## Dr i

Dd2 t dr2 i . . dd t dr ., . d t X y --Y- X At X y -- 'xAt For cash flows, based on the equations 2.45 and 2.48 in the CPR S, A, B, M model, we have dCPR 2 t ,rdRI 2 t dRI t dBM t , 02 AGE t MM t x BM t 0 x 0-' dBM t x dRI t , R t dBM 2 t d0 d0 d02 dRI2 t _d_ _301 5_ drw t -1 x drw t -1 d02 dr10 1 -8.571 430 WAC - r10 t -1 2 d0 d0 1 -8.571 430 WAC - r10 t -1 2 0 M 2 t 07 dB 2 t -1 1 0 2 0 2 B 0 1 _ e_10 5R2 0, t 1 _ e5r 2 t - ---- 1 _ e-l0a dr 2 t 3rl0 t a 39i39j 39t 39 j 10 ' Finally, the...

## Potential New ARM Product

Duration is used to measure the interest risk of a fixed income security. The higher the duration is, the more interest risk that security bears. From the investor's perspective, she will benefit if interest rates fall, and suffer if interest rates climb, if the security is non-callable no prepayment option . From the mortgage borrower's point of view, he will exercise his prepayment option if interest rates drop, and thus reduce the benefit for the investor. He will be able to lock in the low...

## Non Constancy of Credit Spread Sensitivities

The non-constancy of credit spread sensitivities would naturally be embedded in their dependence on state variables in the RCR model. However, we would like to see how they change over time, and compare it to the simple linear regression sensitivity estimators, and find out why the RCR estimators provide better accuracy. Let's take one bond as example, the bond with CUSIP of 001765AE, one of American Airlines' long term bonds, and depict its random coefficients and constant coefficients. The...

## Data Description

We extract data from three databases Warga bond database5, CRSP6, and COMPUSTAT7 for different financial data. Warga database, which is also known as the Lehman Brothers Fixed Income Database, contains the most comprehensive bond data for academia. We only choose those bond that satisfy the following standards 1. Dealer quoted price, instead of matrix price, since it has been pointed out that matrix price could produce some problems Sarig and Warga 1989 2. At least 30 consecutive observations...

## Literature Review

There has been a lot of recent interest on identifying the key factors affecting credit spread. One approach is to add macroeconomic variables into the traditional structural model. However, by adding new state variables, the model not only becomes more complicated in the form, but also harder to identify empirical evidence to improve pricing and hedging practice. Another approach is to concentrate on regression models. Because of the simplicity and convenience in incorporating any new state...

## Introduction

Mortgage-backed securities MBS have become increasingly important fixed income instruments, both because of their volume and the role they play in fund investment and portfolio management. However, there has not been a very comprehensive set of risk indicators to measure and manage the risks involved with MBS. Hedging the interest rate and credit risk of MBS remains a complicated problem in the fixed income industry. This dissertation develops a set of risk measures for interest rate risk and...

## Dt

Where R 0,t the continuous compounding interest rate from now to time t, i.e. the term structure. In order to simplify the simulation process, the model can be re-parameterized from its original to the following dx t -a t x t dt odB t , x 0 0 2.24 a t r t - x t f 0, t 1 - e at 2. 2.25 The process x t is called an Ornstein-Uhlenbeck process, and its solution is given x t oe-at JeaudB u , 2.26 which is a Gaussian Markov process, and can also be represented as where W t , t gt 0 is also a Brownian...

## Introduction to Random Coefficient Model

The most frequently used linear model in statistics might be the following y Xp s, 4.1 where y is the observed response of dependent variables X is the vector of explanatory variables P is the vector of coefficients of the linear model S is the error term, ands N 0, ctS . For time series data, like those we frequently encounter in financial econometrics, it can be written as where yt is the observed random variable, xkt are known explanatory variables, Pk are unknown constants to be estimated,...

## Dependence of Credit Spread Sensitivities to State Variables

In this section, we are going to discuss the regression results of our RCR model, compared with simple linear model. Table 4.2 shows the coefficient estimation for both models, and their t-values. Applying White robust estimator, regression is performed on individual bond and the average statistics9 are reported. In the simple linear regression model, we can find that the sensitivity measures to interest rate change, leverage change, and volatility changes are significant, and the signs and...

## Efficient Sensitivity Analysis of Mortgage Backed Securities

A mortgage-backed security MBS is a security collateralized by residential or commercial mortgage loans. An MBS is generally securitized, guaranteed and issued by three major MBS originating agencies Ginnie Mae, Fannie Mae, and Freddie Mac. The cash flow of an MBS is generally the collected payment from the mortgage borrower, after the deduction of servicing and guaranty fees. However, the cash flows of an MBS are not as stable as that of a government or corporate coupon bond. Because the...

## Motivation

In our previous two chapters, we have assumed that the credit risk of the MBS is totally absorbed by the MBS issuer, and the MBS investor only needs to hedge the interest rate risk due to voluntary prepayment, including housing turnover and refinancing. This assumption is reasonable since in the secondary market for conforming mortgages, the three major MBS issuers, Ginnie Mae, Fannie Mae, Freddie Mac1, all promise that they will guarantee the principal payment when there is a default event...

