On current coupon (cc)

http://www.wisegeek.com/what-is-a-current-coupon.htm

The current coupon is a term used to describe a certain to be announced mortgage backed security with a current delivery month. The security that is trading closest to its par value without going over par is designated as the current coupon. A coupon is the interest payment on a loan, and current refers to an instrument that is at the going market rate. Taken together, this means that the current coupon mortgage backed security has the interest rate that most accurately reflects the current state of the market.

A mortgage backed security is a type of debt instrument issued by three entities: Fannie Mae, Freddie Mac and the Government National Mortgage Association, which is also called Ginnie Mae. These institutions take the money that investors pay for mortgage backed securities and use it to purchase residential mortgages. They form their mortgage holdings into pools, and the securities that they sell guarantee payments to the holders based on the payments made by the borrowers in the underlying mortgages. This allows investors to put funds into residential mortgages without directly negotiating with borrowers. Mortgage-backed securities are easily traded, so they form a secondary market for residential mortgages on which investors can trade the securities in ways that reflect their expectations about the future of the market.

The current coupon security must be a to be announced, or TBA, mortgage backed security. This qualification means that the pool of mortgages that will back the security has not been assigned, even though the contract is about to be made. This is because the mortgage backed security issuing agency can write new loans in between the time of the contract arrangement and the delivery date. The mortgages, however, will be fairly current because of the specification that the current coupon security must be delivered within the month.

To determine which security is the current coupon, it is necessary to know the par value of the mortgages. The par value is the sum of the outstanding principals on the underlying mortgages. With older securities, this can vary based on how many prepayments are made in a given pool. The mortgage backed security is TBA, so the mortgage pool, although still in the planning stages, is full: none of the borrowers have had a chance to prepay. This means that the par value of the security is a predetermined amount which the institution will meet by selecting mortgages with appropriate principal amounts.

The current coupon is of interest to investors because it reflects the state of the mortgage market. It accumulates all of the traders’ knowledge of mortgages to give a succinct report about what they think the fair rate for new mortgages should be. Lenders refer to the current coupon security’s interest rate for a baseline rate when they are writing mortgages.

Motivation for the Large Deviation Principle

This post consists of the material of Scott’s large deviation lecture today, and the point is to motivate the large deviation principle. Think about this: when we evaluate the assymptotics of rare events, why the ratio of {\frac{1}{n}log} comes up, in a natural way? An intuitive answer might be: because it works for averages of i.i.d normals: suppose we have an i.i.d sequence of standard normal random variables {X_{1}, \cdots, X_{n}, \cdots}, then we know that

{\bar{X}_{n} := \frac{1}{n}\sum_{l=1}^{n}X_{l} \sim N(0,\frac{1}{n})}.

So we have

{\mathbb{P}(\bar{X}_{n} > \delta)=\frac{1}{\sqrt{2 \pi n}} \int_{\delta}^{\infty}e^{-\frac{y^{2}}{2n}}dy} = {\frac{1}{\sqrt{2 \pi}}\int_{\sqrt{n} \delta}^{\infty}e^{-\frac{z^{2}}{2}}dz *}

It’s a nice exercise to show that {* \leq \frac{1}{\delta \sqrt{2 \pi n}}e^{-\frac{n \delta^{2}}{2}}}, and  {* \geq \frac{\delta \sqrt{n}}{\sqrt{2 \pi}(1+n \delta^{2})}e^{-\frac{n \delta^{2}}{2}}}. (This is problem 2.9.22 in Karatzas and Shreve.) And from this it’s clear that {\mathbb{P}(\bar{X}_{n} >; \delta) \sim -\frac{\delta^{2}}{2}}.

Now suppose we have a sequence of Borel measures {\mathbb{P}_{n}}, and for each Boreal set {A} we have {\mathbb{P}_{n}(A) \sim e^{-nI(A)}} for some set function {I}. Let’s assume there is some reference measure {\mathbb{P}} such that {\mathbb{P}_{n} <; <; \mathbb{P}}, then formally we have {\frac{d\mathbb{P}_{n}}{d\mathbb{P}}=e^{-nI(x)}}. So we have:

\displaystyle \frac{1}{n}log \mathbb{P}_{n}(A) \approx \frac{1}{n}log[\int_{A}e^{-nI(x)}d \mathbb{P}(x)] \rightarrow \text{esssup}_{x \in A}[-I(x)]=-\text{essinf}_{x \in A}I(x).

(Here we used the fact that {\frac{1}{n}log\mathbb{E}[e^{nX}] \rightarrow \text{esssup} X} for ramdom variable X.)

This suggests that we should have {\lim_{n \rightarrow \infty}\frac{1}{n}log \mathbb{P}_{n}(A)=-inf_{x \in A}I(x)}. This is how the rate function comes into play.

The next question is, if the above statement does hold true, what properties should {I(x)} have? Just by the above limit, it’s clear that we should have:

{I \geq 0}
{inf_{x \in \mathbb{X}}I(x)=0}
I is lower semicontinuous, thus if {F} is compact, {\inf_{x \in F}I(x)} is achieved.

Definition 1 (Rate Function) Let {(X,\tau)} be a topological space, a funtion {I: X \mapsto [0, \infty]} is called a rate function if I is lower semicontinuous. If the sets {\{ I \leq \alpha \}} are compact then I is called a good rate function.

