Sequential Tranche Paydown
• In the sequential tranche paydown structure, for example, Class A receives principal paydown and
prepayments before Class B, which in turn does it before Class C, and so on.
• Each tranche thus has a different effective maturity.
• Each tranche may even have a different coupon rate.
• CMOs were the first successful attempt to alter
mortgage cash flows in a security form that attracts a wide range of investors
An Example
• Consider a two-tranche sequential-pay CMO backed by
$1,000,000 of mortgages with a 12% coupon and six months to maturity.
• The cash flow pattern for each tranche with zero
prepayment and zero servicing fee is shown on p. 1004.
• The calculation can be carried out first for the Total columns, which make up the amortization schedule, before the cash flow is allocated.
• Tranche A is retired after four months, and tranche B starts principal paydown at the end of month four.
CMO Cash Flows without Prepayments
Interest Principal Remaining principal
Month A B Total A B Total A B Total
500,000 500,000 1,000,000
1 5,000 5,000 10,000 162,548 0 162,548 337,452 500,000 837,452
2 3,375 5,000 8,375 164,173 0 164,173 173,279 500,000 673,279
3 1,733 5,000 6,733 165,815 0 165,815 7,464 500,000 507,464
4 75 5,000 5,075 7,464 160,009 167,473 0 339,991 339,991
5 0 3,400 3,400 0 169,148 169,148 0 170,843 170,843
6 0 1,708 1,708 0 170,843 170,843 0 0 0
Total 10,183 25,108 35,291 500,000 500,000 1,000,000
The total monthly payment is $172,548. Month-i numbers reflect the ith monthly payment.
Another Example
• When prepayments are present, the calculation is slightly more complex.
• Suppose the single monthly mortality (SMM) per month is 5%.
• This means the prepayment amount is 5% of the remaining principal.
• The remaining principal at month i after prepayment then equals the scheduled remaining principal as
computed by Eq. (4) on p. 42 times (0.95)i.
• This done for all the months, the total interest payment at any month is the remaining principal of the previous month times 1%.
Another Example (continued)
• The prepayment amount equals the remaining principal times 0.05/0.95.
– The division by 0.95 yields the remaining principal before prepayment.
• Page 1008 tabulates the cash flows of the same two-tranche CMO under 5% SMM.
• For instance, the total principal payment at month one,
$204,421, can be verified as follows.
Another Example (concluded)
• The scheduled remaining principal is $837,452 from p. 1004.
• The remaining principal is hence
837452 × 0.95 = 795579, which makes the total principal payment 1000000 − 795579 = 204421.
• As tranche A’s remaining principal is $500,000, all 204,421 dollars go to tranche A.
• Incidentally, the prepayment is 837452 × 5% = 41873.
• Tranche A is retired after three months, and tranche B starts principal paydown at the end of month three.
CMO Cash Flows with Prepayments
Interest Principal Remaining principal
Month A B Total A B Total A B Total
500,000 500,000 1,000,000
1 5,000 5,000 10,000 204,421 0 204,421 295,579 500,000 795,579
2 2,956 5,000 7,956 187,946 0 187,946 107,633 500,000 607,633
3 1,076 5,000 6,076 107,633 64,915 172,548 0 435,085 435,085
4 0 4,351 4,351 0 158,163 158,163 0 276,922 276,922
5 0 2,769 2,769 0 144,730 144,730 0 132,192 132,192
6 0 1,322 1,322 0 132,192 132,192 0 0 0
Total 9,032 23,442 32,474 500,000 500,000 1,000,000
Month-i numbers reflect the ith monthly payment.
Stripped Mortgage-Backed Securities (SMBSs)
• They were created in February 1987 when Fannie Mae issued its Trust 1 stripped MBS.
• The principal and interest are divided between the principal-only (PO) strip and the interest-only (IO) strip.
• In the scenarios on p. 1003 and p. 1005:
– The IO strip receives all the interest payments under the Interest/Total column.
– The PO strip receives all the principal payments under the Principal/Total column.
Stripped Mortgage-Backed Securities (SMBSs) (concluded)
• These new instruments allow investors to better exploit anticipated changes in interest rates.
