LM1 Pricing and Valuation of Forward Commitments Question Notes

Variant: Tell a Forward, Future, FRA, and Swap Apart

Abstract: All four contracts lock in future economics, but the thing exchanged and the settlement pattern tell you which name belongs on the box.

A contract locks in one stock purchase six months from now and settles only then. A second is exchange-traded and settles gains daily. A third locks a three-month borrowing rate beginning in one month. A fourth exchanges fixed interest for floating interest every quarter. Name each contract.

Forward commitment. What is it? A binding deal today about a later transaction. Forward. What is it? A private, usually one-settlement deal. OTC. What is it? Over the counter: privately negotiated rather than exchange-traded. Future. What is it? A standardized exchange-traded deal settled daily. FRA. What is it? A forward rate agreement: a cash settlement tied to a future market interest rate. Swap. What is it? A series of exchanges, like a bundle of forwards. Counterparty. What is it? The person or institution on the other side.

1. Match the cash-flow pattern

The answers are:

Variant: Separate Price from Value

Abstract: The forward price is the delivery price written into a new fair contract; value is what an existing contract is worth today.

A new six-month forward has a fair delivery price of $104. An old long forward requires payment of $100 at the same date. The discount factor to that date is 0.98. State the new forward price and the old contract's value.

Forward price. What is it? The delivery price that makes a brand-new forward worth zero. Forward value. What is it? Today's worth of an already-existing contract. Delivery price. What is it? The price the long agreed to pay at maturity.

1. Name the price; discount the advantage

The old long gets the asset for 100 when the current fair deal requires 104.

Variant: Find Long and Short Payoffs at Expiration

Abstract: At expiration there is no discounting left: compare the asset price with the agreed delivery price.

A forward's delivery price is $72. At expiration the asset is worth $79. Find the long and short payoffs.

Long forward. What is it? The side forced to buy the asset. Its payoff is \(S_T-F_0\). Short forward. What is it? The side forced to sell. Its payoff is \(F_0-S_T\).

1. Sit in each chair

The short owns the mirror image.

Variant: Price a No-Income Forward with Annual Compounding

Abstract: Buying later must cost the same as buying now with borrowed money and carrying the asset to maturity.

Gold costs $1,900 now, generates no cash flow, and can be financed for nine months at 4% with annual compounding. Find the no-arbitrage forward price.

No-arbitrage price. What is it? The price that prevents a risk-free free lunch. Annual compounding. What is its formula here? \(F_0=S_0(1+r)^T\).

1. Grow spot for the exact fraction of a year

Variant: Back Out the Spot Price

Abstract: If a fair forward is simply spot carried forward, run the same equation backward to recover spot.

A one-year no-income forward price is $84.84 and the annual risk-free rate is 5%. Find the current spot price.

Spot price. What is it? The cash price for buying the asset right now. From \(F_0=S_0(1+r)^T\), the reverse formula is \(S_0=F_0/(1+r)^T\).

1. Discount instead of grow

Variant: Back Out the Financing Rate

Abstract: The spot-to-forward gap is the financing growth embedded in the contract.

A no-income asset is $100 today and its six-month forward price is $102.47. Assume annual compounding. Find the annualized risk-free rate.

Financing rate. What is it? The borrowing or lending rate that carries today's asset price to the delivery date. The equation is \(r=(F_0/S_0)^{1/T}-1\).

1. Undo the half-year power

Variant: Value an Existing Long from the New Forward Price

Abstract: The old long owns the discounted difference between today's fair delivery price and the old locked price.

A long forward locked $105. With three months left, a new matching forward is priced at $111.35. The annual discount rate is 5%. Find the old forward's value.

Matching forward. What is it? A new contract on the same asset with the same remaining maturity. Long-value formula. What is it? \(V_t=(F_t-F_0)/(1+r)^{T-t}\).

1. Find and discount the delivery-price advantage

Variant: Value a Long Directly from Spot

Abstract: For a no-income asset, owning the old long is equivalent to owning spot and owing the present value of the delivery price.

A no-income asset is $110. An old long forward requires $102 in three months. The annual rate is 5%. Find the contract value without first calculating a new forward price.

Direct valuation. What is its formula? \(V_t=S_t-PV_t(F_0)\). Present value. What is it? Money at delivery translated into today's smaller amount.

1. Subtract today's value of the promised payment

Variant: Flip a Long Value into a Short Value

Abstract: A forward is a zero-sum promise, so changing seats changes only the sign.

A long forward is currently worth $3.60 per unit. The contract covers 25,000 units. Find the short position's total value.

Zero-sum. What does it mean? One side's gain is exactly the other side's loss. The invariant is \(V_t^{short}=-V_t^{long}\).

1. Flip the sign, then scale

Variant: Price a Forward with a Known Cash Dividend

Abstract: A dividend belongs to the person holding the stock, not the forward, so remove its present value before carrying the stock price forward.

A stock is $70. It will pay a $2.20 dividend exactly when a one-month forward expires. The annual rate is 1%. Find the forward price.

