AD Prepay Model Meth

MARCH 1999QUAN

TITA

TIVE

PERS

PECT

IVES

..

FIXED RATE MORTGAGE PREPAYMENT MODEL Version 4.1

Eknath Belbase

QUANTITATIVE PERSPECTIVES .. MARCH 1999

FIXED RATE MORTGAGE PREPAYMENT MODEL Version 4.1

Eknath Belbase

Introduction

In 1998, the mortage market experienced dramatic spikes in

prepayments resulting from structural changes in the refinancing

process. This environment created a difficult challenge to prepayment

models. Prepayment analysts made a variety of changes to their models

to account for these high prepayment speeds. Andrew Davidson & Co.,

Inc. has updated our fixed-rate mortgage prepayment models using data

through November 1998. The updated model incorporates the data from

this difficult period.

This article discusses our modeling approach and underlying

philosophy, the model factors, and statistical issues. It also compares the

updated model with the old one and presents some OAS analyses based

on the updated prepayment model.

There are two aims in modeling prepayment behavior that do not necessarily

coincide. The first is to be able to explain and fit observed historical

prepayment behavior in terms of meaningful variables. The second is to be

able to forecast future prepayment behavior under a range of future

environments with the same model.

Historical fit can be increased by increasing the number of factors, from MBS

pool characteristics such as WAC, term, loan type, and age to loan-level

characteristics such as LTV and geographical location. It can also be increased

by adding economic variables, such as GNP growth, unemployment rate and

inflation.

Our Approach

3

Improving historical fit, however, does not imply better forecasts. This can

occur because the perceived relationship between the additional variables and

prepayments is not sufficiently fundamental to act in the future; it can also be

a historical quirk. Furthermore, forecasts often have to be made for

environmental scenarios that simply have not been observed before. Finally,

each variable used in forecasting also has to be predicted. For example, with

recent GNP growth forecasts by economists significantly off the mark even on

a quarterly basis, requiring a GNP forecast input for the next 15 or 30 years

seems undesirable.

The Andrew Davidson & Co., Inc. prepayment model attempts to explain and

forecast prepayment behavior using a small set of pool-level factors, which

have a persistent relationship with prepayments. In this sense it is a

parsimonious model. All of our inputs except for interest rate are deterministic;

hence the level of uncertainty is reduced to one input. The form of the model

has not had to change over time, though parameters are regularly updated with

new data. In general, the parameters have been fairly robust over time.

In 1998, aggressive refinancing campaigns, improved efficiency in the

refinancing process, an increase in the number of alternative loan types and

point-payment combinations made available to borrowers along with falling

rates combined to produce extreme prepayment spikes. Structural changes like

these are very difficult to predict. In addition, after they have occurred, some

time is required to accumulate data and understand which changes are likely to

persist. Simply re-estimating with more historical data can lead to parameters

which are in some sense a weighted average of dynamics before and after the

structural change but which describe neither set of dynamics well.

Prepayment modeling is thus a dynamic problem that requires constant monitoring

and refinement. Though no model is perfect, we believe our prepayment model is

a robust and parsimonious model that captures all the major pool-level factors

affecting MBS prepayments. It is a valuable tool for performing sensitivity analysis

and is an integral component of an option-adjusted risk and valuation framework.

QU A N T I TAT I V E PE R S P E C T I V E S MA R C H 1999

The collateral types supported by the fixed-rate model are:

1. GNMA I and II 15 and 30 year loans

2. Fannie and Freddie Gold 15 and 30 year loans

3. Fannie 20 year loans

4. Fannie and Freddie 5 and 7 year balloons

5. 15 and 30 year Whole loans

6. 30 year Relo loans.

The index rate driving prepayments for a collateral type is the current coupon

rate for that type of collateral. This is defined to be the (net) coupon that would

give a par-priced security. For historical purposes these rates can be gotten

from financial systems (e.g. MTGEGNSF<Index> on Bloomberg for GNMA

30s) and for forecasting are usually modeled in terms of short and long-term

treasury rates with a spread.