Definition 2 (Large Deviation Principle) Let {(\mathbb{P}_{n})} be a family of Borel measures on X. We say that {(\mathbb{P}_{n})} satisfies a LDP with (good) rate function {I} if:

{\varlimsup_{n \rightarrow \infty}\frac{1}{n}log\mathbb{P}_{n}(F) \leq -\inf_{x \in F}I(x)} for {F} closed.
{\varliminf_{n \rightarrow \infty}\frac{1}{n}log\mathbb{P}_{n}(G) \geq -\inf_{x \in G}I(x)} for {G} open.

Definition 3 (Weak LDP) If the upper bound holds just for comapct {F} then we say {(\mathbb{P}_{n})} satisfies a weak LDP.

Theorem 4 (Varadhan) Let {\mathbb{X}} be a Polish space. then we have:

\displaystyle (\mathbb{P}_{n}) satisfies a LDP with rate function I

\displaystyle \iff \lim_{n \rightarrow \infty}\frac{1}{n}\text{log}\int_{\mathbb{X}}e^{nf(x)}d\mathbb{P}_{n}(x)=\sup_{x \in \mathbb{X}}(f(x)-I(x)), \forall f \in C_{b}(\mathbb{X})

Let’s recall the definition of tightness:

Definition 5 A family of probability measures {(\mathbb{P}_{n})} is tight if {\forall \epsilon >;0, \exists} a compact {K_{\epsilon}} s.t. {\sup_{n}\mathbb{P}_{n}(K_{\epsilon}^{c}) \leq \epsilon}.

We have the following

Theorem 6 (Prohorov) {(\mathbb{P}_{n})} tight {\implies} {(\mathbb{P}_{n})} is relatively compact.

Now let’s define exponential tightness:

Definition 7 A family of Borel measures {(\mathbb{P}_{n})} is exponentially tight if {\forall \alpha >; 0}, {\exists K_{\alpha}} compact, s.t. {\varlimsup_{n \rightarrow \infty}\frac{1}{n}log\mathbb{P}_{n}(K_{\alpha}^{c}) < -\alpha}.

Theorem 8 {(\mathbb{P}_{n})} is exponentially tight {\implies} there is {(n_{k})} s.t. {(\mathbb{P_{n_{k}}})} satisfies a large deviation principle.

Theorem 9 Weak LDP + exponential tightness {\implies} a full LDP.

Finally, as we may expect, the Cramer’s theorem:

Theorem 10 (Cramer) Let {X_{l}} be i.i.d random variables in {\mathbb{R}^{d}} with law {\mu}. Set {\bar{X_{n}}=\frac{1}{n}\sum_{l=1}^{n}X_{l}} and let {\mu_{n}} be the law of {\bar{X_{n}}}, then {\lim_{n \rightarrow \infty}\frac{1}{n}log\mu_{n}(x)=-I(X)}. Where {I(x)} is the Legendre-Fenchel transform of the cumulant generation function.

21-260 Recitation Note 11

  • What we covered last week:
    1. We finished the method of undetermined coefficients by studying the case where the nonhomogeneous term is a solution to the corresponding homogeneous equation.
    2. We finished the discuss of mechanical vibrations by studying forced vibrations, i.e. a spring-mass system with external force.
    3. We began to study Laplace transforms (Sec6.1).
  • Example problems:
    1. (3.8.11) A spring is stretched 6 in by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of {0.25 lb\cdot s /ft} and is acted on by an external force of {4cos2t} lb.(a) Determine the steady state response of this system.(b) If the given mass is replaced by a mass {m}, determine the value of {m} for which the amplitude of the steady state response is maximum.
    2. (3.8.12) A spring–mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of {(3cos3t-2sin3t)} N, determine the steady state response. Express your answer in the form {Rcos(\omega t - \delta)} .
    3. (6.1.13) Find the Laplace transform of the function {f(t)=e^{at}sin(bt)}.
    4. (6.1.15) Find the Laplace transform of the function {f(t)=te^{at}}.

Spring in Pittsburgh

The point state park is one of the FEW amazing places here in Pittsburgh.

20120407-184913.jpg

The confluence of the Allegheny and Monongahela rivers, creating the Ohio River, has greatly impacted the history of Point State Park. This confluence was referred to as the Forks of the Ohio, which remains the official NRHP-designated name for the site. It was once at the center of river travel, trade, and even wars throughout the pioneer history of Western Pennsylvania. During the mid-18th century, the armies of France and the Great Britain carved paths through the wilderness to control the point area and trade on the rivers. The French built Fort Duquesne in 1754 on foundations of Fort Prince George, which had been built by the colonial forces of Virginia.

I drove to point state park this afternoon after shopping at the strip district. And it turns out that the most convenient route from home to point state park is Blvd of the Allies–it ends at the park! So just follow the Blvd till the end where it meets the river, then make a right. There you can find public parking. The daily rate for the parking lot is $7.00. It’s actually not bad since the meter rate is $3.00/hour!

I took a walk along the Allegheny River. There were only a few visitors and bikers on this side of the river, and the fountain was under repairs. The water is clear, and trees begin to blossom:

20120407-191201.jpg

20120407-191210.jpg

20120407-191947.jpg

20120407-191954.jpg

It’s always hard to identify spring here in Pittsburgh. It is April now but temperature will drop below zero next Wednesday or so– the weather forecast has always been accurate. It should turn warm sometime if this is not the end of the world, and I guess I may take a road trip then.