Mortgage Backed Securities.

Abdulla Alothman 1

QBS: Asset Pricing Series Abdulla AlOthman

QBS THE SUBPRIME MORTGAGE CRISIS

A STEP BY STEP EXPLANATION

Abdulla Alothman 2

Abstract

No deep understanding of the subprime crisis is possible without an understanding of how Mortgage

Backed Securities (MBS) are priced and accounted for. This paper aims at developing just such an

understanding.

Part1:In order to keep technicalities to a minimum and to obviate the need for introducing heavy

mathematical machinery, a simple - almost cartoon like - setting is used. Nevertheless, it turns out,

that such a setting, is sufficient to provide the reader with the in depth understanding of the key

issues.

Part2: Shows in detail how a simple three period mortgage backed security, with no prepayment

options, can be valued.

Part3: Extends the analysis of part 2 to include - most mortgages at least in the US fall into this

category - prepayment options. It is shown that such an instrument is equivalent to a portfolio

consisting of a long position in a simple MBS and a short position in a call option. It is then shown

how such an option may be valued , what is meant by model risk and finally, how the option's

premium can be incorporated - industry standard practice - into the asset's IRR. An algorithm - in

pseudo code- for doing this is also provided.

Part4: Analyzes the performance of two traders, one of whom invests in an MBS with no prepayment

clause, the other in an MBS with a prepayment option. It is shown how the industry practice of

incorporating the option premiums into the IRR and standard accounting practices, which

subsequently, allow for the accounting of this inflated IRR as income -rather than treat it as the

insurance premium which it is - creates strong incentives for a traders to adopt cavalier strategies at

shareholder expense.

Part5: Extends the analysis of Part3 to allow for the possibility of the borrower defaulting. It is shown

that such a security is equivalent to a portfolio consisting of a long position in a default free MBS

(with or without a prepayment option) and a short position in a put option, the value of which,

depending on the likelihood - represented by an exogenous parameter - of default. The greater this

likelihood, the more valuable the option will be. A model for the underlying physical asset is presented

Using this model, we then show how the default option may be valued and, finally, as in part 3

above, we show how to incorporate this into the asset's IRR using a numeric algorithm.

Part6: Extends the analysis of part4 to the case where one of the assets is no longer default free. The

conclusions are similar to those in part 4, the effects however, are magnified by the inclusion of the

default option into the asset's IRR.

Abdulla Alothman 3

1.The Data

1. We consider a three time period mortgage: t = 0, 1 2

2. The lender's cost of funding is 5.05% per annum

3. Interest for t=0 and t=1 rates are listed below

Table 1.1

Interest Rates Today t=0 Time t=1

Scenario 1

Probability 0.5

Time t=1

Scenario 2

(Probability 0.5)

1 Period 5% 4% 7%

2 Periods 5.5% 5% 8%

Abdulla Alothman 4

2.The Pricing of a Simple Default Free Mortgage Backed Security with

No Prepayment Option (MBS1)

Consider a client wishing to borrow 100 for 2 periods in order to purchase a house,

i.e. a two period mortgage. Assuming the above interest rates hold. The rate on this

mortgage and associated amortization table are calculated as follows:

Step1:

We first need to calculate the payment amount. The client will make two payments -

each including principle + interest - at times t = 1 and t = 2 respectively.

So, we need to solve for x (prepayment amount) in:

2100

(1.05) (1.055)

x x (1.1)

This says that the sum of the payments adjusted for the time value of money (first

payment at the year 1 interest rate, second payment at the year 2 interest rate )

must equal the money advanced.

Solving gives:

10054.0297

1.850833x (1.2)

This represents the amount the borrower needs to repay each period.

Step2: Calculate the IRR

We solve for r in:

2

54.0297 54.0297100

1 / 100 (1 / 100)r r

(1.3)

To get:

r = 5.3269

Abdulla Alothman 5

Step3: Deriving the amortization table

Table 2.1: Interest Income - Cost of Funding

Table 2.1, is what the client sees.