Carry benefit. What is it? Cash or usefulness earned by owning the asset before delivery. Known-income formula. What is it? \(F_0=FV(S_0)-FV(I)\), where \(I\) is the ownership income.

1. Grow spot; remove the dividend at delivery

Variant: Handle a Dividend Paid Before Expiration

Abstract: A dividend paid early must be carried from its payment date to the forward's delivery date, not for the contract's whole life.

A stock is $1,000. A $10 dividend arrives in one month. A three-month forward is priced at a 5% annual rate. Find the forward price.

Future value of income. What is it? The dividend grown only from the day it is received to delivery. The formula is \(F_0=S_0(1+r)^T-D(1+r)^{T-t_D}\).

1. Use two different clocks

Variant: Price an Index Future with Continuous Dividend Yield

Abstract: A continuous dividend yield acts like a continuously flowing benefit that offsets financing cost.

An index is 3,500. Its continuously compounded dividend yield is 3%, the continuously compounded risk-free rate is 0.15%, and maturity is three months. Find the futures price.

Continuous compounding. What is it? Growth represented with \(e^x\). Dividend yield. What is it? Dividends expressed as a rate of index value. The formula is \(F_0=S_0e^{(r_c-q)T}\).

1. Net benefit against financing

Variant: Include Storage Cost and Convenience Benefit

Abstract: Carry costs push the forward up; ownership benefits pull it down.

Copper is $8,000 per tonne. Continuous financing is 4%, storage cost is 1.5%, and the convenience yield is 0.5%. Find the six-month forward price.

Storage cost. What is it? The cost of physically holding the asset. Convenience yield. What is it? The non-cash usefulness of having the physical asset available. The formula is \(F_0=S_0e^{(r_c+CC-CB)T}\).

1. Add costs and subtract benefits

Variant: Execute Carry Arbitrage on an Overpriced Forward

Abstract: If the market forward is too expensive, manufacture delivery cheaply and sell the expensive promise.

A no-income asset is $100, the one-year rate is 5%, and the market forward price is $110. Show the locked arbitrage profit at maturity.

Carry arbitrage. What is it? Borrow, buy the asset, and short an overpriced forward. Short the forward. What does that mean? Promise to deliver the asset and receive the delivery price.

1. Build the cheap synthetic delivery

Borrow 100, buy the asset, and short at 110. At maturity deliver the asset for 110 and repay \(100(1.05)=105\).

Variant: Execute Reverse Carry on an Underpriced Forward

Abstract: If the forward is too cheap, buy it and fund the later payment with proceeds from short-selling the asset now.

A shortable no-income asset is $100, the one-year rate is 5%, and the market forward is $101. Find the locked maturity profit.

Reverse carry arbitrage. What is it? Short the asset, invest the proceeds, and go long an underpriced forward. Shortable. What does it mean? The asset can be borrowed and sold now.

1. Lock both ends

Short the asset for 100 and invest it. The investment becomes 105. Pay 101 under the long forward, receive the asset, and return it to the lender.

Variant: Revalue after a New Dividend Is Announced

Abstract: A newly announced dividend lowers the new fair forward price, which hurts an existing long if spot itself does not move.

A long forward locked $102. With three months left, spot is $110 and the rate is 5%. A $2 dividend is newly announced for delivery day. Find the long value.

Revaluation. What is it? Recalculating an old contract using today's inputs. New forward price. What is it here? \(F_t=S_t(1+r)^{T-t}-D_T\).

1. Reprice, then discount the gap

Now compare that fair price with the old locked price.

Variant: Explain Why a Settled Future Has Zero Value

Abstract: Daily settlement pays yesterday's gain or collects yesterday's loss, leaving a fresh contract at today's futures price.

A long futures position was entered at 102. Today's settlement price is 112.35. After today's variation margin is paid, what is the contract's value?

Mark to market. What is it? Settle the day's gain or loss in cash. Variation margin. What is it? That daily cash transfer. Immediately after settlement, futures value is reset to zero.

1. Separate cash already paid from value still inside

The price move created cash, but that cash has already left the contract through margin.

Variant: Decide When Forward and Futures Prices Can Differ

Abstract: Daily futures cash flows can be reinvested, so their timing matters when price changes and interest rates move together.

Interest rates tend to rise on days when an asset's futures price rises. Relative to a forward, which contract is likely more valuable to the long and why?

Positive correlation. What is it? Two things tend to rise and fall together. Daily settlement effect. What is it? Futures gains arrive early and can be reinvested at then-current rates.

1. Follow the cash timing

The futures long receives gains when rates are high and can reinvest them well. Losses tend to arrive when rates are low.

Variant: Revalue a Forward with Remaining Costs and Benefits

Abstract: At the valuation date, rebuild today's fair delivery price using only costs and benefits that remain from today to expiration.

An old long forward locked $1,000. Seven months later, the asset is $1,050. Present value of remaining ownership costs is $4 and remaining benefits is $28. Five months remain and the annual rate is 2%. Find today's fair forward price and the old long's value.