• The collateral for an SMBS is a pass-through,.
• CMOs and SMBSs are usually called derivative MBSs.
Prepayments
• The prepayment option sets MBSs apart from other fixed-income securities.
• The exercise of options on most securities is expected to be “rational.”
• This kind of “rationality” is weakened when it comes to the homeowner’s decision to prepay.
• Even when the prevailing mortgage rate exceeds the mortgage’s loan rate, some loans are prepaid.
Prepayment Risk
• Prepayment risk is the uncertainty in the amount and timing of the principal prepayments in the pool of
mortgages that collateralize the security.
• This risk can be divided into contraction risk and extension risk.
• Contraction risk is the risk of having to reinvest the
prepayments at a rate lower than the coupon rate when interest rates decline.
• Extension risk is due to the slowdown of prepayments when interest rates climb, making the investor earn the security’s lower coupon rate rather than the market’s
Prepayment Risk (concluded)
• Prepayments can be in whole or in part.
– The former is called liquidation.
– The latter is called curtailment.
• Prepayments, however, need not always result in losses.
• The holder of a pass-through security is exposed to the total prepayment risk associated with the underlying pool of mortgage loans.
• The CMO is designed to alter the distribution of that risk among the investors.
Other Risks
• Investors in mortgages are exposed to at least three other risks.
– Interest rate risk is inherent in any fixed-income security.
– Credit risk is the risk of loss from default.
∗ For privately insured mortgage, the risk is related to the credit rating of the company that insures the mortgage.
– Liquidity risk is the risk of loss if the investment must be sold quickly.
Prepayment: Causes
Prepayments have at least five components.
Home sale (“housing turnover”). The sale of a home generally leads to the prepayment of mortgage because of the full payment of the remaining principal.
Refinancing. Mortgagors can refinance their home
mortgage at a lower mortgage rate. This is the most volatile component of prepayment and constitutes the bulk of it when prepayments are extremely high.
Prepayment: Causes (concluded)
Default. Caused by foreclosure and subsequent liquidation of a mortgage. Relatively minor in most cases.
Curtailment. As the extra payment above the scheduled payment, curtailment applies to the principal and
shortens the maturity of fixed-rate loans. Its contribution to prepayments is minor.
Full payoff (liquidation). There is evidence that many mortgagors pay off their mortgage completely when it is very seasoned and the remaining balance is small. Full payoff can also be due to natural disasters.
Prepayment: Characteristics
• Prepayments usually increase as the mortgage ages — first at an increasing rate and then at a decreasing rate.
• They are higher in the spring and summer and lower in the fall and winter.
• They vary by the geographic locations of the underlying properties.
• They increase when interest rates drop but with a time lag.
Prepayment: Characteristics (continued)
• If prepayments were higher for some time because of high refinancing rates, they tend to slow down.
– Perhaps, homeowners who do not prepay when rates have been low for a prolonged time tend never to prepay.
• Plot on p. 1019 illustrates the typical price/yield curves of the Treasury and pass-through.
0.05 0.1 0.15 0.2 0.25 0.3Interest rate 50
100 150 200
Price
The cusp
Treasury MBS
Price compression occurs as yields fall through a threshold.
The cusp represents that point.
Prepayment: Characteristics (concluded)
• As yields fall and the pass-through’s price moves above a certain price, it flattens and then follows a downward slope.
• This phenomenon is called the price compression of premium-priced MBSs.
• It demonstrates the negative convexity of such securities.
Analysis of Mortgage-Backed Securities
Oh, well, if you cannot measure, measure anyhow.
— Frank H. Knight (1885–1972)
Uniqueness of MBS
• Compared with other fixed-income securities, the MBS is unique in two respects.
• Its cash flow consists of principal and interest (P&I).
• The cash flow may vary because of prepayments in the underlying mortgages.
Time Line
-
Time 0 Time 1 Time 2 Time 3 Time 4 Month 1 Month 2 Month 3 Month 4
• Mortgage payments are paid in arrears.
• A payment for month i occurs at time i, that is, end of month i.
• The end of a month will be identified with the beginning of the coming month.
Cash Flow Analysis
• A traditional mortgage has a fixed term, a fixed interest rate, and a fixed monthly payment.