Starting

Balance

Payment Interest

Income

Cost of

Funding

Net

Income*

Ending

Balance

0 -100

1 -100 54.03 5.33 5.05 0.28 -51.30

2 -51.30 54.03 2.73 2.59 0.14 0

Abdulla Alothman 6

3.The Pricing of a Default Free Mortgage with an Option to Prepay

(MBS2)

Step 1: Analysis

At time t = 1, the borrower can choose to refinance. Whether or not he chooses to

do so will depend on which of the above scenario's prevails:

At time 1, after making his payment, the mortgage holder has a one period note

outstanding with a face value of 51.374 (see table 2.1 above) and a market value of:

Scenario1

Rates Fall

54.0297/1.04 = 51.9516

Scenario2

Rates Rise

54.0297/1.07= 50.4950

100

1/2

1/2

Figure 3.2: Time 1 Scenarios

t=0 5% 5.5%

Scenario1 4% 5%

Scenario2 7% 8%

1/2

1/2

Figure 3.1: Interest Rate Scenarios

Abdulla Alothman 7

The gain from refinancing is:

Clearly a rational borrower will only refinance if the gain is positive, so his actual

time 1 refinancing payoffs will look like:

Figure 3.4, is the payoff of a one period call option on a bond with one period left to

maturity and a strike price of 51.297.

Scenario1

Rates Fall

(51.952 -51.297)= 0.655

Scenario2

Rates Rise

(50.495 - 51.297) = -0.802

V

1/2

1/2

Figure 1.3: Gain from refinancing

Scenario2

Rates Rise

0

Scenario1

Rates Fall

(51.952 -51.297)= 0.655

V

1/2

1/2

Figure3.4: Gain from rational refinancing

Abdulla Alothman 8

Step 2: Pricing the Option

1. If we look at the payoffs in figure 3.4 above, it seems clear that such an asset

cannot have a value greater than its maximum payoff of 0.655 nor a value less

than its minimum payment of 0. So the time t=0 price will perforce lie in the

range (0,0.655)

2. Noting that the average payoff is 0.3275 - if we assume that investors on average

require to be compensated for bearing risk (i.e. are risk averse) the above range

can further be restricted to (0,0.3275).

3. Where exactly in this range the price should lie, will depend on how risk averse

the market actually is. More risk averse and the value will be closer to 0, less risk

averse and the value will be closer to 0.3275. At this point, there are two ways to

proceed:

Try and estimate the level of the risk adjustment process directly

Build a model for the underlying asset process( most canned software use

one of standard models e.g. Black and Scholes, Hull and White, CIR etc) ,

and calibrate its parameters using market data (in fact, though this is far

from obvious, it turns out that this approach is equivalent to assuming a

specific (parameter dependent) form for the risk adjustment process)

To keep the analysis as simple as possible, I will assume an adjustment for risk of

(0.62148,1.2833). This means that the average investor values a dollar less in an

upstate (when the economy is booming for example) and more in a down state

(during a recession). These risk adjustments can be factored in to the probability

assumptions of (1/2, 1/2) to give the following pricing model :

Abdulla Alothman 9

Using the above model we can value the refinancing option as follows:

1. Multiply each payoff by its risk adjusted probability

2. Add these, to get the risk adjusted expected payoff

3. Divide by the one period interest rate (1.05) to adjust for the time value of

money

The value of the refinancing option is:

(0.32625 0.655 0.67375 0)/1.05 0.2035V (2.1)

Digression - A Note on Model Risk

Suppose we had picked a different risk adjustment factor, One consistent with a

higher degree of average risk aversion. For concreteness suppose we had chosen to

adjust for risk using (0.6, 1.304) instead*. This would have resulted in a model with

scenario probabilities of - see appendix - (0.315,0.685) and an option value of:

(0.315 * 0.815 0.685 * 0)/1.05 0.2445V (2.2)

So, the option value we obtain, depends on which model we decide to use. So long

as the risk adjustment process implied by the model differs (and this will almost

always be the case) from the underlying true process, the price implied by the model

X1=Max(Aup-K,0)=0.655 0.32625

0.67375

V=(0.32625X1+0.67

375X2)/1.05 X2=Max(Adown-K,0)=0

Table 3.5 Option Model

Abdulla Alothman 10

will differ from the true price needed to replicate the option. This is is known as

model risk.