The inputs required by the model are loan type, WAC, age, monthly current

coupon forecast, and a history of monthly current coupon rates. The historical

data is provided by Andrew Davidson & Co., Inc. and is updated monthly. The

functional forms for all the sub-models are very similar, with slight differences

for balloons. The parameters were estimated using historical pool-level

prepayment data from 1987 through November 1998. The newer collateral

types, such as balloons, were estimated with considerably less historical data,

though in all cases at least five years of data were used. In some cases where a

majority of the data was old, some lags were estimated using new data to

reflect recent structural changes in the refinancing process. In general, the

updated model shows faster aging, higher prepayment speeds and shorter lags

between interest rate movements and changes in prepayment rates.

4 QU A N T I TAT I V E PE R S P E C T I V E S MA R C H 1999

Model Overview


Figure 1 shows a scatter plot of age versus observed prepayment speed in CPR for

some GNMA 1 30 pools. Such an aggregate graph contains a wide range of

prepayment speeds at any given age. This is partly because we are putting loans that

were premiums and discounts together, but is also the result of some inherent noise

in pool behavior.

Figure 2 shows the current coupon rate graphed together with the prepayment

speeds for a relatively “clean” Fannie Mae pool with a WAC of approximately

8.7.

Figure 1

GNMA 1 30

Prepayments

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120

Age

CP

R

Figure 2

Current Coupon

vs. CPR

4

5

6

7

8

9

10

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81

Age(Months)

Cur

rent

Cou

pon

(%)

0

10

20

30

40

50

60

70

80C

PR

(%)

Coupon CPR

The Historical Data

In general, prepayments rise as the current coupon rate falls, with a short lag

between the troughs for rates and the peaks for prepayments. However, there

also seem to be minor fluctuations in prepayment speeds, which are not

mirrored in the current coupon movements. In contrast, Figure 3 graphs

current coupon against CPR for a very noisy GNMA pool with WAC of 8.5.

The CPR stays mostly at zero with occasional leaps.

There are several periods when the spikes in prepayment rates correspond to

rises in current coupons instead of falls and in general the pool behaves

erratically. This could be due to small pool size, bad data, or a number of other

factors.

For modeling purposes, the first type of pool is very useful, and leads to fast

parameter convergence due to the lack of noise. The second type of pool makes

it difficult to obtain parameter estimates, displays what seems to be

occasionally irrational behavior with respect to current coupon rates, and

changes the parameters which would be estimated with the first type of pool.

Because of this, model estimation is always done with a clean sub-sample of

the original data.


Figure 3

A Noisy GNMA

Pool

4

5

6

7

8

9

10

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Age(Months)

Cur

rent

Cou

pon(

%)

0

10

20

30

40

50

60

70

CP

R (%

)

Coupon CPR

We give a brief overview of the components before going over each one in

detail. The Andrew Davidson & Co., Inc. fixed-rate mortgage prepayment

model has four main factors that interact with each other for all collateral

types: the interest-rate, age, burnout,andseasonality factors. It also has a

points effectfor some collateral types, which acts by modifying the first three

of these four factors.

The interest-rate factor models the effect of current coupon rates as they

affect refinancing as well as a base prepayment speed due to turnover. The

age (or seasoning) factor models the observed tendency of prepayment

speeds to initially rise and then level off regardless of the level of the current

coupon (though how fast they age may depend on coupon level). The

burnout factor adjusts for the fact that as fast prepayers leave a pool during

times when it is a premium, it will prepay more slowly during subsequent

such periods. The seasonality effect takes into account the observed rise in

summer and fall in winter of prepayments. Finally, the points effect adjusts

the interest rate, aging and burnout effect for loans which were originated at

a high WAC relative to rates prevalent at origination.

The number of months a given coupon takes to reach its peak depends on the

collateral type and whether it is a discount, current coupon or premium.

Whether something is a premium or discount at any month is measured by

taking the ratio of the WAC of the pool to the weighted average of a number

of lagged current coupon rates.

Interest Rate Factor

Figure 4 displays the relationship between the ratio (of WAC to weighted

lagged current coupon rates) and the strength of the interest rate effect. When

the ratio is very low, we have a deep discount, which has some base

prepayment speed due to turnover.

Model Features in Depth


As the ratio increases past one, we have a premium and the refinancing

incentive produces prepayments at an increasing rate. At some fairly high ratio

the interest rate effect caps at its maximum value. The exact shape of this curve

is determined from the historical data for each collateral type using appropriate

statistical techniques.

Aging

Figure 5 compares two hypothetical aging ramps, one for a GNMA pool with net

coupon 6.0 and another for a GNMA with net coupon 8.5. The premium takes

about 10 months for its age effect to reach peak level while the discount takes

about 50.


