Agency Risk Model

Fundamental Factor Model for US Agency Stripped MBS and CMO Products Sunny Wei Zhao April 2014 Version 3.0

Bloomberg Fundamental Factor Models for SMBS and CMO

2

Table of Contents

1. Introduction 3

2. The Stripped MBS and CMO Market 3

3. Estimation and Data Universes 4

4. Strip IO Model 6

5. Strip PO and Agency CMO Model 12

6. Summary 15

7. Appendix I. A Short Summary of RMBS Risk Model 15

8. Appendix II. A Hedging Example 16

9. References 16


3

INTRODUCTION

This article introduces the Bloomberg fundamental factor risk model for US Agency stripped

mortgage-backed securities (SMBS) and collateralized mortgage obligation (CMO) bonds (for

factor models of general fixed-income products and other securitized products, refer to [1, 2]).

THE STRIPPED MBS AND CMO MARKET

Stripped MBS and CMO bonds are derivative types of MBS securities backed by Agency residential mortgage pass-throughs. Typically each SMBS or CMO deal is backed by a group of pools of the same Agency, program, and coupon, and usually a narrow range of issuance time.

AGENCY STRIPS Stripped MBS are created by stripping the interest and principal payments from a collection of Agency pools and distributing these payments to separate classes. IO Strips receive all of the interest payments from the underlying collateral and none of the principal. PO Strips receive the entire principal payment and none of the interest. Typically additional classes are also issued in the same deal that receive various proportions of interest and principal, forming a whole stack of SMBS with coupon ladder from 0 to twice the collateral coupon rate. The majority of SMBS are backed by conventional (FNMA or FHLMC) 30 Year MBS pools. The cash flows of an IO or a PO strip are usually much more sensitive to prepayment projections than the cash flows of the underlying collateral. As interest rates decline, prepayment speeds on the collateral begin to accelerate. For the PO, higher speeds combined with lower discount rates boost the price, leading to a high positive duration and also giving the PO positive convexity. For the IO, the price actually falls as interest rates decline, as higher prepayment speeds (which means shorter average life and shorter coupon-receiving period) usually overwhelm the effect of lower discount rates; hence, IO typically displays negative duration. The unusual duration and convexity characteristics of IOs and POs make them among the most versatile instruments in the fixed-income markets, appealing to a wide range of market participants with different investment motives. They are often used by investors to express a leveraged view on prepayments relative to the collateral. Stripped IO and PO are also used for effectively hedging prepayment or interest rate risk. With its highly negative effective duration profile, IO is often used by investors as a hedge for increasing interest rates. Similarly, PO responds favorably to increased prepayment risk and can be used as hedges for decreasing rates and increasing prepayment risk. Later in this paper we will use Strip IOs and MBS Generics to construct hypothetical hedging portfolios in order to measure the spread risk of CMO bonds.


4

CMO The cash-flow characteristics of pass-through securities do not meet the needs of many institutional investors due to their long maturity and uncertain nature. CMOs were first designed to make a better match between the requirements of a wide range of investors and the expected cash flows from a pool of mortgages and to redistribute prepayment risk to different classes. Since Freddie Mac issued the first CMO deal in 1983, this market has grown rapidly and trillions of dollars of CMOs have been brought to the market. An Agency CMO deal is typically collateralized by hundreds of Agency pools, which leads to geographical diversification and reduction of payment noise compared to individual pools. Typical Agency MBS-backed CMO structures include: 1) Sequential Bonds: Sequential bonds within a deal all receive interest payments, but all

principal payments are directed to the bonds with the shortest maturity, till it is retired, and then all principal payments are directed to the next shortest bond, etc. This process continues until all bond classes have been paid off. This structure creates MBS bonds of different expected maturity from the underlying collateral.

2) Z-Bonds: Z-bonds receive no interest payment until their principal payment window starts. The interest accrued each month is added to the principal balance, until earlier classes are retired, and then collateral cash flows are directed to pay down the principal balance of the Z-bond. Z-bonds provide a beneficial effect on the cash flow stability of earlier bonds when its accrued interest is used to pay down earlier bonds.

