for USD 31.00 to close short. Arbitrage profit = USD 31.50 − USD 31.00 = USD 0.50.

How do holding costs (storage, insurance) and holding benefits (dividends, interest, convenience yield) affect the forward price of an asset?

Holding costs increase the forward price relative to spot. Holding benefits reduce the forward price relative to spot.

Define convenience yield for commodity derivatives.

Convenience yield is the non-monetary benefit of physically holding a physical commodity asset (e.g., advantage of preventing production disruptions during supply shortages).

A stock trades at S0 = EUR 125 and pays a cash dividend of EUR 2.50 at T = 1. Risk-free rate is 1%. Calculate the 1-year forward price.

F0(1) = (S0 − PV(Dividend)) × (1 + r)^1 = (EUR 125 − EUR 2.50 / 1.01) × 1.01 = EUR 125 × 1.01 − EUR 2.50 = EUR 126.25 − EUR 2.50 = EUR 123.75.

What is the value of a forward contract at initiation, and why?

Zero to both parties. The forward price F0(T) is set at the no-arbitrage level so that S0 = F0(T)(1 + r)^(−T): the PV of what the long will pay equals what the asset is worth today, so no money changes hands.

Contrast the price and the value of a forward contract over its life.

The price, F0(T), is fixed at initiation and never changes. The value, Vt(T), starts at zero and then moves as the spot price, interest rates, and time to maturity change. It is a mark-to-market gain to one side and an equal loss to the other.

State the formula for the value of a long forward position at time t during its life (no costs or benefits of carry), and define each term.

Vt(T) = St − F0(T)(1 + r)^(−(T−t)). St = current spot price; F0(T) = forward price agreed at t = 0; r = risk-free rate; T − t = time remaining to maturity. The long's value is the spot price minus the PV of the forward price.

What is the value of a forward contract to the SHORT at time t during its life (no carry)?

Vt(T) = F0(T)(1 + r)^(−(T−t)) − St: the PV of the forward price minus the current spot price. It is exactly the negative of the long's value.

What is the value of a forward contract at expiration to the long and to the short?

Long: VT(T) = ST − F0(T). Short: VT(T) = F0(T) − ST. At maturity the value equals the settlement amount, since there is no time left to discount.

VFO agreed to SELL Biomian shares forward at F0(T) = INR 300.84 in six months. At maturity, ST = INR 287. What is the contract value per share to the buyer and to VFO?

Buyer (long): ST − F0(T) = 287 − 300.84 = −INR 13.84. VFO (short): F0(T) − ST = 300.84 − 287 = +INR 13.84. The short gains because it sells above the market price.

VFO is short a Biomian forward at INR 300.84, but at maturity ST = INR 312. What are the values to each side?

Buyer (long): 312 − 300.84 = +INR 11.16. VFO (short): 300.84 − 312 = −INR 11.16. Gains and losses are equal and opposite.

Biomian trades at S0 = INR 295, and its six-month forward price is INR 300.84 (no dividends). What risk-free rate is implied?

F0(T) = S0(1 + r)^T, so 300.84 = 295(1 + r)^0.5. Step 1) 300.84 / 295 = 1.019797. Step 2) Square it: 1.0400. Answer: r = 4%.

Right after VFO agrees to sell Biomian forward at INR 300.84 (S0 = INR 295), the share price instantly jumps to INR 325. What is VFO's MTM value, and what shortcut applies?

At t = 0, F0(T)(1 + r)^(−T) = S0, so the short's value simplifies to S0 − St = 295 − 325 = −INR 30 per share. VFO (short) has an MTM loss of INR 30.

VFO is short a six-month Biomian forward at F0(T) = INR 300.84 with r = 4%. Three months later (t = 0.25), St = INR 285. What is VFO's MTM value?

Short value = F0(T)(1 + r)^(−(T−t)) − St. Step 1) PV of F0(T) = 300.84 × 1.04^(−0.25) = INR 297.90. Step 2) 297.90 − 285 = INR 12.90 MTM gain per share.

In the Biomian example (VFO short at INR 300.84, t = 0.25, St = INR 285), what happens to VFO's MTM value if the risk-free rate doubles from 4% to 8%, and why?

