Since July 2007, we have witnessed a growing number of mortgages

going into default and eventually being foreclosed. Mortgage servic-

ing rights are fees collected by institutions managing mortgages. The

yearly fees correspond to a percentage of the outstanding balance

of individual mortgages. Typically, an institution collects between 25

and 50 basis points per year (.25%-.5% of the outstanding balance

each year). The servicing is a compensation that institutions receive

for services that include the collection of the monthly payments, for

making sure that the monthly payments are paid on time, and, when

necessary, for foreclosing the property when a mortgagor defaults

on the property. Banks have been confronted with major losses

related to mortgage-backed securities and to the heavy reduction in

the volume of mortgage servicing rights.

Ortiz et al. (2008) develop a dynamical hedge ratio for portfolios of

mortgage servicing rights (MSR) and U.S. Treasury securities, such

that it is readjusted for changes in market rates and prepayment

rates. They develop a delta-hedge ratio rebalancing function for

three different portfolios, compare the three dynamic hedge ratios,

and rank them with respect to the gamma hedge ratio.

In this paper we develop a general model to obtain the optimal delta

hedge for a portfolio of two-fixed-income-securities (a1, a2), each

a function of the market interest rate y, such that when the value

of each of the individual securities changes up or down, because

of changes in market rates y, the total value of the portfolio is

unchanged. We develop the delta hedge under the constraint of a

zero-gamma in order to avoid costs related to the rebalancing of

such portfolio.

We first describe in details mortgage servicing rights (MSR), then

we develop the model, and finally discuss how our model can be

applied to MSR.

Mortgage servicing rightsBecause MSRs correspond to a percentage of the mortgages’

outstanding balance their characteristics are the same as those of

interest only (IOs). IO securities have values that are both affected

by interest rates and prepayment rates. It is interesting to analyze

the price/yield relationship of an IO security in order to understand

MSRs.

Figure 1 shows the projected scheduled principal (at the very bot-

tom), the unscheduled principal, the interest, and servicing (top)

to be paid over time for a 5.5%-FNMA pass-through security. The

pass-through security is backed by a pool of mortgages with a cur-

rent balance of U.S.$52 million, a weighted average maturity of

28.8 years, and a weighted average coupon rate of 6.243%. The

cash flows are based on a projected PSA of 50, which corresponds

to an annual prepayment rate of 3% (cpr = 3%).

Because the graph in Figure 2 is compressed, it is important, when

comparing it to Figure 1, to read the numbers on the vertical axis.

Figure 2 shows what happens to the different projected cash flows

for a projected 1000 PSA (60% annual prepayment rate). Clearly,

the total principal being repaid overtime remains the same (same

area under the curve) under the two different prepayment scenari-

os, however, both the interest and servicing are being significantly

reduced for faster prepayment rate. When principal is refinanced,

no more interest is paid by mortgagors and banks cannot collect

servicing fees for mortgages that do not exist anymore.

Prepayment rates are a function of interest rates. When interest

rates decrease, prepayment rate increases and vice versa. Figure

3 graphs the projected cash flows of an IO security over time, for

different prepayment scenarios.

Figure 3 is taken from Bloomberg and shows the cash flows of a 9%

FHS IO under four different PSA scenarios. CPR stands for constant

prepayment rate. A CPR of 5% indicates that mortgage principal

is being prepaid at an annual rate of 5%. The prepayment model

developed by the Public Securities Association (PSA) is also widely

used for a basis of quoting prices on mortgage-backed securities.

A 100 PSA corresponds to a 6% cpr, a 200 PSA corresponds to a

12% cpr, and a 500 PSA corresponds to a 30% cpr. In the Figure,

PSA ranges from 0% to 395%. When PSA rate is high, more of the

outstanding balance is being prepaid, so less IO or MSR is available

as a percentage of outstanding remaining balance. This can be

Delta hedging a two-fixed-income-securities portfolio under gamma and vega constraints: the example of mortgage servicing rights Carlos E. Ortiz, Chairman, Department of Computer Science and Mathematics, Arcadia University

Charles A. Stone, Associate Professor, Department of Economics, Brooklyn College, City

University of New York

Anne Zissu, Chair of Department of Business, CityTech, City University of New York, and Research

Fellow, Department of Financial Engineering, The Polytechnic Institute of New York University

8 – The journal of financial transformation

observed for example under PSA 395, the lowest curve in Figure 3.