3) PAC Bonds: Planned amortization class (PAC) bonds play a central role in the Agency CMO

market. PAC bonds have a high degree of cashflow certainty provided prepayment speeds stay within a given range (called the PAC band). A PAC bond is characterized by a specified principal payment schedule, defined by the minimum of the collateral principal payments at the two PAC-band speeds. In allocating principal paydown from the collateral to the CMO bonds, priority is given to meeting the PAC principal schedule; thus the support bonds absorb prepayment variations as much as possible. Support bonds typically have a high degree of WAL sensitivity to prepayment change and tend to be priced at higher yields as compensation for absorbing extra prepayment risk.

Development in structured MBS market has produced many other different types of CMO structures, like TACs, PAC POs, PAC IOs, PAC-2, PAC-3, PAC Zs, companion Zs, VADMs, etc. Agency CMOs carry the same explicit or implicit government guarantee as the underlying pools. CMO classes often offer attractive yields relative to other fixed-income instruments of similar credit quality.

ESTIMATION AND DATA UNIVERSES

The estimation universe for Stripped MBS and CMO model consists of all outstanding Strip IO certificates that are covered by Bloomberg Valuation Service (BVAL). BVAL covers a majority of Stripped MBS deals and it adds new issuance into its coverage promptly. Figure 1(a) demonstrates the number of Strip IOs in the estimation universe. The number of IO certificates is around 300-400 in the last few years, and continue to grow as new deals are issued.


5

Figure 1. Estimation universe

For the data universe, the minimum requirements for coverage for a Stripped MBS or CMO security

include:

Security prices, provided by Bloomberg or by the client;

Risk exposures (e.g., key rate durations, vega, spread duration, etc.) based on the above prices;

Descriptive information, such as ticker, issuance date, and underlying collateral information,

which allow us to map the securities to the right model factors and define their exposures.

Figure 2 shows the statistics of the Bank of America – Merrill Lynch (BAML) CMOS index and a measure of the trend of this market. From figure 2(a) we see that the outstanding market value of Stripped MBS and CMO has been around $200 billion, greater in the housing boom years but has shrunk since the subprime mortgage crisis. The same trend is also observed in figure 2(b), which shows the ratio between the market values of MBS derivatives and fixed-rate pass-through. This ratio used to be around 10%, but has gradually shrunk to less than 5% after the subprime crisis, reflecting a trend that investors have become more cautious about these sophisticated products.

0

50

100

150

200

250

300

350

400

450

Jan-09 Jul-09 Jan-10 Jul-10 Jan-11 Jul-11 Jan-12 Jul-12 Jan-13 Jul-13

(a) Number of Strip IO in Estimation Universe


6

Figure 2. Statistics of the CMO market, demonstrated using the BAML CMOS index.

STRIP IO MODEL Our objective is to construct a linear risk factor model that adequately characterizes the covariance structure of total returns. In general, the total return of a security over time period can be written as the sum of several return components. The stochastic return, which is the difference between total return and time return, can be written as:

0

50

100

150

200

250

300

350

Jan

-00

Jul-

00

Jan

-01

Jul-

01

Jan

-02

Jul-

02

Jan

-03

Jul-

03

Jan

-04

Jul-

04

Jan

-05

Jul-

05

Jan

-06

Jul-

06

Jan

-07

Jul-

07

Jan

-08

Jul-

08

Jan

-09

Jul-

09

Jan

-10

Jul-

10

Jan

-11

Jul-

11

Jan

-12

Jul-

12

Jan

-13

(a) Market Value of Asset Classes ($bln)

Trust IO Trust PO CMO

0%

2%

4%

6%

8%

10%

12%

Jan

-00

Jul-

00

Jan

-01

Jul-

01

Jan

-02

Jul-

02

Jan

-03

Jul-

03

Jan

-04

Jul-

04

Jan

-05

Jul-

05

Jan

-06

Jul-

06

Jan

-07

Jul-

07

Jan

-08

Jul-

08

Jan

-09

Jul-

09

Jan

-10

Jul-

10

Jan

-11

Jul-

11

Jan

-12

Jul-

12

Jan

-13

(b) Market Value of CMOS Index as a Fraction of Agency Fixed-rate RMBS


7

∑

(1)

where is the total return of security during time period t,

the time return, a

pre-defined factor exposure, is the factor k at time t to be derived from running the above

regression, and the non-factor return.