• Page 1026 illustrates the scheduled P&I for a 30-year, 6% mortgage with an initial balance of $100,000.
• Page 1027 depicts how the remaining principal balance decreases over time.
Scheduled Principal and Interest Payments
50 100 150 200 250 300 350 Month 100
200 300 400 500 600
Principal Interest
Scheduled Remaining Principal Balances
50 100 150 200 250 300 350 Month 20
40 60 80 100
Cash Flow Analysis (continued)
• In the early years, the P&I consists mostly of interest.
• Then it gradually shifts toward principal payment with the passage of time.
• However, the total P&I payment remains the same each month, hence the term level pay.
• Identical characteristics hold for the pool’s P&I
payments in the absence of prepayments and servicing fees.
Cash Flow Analysis (continued)
• From Eq. (4) on p. 42 the remaining principal balance after the kth payment is
C 1 − (1 + r/m)−n+k
r/m . (116)
– C is the scheduled P&I payment of an n-month mortgage making m payments per year.
– r is the annual mortgage rate.
• For mortgages, m = 12.
Cash Flow Analysis (continued)
• The remaining principal balance after k payments can be expressed as a portion of the original principal
balance:
Balk ≡ 1 − (1 + r/m)k − 1 (1 + r/m)n − 1
= (1 + r/m)n − (1 + r/m)k
(1 + r/m)n − 1 . (117)
• This equation can be verified by dividing Eq. (116) (p. 1029) by Bal0.
Cash Flow Analysis (continued)
• The remaining principal balance after k payments is RBk ≡ O × Balk,
where O will denote the original principal balance.
• The term factor denotes the portion of the remaining principal balance to its original principal balance
expressed as a decimal.
• So Balk is the monthly factor when there are no prepayments.
• It is also known as the amortization factor.
Cash Flow Analysis (concluded)
• When the idea of factor is applied to a mortgage pool, it is called the paydown factor on the pool or simply the pool factor.
An Example
• The remaining balance of a 15-year mortgage with a 9%
mortgage rate after 54 months is
O × (1 + (0.09/12))180 − (1 + (0.09/12))54 (1 + (0.09/12))180 − 1
= O × 0.824866.
• In other words, roughly 82.49% of the original loan amount remains after 54 months.
P&I Analysis
• By the amortization principle, the tth interest payment equals
It ≡ RBt−1 × r
m = O × r
m × (1 + r/m)n − (1 + r/m)t−1 (1 + r/m)n − 1 .
• The principal part of the tth monthly payment is Pt ≡ RBt−1 − RBt
= O × (r/m)(1 + r/m)t−1
(1 + r/m)n − 1 . (118)
P&I Analysis (concluded)
• The scheduled P&I payment at month t, or Pt + It, is (RBt−1 − RBt) + RBt−1 × r
m
= O ×
· (r/m)(1 + r/m)n (1 + r/m)n − 1
¸
, (119)
indeed a level pay independent of t.
• The term within the brackets, called the payment factor or annuity factor, is the monthly payment for each
dollar of mortgage.
An Example
• The mortgage on pp. 38ff has a monthly payment of
250000 × (0.08/12) × (1 + (0.08/12))180
(1 + (0.08/12))180 − 1 = 2389.13 by Eq. (119) on p. 1035.
• This number agrees with the number derived earlier.
Pricing Adjustable-Rate Mortgages
• We turn to ARM pricing as an interesting application of derivatives pricing and the analysis above.
• Consider a 3-year ARM with an interest rate that is 1%
above the one-year T-bill rate at the beginning of the year.
• This 1% is called the margin.
• Assume this ARM carries annual, not monthly, payments.
• The T-bill rates follow the binomial process, in boldface, on p. 1038, and the risk-neutral probability is 0.5.
*
j
A
4.000%
5.000%
0.36721
*
j
B
3.526%
4.526%
0.53420 *
j
C
5.289%
6.289%
0.54765
-
1.0 D
2.895%
3.895%
1.03895
-
1.0 E
4.343%
5.343%
1.05343
-
1.0 F
6.514%
7.514%
1.07514
year 1 year 2 year 3
Stacked at each node are the T-bill rate, the mortgage rate, and the payment factor for a mortgage initiated at that node and ending at the
Pricing Adjustable-Rate Mortgages (continued)
• How much is the ARM worth to the issuer?