Remaining carry. What is it? Only costs and benefits occurring after today's valuation date. Productive asset. What is it? An asset that gives a non-cash ownership benefit. The formulas are \(F_t=FV_t(S_t+CC_t-CB_t)\) and \(V_t=PV_t(F_t-F_0)\).

1. Reprice the remaining package

Now discount the advantage over the old price.

Variant: Read FRA Tenor Notation

Abstract: The two numbers tell you when the protected loan starts and ends; their difference gives the loan length.

Interpret a \(2\times5\) FRA using 30-day months. State the FRA expiration, underlying loan end, and loan length.

Tenor notation. What is it? In \(h\times T\), \(h\) is months until the loan starts and \(T\) is months until it ends. Underlying loan. What is it? The imagined deposit or borrowing whose rate drives the FRA cash settlement.

1. Read left, right, difference

In words, that means:

Variant: Calculate MRR Interest on Add-On Basis

Abstract: Add-on interest is principal times annual rate times the fraction of a year.

A £10 million 90-day deposit earns a 2.55% market reference rate using a 360-day year. Find interest and terminal amount.

MRR. What is it? Market reference rate: the observable money-market rate used in the contract. ACT/360. What is it? Actual days in the period divided by a 360-day rate year. Add-on basis. What is its formula? \(I=NA\times L_m\times t_m\) and \(TA=NA(1+L_mt_m)\).

1. Shrink the annual rate to 90 days

Add interest back to principal for the amount returned.

Variant: Derive a Fair FRA Rate from Two Spot Rates

Abstract: Two ways of investing to the same final date must grow to the same amount, so the missing middle rate is forced.

The 90-day MRR is 0.90% and the 180-day MRR is 0.95%, both add-on with a 360-day year. Find the fair \(3\times6\) FRA rate for the 90-day period beginning on day 90.

Implied forward rate. What is it? The future-period rate forced by today's two spot rates. Its formula is \(FRA=[(1+L_Tt_T)/(1+L_ht_h)-1]/t_m\).

1. Divide long growth by short growth

Variant: Price a 1x4 FRA

Abstract: Use the spot rate to month one, the spot rate to month four, and a three-month underlying period.

The 30-day MRR is 0.75% and the 120-day MRR is 0.92%. Using a 360-day year, find the fair \(1\times4\) FRA rate.

\(1\times4\) FRA. What is it? A rate fixed now for a three-month loan that begins one month from now. The formula is \(FRA=[(1+L_{120}120/360)/(1+L_{30}30/360)-1]/(90/360)\).

1. Put every day count in its own slot

Variant: Back Out a Missing Long Spot Rate from an FRA

Abstract: A quoted FRA plus the short spot investment must reproduce the long spot investment.

The 90-day rate is 1.00% and the fair \(3\times6\) FRA rate is 1.40%. All rates use add-on basis and a 360-day year. Find the 180-day spot rate.

Long spot rate. What is it? The rate running from today all the way to the later date. Rearrangement gives \(1+L_Tt_T=(1+L_ht_h)(1+FRA\,t_m)\).

1. Chain the two shorter growth pieces

Variant: Count All Possible FRAs

Abstract: Each FRA uses two distinct spot maturities: one start date and one later end date.

A curve contains 7 usable spot maturities. How many distinct FRAs can be implied?

Distinct FRA. What is it? One unique choice of an earlier start maturity and a later end maturity. The count is \(\binom{n}{2}=n(n-1)/2\).

1. Choose two ordered-by-time endpoints

Variant: Value a Long FRA Before Expiration

Abstract: A pay-fixed FRA gains when today's fair FRA rate rises above the old locked rate.

A $20 million pay-fixed/receive-floating FRA locked 0.70%. Today the matching FRA rate is 0.9978%, the underlying period is 90 days, and the discount rate to the loan end is 0.95% for 180 days. Find its value.

Long FRA. What is it? Pay fixed and receive floating; it benefits when rates rise. Interim value formula. What is it? \(V_g=NA(FRA_g-FRA_0)t_m/[1+D_{T-g}t_{T-g}]\).

1. Price the rate advantage, then discount it

Variant: Value the Short Side of an FRA

Abstract: The short FRA is the receive-fixed side, so its value is exactly the long side with the sign flipped.

The pay-fixed side of an FRA is worth +$14,820. Find the value to the receive-fixed/pay-floating side.

Short FRA. What is it? Receive fixed and pay floating; it benefits when rates fall. Symmetry formula. What is it? \(V_g^{short}=-V_g^{long}\).

1. Change the chair, not the economics

Variant: Settle a Pay-Fixed FRA at Expiration

Abstract: The loan interest difference belongs at the loan end, but an advanced-settled FRA pays its discounted value at the loan start.

A $20 million pay-fixed FRA locked 0.70%. At FRA expiration, the 90-day MRR is 1.10%. The settlement discount rate is 1.10%. Find the cash received.

Advanced settled. What does it mean? Cash changes hands when the underlying loan begins, before its interest would normally be paid. The formula is \(NA(L_m-FRA_0)t_m/[1+D_mt_m]\).

1. Find end-date interest difference; bring it to the start

Variant: Settle the Receive-Fixed FRA

Abstract: A receive-fixed FRA gains when the observed market rate finishes below the locked fixed rate.