*Choosing a different risk in this case (recall we already specified the future movement of rates, and

the spot rates - and in so doing implicitly pined down the market price of risk -would result in an

arbitrage opportunity). With the above choice - one way to insure we do not introduce arbitrage, is to

change the two year spot rate to 5.516, which would result in the payment being 53.86, the IRR

5.1048, the strike 51.2444 and the option value 0.1633.

End of Digression

Step 3: Incorporating the Option Price into the Asset's IRR

In practice, this option is not paid for at time 0, rather, it is build into the IRR of

the bond. The following algorithm - in pseudo code - shows how to do this:

The Algorithm{

Set Quit = NO

Set V = 0.2035( The option value from (2.1))

Do While (Quit = NO)

{

a)Solve for mortgage payment amount PMT:

2(1.05) (1.055)0.2035 (1.8508333))

0.10995

PMT PMTV

PMT

PMT

b) Calculate the IRR

2 2

54.0297 54.0297 54.0297 0.10995 54.4462 0.10995100

1 / 100 1 / 100(1 / 100) (1 / 100)5.47097

PMT PMT

r rr rr

c) Use the this IRR to calculate the strike price:

54.1396551.3313

(1 5.47097 /100)K

d) Use the option model in (2.1) to value the option for this new value of K

Abdulla Alothman 11

54.13965(0.32625 * ( 51.3313) 0.67373 * 0) / 1.05 0.22559

1.04nextV

e) (Test if we are ready to quit)

if ( 0.0000003nextV V )

nextV V

Else (If we are, then stop)

Quit = YES

Return nextV

}

PrintResults

Iteration Strike Option Value Period Payment

Equivalent

Mortgage

IRR

1 51.29715 0.203355 0.1098725 5.326890

2 51.33130 0.22557 0.1218958 5.470874

3 51.335033 0.2280049 0.123190438 5.48662886

4 51.335434348 0.228266912 0.123331967 5.488325249

5 51.33547873 0.22829555 0.1233474 5.488510698

6 51.33548351 0.228298692 0.1233491 5.488530969

7 51.33544841 0.228299018 0.1233493 5.488533144

8 51.335484 0.228299061 0.1233493 5.4885334

Table 3.1: Option Premium Payment

Comment:

2 1

54.03 0.123 54.03 0.123100

(1 / 100) (1 / 100) (1 / 100)(1 / 100)5.49

PMT PMT

IRR IRR IRRIRRIRR

(2.3)

}End of Algorithm

Abdulla Alothman 12

Step 4: Deriving the amortization table

t Beginning

Balance

Payment

Interest

Income

Cost of

Funding

Income

Net

Income**

Ending

Balance

0

1 -100 54.15 5.49* 5.05 0.44 51.34

2a1 -51.34 54.16 2.82 2.59 0.23 0

2b2 -51.34 53.39 2.05 2.59 -0.54 0

Table 3.2: Amortization

1 No prepayment scenario

2. Mortgage is prepaid monies received are reinvested in the market

Beginning

Balance

Payment Interest

Income

Cost of

Funding

Net

Income*

Ending

Balance

0 -100

1 -100 54.03 5.33 5.05 0.28 -51.30

2 -51.30 54.03 2.73 2.59 0.14 0

Table 2.1 Reproduced for ease of comparison

Abdulla Alothman 13

4. A Story of Two Traders:

Consider two traders, A and B, each with 1000 million of his institutions money to

invest. Assume each is paid 20% of net annual income as an end of year bonus.

Assume further that each can invest in only one of the above securities. Trader A

chooses to invest in the no prepayment MBS (MBS1), Trader B chooses to invest in

the MBS with the prepayment option (MBS2). Based on industry standard

accounting practices, the profit that will accrue to their respective institutions is as

follows:

Table 4.1

Trader A will receive a bonus of 560,000 in year 1, and, assuming he does not get

fired for "poor performance", a 280,000 bonus in the following year. Based on this

performance, the markets view of him -especially if rates remain high, a random

outcome- will probably be that of a "mediocre performer".