There can be explicit or implicit factors contributing to the return of a security. Explicit factors, such as shift of the yield curve or volatility surface and change of current coupon spread are observable in the market. Implicit factors capture the change of spread of securities and are closely related to the product type and individual characteristics of each security. They are not observable in the market but instead are obtained via a cross-sectional regression, with pre-assigned factor loadings. Details on model factors will be given in later sections of this paper.

EXPLICIT FACTORS Like MBS pass-throughs, prices of Stripped MBS and CMO bonds have exposures to shift of yield curve, interest-rate volatility, and change of current coupon rate. These return components can be express as, respectively,

∑

(2)

(3)

(4)

In the above expressions, , , and denote curve return, volatility return, and current coupon return, respectively; , , , and are key-rate durations, effective convexity, volatility duration, and current coupon duration, respectively; is the change of i-th key swap rate (i = 1 – 9 for 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, and 30Y tenors, respectively), and is their simple average; is the change in volatility in the period. is the change in current coupon spread in the period. Each CMO bond, according to its collateral type, is mapped to one of the three current coupon indexes: FNMA 30-year, GNMA 30-year, and FNMA 15-year. The key-rate durations, effective convexity, volatility duration, and current coupon duration are generated using the Bloomberg mortgage prepayment model for each security and at each time point. Subtracting the above components from the stochastic return, we get the excess return, which is to be modeled by implicit factors in the next step:


8

(5)

IMPLICIT FACTORS The implicit factor model is presented as a multivariate linear equation, with pre-defined factor loadings multiplied by factor returns to be determined in a multivariate regression process. The dependent variable of the cross-sectional regression is the excess return divided by spread duration, which measures the change of spread of the bonds. The level of the spread of fixed-income products reflects additional premium an investor requires for taking on the additional prepayment risk, liquidity risk, model risk, and other risks. The Strip IO certificates are usually created based on a mega pool of Agency Fixed-rate pass-through, of the same Agency, program (the majority are backed by FNMA or FHLMC 30-Year), coupon, and a narrow range of WALA. Noting the fact that the overwhelming majority of Stripped MBS are backed by conventional 30-year pools and the substantial similarity between FNMA and FHLMC corresponding programs, we ignore Agency and program and construct the implicit factors using price premium-ness and collateral WALA. The Bloomberg Strip IO spread model has three price-WALA factors:

∑

(6)

where

is the spread return of security i during period t-1 to t, and the spread duration

of the strip IO; , and are the implicit factors, loadings, and the residuals, respectively. The

loading of each factor, , is the product of two variables, one a function of price distinguishing

premium and non-premium collaterals, and the other a function of WALA distinguishing new and

seasoned collaterals. The fundamental reason behind this model structure is the distinctive

prepayment behaviors of these different collaterals. These functions are defined by the piece-wise

linear curves as shown in figure 3 and 4, respectively.

Figure 3. Price Functions. The turning points are at Price = 100 and 104.

0

0.2

0.4

0.6

0.8

1

1.2

96 98 100 102 104 106

Pri

ce f

acto

rs

Price

Non_Premium Premium


9

Figure 4. WALA Functions. The turning points of these function are at WALA = 18 and 48.

We define three factors using the product of the price and WALA functions, as demonstrated in

figure 5. Note that the supposed two discount factors, for new and seasoned respectively, are merged

into a single non-premium factor, in consideration of the lack of discount stripped MBS in the last

years.

Figure 5. The Price–WALA factor diagram

Non-Premium Premium

New USD_sio_D

USD_sio_P_N

Seasoned USD_sio_P_S

NON-FACTOR RISK The Strip IO residual model contains only one factor, which represents the average level of the residual absolute value:

(7)

In use of the non-factor risk model, the non-factor risk of a Strip IO position can be estimate as:

[ ]

{ }

(8)

0

0.2

0.4

0.6

0.8

1

1.2

0 6 12 18 24 30 36 42 48 54 60 66 72

WA

LA F

acto

rs

WALA

New Seasoned


10

where the constant

comes from the ratio between the variance and square of expectation of

absolute value for a normal variable.