0

10

20

30

40

50

60

0.7 1.2 1.7

Ratio(WAC/Lagged CC)

CP

R(%

)

Figure 4

Interest Rate

Factor

0

20

40

60

80

100

1 7 13 19 25 31 37 43 49

Age(Months)

Per

cent

Premium Discount

Figure 5

Aging Factor

Burnout

The burnout factor adjusts prepayment speeds to take into account the fact that

after each wave of refinancing, the pool becomes slightly less sensitive to

subsequent interest rate drops. This is because every pool consists of borrowers

with different degrees of interest rate sensitivity; when rates drop, the more rate-

sensitive ones leave, leaving the pool less rate-sensitive for future rate drops.

The burnout effect can be conceptualized as in Figure 6 via the aging effect.

Having a burnout effect decreases the strength of the aging effect some time after

its peak occurs. Implementing the burnout effect via the aging curve as in the

figure would lead to some decrease in interest rate sensitivity after age 36.

However, the amount of burnout in a pool depends on the entire history of its

prepayment behavior, and thus on the entire past path of current coupons.

Implementing burnout via the aging curve would not capture the path-dependent

nature of burnout. Rather than implement it through the aging curve, the Andrew

Davidson & Co., Inc. fixed-rate prepayment model computes burnout using a

path-dependent, non-linear method which is more flexible than the approach

displayed in Figure 6. In addition, the burnout factor has a reversion component,

which takes into account the tendency of burnout todecrease due to additional

factors.

9 QU A N T I TAT I V E PE R S P E C T I V E S MA R CH 1999

0

20

40

60

80

100

120

1 6 11 16 21 26 31 36 41 46

Age(Months)

Per

cent

Without Burnout With Burnout

Figure 6

Burnout Factor

Seasonality

There are changes in prepayment speeds that depend on the time of year but not on

the interest rate, aging or burnout effects. These are modeled using procedure X11

in SAS by only using discounts to remove the interest rate effect. Some sample

seasonality effects are graphed in Figure 7. In general, speeds tend to rise during

spring months, peak over the summer, drop over the fall and are at their lowest over

the winter.

As with all the model factors, each collateral type has its own characteristics,

obtained from the historical data.

The Points Effect

The points effect was added to the Andrew Davidson & Co., Inc. fixed-rate

prepayment model in a previous release to account for observed differences in

the behavior of high relative WAC pools. The relative WAC is the difference

between the WAC of the pool and the prevailing mortgage rate for that type of

collateral at the time of and just prior to the time that the pool was originated.

A high relative WAC would indicate that the mortgage holders in the pool

obtained their mortgages at higher than market rates. This serves as a proxy for

a low point loan, credit constraints or poor documentation availability on the

part of the mortgage holder.

Figure 7

Seasonality

10

0.4

0.6

0.8

1

1.2

1.4

1.6

Jan

Feb

Mar

chApr

ilMay

June Ju

lyAug

Sept

OctNov Dec

Month

GN1-30 Conventional 20 7-year Balloon


The shapes of the interest rate and aging curves, as well as the burnout effect,

are modified for high relative WAC pools. For some collateral types the points

effect does not exist in the data and hence was not modeled; for a few the

amount of data available did not allow accurate estimation of this affect due to

the paucity of both high relative WAC data and clean data. Figure 8 illustrates

how the points effect reshapes the interest rate factor. Low point loans

generally have higher base speeds but less of a refinancing component until

they become seasoned.

In general, the updated model exhibited faster aging ramps than the old one,

particularly for premiums. Figure 9 compares the two versions for a

conventional 30 year loan with a WAC of 8. The graph compares projected

monthly CPR over the life of the loans assuming that the current coupon stays

at 6.3.

The updated model reaches its peak prepayment speed considerably earlier

than the previous version. It also remains at a higher speed for most of the life

of the loan. The periodic wave-like changes come from the seasonality factor.

Finally, the steady decline due to burnout is curved in a way that modeling

burnout by modifying the aging ramp (as in Figure 6) does not allow.

Another significant change in the updated model is in the lag structure. Figure 10

shows the response to an immediate 100 bp drop in rates a year from now for a

GNMA 30 year loan with a WAC of 7 from the old and updated version of the

model.