On the other hand, for low PSA, the outstanding balance remains

at a greater level, and the derived cash flows stay higher (see top

curve for extreme scenario of 0 PSA).

It is important that the reader understands that the value of an IO,

or of a MSR, is the present value of the cash flows received over

time. Clearly, the lower the cash flows are (due to high prepayment

rates), the lower the value is. On the other hand, the higher the cash

flows are, due to high market interest rates and therefore low pre-

payment rates, the higher the value is. This is true only over a range

of interest rates (prepayment effect). At some point, however, the

discount effect takes over the prepayment effect, and the value of

IOs or MSRs will start to decrease. We show the typical value of an

IO or MSR as a function of interest rates in Figure 4.

The value of an IO security or a MSR increases when interest rates

(y) increase as long as the prepayment effect is greater than the

discount effect. When the discount effect takes over, we observe

the value decreasing for increases in interest rates. Both IOs and

MSRs are very interesting types of securities. Most fixed income

securities decrease in value when yields increase. IOs and MSRs do

the opposite for a wide range of yields.

s-curve prepayment functionIt is important to understand the prepayment behavior in order to

value MSR. Periodically, investment banks submit to Bloomberg

their expected PSA level for corresponding changes in yield.

Figure 5 shows the data submitted by twelve investment banks

on their expected PSA levels for corresponding changes in yields

ranging from -300 bps to +300 bps, for the same 5.5% FNMA

pass-through security we described in the beginning of the paper

with Figures 1 and 2. It is interesting to observe how different

these projections can be. For a decrease of 300 bps, the projected

PSA ranges from 884 to 3125 (a corresponding cpr ranging from

53% to 187.6%).

9

Figure 1 Figure 3

Figure 2 Figure 4

10 – The journal of financial transformation

Figure 6 plots for different changes in yield, the corresponding

expected PSA level for each investment bank that submitted

information to Bloomberg on October 2008 for a specific pool of

securitized mortgages. The changes in yield are represented on the

horizontal axis, with expected PSA levels on the vertical axis. The

curve represents the median for the twelve banks that submitted

the data. One can see that the prepayment curve, as a function of

changes in yield, has an S-shape. The S-shape prepayment function

is common to all banks, but each differs in projecting its magnitude/

steepness.

The S-curve shows a steeper slope around the initial market rate

(at 0% change in yield), with prepayment rate increasing as mar-

ket rates continue to drop, until a burnout effect is reached, and

the curve flattens, meaning that prepayment rate is no longer

increasing as it did in the middle range of market rates decrease.

The S-curve also flattens in the high range of market rates. What

happens is that prepayment rate decreases with increase in market

rates until it reaches a minimum beyond which no further decrease

in prepayment rates is observed. The natural level of prepayment

rate is reached, that is, the prepayment rate that is independent of

market rates, but is a function of mortgagers’ personal events.

We next, express the relationship between prepayment rate (we

use cpr for prepayment rate instead of PSA) and the change in

basis points. The prepayment rate, cpr, is mainly a function of the

difference between the new mortgage rate y in the market and the

contracted mortgage rate r. cpr = a ÷ [1 + eb(y-r)].

We now develop a general model for a two-fixed-income-security

portfolio that is delta and gamma hedged against interest rate

changes.

Model for portfolio optimizationWe have a portfolio of two securities (a1, a2). Each security's value

is a function of the market interest rate y. We want to find the opti-

mal share of each security in the portfolio such that: Σi=k→2 αiai = K

(1), where αi is the weight of security ai. In order for the value K of

the portfolio, at a specific yield y, to be hedged against any move-

ments in interest rate we need to find the optimal weights for each

security in the portfolio, such that when there is a change in market

rates the sum of the change in value of each security times its cor-

responding weight in the portfolio is equal to zero.

We next, develop a general model and apply it to the particular case

of portfolios of mortgage servicing rights (MSRs). Consider two

functions ai(y), i = 1,2 and two coefficients αi, i = 1,2. We want the

following to hold: Σi=1→2 αiai(y) = K (2).

The values of the functions ai(y) are given for every value of y and

we want to find the values of αi that will satisfy equation (2), and

other conditions of ‘stability’ that we will describe later.

notation: For any ƒ(y) let us denote by ƒ(n)(y) the n-th derivative

of ƒ.