A $10 million receive-fixed FRA locked 2.60% for 90 days. At expiration the 90-day MRR is 2.55%, and the settlement discount rate is 2.40%. Find the settlement cash flow to the receive-fixed side.

Receive-fixed settlement formula. What is it? \(NA(FRA_0-L_m)t_m/[1+D_mt_m]\). Settlement cash flow. What is it? The one net amount paid between the two parties.

1. Fixed beats floating by five basis points

Variant: Catch Zero and Negative FRA Settlements

Abstract: Settlement direction comes entirely from the market-minus-contract rate spread.

A pay-fixed FRA locked 1.20%. State its settlement sign if the observed MRR is (a) 1.20% and (b) 0.90%.

Settlement sign. What is it? Positive means cash received by the named party; negative means cash paid. For pay-fixed, the numerator is \(L_m-FRA_0\).

1. Compare, without unnecessary arithmetic

Below the locked rate, the pay-fixed side loses.

Variant: Choose the Correct FRA Hedge

Abstract: A future borrower fears higher rates; a future lender fears lower rates.

A company will borrow for three months beginning six months from now. Which \(6\times9\) FRA side hedges it?

Hedge. What is it? A position designed to offset an unwanted risk. Future borrower. What is the danger? Market rates may rise. Pay-fixed FRA. What is it? Pay the locked rate and receive floating, producing a gain when rates rise.

1. Choose the payoff that fights the pain

Variant: Calculate Accrued Interest

Abstract: Accrued interest is the coupon earned since the last coupon date, even though it has not yet been paid.

A 1.5% semiannual Treasury note has $100 par. Sixty days have passed in a 180-day coupon period. Find accrued interest per $100 par.

Accrued interest. What is it? The seller's earned slice of the next coupon. Its formula is \(AI=(NAD/NTD)(C/n)\), where \(NAD\) is days accrued, \(NTD\) is total days, \(C\) is annual coupon per 100, and \(n\) is payments per year.

1. Earn one-third of the half-year coupon

Variant: Convert Clean Price to Full Price

Abstract: The invoice-like full price includes accrued interest; the quoted clean price does not.

A bond's clean price is 101.00 and accrued interest is 0.25. Find its full spot price.

Clean price. What is it? The quoted bond price excluding accrued interest. Full price. What is it? Clean price plus accrued interest; it is also called dirty price. The formula is \(S_0=B_0+AI_0\).

1. Put the earned coupon slice back in

Variant: Price a Bond Forward with No Interim Coupon

Abstract: Start with the bond's full price and finance it to delivery.

A bond's clean price is 104.00, current accrued interest is 0.17, no coupon arrives before a three-month forward expires, and the annual rate is 1.65%. Find the full forward price.

Full forward price. What is it? The all-in delivery value before converting it into a quoted futures price. With no interim coupon, \(F_0=(B_0+AI_0)(1+r)^T\).

1. Carry today's full bond price

Variant: Remove a Coupon Paid Before Bond-Forward Delivery

Abstract: A coupon collected by the spot owner before delivery is a carry benefit and must be removed from the forward price.

A bond's full spot price is 102. A coupon of 2 is paid in six months, delivery is in one year, and the annual rate is 4%. Find the bond forward price.

Coupon income. What is it? Cash paid by the bond before delivery. The formula is \(F_0=FV(B_0+AI_0)-FV(CI)\).

1. Grow the bond for a year; grow the coupon for only six months

Variant: Turn Full Forward Price into a Quoted Bond Futures Price

Abstract: The exchange quote removes delivery-date accrued interest and divides by the bond's conversion factor.

A bond's full forward price is 104.60, accrued interest at futures delivery will be 0.67, and its conversion factor is 0.7025. Find the quoted futures price.

Conversion factor. What is it? An exchange adjustment that makes deliverable bonds with different coupons and maturities comparable. Quoted futures price. What is its formula? \(Q_0=(F_0-AI_T)/CF\).

1. Strip delivery accrued interest; scale by the factor

Variant: Recover Full Delivery Price from a Futures Quote

Abstract: Reverse the quoting convention: multiply by the conversion factor and then restore accrued interest.

A bond futures quote is 125.00, the conversion factor is 0.90, and delivery-date accrued interest is 0.20. Find the full delivery invoice per $100 par.

Invoice price. What is it? The actual amount the futures long pays for the delivered bond, before any contract multiplier. Its formula is \(Invoice=Q\times CF+AI_T\).

1. Undo the quote adjustment

Variant: Value a Bond Forward Position with Quoted Prices

Abstract: Discount the price-point gain, convert points to a fraction of par, then multiply by contract notional and count.

Eight JGB forwards each cover JPY100 million par. The long locked 153; the matching six-month forward is now 155. The annual rate is 0.12%. Find total value.

Price point. What is it? One percent of par because bond prices are quoted per 100. Contract notional. What is it? The face amount used to scale the quoted-price result.

1. Discount the two-point advantage

Now translate points into yen across all contracts.