Trader B will receive a bonus of 880,000 in year 1 and, assuming rates stay high - i.e.

no refinancing takes place - a bonus of 446,000 in the following year. Moreover, after

year1's results his market reputation will be that of a "star" performer If, after

collecting his year 1 bonus rates drop, he can simply - something only too easy for a

"star" trader to do - "jump ship" Leaving his institution and ultimately the

shareholders to foot a loss of 5,400,000 to be realized at the end of year 2!!

The problem in the above example, is that all of the IRR - this is standard accounting

practice - including the part representing the option premium - is being booked as

interest income!! This tantamount to an insurance company, booking all the

premiums it receives on policies it writes, as profit!! From the perspective of standard

t Trader A

Income

Trader A

Bonus

Trader B

(Scenario 1)

Income

Trader B

Bonus

Trader B

Scenario 2

Income

Trader B

Bonus

0

1 2,800,000 560,000 4,400,000 880,000 8,800,000 880,000

2 1,400,000 280,000 2,230,000 446,000 -5,400,000 0

Abdulla Alothman 14

accounting practices however, trader B, is simply long a risky bond, and the rules

governing the book keeping of such an instrument are clear.

Analysis

Pausing for a moment and comparing (1.3) with (2.3), which for the readers

convenience have been reproduced in modified form below:

2

54.03 54.03100

1 / 100 (1 / 100)r r

(2.4)

2

54.03 0.123 54.03 0.123100

(1 / 100) (1 / 100)r r

(2.5)

We see that the extra yield accruing to trader B is a result of the option premium

payment. The present value of this is - see table 3.1 - 0.2283 per 100 dollars invested

i.e. 2.283 million. A natural question to ask at this point is, what does that premium

really represent? To understand what is really going on, let is consider the following,

time t = 0, portfolio:

1. An investment of 272.533 million in two period zero coupon bonds

(B(0,2))

2. A one year loan of 242.575 million (at 5%)

3. A loan of 2.283 million, the present value of the option premiums -

currently being accounted for as income, to be repaid in 2 instalments

of 1.233 million each.

The value of this portfolio is:

0 2

1272.533 242.575-2.283 0

(1.055)V

Abdulla Alothman 15

It's payoff, excluding the loan repayment amounts, in millions, is:

Now consider a third trader C, who invests 1000 million in MBS2 and in addition

invests in the above portfolio:

Table 4.2

The payoff to Trader C, is almost identical to trader A's. Which shows that the

extra payoff to trader B was not the result of superior performance. But rather, a

t Beginning

Balance

Payment

/Principle

Interest

Income

Cost of

Funding

Income

Loan

Payment

Portfolio Net

Income**

Ending

Balance

0

1 -100 54.15 5.49* -5.05 -0.12 0.32 51.34

2a1 -51.34 54.16 2.82 -2.59 -0.12 0 0.11 0

2b2 -51.34 51.34 2.05 -2.59 -0.12 0.7 0.11 0

Beginning

Balance

Payment Interest

Income

Cost of

Funding

Net

Income*

Ending

Balance

0 -100

1 -100 54.03 5.33 5.05 0.28 -51.30

2 -51.30 54.03 2.73 2.59 0.14 0

X1**=7.347235

0.5

0.5

V=0

X2**=0

*

1*

2

(272.5334379 *1/ 1.04) 242.5753787*(1.05)

(272.53344379 *1/ 1.07) 242.5753787*(1.05)

X

X

Abdulla Alothman 16

direct result of being able to book option premiums - needed to create the necessary

replicating portfolios to protect institutions from market risks associated with

prepayment - as income. To protect against such behaviour, a simple change in

accounting rules, is all that is needed. To insure that such new rules are not

violated, accountants need to be able to recognize cases - such as in the above case -

when they apply. This in general - especially with complex structures - will not be

possible without at least some advanced training in the theory of asset pricing.

Abdulla Alothman 17

5. The Pricing of a Mortgage with Prepayment Option and Risk Of

Default. (MBS3)

Continuing with the framework above. Suppose we allow for the possibility of default

in period 2 on MBS2. That is, the possibility the borrower will not make the final

payment.

Step1: Analysis

The payoff structure is represented below:

What the above shows is that:

MBS3 = MBS2 - Put Option(K,2,)

Here is a proxy for the cost of default ( credit rating, social stigma etc). It is high

(the option less valuable) for prime mortgage holders and lower (the option more

valuable) for subprime borrowers.