Short value = 300.84 × 1.08^(−0.25) − 285 = INR 10.11, down INR 2.79 from INR 12.90. A higher r lowers the PV of the forward price the short will receive, so the short's gain shrinks (and the long's value rises).

All else equal, how does an increase in the risk-free rate during the life of a forward affect its value to the long and to the short, and why?

The long's value rises and the short's value falls. A higher r lowers the PV of the fixed forward price, F0(T)(1 + r)^(−(T−t)), which the long pays and the short receives.

Hightest Capital is LONG a six-month Unilever forward at EUR 51.23 (no dividends, r = 5%). Three months later, St = EUR 50.50. What is Hightest's MTM value?

Vt(T) = St − F0(T)(1 + r)^(−(T−t)). Step 1) PV of F0(T) = 51.23 × 1.05^(−0.25) = EUR 50.6089. Step 2) 50.50 − 50.6089 = −EUR 0.11 per share, an MTM loss.

A trader is long a one-year forward at USD 105 (no carry). Six months later, St = USD 108 and r = 3%. What is the value to the long and to the short?

Step 1) PV of F0(T) = 105 × 1.03^(−0.5) = USD 103.46. Step 2) Long: 108 − 103.46 = +USD 4.54. Short: −USD 4.54.

State the value of a long forward at time t when the underlying has known costs (C) and benefits or income (I).

Vt(T) = [St − PVt(I) + PVt(C)] − F0(T)(1 + r)^(−(T−t)). The spot price is adjusted for the PV of the income and costs remaining from t to T, and then the PV of the forward price is subtracted.

At time t, the PV of the remaining benefits of owning the underlying exceeds the PV of the remaining costs. Is the long forward's value higher or lower than St − PV of F0(T)?

Lower. With PVt(C) − PVt(I) < 0, the value [St − PVt(I) + PVt(C)] − F0(T)(1 + r)^(−(T−t)) falls below St − PV of F0(T). If costs exceed benefits, the value is higher; if they offset, it is equal.

Hightest is short a six-month Unilever forward at F0(T) = EUR 50.6311 per share. Unilever pays EUR 0.30 dividends at 3 months and at T, and r = 5%. Four months after inception, what spot price makes Vt(T) = 0?

Only the dividend at T (2 months away) remains. Step 1) PVt(I) = 0.30 × 1.05^(−0.167) = EUR 0.2976. Step 2) PV of F0(T) = 50.6311 × 1.05^(−0.167) = EUR 50.2202. Step 3) Set (St − 0.2976) − 50.2202 = 0, so St = EUR 50.5178.

TRAP: "The MTM value of a forward to the seller at time t is St − PV of F0(T)." What is wrong?

That is the long's value. The seller's (short's) value is PV of F0(T) − St: the PV of what it will receive minus what the asset is worth now.

How can the long lock in the current value of a forward, St − F0(T)(1 + r)^(−(T−t)), before maturity?

Short-sell the asset at St and invest F0(T)(1 + r)^(−(T−t)) in a risk-free zero-coupon bond. At T, the bond pays F0(T), which buys the asset under the forward, and that asset is delivered to close the short. The cash left over today is the forward's value.

State the formula for the MTM value of a long FX forward at time t (continuous compounding), and define the terms.

Vt(T) = St,f/d − F0,f/d(T) × e^(−(rf − rd)(T − t)). St,f/d = current spot rate (units of price currency f per unit of base currency d); F0,f/d(T) = contracted forward rate; rf and rd = risk-free rates of the price and base currencies.

Rook Point is long a one-year forward to buy EUR 1,000,000 for USD 1,201,000. Spot USD/EUR = 1.192 and rEUR = −0.25%. Right after initiation, the USD rate rises from 0.50% to 0.75%. What is Rook Point's MTM value?

Vt(T) = S0 − F0 × e^(−(rUSD − rEUR)T) = 1.192 − 1.201 × e^(−(0.0075 + 0.0025)) = 0.00295 USD/EUR. On EUR 1,000,000, that is about a USD 2,950.15 MTM gain. A wider rate gap lowers the PV of the forward price the long pays.