• Each new coupon rate at the reset date determines the level mortgage payment for the months until the next reset date as if the ARM were a fixed-rate loan with the new coupon rate and a maturity equal to that of the ARM.
• For example, for the interest rate tree on p. 1038, the scenario A → B → E will leave our three-year ARM
with a remaining principal at the end of the second year different from that under the scenario A → C → E.
Pricing Adjustable-Rate Mortgages (continued)
• This path dependency calls for care in algorithmic design to avoid exponential complexity.
• Attach to each node on the binomial tree the annual payment per $1 of principal for a mortgage initiated at that node and ending at the end of year three.
– In other words, the payment factor.
• At node B, for example, the annual payment factor can be calculated by Eq. (119) on p. 1035 with r = 0.04526, m = 1, and n = 2 as
0.04526 × (1.04526)2
= 0.53420.
Pricing Adjustable-Rate Mortgages (continued)
• The payment factors for other nodes on p. 1038 are calculated in the same manner.
• We now apply backward induction to price the ARM (see p. 1042).
• At each node on the tree, the net value of an ARM of value $1 initiated at that node and ending at the end of the third year is calculated.
• For example, the value is zero at terminal nodes since the ARM is immediately repaid.
¼Y
A
0.0189916
¼Y
B
0.0144236
¼Y
C
0.0141396
¾
0 D
0.0097186
¾
0 E
0.0095837
¾
0 F
0.0093884
year 1 year 2 year 3
Pricing Adjustable-Rate Mortgages (continued)
• At node D, the value is 1.03895
1.02895 − 1 = 0.0097186,
which is simply the net present value of the payment 1.03895 next year.
– Note that the issuer makes a loan of $1 at D.
• The values at nodes E and F can be computed similarly.
Pricing Adjustable-Rate Mortgages (continued)
• At node B, we first figure out the remaining principal balance after the payment one year hence as
1 − (0.53420 − 0.04526) = 0.51106,
because $0.04526 of the payment of $0.53426 constitutes the interest.
• The issuer will receive $0.01 above the T-bill rate next year, and the value of the ARM is either $0.0097186 or
$0.0095837 per $1, each with probability 0.5.
Pricing Adjustable-Rate Mortgages (continued)
• The ARM’s value at node B thus equals 0.51106 × (0.0097186 + 0.0095837)/2 + 0.01
1.03526 = 0.0144236.
• The values at nodes C and A can be calculated similarly as
(1 − (0.54765 − 0.06289)) × (0.0095837 + 0.0093884)/2 + 0.01 1.05289
= 0.0141396
(1 − (0.36721 − 0.05)) × (0.0144236 + 0.0141396)/2 + 0.01 1.04
= 0.0189916,
respectively.
Pricing Adjustable-Rate Mortgages (concluded)
• The value of the ARM to the issuer is hence $0.0189916 per $1 of loan amount.
• The above idea of scaling has wide applicability in pricing certain classes of path-dependent securities.
More on ARMs
• ARMs are indexed to publicly available indices such as:
– libor
– The constant maturity Treasury rate (CMT) – The Cost of Funds Index (COFI).
• If the ARM coupon reflects fully and instantaneously current market rates, then the ARM security will be priced close to par and refinancings rarely occur.
• In reality, adjustments are imperfect in many ways.
• At the reset date, a margin is added to the benchmark index to determine the new coupon.
More on ARMs (concluded)
• ARMs also often have periodic rate caps that limit the amount by which the coupon rate may increase or
decrease at the reset date.
• They also have lifetime caps and floors.
• To attract borrowers, mortgage lenders usually offer a below-market initial rate (the “teaser” rate).
• The reset interval, the time period between adjustments in the ARM coupon rate, is often annual, which is not frequent enough.
• These terms are easy to incorporate into the pricing
Expressing Prepayment Speeds
• The cash flow of a mortgage derivative is determined from that of the mortgage pool.