Model Factors: Old vs. New


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

Ratio (WAC/CCRate)

SM

M

No point High Point

Figure 8

Impact of Points

The models predict somewhat similar behavior (with the new model predicting

slightly higher speeds) until the rate drop occurs in month 12. At that point, the

subsequent rise in prepayments due to increased refinancing occurs earlier in

version 4.1 due to the change in lag structure.

During our initial qualitative analysis of the aggregate data, we found an aging

pattern that differs from that of non-balloons. This is shown in Figure 11 for 7-year

balloons.

Figure 10

GNMA 30

Lagged Response


0

10

20

30

40

50

1 6 11 16

Month

CP

R(%

)

4

4.5

5

5.5

6

6.5

7

Cur

rent

Cou

pon

%

version 4.0 version4.1 coupon

Change in Balloon Aging

0

10

20

30

40

50

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353

MonthC

PR

(%)

4

4.5

5

5.5

6

6.5

7

Cur

rent

Cou

pon%

version 4.0 version 4.1 Coupon

Figure 9

A Conventional 30

with WAC=8

Somewhere between month 50 and 60 prepayments begin to pick up. This

behavior was not taken into account in version 4.0 of the model because at the

time that model was estimated, there were not enough balloons of age greater

than 40 to exhibit a trend. Closer statistical analysis revealed that this behavior

is almost independent of the ratio of WAC to current coupon rate. This indicates

that the decision to refinance as the age approaches the balloon date becomes

less and less dependent on the interest rate and deterministically dependent on

the age alone. To take into account this observed behavior, the aging component

of our functional form was modified. The basic idea is illustrated in Figure 12

for a hypothetical example.


0

20

40

60

80

100

0 10 20 30 40 50 60 70 80

Age(Months)

CP

R(%

)

Figure 11

Seven-Year

Balloons

0

5

10

15

20

25

30

35

40

45

50

1 6 11 16 21

Age(Months)

CP

R(%

)

Non-Balloons Balloons

Figure 12

Balloon Aging

Both balloons and non-balloons have an initial rise in prepayments due to age (the

slope of which depends on the WAC ratio). While non-balloons flatten out thereafter,

there is a later point at which balloons rise once again as their age approaches the

balloon date. The exact point at which this change occurs depend on the type of

balloon, as does the slope with which it rises.

Table 1 shows some fit statistics for each model subtype: R-squared with and

without the points effect, the regression sum of squares (SSR), the sum of squares

from the error (SSE) and the total degrees of freedom (number of observations).

One interpretation of the R-squared numbers is that they measure the percent of

variation in the observed data that is explained by the independent variables using

the Andrew Davidson & Co., Inc. prepayment model with the optimized

parameters. Thus, with an R-squared of 80%, the fitted model cannot account for

20% of the observed variation in actual prepayments. One reason for this residual

component of variation is usually due to the presence of loan-level variables (such

as LTV, geographical location, etc.) which were not taken into account.

The R-squared numbers range from 77 to 94 percent without the points

effect and the points effect typically adds 2 percent. The points effect was

not estimated for GNMA II 15s and Conventional 20 year loans because

there was not enough data. For the others it was not found to improve the fit.


Model Fit: Actual vs. Fitted

Collateral Type without points with points SSR SSE DF

GNMA I 30 year 89 % 90 % 3.9185 0.5023 11530

GNMA II 30 year 85 % 86 % 3.1837 0.5661 8642

Conventional 30 year 89 % 91 % 3.999 0.4780 6971

GNMA I 15 year 87 % 89 % 3.0940 0.4774 12654

GNMA II 15 year 77 % -- 0.6867 0.2105 2707

Conventional 15 year 88 % 90 % 2.5400 0.3360 9050

Conventional 20 year 84 % -- 0.2285 0.0427 1999

Conventional 10 year 86 % -- 2.5000 0.3860 9081

5 Year Balloon 77 % -- 8.3632 2.4460 4132

7 Year Balloon 94 % -- 4.1459 0.2822 5612

Table 1

Fit Statistics by

Sub-Model

Figures 13-18 display some actual vs predicted graphs for version 4.1.