## Hedging Credit Risk of Mbs A Random Coefficient Approach

In order to hedge the credit risk of MBS, the MBS issuer sometimes needs to purchase pool insurance from a third party, beyond the protection of mortgage collateral, and primary mortgage insurance. In this case, it is important to model the credit risk of the third party. Recently there has been increased interest in some research papers to use regression method to determine what factors affect credit spread. Most of the papers, which use simple linear regression, found that variables in...

## Random Coefficient Model for Credit Spread Changes

Huang and Kong 2003 mention that the low explanatory power of theoretical determinants, documented in Collin-Dufresne et al. 2001 , could be due to two reasons. The first reason is that the explanatory variables may not be the best proxies to measure the changes in default risk. The second reason is that the current existing corporate bond pricing model might miss some important systematic risk factors. We have a different opinion as to why the simple linear regression model lacks explanatory...

## Hedging MBS in HJM Framework

There is a large body of literature on hedging with different interest risk measures, like first-order hedging with duration Ilmanen 1992 , second order hedging with convexity Kahn and Lochoff 1990 , Lacey and Nawalkha 1993 , principal components hedging Golub and Tilman 1997 , key rates hedging Ho 1992 , level slope curvature hedging Willner 1996 , etc. Yet there has not been a unifying effort in combining hedging the term structure together with hedging volatility factors. This essay tries to...

## Derivation of General PA Estimators

If P, the price of the MBS, is a continuous function of the parameter of interest, say 0, and assuming the interchange of expectation and differentiation is permissible4, we have the following PA estimator by differentiating both sides of 2.1 d PV t,0 dd t0 C t ,0 M0 d t ,0 . Now we have reduced the original problem from estimating the gradient of a sum function to estimating the sum of a bunch of gradients. Actually now we only need to dc t,0 J dd t,0 , . estimate two gradient...

## Conclusion

In this essay, we proposed a new method to hedge the interest risk of MBS, based on PCA factors estimated from historical interest rate data. We estimated the PA estimators for hedging MBS, and implemented the hedging with a dynamically rebalancing portfolio of MBS and Treasury bonds. We achieved much better hedging efficiency, compared with traditional hedging, not only in the measure of mean hedging error, but also in the standard deviation of hedging error. We made the following contribution...

## Deriving PA estimators in HJM Framework

Following the logic in Chapter 2, we only need to derive the PA estimator for short rate r t and 10-year rate rio t , since our prepayment model and valuation model are totally dependent on these two factors. If we assume that in a short period of time, the principal components for yield curve volatility are going to be constant, then any interest rate yield curve shift can be decomposed of these principal components, which is to say which is analogous to 2.35 . Following the same logics as in...

## Hedging Performance Analysis

In this section, we compare the hedging performance of our PCA-based hedging and traditional duration and convexity based hedging for a FRM30 MBS instrument. The principal balance of the MBS is 4 million. We are selling short this MBS at the market price, and use the proceeds to buy treasury bonds. Initial net present value of the hedging portfolio is zero. Every month, we try to rebalance the portfolio, and we sell part of our bonds to meet the payment obligation of the MBS. Hedging error is...

## Simulation in HJM Framework

This section gives the detailed implementation of HJM model, using the volatility factors identified in PCA analysis. We know that, in a multifactor HJM framework, the dynamics of instantaneous forward rate looks like df t, T m t, T, a t dt t,T,Q t dZk t , 3.1 where under no arbitrage assumption, the drift term is determined by volatility structure. where PCi t, T is the principal components we get in last section Pi is a parameter to be calibrated to market price of interest rate derivatives....

## Rt df t t dT drwt Jt dA k dA k

And follow the same logic, we can get the gradients of discounting factors, prepayment rate, cash flows, present values, etc. 1 Although observed Ak and fik might have a positive correlation, i.e., when the volatility is high, the observed shift also might have bigger magnitude. But they have total different meaning, fik is the parameter to calibrate to market price, and A k is the observed shift in yield curve.

## Numerical Example

2.4.1 Specification of Numerical Example We need to specify two sets of data to price the mortgage the mortgage data and the interest rate data, which includes the initial term structure and parameters for the interest rate model. We price different mortgages to examine the different impacts that a term structure shift or change in volatility may have on different mortgage products. The following data are fixed for all products Table 2.1 shows the difference between all the products. All the...

## Estimation of Volatility Factors via PCA

The Principal Components Analysis method is generally used to find the explanatory factors that maximize successive contributions to the variance, effectively explaining variations as a diagonal matrix. This method has been used in yield curve analysis for more than 10 years, see Litterman and Scheinkman 1991 , Steeley 1990 , Carverhill and Strickland 1992 . Here we give a brief description of PCA method applied in yield curve analysis 1. Suppose we have observation of interest rates rt Tj at...