Variant: Find Bond-Futures Arbitrage Profit

Abstract: Compare the full futures invoice with the carried full spot bond price in the same units, then discount the locked delivery-date gap.

A bond has clean price 112.00 and current accrued interest 0.08. No coupon is paid before three-month delivery. The rate is 0.30%. Futures quote is 125, conversion factor 0.90, and delivery accrued interest 0.20. Find arbitrage profit per $100 today.

Mispricing. What is it? The market invoice minus the no-arbitrage full forward price. Present arbitrage profit. What is it? That locked delivery gap discounted to today.

1. Put both choices on a full-price basis

The market futures contract instead demands this full invoice:

Discount the excess market invoice to today.

Variant: Identify the Cheapest-to-Deliver Bond

Abstract: The futures short chooses the eligible bond that is cheapest after the exchange's conversion adjustment.

Bond A has full forward price 112.0, delivery accrued interest 0.2, and conversion factor 0.90. Bond B has 118.0, 0.4, and 0.95. Which is cheapest to deliver based on implied quoted cost?

Cheapest-to-deliver bond. What is it? The eligible bond that costs the short least after conversion-factor adjustment. Compare \((F-AI_T)/CF\) across bonds.

1. Translate both bonds into comparable quotes

Apply the same quote translation to Bond B.

The smaller adjusted cost wins.

Variant: Price a Discount Factor from a Spot Rate

Abstract: A present value factor tells you today's value of one currency unit paid at a specific future date.

The 180-day add-on spot rate is 2.00% on a 360-day year. Find the present value factor for $1 paid on day 180.

Present value factor. What is it? The amount today that grows to 1 at the stated spot rate. The formula is \(PV_i(1)=1/[1+R_i(NAD_i/NTD)]\).

1. Discount one future dollar

Variant: Price a Plain-Vanilla Interest-Rate Swap

Abstract: Choose the fixed rate that makes the fixed-rate bond worth par, because the floating-rate bond is worth par on a reset date.

A three-year annual-pay swap has discount factors 0.990099, 0.977876, and 0.965136. Find the par fixed swap rate.

Plain-vanilla interest-rate swap. What is it? One party pays a fixed rate and the other pays a market floating rate in the same currency. Par swap rate. What is it? The fixed rate making the swap worth zero initially.

1. Make fixed coupons fill the gap to par

Variant: Convert an Annual Swap Rate into a Payment

Abstract: The fixed leg pays notional times annual rate times the accrual-period fraction.

A $50 million quarterly swap has a fixed rate of 2.20% and uses 90/360. Find each fixed payment.

Accrual period. What is it? The fraction of a rate year covered by one payment. Fixed swap amount. What is its formula? \(FS=NA\times AP\times r_{FIX}\).

1. Pay one quarter of the annual rate

Variant: Value a Receive-Fixed Interest-Rate Swap

Abstract: Receiving an old fixed rate above today's fair fixed rate is valuable; discount every remaining rate advantage.

A €100 million receive-fixed swap pays annually. Its old fixed rate is 2.00%, today's matching fixed rate is 1.30%, and remaining discount factors sum to 4.822107. Find value.

Receive-fixed swap. What is it? Receive the contractual fixed payments and pay floating. Its reset-date formula is \(V=NA(FS_0-FS_t)\sum PV_i\).

1. Price seven-tenths of a percent across all remaining dates

Variant: Value the Pay-Fixed Side

Abstract: The pay-fixed party owns the exact opposite cash flows, so its value is the negative of receive-fixed value.

The receive-fixed side of a swap is worth €3,375,475. Find the pay-fixed side's value.

Pay-fixed swap. What is it? Pay the contractual fixed rate and receive floating. Counterparty symmetry. What is it? \(V_{pay-fixed}=-V_{receive-fixed}\).

1. Flip the viewpoint

Variant: Predict Swap Value from a Rate Move

Abstract: A fixed payment becomes attractive when new market fixed rates fall and unattractive when they rise.

Market swap rates fall from 3% to 1.5%. Without calculating, state the sign of a receive-fixed position and a pay-fixed position entered at 3%.

Market swap rate. What is it? The fixed rate on a new zero-value swap today. Receive-fixed position. What is it? Receive the old contractual rate and pay floating.

1. Compare old fixed with replacement fixed

Variant: Back Out the Current Swap Rate from Value

Abstract: Divide the swap's value by notional and the discount-factor sum to recover the fixed-rate advantage.

A $40 million receive-fixed swap has value $600,000, an old annual fixed rate of 2.50%, and remaining discount factors summing to 3. Find today's matching fixed rate.

Matching fixed rate. What is it? The par rate on a new swap with the same remaining payment dates. Rearrangement gives \(FS_t=FS_0-V/(NA\sum PV_i)\).

1. Peel value back into a rate spread

Variant: See an Interest-Rate Swap as Two Bonds

Abstract: Receive fixed/pay floating has the same cash-flow value as long a fixed-rate bond and short a floating-rate bond.

A fixed-rate bond leg is worth 101.4 per 100 notional and the floating-rate bond leg is worth 100 on a reset date. Find the receive-fixed swap value per 100.