• The single most important factor complicating this endeavor is the unpredictability of prepayments.
• Recall that prepayment represents the principal payment made in excess of the scheduled principal amortization.
• Compare the amortization factor Balt of the pool with the reported factor to determine if prepayments have occurred.
Expressing Prepayment Speeds (concluded)
• The amount by which the reported factor exceeds the amortization factor is the prepayment amount.
Single Monthly Mortality
• A SMM of ω means ω% of the scheduled remaining balance at the end of the month will prepay.
• In other words, the SMM is the percentage of the remaining balance that prepays for the month.
• Suppose the remaining principal balance of an MBS at the beginning of a month is $50,000, the SMM is 0.5%, and the scheduled principal payment is $70.
• Then the prepayment for the month is 0.005 × (50,000 − 70) ≈ 250 dollars.
Single Monthly Mortality (concluded)
• If the same monthly prepayment speed s is maintained since the issuance of the pool, the remaining principal balance at month i will be RBi × (1 − s/100)i.
• It goes without saying that prepayment speeds must lie between 0% and 100%.
An Example
• Take the mortgage on p. 1033.
• Its amortization factor at the 54th month is 0.824866.
• If the actual factor is 0.8, then the SMM for the initial period of 54 months is
100 ×
"
1 −
µ 0.8 0.824866
¶1/54 #
= 0.0566677.
• In other words, roughly 0.057% of the remaining principal is prepaid per month.
Conditional Prepayment Rate
• The conditional prepayment rate (CPR) is the annualized equivalent of a SMM,
CPR = 100 ×
"
1 − µ
1 − SMM 100
¶12 # .
• Conversely,
SMM = 100 ×
"
1 − µ
1 − CPR 100
¶1/12 # .
Conditional Prepayment Rate (concluded)
• For example, the SMM of 0.0566677 on p. 1053 is equivalent to a CPR of
100 ×
"
1 − Ã
1 −
µ0.0566677 100
¶12!#
= 0.677897.
• Roughly 0.68% of the remaining principal is prepaid annually.
• The figures on 1056 plot the principal and interest cash flows under various prepayment speeds.
• Observe that with accelerated prepayments, the principal cash flow is shifted forward in time.
50 100 150 200 250 300 350 Month 200
400 600 800 1000 1200 1400
2%
4%
6%
10%
15%
50 100 150 200 250 300 350 Month 100
200 300 400 500 600 700 800
15%
10%6%
4%
2%
Principal (left) and interest (right) cash flows at various CPRs. The 6% mortgage has 30 years to maturity and an original loan amount of $100,000.
PSA
• In 1985 the Public Securities Association (PSA) standardized a prepayment model.
• The PSA standard is expressed as a monthly series of CPRs.
– It reflects the increase in CPR that occurs as the pool seasons.
• The PSA standard postulates the following prepayment speeds: The CPR is 0.2% for the first month, increases thereafter by 0.2% per month until it reaches 6% per year for the 30th month, and then stays at 6% for the remaining years.
PSA (continued)
• At the time the PSA proposed its standard, a seasoned 30-year GNMA’s typical prepayment speed was
approximately 6% CPR.
• The PSA benchmark is also referred to as 100 PSA.
• Other speeds are expressed as some percentage of PSA.
– 50 PSA means one-half the PSA CPRs.
– 150 PSA means one-and-a-half the PSA CPRs.
0 50 100 150 200 250 300 350 Mortgage age (month)
0 2 4 6 8 10
CPR (%)
100 PSA 150 PSA
50 PSA
Figure 1:
PSA (concluded)
• Mathematically,
CPR =
6% × PSA100 if the pool age exceeds 30 months
0.2% × m × PSA100 if the pool age m ≤ 30 months
• Conversely,
PSA =
100 × CPR6 if the pool age exceeds 30 months
100 × 0.2×mCPR if the pool age m ≤ 30 months
Cash Flows at 50 and 100 PSAs
50 100 150 200 250 300 350 Month 100
200 300 400 500
Principal
Interest
50 100 150 200 250 300 350 Month 100
200 300 400 500
Principal
Interest
The 6% mortgage has 30 years to maturity and an original loan amount of $100,000.