Figure 13

Sept 93

Origination FNCL

30-Year with

Gross WAC=7.5

and

Net Coupon=7 0

10

20

30

40

50

60

70

Sep-93 Sep-94 Sep-95 Sep-96 Sep-97 Sep-98

DateC

PR

(%)

Actual Fitted

0

10

20

30

40

50

60

70

Mar-96 Mar-97 Mar-98

Date

CP

R(%

)

Actual Fitted

Figure 14

May 96

Origination FNCL

30-Year with

Gross WAC=9.1

and

Net Coupon=8.5 0

10

20

30

40

50

60

70

May-95 May-96 May-97 May-98

Date

CPR

(%)

Actual Fitted

Figure 15

Mar 96

Origination GNMA

30-Year

with

Gross WAC=7.5

and

Net Coupon=7

In general, the model captures the average behavior of these cohorts quite well.

16QU A N T I TAT I V E PE R S P E C T I V E S MA R C H 1999

0

10

20

30

40

50

60

70

Sep-92 Sep-93 Sep-94 Sep-95 Sep-96 Sep-97 Sep-98

Date

CP

R(%

)Actual Fitted

Figure 16

Sept 92

Origination

7-Year FRDG

Balloon with

Gross WAC=7.6

and

Net Coupon=7

0

10

20

30

40

50

60

70

Mar-97 Mar-98

Date

CP

R(%

)

Actual Fitted

Figure 17

Mar 97

Origination

15-Year GNMA

with Gross WAC=7

and

Net Coupon=6.5

0

10

20

30

40

50

60

70

Sep-95 Sep-96 Sep-97 Sep-98

Date

CP

R(%

)

Actual Fitted

Figure 18

Sept 95

Origination

15-Year FNMA

with

Gross WAC=7.5

and

Net Coupon=7

However, it is clear that the model does not precisely predict all the monthly

variation in prepayments. Some of these differences are a result of sampling error,

as prepayments vary from month to month. Other errors are a result of the model’s

inability to capture certain effects. Of particular note is that not all of the increase

in prepayment speeds in 1998 is reflected by the model. We believe some of the

prepayments during this period reflected unique events which are not likely to

reoccur in a predictable manner. Therefore, altering the model to capture those

spikes would reduce the predictive power of the model.

In particular, we believe that the fast prepayment speeds during early 1998, were

the result of several factors. Introduction of streamlined refinancing programs by

Fannie Mae and Freddie Mac may have contributed to increased prepayment

speeds. Very low intra-month interest rates in January may have also led to faster

prepayments during the period. While these particular factors may not lead to

prepayment spikes in the future, this date clearly indicates the potential volatility

of prepayments.

All parameter estimation was done using the procedure NLIN (non-linear) in

SAS. Since the model is non-linear in both the factors and parameters, as well

as path-dependent, standard regression methodology is insufficient for model

estimation. Grids were formed in the parameter space around suitable values to

obtain initial fits which were then optimized by the NLIN procedure using an

iterative empirical derivative based search technique. Some parameters have to

be constrained in order that factor effects and the final output remain in

reasonable ranges. This methodology leads to finding values of the parameters

which are both meaningful and which minimize the sum of square errors

between predicted and observed values.

The seasonality weights were estimated by eliminating premiums and hence

the interest rate effect. This was done using PROC X11 in SAS (which was

developed by the U.S. Census Bureau). Lags and the rest of the parameters

were estimated using NLIN. In this update, separate models were estimated

with and without the points effect even for those collateral types where the

points effect exists in the data and can be estimated. This is because some

systems on which the model is implemented do not have the ability to

implement the points effect. On such systems, it is better to have the model

parameters optimized without the points effect.

Statistical Issues and Techniques


The Lag Structure

Table 2 shows lag weights for some collateral types:

The lag weights determine what percent of the current coupon used in the ratio

(which determines premium or discount status) come from lags of one, two and

three months respectively. This ratio affects how the aging, interest rate and

burnout factors behave; the lags determine how quickly rate changes lead to

prepayment changes.

Tuning

The tuning feature allows users to change components of our prepayment

model to examine how altering assumptions affects prepayment behavior.

Version 3.0 allowed access to the slide, steepness, burnoutand scale

parameters. In version 4.1 we also allow tuning of the lag. In general,

increasing the tuning parameters increase prepayment speeds.

The slide tuning parameter allows the user to shift the S-shaped interest rate

curve to the right or left. Steepallows the curve to be made steeper or flatter.