State the no-arbitrage forward price for an asset with costs and benefits, in discrete terms (known amounts) and in continuous terms (rates).

Discrete: F0(T) = [S0 − PV0(I) + PV0(C)](1 + r)^T. Continuous: F0(T) = S0 × e^((r + c − i)T), where c = cost rate and i = income rate. With no carry, these become S0(1 + r)^T and S0 × e^(rT).

An index trades at 100. The continuously compounded risk-free rate is 3%, the cost rate is 1%, and the income (dividend) rate is 2%. What is the one-year forward price?

F0(T) = S0 × e^((r + c − i)T) = 100 × e^((0.03 + 0.01 − 0.02) × 1) = 100 × e^0.02 = 102.02.

Define an implied forward rate (IFR) in terms of two equivalent investment strategies.

It is the breakeven reinvestment rate that makes these two strategies earn the same return: (1) invest from today to the forward start date and roll the proceeds over at the forward rate, or (2) invest today through the final maturity. This equality is the no-arbitrage condition.

What do the notations "2y3y" (F2,3) and "F3m,6m" mean?

2y3y is a 3-year rate starting 2 years from now (ending at year 5). F3m,6m is a 6-month MRR starting in 3 months (ending at month 9). The first number is when the rate starts; the second is its tenor.

State the general formula linking zero rates zA and zB to the implied forward rate IFR A,B−A.

(1 + zA)^A × (1 + IFR A,B−A)^(B−A) = (1 + zB)^B. zA is the zero rate to time A, zB is the zero rate to time B, and IFR A,B−A is the rate for the B − A periods that start at time A.

Zero rates are z1 = 2.3960% and z2 = 3.4197%. Calculate IFR1,1.

(1.02396)(1 + IFR1,1) = (1.034197)^2. Step 1) 1.034197^2 = 1.069563. Step 2) 1.069563 / 1.02396 = 1.044536, so IFR1,1 = 4.4536%. Check: USD 100 × 1.02396 × 1.044536 = USD 106.96 = USD 100 × 1.034197^2.

Zero rates are z2 = 3.4197% and z3 = 4.0005%. Calculate IFR2,1.

(1.034197)^2 × (1 + IFR2,1) = (1.040005)^3. Step 1) 1.040005^3 = 1.124880. Step 2) 1.124880 / 1.069563 = 1.051719, so IFR2,1 = 5.1719%.

Z2 = 2% and Z3 = 3%. Calculate F2,1, the implied 1-year rate two years from now.

Step 1) USD 100 × 1.02^2 = USD 104.04 at t = 2. Step 2) USD 100 × 1.03^3 = USD 109.27 at t = 3. Step 3) F2,1 = 109.27 / 104.04 − 1 = 5.03%.

Zero rates are z1 = 3% and z2 = 4%. Calculate IFR1,1.

1.03 × (1 + IFR1,1) = 1.04^2. Step 1) 1.04^2 = 1.0816. Step 2) 1.0816 / 1.03 = 1.050097, so IFR1,1 = 5.0097%.

Three-year and four-year zero-coupon government bonds trade at 93 and 90 per 100 face value. Calculate IFR3,1, and give the shortcut.

Step 1) 0.93 = 1 / (1 + z3)^3, so z3 = 2.4485%. Step 2) 0.90 = 1 / (1 + z4)^4, so z4 = 2.6690%. Step 3) (1.024485)^3 × (1 + IFR3,1) = (1.02669)^4, so IFR3,1 = 3.3333%. Shortcut: IFR3,1 = DF3 / DF4 − 1 = 0.93 / 0.90 − 1 = 3.3333%.

Define a discount factor, give its formula, and say what it represents.

DFi = 1 / (1 + zi)^i. It is the PV of one currency unit received at time i: the price equivalent of the zero rate, which can be read as the price of a zero-coupon bond paying 1 at time i.

If the zero (spot) curve is upward sloping, do implied forward rates lie above or below the zero rates? Give an example.

Above. For example, z1 = 2.396% and z2 = 3.4197%, but IFR1,1 = 4.4536%. To earn the higher 2-year zero rate overall, the rate for the second year must be higher than the 2-year rate itself.

