FRM 2006-Market risk
發布時間：2010-01-19 共1頁
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:
Add comment July 15th, 2006<!-- at 09:16am--> <!--David Harper-->
Cost of CarryForpreorder customers, today we released a movie (Derivatives - Part I)that reviews forwards pricing and swaps.At the heart of forward pricingis the “cost-of-carry model.” This model summarizes a relationshipbetween spot (S0) and forward (F0) prices
There are two steps to understanding this model. First, see that the forward price is an unbiased estimate of the future spot price.If you agree to pay me next month for one bushel of corn, mostly likelyyou’ll want to pay me what you think that bushel will cost in one month(i.e., the spot rate in the future). If you lock in anything more orless, one of us will profit.But what is an estimate of the future spotrate? It is today’s spot rate compounded forward. In the case of afinancial asset?Clike a contract on the S&P 500 Index?Cthe expectedfuture spot price is today’s index price compounded forward. In words,the above says, the forward price is what we expect of the future spotprice: today’s spot price (continuously) compounded forward at the riskfree rate (r) over time (t). (this is a theoretically clean formulathat omits the role of uncertainty, which would effectively “plus-up”the rate).
Asillustrated here, as time marches forward toward the delivery date, theforward/futures price converges toward the spot rate. The cost of carrymodel introduces a few realities into the equation. Consider that youhave two choices if you want corn next month. You can buy corn today orgo long a corn futures contract. If you go long the futures contract,you don’t own the corn in the meantime. Somebody else owns and storesthe corn and you’ll buy it from them. Whether ownership is a good orbad thing depends on the underlying asset and can be complicated (e.g.,seasonality). The essence of the cost-of-carry model is this: costs and benefits to the asset owner will be recognized in the forward price.So consider all possible costs/benefits to the asset owner (we aren’tdistinguishing between physical and financial assets here):
There are two things that make the forward/futures contract more expensive:the cost to finance the asset (r) and storage costs. If somebody incursa cost to store the corn until you buy it, you are going to need toreimburse them in the form of a higher futures price. There are two offsetting “benefits” to ownership that make the forward less expensive:income (or dividends, denoted by ‘q’) and the so-called convenienceyield. These accrue benefits to the owner so they reduce the futuresprice. The convenience yield is a catch-all: it is the quantificationof ownership benefits. These forces can be summarized in a universalcost of carry equation:
Whichis not quite realistic because we included all factors so we are mixingphysical and financial asset characteristics; i.e., a commodity(physical asset) would not pay a dividend (q) nor would a financialasset incur a storage cost (u). But you could start here and excludethose costs/benefits that don’t apply. The main thing to remember is:the spot rate will be continuously compounded, over time (T), by afunction of the financing rate (+r), the storage costs (+u), incomereceived (-q) and any convenience yield (-y).
Add comment June 23rd, 2006<!-- at 02:06am--> <!--David Harper-->
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:
Add comment July 15th, 2006<!-- at 09:16am--> <!--David Harper-->
Cost of CarryForpreorder customers, today we released a movie (Derivatives - Part I)that reviews forwards pricing and swaps.At the heart of forward pricingis the “cost-of-carry model.” This model summarizes a relationshipbetween spot (S0) and forward (F0) prices
There are two steps to understanding this model. First, see that the forward price is an unbiased estimate of the future spot price.If you agree to pay me next month for one bushel of corn, mostly likelyyou’ll want to pay me what you think that bushel will cost in one month(i.e., the spot rate in the future). If you lock in anything more orless, one of us will profit.But what is an estimate of the future spotrate? It is today’s spot rate compounded forward. In the case of afinancial asset?Clike a contract on the S&P 500 Index?Cthe expectedfuture spot price is today’s index price compounded forward. In words,the above says, the forward price is what we expect of the future spotprice: today’s spot price (continuously) compounded forward at the riskfree rate (r) over time (t). (this is a theoretically clean formulathat omits the role of uncertainty, which would effectively “plus-up”the rate).
Asillustrated here, as time marches forward toward the delivery date, theforward/futures price converges toward the spot rate. The cost of carrymodel introduces a few realities into the equation. Consider that youhave two choices if you want corn next month. You can buy corn today orgo long a corn futures contract. If you go long the futures contract,you don’t own the corn in the meantime. Somebody else owns and storesthe corn and you’ll buy it from them. Whether ownership is a good orbad thing depends on the underlying asset and can be complicated (e.g.,seasonality). The essence of the cost-of-carry model is this: costs and benefits to the asset owner will be recognized in the forward price.So consider all possible costs/benefits to the asset owner (we aren’tdistinguishing between physical and financial assets here):
There are two things that make the forward/futures contract more expensive:the cost to finance the asset (r) and storage costs. If somebody incursa cost to store the corn until you buy it, you are going to need toreimburse them in the form of a higher futures price. There are two offsetting “benefits” to ownership that make the forward less expensive:income (or dividends, denoted by ‘q’) and the so-called convenienceyield. These accrue benefits to the owner so they reduce the futuresprice. The convenience yield is a catch-all: it is the quantificationof ownership benefits. These forces can be summarized in a universalcost of carry equation:
Whichis not quite realistic because we included all factors so we are mixingphysical and financial asset characteristics; i.e., a commodity(physical asset) would not pay a dividend (q) nor would a financialasset incur a storage cost (u). But you could start here and excludethose costs/benefits that don’t apply. The main thing to remember is:the spot rate will be continuously compounded, over time (T), by afunction of the financing rate (+r), the storage costs (+u), incomereceived (-q) and any convenience yield (-y).
Add comment June 23rd, 2006<!-- at 02:06am--> <!--David Harper-->