When the αi’s are assumed to be constant, the objective is to find

for a fixed y0, hence for any given value of the ai(y0), values of the

constants αi, such that: Σi=1→2 αiai(y0) = K (3).

[Σi=1→2 αiai(y0)]1 = [Σi=1→2 αiai(1)(y0)] = 0 (4), where the second con-

dition represents the constraint that the total sum of the αiai(y0)

will not change for small changes in y from the initial value y0.

More generally, to obtain an optimal portfolio we need to find the

values of the αi for i = 1,2 such that for small changes in y from

the original y0, the value of Σi=1→2 αiai(y) will not change from the

original value of Σi=1→2 αiai(y0) = K.

Figure 5

Figure 6

11

For a fixed y0 there exists a unique solution that optimizes the port-

folio of two fixed income securities if: a1(y0)a2(1)(y0) – a1

(1)(y0)a2(y0)

≠ 0. As expected, this condition depends only on the values of the

functions a1, a2 and their derivatives at point y0. The values for α1

and α2, the optimal weights for the two securities that constitute a

delta-gamma hedged portfolio, are:

α1 = [Ka2(1)(y0)] ÷ [a1(y0)a2

(1)(y0) – a1(1)(y0)a2(y0)]

α2 = [-Ka1(1)(y0)] ÷ [a1(y0)a2

(1)(y0) – a1(1)(y0)a2(y0)]

If: a1(y0)a2(1)(y0) – a1

(1)(y0)a2(y0)] = 0

Then we have no solutions or there exist infinitely many solutions.

Examples of no solutionsIt is not unusual to have a portfolio of fixed-income securities that

increases in value when market rates decline and decreases in

value when market rates rise. We could have, for example, a portfo-

lio composed of a 30-year 8% coupon bond with a 10-year Treasury

note. This is a case when it is impossible to delta hedge the portfolio

under gamma and vega constraints.

The case of mortgage servicing rightsA delta-hedged portfolio could have a combination of bonds and MSR.

We next present the valuation approach developed by Stone and

Zissu (2005) that incorporates the prepayment function (S-curve).

Valuation of Msr

The cash flow of a MSR portfolio at time t is equal to the servicing

rate s times the outstanding pool in the previous period: MSRt =

(s)m0(1 – cpr)t-1Bt-1 (5), where m0 is the number of mortgages in

the initial pool at time zero, B0 is the original balance of individual

mortgage at time zero, r is the mortgage coupon rate, cpr is the

prepayment rate, m0(1-cpr)t is the number of mortgages left in pool

at time (t), Bt is the outstanding balance of individual mortgage at

time (t), and s is the servicing rate.

V(MSR) = (s)m0 [Σ(1 – cpr)t-1Bt-1] ÷ [(1+y)t] (6), with t = 1,…..n (through

the entire paper)

Equation (6) values a MSR portfolio by adding each discounted cash

flow generated by the portfolio to the present, where n is the time

at which the mortgages mature, and y is the yield to maturity.

After replacing the prepayment function in equation (6) we obtain

the MSR function as:

V(MSR) = (s)m0 [Σ(1 – (a/(1+expb(y-r)))t-1Bt-1] ÷ [(1+y)t] (6a)

Valuation of a bond

The valuation of a bond with yearly coupon and face value received

at maturity is represented in equation (4): V(B) = c Σ1/(1+y)t + Face/

(1+y)t (7), where V(B) is the value of a bond, c is the coupon, Face

is the face value, n is the time at which the bond matures, and y is

the yield to maturity.

If we now relabel equation (6a) and equation (7) as a1 and a2 respec-

tively, and replace them in the equations we derived previously for

the optimal α1 and α2 respectively, we obtain a portfolio of bonds

and of MSR that is delta- and gamma-hedged against small changes

in interest rates and corresponding changes in prepayment rates.

conclusionWith an estimated $10 trillion in outstanding mortgages, MSR gen-

erate a significant source of income for banks. This is a significant

market with risks that need to be addressed. If not managed prop-

erly, banks will have important losses to report. Prepayment risk

and interest rate risks need to be carefully evaluated when creating

a portfolio of fixed income securities. We have developed a gen-

eral portfolio of two-fixed-income securities, each with the optimal

weight, in order for the portfolio to be delta- and gamma-hedged

against small changes in interest rates. We have shown how this

model can be applied to portfolios’ MSR.

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