Bond replication. What is it? Replacing a derivative with ordinary positions that create identical cash flows. Floating-rate bond at par. What does it mean? Immediately after reset, its coupon matches the market, so value is 100.

1. Long incoming leg, short outgoing leg

Variant: Set Currency-Swap Notionals

Abstract: The two principals must be equal in value at the opening spot exchange rate.

A currency swap exchanges A\(100 million against US dollars at A\)1.14 per US$1. Find the US-dollar notional.

Currency-swap notional. What is it? The principal amount used to calculate payments in each currency. Spot quote A\(/US\). What does it mean? Australian dollars per one US dollar. The identity is \(NA_A=S_0NA_B\).

1. Divide A$ by A$ per US$

Variant: Price Both Fixed Rates in a Currency Swap

Abstract: Each currency has its own yield curve, so each leg gets its own par swap rate.

Quarterly A$ discount factors sum to 3.933870 with final factor 0.972763. US$ factors sum to 3.995009 with final factor 0.997506. Find both annual fixed rates using \(AP=0.25\).

Fixed-for-fixed currency swap. What is it? Exchange fixed-rate payments and principal in one currency for fixed-rate payments and principal in another. For currency \(k\), \(r_k=(1-PV_{n,k})/(AP\sum PV_{i,k})\).

1. Price each bond leg separately

Repeat using only the US-dollar curve.

Variant: Find Currency-Swap Periodic Payments

Abstract: Once each annual fixed rate and notional are known, each leg's payment is ordinary fixed interest in its own currency.

An A$ leg has A\(100 million notional at 2.7695%; a US\) leg has US$87,719,298 at 0.2497%. Payments are quarterly. Find both payments.

Swap leg. What is it? One stream of payments inside a swap. Periodic fixed payment. What is its formula? \(FS_k=NA_k\times AP\times r_k\).

1. Keep the currencies separate

The US-dollar leg uses its own notional and rate.

Variant: Value a Fixed-for-Fixed Currency Swap

Abstract: Value the bond you receive, subtract the spot-converted bond you pay, and keep everything in one currency.

A dealer receives A$ and pays US$ in a currency swap. The A$ notional is 100 million, its periodic fixed rate is 0.692375%, its discount-factor sum is 3.967683, and its final factor is 0.986031. The US$ notional is 87,719,298, its periodic rate is 0.062425%, its factor sum is 3.994841, and its final factor is 0.998336. Spot is A\(1.13/US\)1. Find value in A$.

Receive-currency-a value. What is it? The present value of currency-a inflows minus the spot-converted present value of currency-b outflows. The formula is \(V_{CS}=NA_a[c_a\sum PV_{i,a}+PV_{n,a}]-S_tNA_b[c_b\sum PV_{i,b}+PV_{n,b}]\).

1. Price each bond; convert only the US$ bond

Price the US-dollar bond on its own curve.

Convert that bond to A$ and subtract it.

Variant: Convert Currency-Swap Value to the Other Party and Currency

Abstract: First flip the counterparty sign; then divide by an A\(-per-US\) quote to convert A$ into US$.

A receive-A$ dealer's currency swap is worth A\(2,145,167. Spot is A\)1.13 per US$1. Find the value to the opposite party in US dollars.

Opposite party. What does it mean? The counterparty paying A$ and receiving US\(. **Currency conversion. What is it here?** Divide A\) by A\(/US\) to obtain US$.

1. Flip, then convert

Now cancel A$ through the quoted exchange-rate units.

Variant: Predict Currency-Swap Value from FX Movement

Abstract: If the currency you must pay becomes more expensive, your swap position gets worse, all else equal.

A US firm receives US$ and pays A$ under a swap. The quote falls from A\(1.14/US\) to A\(1.05/US\), with yield curves unchanged. What happens to the firm's value?

A$ strengthens. What does it mean? One US dollar buys fewer Australian dollars. FX risk. What is it? Swap value changes because the two future currency streams translate at a new spot rate.

1. Follow the payment burden

The firm owes A\(. Each US\) now buys fewer A\(, so those A\) payments are more expensive in US$.

Variant: See a Currency Swap as Two Bonds

Abstract: A fixed-for-fixed currency swap is long one fixed-rate bond and short another after translating them into one reporting currency.

In A$ terms, the received A$ bond is worth A\(102 million. The US\) bond owed is worth US\(88 million, and spot is A\)1.10/US\(. Find the swap value to the A\) receiver.

Reporting currency. What is it? The single currency used to state the answer. Bond-difference formula. What is it? \(V_{CS}=V_a-S_tV_b\) for the party receiving currency \(a\).

1. Translate before subtracting

Variant: Calculate a Positive Equity-Swap Cash Flow

Abstract: The receive-equity side gets the equity return and pays the fixed slice for the same period.

A €5 million quarterly receive-equity/pay-fixed swap has a 1.6% annual fixed rate. The equity index returns +4.0% during the quarter. Find the net cash flow to the receive-equity side.