54.15*

51.34

100

53.38

du

du

154.16 max(54.16 , 0)

54.16A

A

Borrower Prepays

51.34

54.15*

*First Payment

1.04

dd

dd

154.16 max(54.16 ,0)

54.16A

A

Fig 5.1 MBS3 payoff diagram

Abdulla Alothman 18

Step2: Building the Property Model

Let's assume (this will be our choice of model) that market adjusts for risk and time

value of money on real estate assets according to:

0 1 2

0

1

2

{ , , }

:

{1}

{1.3333,0.571428}

{1.79442,0.7690077,0.91268,0.15541}

where

Given our interest model in part1 above, and a spot property price of 100, this

implies the following real estate pricing model:

Step2: Estimating the Default Cost Proxies

Let us assume, for simplicity, that these are exogenously given:

Prime Borrower 15

Subprime Borrower 5

100

Aud=108

Auu=132

100

Adu=80.5

Add=42

0.7

q=0.7

1-q=0.3

1-q=0.3

1-q=0.1455

q=0.8545

r=7%

r=4%

r=5%

Fig 5.2. Real Estate Asset Model

Abdulla Alothman 19

Step 3: Valuing the Default Option

Step 3: Incorporating the Option Price into the Asset's IRR

a)Solve for option payment amount PMT:

2(1.05) (1.055)0.4724379 (1.850833))

0.25525685

PMT PMTV

PMT

PMT

b) Calculate the IRR

2

54.0297 0.10995 0.255256 * 54.0297 0.10995 0.255256100

1 / 100 (1 / 100)5.805291188

r rr

*The sum of the payment on MBS1+montly Call Option Premium + Monthly

Subprime Put Option Premium

c) The same algorithm as in part 3 above, then yields:

1. K=54.40 (Strike)

2. PMT = 54.40

3. P(5)=0.481739

4. MBS3 IRR = 5.81188

P(15)=0

P(5)=0.4724379 Pud=0

Puu=0

0

Pdu=0

Pdd(15)=0

Pdd(5)=12.16

0.7

q=0.7

1-q=0.3

1-q=0.3

1-q=0.1455

q=0.8545 Pd(5)=1.769

r=4%

r=5%

r=7% Pd=q Pdu+1-q Pdu/1+r

The other nodes are calculated similarly

Fig 5.3 Valuing the Default Option

Abdulla Alothman 20

.Step 4: Deriving the Amortization Table

Table 5.1

1) No exercise

2) Borrower refinances, monies reinvested at lower rate of 4%

3) Borrower defaults

t Beginning

Balance

Payment

Interest

Income

Cost of

Funding

Income

Net

Income**

Ending

Balance

0

1 -100 54.40 5.81* 5.05 0.76 51.41

2a1 -51.41 54.40 2.99 2.69 0.30 0

2b2 -51.41 53.27 2.06 2.60 -0.54 0

2c3 -51.41 42.00 -9.41 2.60 -12.01 0

Beginning

Balance

Payment Interest

Income

Cost of

Funding

Net

Income*

Ending

Balance

0 -100

1 -100 54.03 5.33 5.05 0.28 -51.30

2 -51.30 54.03 2.73 2.59 0.14 0

Abdulla Alothman 21

6. The Story of Two Traders Revisited:

Analysis: The analysis here is -mutatis mutandis - exactly the same as in Part 4. The

only differences being:

1. Trader B has even a bigger incentive to invest in the high yielding

asset

2. The Shareholders are left with a larger bill to foot!! Their time 2 payoff

distribution is:

Payoff Probability

2,400.000 25.63%

-5,400,000 70%

-120,100,000 4.37%

t Trader

A

Income

Trader

A

Bonus

TraderB

Income

Case1

Trader

Bonus

TraderB

Income

Case2

Trader

B

Bonus

TraderB

Income

Case3

TraderB

Bonus

0

1 2,800,00

0

560,000 7,600,00

0

1,520,00

0

7,600,000 1,520,0

00

7,600,00

0

1,520,00

0

2

1,400,00

0

280,000 3,000,00

0

600,000 (5,400,000) 0 120,100,

000!!!

Sub

Prime

Crisis