Burn is used to make burnout occur faster or slower. Scalecan be used to

multiply the output SMMs by a constant factor, magnifying or damping

speeds. Finally,lag can be used to increase or decrease the amount of time

taken for prepayments to respond to interest rate changes.

OAS Analysis

The Andrew Davidson & Co., Inc. prepayment model is used to obtain option-

adjusted value and risk measures for MBS and CMOs using our proprietary OAS

tool. The two main components of the OAS methodology for MBS are the interest

rate generation process and the prepayment model. Changes to either will affect all

option-adjusted measures, including average WAL, OAS, option cost, duration and

convexity.

18

Additional Features

Collateral One Month Two Months Three Months

GNMA 1 - 30 15 % 40 % 45 %

GNMA 1 - 15 30 % 45 % 25 %

Conventional 30 15 % 30 % 55 %

5-year Balloon 0 % 85 % 15 %

Table 2

Lag Weights by

Month


To produce these measures, the required inputs are the yield curve, volatility

and mean reversion assumptions along with the type and age of collateral.

Based on these inputs, 512 arbitrage free short rates and 2 and 10 year rates are

forecast monthly out to the maturity of the collateral using our Monte Carlo

interest rate process. The mortgage current coupon is computed as a weighted

average plus spread of the 2 and 10 year rates. Prepayment vectors (of SMMs)

are forecast for each current coupon path and these give us 512 cash flow

paths. The short rates are used to discount cashflows along each rate path.

Table 3 displays WAL, OAS, duration and convexity for 4 securities based on

their prices on February 22, 1999 and the yield curve and volatility

assumptions.

Because running an OAS analysis with a large number of paths can be

potentially time-consuming, it is important that each component work as

efficiently as possible. Our prepayment model version 4.1 has several

enhancements to improve its performance speed.

19

Security Coupon Age price OAS WAL duration convexity

GNI 30 8 21 104-10 105 3.7 1.4 -1.3

FN15 7 24 102-09 86 3.8 1.8 -1.4

FN7 Balloon 6.5 8 100-16 114 1.9 1.6 -0.6

FN20 6.5 10 102-31 103 2.8 1.9 -0.6

Table 3

Some Value And Risk

Measures Calculated

Using the Prepayment

Model

Rate Process On-the-Run Treasuries

Volatility Reversion 3 mo 6 mo 1 yr 2 yr 5yr 10 yr 30 yr

15 % 3% 4.56 4.57 4.65 4.91 4.94 5.02 5.35


The prepayment model is available under the product name VectorsTM for

analyzing mortgage-backed securities through a number of vendor systems

including: Algorithmics, ALLTEL/CPI/Busch Analytics, Bloomberg, Carolina

Capital, Derivative Solutions, Intex, MIAC, Polypaths, QRM, Reuters,

RF/ Spectrum, SS&C Technologies and Wall Street Analytics. The model is

also available in a stand-alone format and is callable from ExcelTM and

MATLAB.

For Windows 95,98 or NT systems, it is available as a dynamically linked

library (dll) which plugs into these vendor systems. It is also available as a

shared object for HP/UX, OS/2 and Sun/Solaris environments. Finally, a

standard subroutines version is available which can be called from users’ in-

house analytical systems on any of these platforms.

The Andrew Davidson & Co. fixed-rate MBS prepayment model is a

robust and parsimonious pool-level model. It has been updated with

new data taking into account recent behavior and structural changes.

There have been significant changes to the lags, aging structure, and

functional form for balloons. In addition, there have been programming

changes, which will speed up OAS analysis. The model has been tested

across a range of scenarios and extreme behavior analyses and works

with a number of vendor systems under different platforms and

operating systems. We continually monitor its behavior and market

dynamics to keep it up to date. Our client input is an integral

component of this process and we welcome your comments.

ANDREW DAVIDSON & CO., INC.

588 BROADWAY, SUITE 610NEW YORK, NY 10012


Implementations

Conclusion

212•274•9075

FAX 212•274•0545

EMAIL [email protected]

QUANTITATIVE PERSPECTIVES is available via Bloomberg and the internet.BLOOMBERG ADQP<GO>

WEBSITE www.ad-co.com

We welcome your comments and suggestions.

Contents set forth from sources deemed reliable, but Andrew Davidson & Co., Inc. does notguarantee its accuracy. Our conclusions are general in nature and are not intended for use asspecific trade recommendations.

Copyright 1999Andrew Davidson & Co., Inc.

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