Equity swap. What is it? A contract exchanging an equity return for fixed, floating, or another equity return. Receive-equity cash flow. What is its formula? \(NA(RE-AP\,r_{FIX})\) when \(RE\) is already the period return.

1. Compare two quarterly returns

Variant: Calculate a Negative Equity-Swap Cash Flow

Abstract: If the equity leg loses money, the receive-equity party may owe both the equity loss and the fixed payment.

Use the same €5 million quarterly swap and 1.6% annual fixed rate, but the quarterly equity return is -6%. Find the receive-equity side's cash flow.

Negative equity return. What does it mean in a swap? The receive-equity side pays the loss rather than receiving a gain. The formula remains \(NA(RE-AP\,r_{FIX})\).

1. Keep the negative sign

Variant: Price the Fixed Leg of an Equity Swap

Abstract: With equity notional equal to bond par, the fixed rate is the same par rate used for an interest-rate swap.

A five-year annual equity swap has discount factors 0.990099, 0.977876, 0.965136, 0.951529, and 0.937467. Find its fair annual fixed rate.

Fair equity-swap fixed rate. What is it? The fixed rate making a new fixed-versus-equity swap worth zero. The formula is \(r_{FIX}=(1-PV_n)/(AP\sum PV_i)\) when equity notional equals par.

1. Fill the discounted gap to par

Use the sum with the final factor.

Variant: Value a Receive-Fixed, Pay-Equity Swap

Abstract: Compare the current value of the promised fixed bond with the equity notional grown by the index since the last reset.

A €10 million receive-fixed/pay-equity swap was entered when the index was 100. It is now 105. The old fixed leg is currently worth €10,216,019 and bond par equals equity notional. Find the swap value.

Last reset price. What is it? The equity level from which the current swap-period return is measured. Valuation formula. What is it? \(V_{EQ,t}=V_{FIX}(C_0)-(S_t/S_{t-1})NA_E-PV(Par-NA_E)\).

1. Value the equity obligation

Subtract that obligation from the fixed-leg value.

Variant: Find the Break-Even Equity Index Level

Abstract: Set swap value to zero and solve for the index level that makes the equity leg exactly equal the fixed-leg value.

A receive-fixed/pay-equity swap has fixed-leg value €10,216,019, notional €10 million, last-reset index 100, and \(Par=NA_E\). Find the current index that makes value zero.

Break-even index. What is it? The current equity level that makes neither party's position valuable. With \(Par=NA_E\), \(S_t=S_{t-1}V_{FIX}/NA_E\).

1. Solve the zero-value equation

Rearrange for the unknown index level.

Variant: Include a Par-Notional Mismatch in Equity-Swap Value

Abstract: If the fixed bond's par and equity notional differ, finance the terminal difference instead of silently dropping it.

A receive-fixed/pay-equity swap has \(V_{FIX}=10.4\) million, current equity-leg value \(10.2\) million, bond par \(10.5\) million, equity notional \(10.0\) million, and the discount factor to maturity is 0.96. Find swap value.

Par-notional mismatch. What is it? The fixed bond repays a different terminal principal from the equity position. The adjustment is \(PV(Par-NA_E)\) in \(V_{EQ}=V_{FIX}-V_{equity}-PV(Par-NA_E)\).

1. Price the terminal mismatch

Subtract it along with the equity obligation.

Variant: Net an Equity-for-Equity Swap Cash Flow

Abstract: An equity-for-equity swap simply pays one equity return and receives another on the same notional.

A $12 million swap receives Index A, which returns 3%, and pays Index B, which returns 5% during the period. Find the net cash flow to the receive-A side.

Equity-for-equity swap. What is it? Exchange one equity return for another without a fixed-rate leg. Its cash flow is \(NA(RE_A-RE_B)\).

1. Net the two period returns

Variant: Decide Whether Dividends Belong in an Equity Leg

Abstract: The contract definition decides whether the equity return is price-only or total return; never assume.

An index rises from 200 to 206 and distributes dividends worth 2 index points. Find the period return for (a) a price-return equity leg and (b) a total-return equity leg.

Price return. What is it? Return from the index-level change only. Total return. What is it? Price change plus dividends, assuming dividends are included or reinvested.

1. Read the contract's return definition

Now include the dividend points for total return.

Variant: Appendix — Derive the No-Income Forward Price

Abstract: Two routes to owning the asset at delivery must have the same cost or a risk-free trade appears.

Derive the no-income forward-price formula under annual compounding.

Law of one price. What is it? Two strategies with identical future cash flows must have the same price. Invariant. What is it here? Delivery through spot-and-carry must cost the same as delivery through the forward.

1. Build the replicating route

Borrow \(S_0\), buy the asset, and hold it. At \(T\), you own the asset and owe:

A long forward also gives the asset at \(T\) for \(F_0\). Equal future assets force equal future costs:

Variant: Appendix — Derive Forward Value

Abstract: Cancel an old forward with an opposite new forward; the only cash left is the delivery-price difference.

Derive the value of an existing long forward using today's matching forward price.

Offsetting forward. What is it? A new opposite contract with the same asset and maturity. Value additivity. What is it? The value of combined positions equals the sum of their values.