Today's 3-month MRR is 1.25% and the 6-month MRR is 1.75% (both annualized). Calculate the implied 3-month rate in 3 months, IFR3m,3m.

Match the periods: (1 + 0.0125/4)(1 + IFR/4) = (1 + 0.0175/2). Step 1) 1.00875 / 1.003125 = 1.0056075. Step 2) (1.0056075 − 1) × 4 = 2.2430%. Check: CNY 100m × 1.003125 × 1.0056075 = CNY 100.875m = CNY 100m × 1.00875.

The current 3-month MRR is 1.0% and the 9-month MRR is 1.2% (annualized). Calculate the implied 6-month rate starting in 3 months, F3m6m.

[1 + 0.012(9/12)] = [1 + 0.01(3/12)] × [1 + F3m6m(6/12)]. Step 1) 1.009 / 1.0025 = 1.006484. Step 2) 0.006484 × 12/6 = 1.30% (annualized).

What is a forward rate agreement (FRA), and what do the long and the short do?

An OTC contract that applies a fixed rate to a hypothetical future deposit (the notional) for a future period. The FRA buyer (long) pays the fixed rate and receives the MRR, which is set at the start of the period (time A). The short pays MRR and receives fixed. Only the net amount is exchanged.

State the FRA net payment formula from the long's (fixed-rate payer's) perspective, and explain how it is actually settled.

Net payment = (MRR B−A − IFR A,B−A) × notional × period, which is owed at the end of the interest period (B). FRAs usually cash-settle at the start of the period (A), so the amount is discounted at the MRR: settlement = net payment / (1 + MRR × period).

Yangzi Bank is long a CNY 100m FRA at 2.24299% on 3-month MRR (90/360). MRR sets at 2.15%. Calculate the net payment and the actual cash settlement.

Step 1) (2.15% − 2.24299%) × 100,000,000 × 90/360 = −CNY 23,247.50 (end of period). Step 2) Settled at the start of the period: −23,247.50 / (1 + 0.0215/4) = −CNY 23,123.21. Yangzi pays because MRR set below the fixed rate.

Yangzi Bank uses a long (pay-fixed) FRA to hedge a liability on which it owes MRR. MRR sets below the FRA rate, so Yangzi makes a net FRA payment. Is the hedge failing?

No. The FRA payment is offset by the lower MRR it now pays on the liability, so its all-in cost stays near the FRA rate. Locking in that rate is the purpose of the hedge.

An F3m6m FRA has a fixed rate of 1.3% on USD 1 million notional. The 6-month MRR sets at 1.5% in three months. What does the fixed-rate payer receive at settlement?

Step 1) Interest difference = 1,000,000 × (0.015 − 0.013) / 2 = USD 1,000 (owed at the end of the 6 months). Step 2) Discount to settlement: 1,000 / (1 + 0.015/2) = USD 992.56 received by the fixed-rate payer.

A bank is long a USD 10m FRA at 3.00% on 3-month MRR (90/360). MRR sets at 3.40%. What are the net payment and the cash settlement?

Step 1) (3.40% − 3.00%) × 10,000,000 × 90/360 = +USD 10,000 owed to the bank at the end of the period. Step 2) Cash settlement at the start of the period = 10,000 / (1 + 0.034/4) = USD 9,915.72 received.

Who mainly uses FRAs, and how do FRAs relate to interest rate swaps?

FRAs are used almost exclusively by financial intermediaries to manage rate-sensitive assets and liabilities. An FRA is equivalent to a single-period swap, and FRAs are the building blocks of multi-period interest rate swaps, which issuers and investors use more often.

The 2-year GBP zero rate is 3.42%, but a 2-year GBP 100 zero-coupon bond trades at GBP 92.45. What is the no-arbitrage price, and what is the arbitrage?

No-arbitrage price = 100 / 1.0342^2 = GBP 93.50, so the bond is cheap. Borrow GBP 92.45 at 3.42% and buy the bond. At T, receive GBP 100 and repay 92.45 × 1.0342^2 = GBP 98.88, for a riskless profit of GBP 1.12.