EXPLANATORY POWER The explanatory power of Strip IO factors is demonstrated in Figure 5 using the 6 month rolling average of adjusted r-squared. The blue line shows the effect of explicit factors (time return, yield curve return, volatility return, and current coupon return), and the pink curve demonstrates the joint effect of explicit and implicit factors of the Strip IO model. On average, the explicit factors explained 10-30% of the variance of total return, while adding the implicit spread factors improves this ratio to 20-60%.

Figure 6. Contribution of Factors to Explanatory Power of Strip IO model

BIAS TEST The bias of a model is measured by the standard deviation of normalized portfolio returns,

(

) (9)

If the risk model forecasts the volatility of realized return properly, the expectation of this bias statistic should be 1. In practice, we calculate this statistic for a rolling 10-month period, and

count what fraction of this quantity falls within its 95% confidence interval ( √

√

),

where T=10 is the rolling period. The Strip IO model is tested on numerous portfolios. As

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Aug-10 Feb-11 Sep-11 Mar-12 Oct-12 Apr-13 Nov-13

Explanatory Power of Strip IO Model

Explicit Factors + Implicit Factors


11

shown in Figure 7, on average, 75% of time the portfolios bias measure falls in this confidence interval. This is satisfactory for a simple model of only three factors.

Figure 7. Percentage of time the risk estimate was within confidence interval.

PORTFOLIO RETURNS VS. FORECAST RISK

00.10.20.30.40.50.60.70.80.9

1

Agency Strip IO - Bias Test


12

Figure 8 compares the realized returns of market portfolio for Strip IO to model-predicted two standard deviation bands. During the 22 months back-testing period, two realized returns fell out of the band. This is a little higher than the expected (1 outlier out of 22), but still acceptable for this simple model.


13

Figure 8. Realized Return vs. Predicted Risk.

STRIP PO AND AGENCY CMO MODEL No new implicit factors are introduced for Strip PO and CMO bonds. Instead, we construct a

hedging portfolio for each Strip PO or CMO bond, using the corresponding Strip IO and MBS

Generic of its underlying collateral. The hedge ratios (for collateral) and (for strip IO) are

determined by matching the dollar prepayment and spread durations of the hedging portfolio with

that of the hedged security, as shown in the Eq. (10):

(10)

where , , and are the prepayment duration, spread duration, and price,

respectively. The subscripts CMO, C, and IO stand for CMO bond, collateral MBS Generic, and a

corresponding Strip IO, respectively. With the above-defined hedge ratios, it is assumed that the

dollar spread return of a PO or CMO bond is approximated by that of the hedging portfolio:

(

) (11)

Inserting the formulas for MBS (see Appendix I) and strip IO models into Eq. (11), and

incorporating the subscripts (for security) and (for time), we have the following expression for

CMO spread return:

-20

-15

-10

-5

0

5

10

15

Strip IO: Realized market portfolio return vs. 2 StDev band

Realized market portfolio return two standard deviation band


14

( ∑

∑

)

(

)

(12)

In the above equations, expressions for Strip PO are obtained by simply replacing subscript “CMO”

with “PO”. In this hedging strategy, PO securities are not simply treated as longing a unit amount of

collateral MBS and shorting a unit amount of the corresponding IO, as this simple hedge is only

accurate when two preconditions are satisfied: (a) the collateral of the trust is exactly the same as the

MBS generic; (b) the IO and PO are priced at their break-even OAS. These are usually untrue in

reality. However, a detailed look at the calculated hedge ratios reveals that for a PO, typically the

MBS hedging coefficients are scattering around 1 while the IO coefficients are close to -1.

For an example, see Appendix II.

NON-FACTOR RISK

The above analysis demonstrates that the residual return of PO and CMO is approximated by a linear

combination of the residual returns of hedging MBS and strip IO. Ignoring the correlation between

the residual returns of MBS and IO, we have

[

]

(

)

( )

(

)

(13)

Therefore, the non-factor risk of CMO and PO are estimated by combining the MBS and Strip IO non-factor risks.

