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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