1. Lock the remaining cash flow

Old long: receive asset and pay \(F_0\). New short: deliver that asset and receive \(F_t\). The asset cancels, leaving \(F_t-F_0\) at \(T\).

For annual compounding:

Variant: Appendix — Derive the General Carry Formula

Abstract: Start with spot, add every ownership cost, subtract every ownership benefit, and carry the net amount to delivery.

Derive the forward formula for an asset with present-value carry costs \(CC_0\) and benefits \(CB_0\).

Carry balance. What is it? The net cost of owning the asset until delivery. Carry cost. What is it? Storage, insurance, or other ownership expense. Carry benefit. What is it? Income or convenience received only by the owner.

1. Price the complete spot-and-carry package

Today's funded amount is:

Grow it to delivery:

With continuous rates:

Variant: Appendix — Derive the FRA Rate

Abstract: Investing to the long date directly must equal investing to the short date and then rolling at the FRA rate.

Derive the no-arbitrage FRA rate between times \(h\) and \(T\), where \(m=T-h\).

Roll investment. What is it? Invest to \(h\), then reinvest from \(h\) to \(T\). Growth-factor invariant. What is it? Direct and rolled investments ending at \(T\) must produce the same cash.

1. Equate the two routes

Divide by the short growth factor and solve:

Variant: Appendix — Derive Advanced FRA Settlement

Abstract: Compute the interest difference due at the loan end, then discount it one loan period because the FRA pays at the loan start.

Derive expiration settlement to the pay-fixed/receive-floating FRA side.

Advanced set. What is it? The floating rate is observed at the loan's start. Advanced settled. What is it? The FRA cash is paid then too. Arrears interest. What is it? Ordinary loan interest paid at the loan's end.

1. Form the arrears-date difference

Bring it back one loan period:

Variant: Appendix — Derive the Bond-Futures Quote

Abstract: Carry the full bond, remove coupons received before delivery, strip delivery accrued interest, then apply the conversion factor.

Derive the quoted bond-futures price from a clean spot bond price.

Full spot bond. What is it? \(B_0+AI_0\). Interim coupon income. What is it? Coupons received before delivery. Quoted futures price. What is it? The standardized exchange quote after delivery adjustments.

1. Build the full forward value

The delivery invoice identity is \(F_0=Q_0CF+AI_T\). Solve for the quote:

Variant: Appendix — Derive the Par Swap Rate

Abstract: At inception, discounted fixed coupons plus discounted principal must equal par.

Derive the fixed rate on an at-market interest-rate swap with constant accrual period \(AP\).

At-market swap. What is it? A new swap with zero value. Par replication. What is it? Treat the fixed leg as a bond priced at 1 and the floating leg as a par bond worth 1 on a reset date.

1. Set the fixed bond equal to one

Move the final principal's present value and divide:

Variant: Appendix — Derive Interest-Rate Swap Value

Abstract: Offset the old fixed coupons with today's fair fixed coupons; the floating pieces cancel on a reset date.

Derive the reset-date value of a receive-fixed swap.

Offsetting swap. What is it? A new opposite swap using today's fair fixed rate. Coupon-difference invariant. What is it? After floating legs cancel, only old fixed minus new fixed remains on every payment date.

1. Present-value every remaining fixed-rate difference

Each per-unit difference is \(FS_0-FS_t\), so:

The opposite position is:

Variant: Appendix — Derive Currency-Swap Value

Abstract: Every received currency stream is a long bond; every paid stream is a short bond; convert them to one unit before subtracting.

Derive the value in currency \(a\) of receiving fixed currency \(a\) and paying fixed currency \(b\).

Currency-unit invariant. What is it? Quantities may be added or subtracted only after they are expressed in the same currency. Currency-\(a\) bond value. What is it? Fixed coupons plus principal discounted on the currency-\(a\) curve.

1. Price and translate the two bonds

Price the paid bond on currency \(b\)'s curve.

Therefore:

Variant: Appendix — Derive Equity-Swap Value

Abstract: Replicate receive-fixed/pay-equity with a long fixed bond and a short reset equity position, then correct any terminal principal mismatch.

Derive the value of a receive-fixed/pay-equity swap between reset dates.

Reset equity position. What is it? The notional grown by the equity price ratio since the last reset, \((S_t/S_{t-1})NA_E\). Terminal mismatch. What is it? Bond par minus equity notional, discounted from maturity.

1. Add the replicated pieces

Long fixed bond contributes \(V_{FIX}(C_0)\). The equity obligation subtracts \((S_t/S_{t-1})NA_E\). Financing the terminal mismatch subtracts \(PV(Par-NA_E)\).

Variant: Appendix — Use the Seven Invariants as an Error Check

Abstract: When a long formula sheet feels slippery, test the answer against rules that cannot change.

List the core invariants that should survive every pricing or valuation problem in this module.

Invariant. What is it? A relationship that must remain true even when the numbers or contract type change. Error check. What is it? A fast test that catches a wrong sign, unit, timing, or price basis.

1. Keep these seven locks on the answer