Today we published a 1 hour 20 minute movie on *delta normal VAR* and *Monte Carlo simulation methods*.These movies focus on two Jorian chapters that are *critical* toNovember’s FRM exam. We hope to help especially if you are confused bycovariance and correlation matrices (see, we read our mail). Plus,we’ve got a new set of *Quantitative* questions. It’s summer and you should have already started your preparation. You’ve seen the *Pirates*sequel, now view a movie with real production value. This one is adramatic nail-biter. It gets so dicey we have to open Excel (twice) tocope with matrix math. Can Geometric Brownian Motion (GBM) be appliedto interest rate dynamics (hint: can a mean-reverting series *really* be a random walk?). Sorry, we won’t give away the ending…

SwapsThereare several swap instruments: equity swaps (you pay me LIBOR and inexchange I’ll pay you the S&P 500 return), commodity swaps (pay methe floating price of oil and I’ll pay you a fixed price), currencyswaps (pay me interest in German marks and I will pay you interest inAmerican dollars). The largest category is interest rates swaps and a"plain vanilla" interest rate swap is where the company pays a fixedrate (*fixed rate payer*) in exchange for receiving a floating rate (*floating-rate receiver*).

Of course, the counterparty to this swap is the opposite position. The counterparty is a*floating-rate payer, fixed-rate receiver*. A few things to keep:

Theimplicit floating-rate bond isn’t too much trouble after all, if weremember that its price is $100 (the principal) immediately after itpays the next coupon! Why? At that point, the coupon is paying LIBORand rates are LIBOR, so you’d pay $100 for such a bond because bydefinition it earns market rate. So will only need to present valuethis implicit bond at the next coupon date; i.e., in three months itwill pay 1/2 (semi-annual) of 5% (the six month rate) of the $100notional. In three months, the future value of this bond is $2.5 plusthe principal:

Thevalue of this swap is therefore $101.40 - $101.35 = $0.05. That’spositive five cents for one counterparty and negative five cents forthe other.

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Section V Notes ReadyPosted by David Harper on 15th July 2006

Yesterday we posted the**Section V Study Notes** (Investments), a question set on **Investments**, and a new movie tutorial, *Introduction to Value at Risk*(VAR). We hope you agree this is the best way to learn about VAR. Thisweek’s movie is introductory: We review one-period VAR, absolute versusrelative VAR, and n-period VAR.

Interest rate parity (IRP) Theinterest rate parity (IRP) formula is a just a flavor of thecost-of-carry model that we reviewed in the last two posts. Thecost-of-carry model says the forward rate is a function of thecompounded spot rate. The difference is that, instead of an underlyingphysical commodity (e.g., corn, oil futures), we are dealing withforeign currency. So the*forward exchange rate *is a function of the *spot exchange rate*:

Inthe IRP, the spot exchange rate is simply the result of (continuously)compounding the difference between the domestic riskless rate and theforeign country riskless rate (r - rf). What if they happened to beequal? Then exp(0) = 1 and the forward exchange rate would equal thespot exchange rate. Why? Because if the country rates are equal, you’llend up at the same place regardless of whether you hold home currencyor covert immediately to foreign currency. The IRP, as a flavor of thecost-of-carry model, depends on the "no arbitrage" assumption: you needto be roughly indifferent to holding domestic or foreign currency.

Nowassume a 1.2 spot exchange rate, a domestic riskless rate of 5% and aforeign riskless rate is 2.75%. For a three month period (t=0.25), theIRP says the forward exchange rate must be 1.207:

Theforward rate must be higher. If it were not, you would always hold thedomestic currency and an arbitrage opportunity would (temporarily)exist.

Normal BackwardationKeep in mind that contango is__not__ when the forward rate is greater than the spot rate. Contango is when the forward rate exceeds the *expected *spot rate:

Itis not obvious why the forward rate would be different from theexpected (future) spot rate. Contango is not what we expect; after all,why should we expect speculators to pay more for a futures contractthan its expected spot price. Normal backwardation, however, isreasonable when we consider that*speculators (buyers of the foward contract) expect a profit*.If speculators expect a profit, then they will pay something less thanthe expected (future) spot price. Therefore, normal backwardation is areasonable phenomenon:

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SwapsThereare several swap instruments: equity swaps (you pay me LIBOR and inexchange I’ll pay you the S&P 500 return), commodity swaps (pay methe floating price of oil and I’ll pay you a fixed price), currencyswaps (pay me interest in German marks and I will pay you interest inAmerican dollars). The largest category is interest rates swaps and a"plain vanilla" interest rate swap is where the company pays a fixedrate (

Of course, the counterparty to this swap is the opposite position. The counterparty is a

- Most of them do not require an exchange principal at inception (that’s why it’s called
*notional*amount). The notable exception is currency swaps: in general, currency swaps exchange principal at inception and at maturity. - Atinception, the value of the swap is typically zero (why would acounterparty enter into a negative value?). Subsequent to inception, asthe truth is revealed on the floating side, the swap becomes netpositive or negative. That’s why the pricing problems must besubsequent to inception.
- For the interest rate swap,remember it breaks down into a floating-rate bond on one side and afixed-rate bond on the other side. Most of the examples in Hull arevariations on the simple idea that the Value [swap] = Present Value[fixed-rate bond] - Present Value [floating-rate bond].
- The key simplification in the pricing of the plain-vanilla interest rate bond is this: the value of the floating-rate bond is
*exactly its notional immediately after a coupon (interest) payment*.

Theimplicit floating-rate bond isn’t too much trouble after all, if weremember that its price is $100 (the principal) immediately after itpays the next coupon! Why? At that point, the coupon is paying LIBORand rates are LIBOR, so you’d pay $100 for such a bond because bydefinition it earns market rate. So will only need to present valuethis implicit bond at the next coupon date; i.e., in three months itwill pay 1/2 (semi-annual) of 5% (the six month rate) of the $100notional. In three months, the future value of this bond is $2.5 plusthe principal:

Thevalue of this swap is therefore $101.40 - $101.35 = $0.05. That’spositive five cents for one counterparty and negative five cents forthe other.

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Section V Notes ReadyPosted by David Harper on 15th July 2006

Yesterday we posted the

Interest rate parity (IRP) Theinterest rate parity (IRP) formula is a just a flavor of thecost-of-carry model that we reviewed in the last two posts. Thecost-of-carry model says the forward rate is a function of thecompounded spot rate. The difference is that, instead of an underlyingphysical commodity (e.g., corn, oil futures), we are dealing withforeign currency. So the

Inthe IRP, the spot exchange rate is simply the result of (continuously)compounding the difference between the domestic riskless rate and theforeign country riskless rate (r - rf). What if they happened to beequal? Then exp(0) = 1 and the forward exchange rate would equal thespot exchange rate. Why? Because if the country rates are equal, you’llend up at the same place regardless of whether you hold home currencyor covert immediately to foreign currency. The IRP, as a flavor of thecost-of-carry model, depends on the "no arbitrage" assumption: you needto be roughly indifferent to holding domestic or foreign currency.

Nowassume a 1.2 spot exchange rate, a domestic riskless rate of 5% and aforeign riskless rate is 2.75%. For a three month period (t=0.25), theIRP says the forward exchange rate must be 1.207:

Theforward rate must be higher. If it were not, you would always hold thedomestic currency and an arbitrage opportunity would (temporarily)exist.

Normal BackwardationKeep in mind that contango is

Itis not obvious why the forward rate would be different from theexpected (future) spot rate. Contango is not what we expect; after all,why should we expect speculators to pay more for a futures contractthan its expected spot price. Normal backwardation, however, isreasonable when we consider that

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