BIAS TEST Figure 9 shows the bias test results on various PO and CMO portfolios, including market portfolios and spread duration quartile portfolios. On average, the model performs better on PO portfolios than on CMO.


15

Figure 9. Percentage of time the risk estimate was within confidence interval.

PORTFOLIO RETURNS VS. FORECAST RISK

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

PO: market PO:oasd.bot

PO:oasd.top

PO:oasd.mid

CMO:market

CMO:oasd.bot

CMO:oasd.top

CMO:oasd.mid

Agency Strip PO & CMO - Bias Test


16

Figure 10 compares the realized returns of market portfolios for Strip PO and CMO to their model-predicted two times standard deviation bands, respectively. During the 22 months period, one realized return for PO portfolio and 3 realized returns for CMO portfolio fell out of the bands, respectively. Again, the model performs on PO portfolio better than on the CMO portfolio.

Figure 10. Realized Return vs. Predicted Risk.

Summary

In this paper, we have presented the Bloomberg fundamental factor model for the US Agency Strip MBS and CMO bonds. The model is carefully constructed by choosing intuitive factors and using fully transparent statistical methodologies. The model passes bias testing with no significant under- or over-forecasting of risk for a broad variety of portfolios.

-3

-2

-1

0

1

2

3

(a) Strip PO: Realized market portfolio return vs. 2 StDev band


-2

-1.5

-1

-0.5

00.5

1

1.5

2

(b) CMO: Realized market portfolio return vs. 2 StDev band



17

Appendix I. A Short Summary of RMBS Risk Model

The Bloomberg RMBS fundamental factor model consists of 6 price-WALA factors (new-premium, seasoned-premium, new-current, seasoned-current, new-discount, and seasoned-discount) and 5 program factors (Conv15, Conv20, GNMA30, GNMA15, and IO):

∑

∑

(14)

For non-factor risk, we subsequently fit a model for the absolute value of residuals :

(15)

where classifies each security into a certain MBS group, and are the model

coefficients. Essentially, these coefficients represent the average absolute magnitude of errors in each group. There are six groups in the non-factor model (Conv30, Conv15, Conv20, GNMA30, GNMA15, and IO). The non-factor risk of an MBS position is estimated as:

[ ]

{ }

(16)

where the constant

comes from the ratio between the variance and square of expectation of

absolute value for a normal variable. More details can be found in [2].

Appendix II. A Hedging Example

As an example, on April 1st, 2014, CMO Z-bond FNR 2003-32 ZT (CUSIP 31393BKU8) is

hedged by MBS Generic FNCL 6 2003 and Strip IO certificate FNS 344 2 (CUSIP 3136FCAB2).

On this date, we have the following data

FNR 2003-32 ZT FNCL 6 2003 FNS 344 2

price 111.651 111.146 16.680

OASD 4.729 4.078 2.692

ppayDur 0.035 0.043 0.452

Inserting the above values into Eq. (10), we have and . This means

we assume the risk characteristics of the Z-bond is approximated by a hedging portfolio

longing 1.188 times face value of MBS and shorting 0.237 times face value of Strip IO. On this

date, the underlying MBS is exposed to the RMBS Seasoned Premium factor, while the Strip IO


18

is exposed to the Strip IO Premium Seasoned factor. Putting these facts together, and calling

Eq. (12), we find that this CMO Z-bond is exposed to implicit risk factors as follows:

Exposure

RMBS USD: Seas Prem -4.825

Agency CMO: SIO Prem Seas 0.095

The signs of the exposures shown on PORT are flipped, as we do for all factor exposures

including yield curve.

References:

[1] Yingjin Gan and Luiza Miranyan, “Fixed Income Fundamental Factor Model”, February,

2012.

[2] Sunny Wei Zhao, Yingjin Gan, Luiza Miranyan, Hui Zhang, and Nick Baturin, “Fundamental Factor Model for Securitized Products”, March, 2012.

Agency Risk Model

Documents

Transcript of Agency Risk Model