Introduction

• Mortgages are used to finance residential and commercial property. Mortgage-backed securities (MBSs) are investments created from the cash flows provided by portfolios of mortgages. The United States mortgage market is a large market that is very important to fixed-income investors.
• Residential mortgages in the United States typically last 15 or 30 years. Their interest rates can be fixed or variable.
• Variable-rate mortgages are termed adjustable-rate mortgages (ARMs). In an ARM, the interest rate is typically fixed for several years and is then tied to an interest rate index. Among the most common indices are the one-year Treasury rate (which is referred to as the constant maturity Treasury rate), a cost of funds index (which is the average interest expense incurred by financial institutions in a region), and Libor. ARMs are less risky than fixed-rate mortgages for lenders and riskier than fixed-rate mortgages for borrowers. For this reason, ARMs typically have lower initial interest rates than comparable fixed-rate mortgages.
• Fixed-rate mortgages in the United States have an American-style option for borrowers to pay off their outstanding mortgage balances. This is referred to as the borrower’s prepayment option. Sometimes a mortgage is prepaid because the mortgaged property is being sold. Often, it will be prepaid because interest rates have declined and thus the property can be refinanced at a lower interest rate. There is usually no penalty for exercising the prepayment option, and it can be quite valuable to the borrower. On the other hand, the prepayment option can be quite costly to those who invest in MBSs. This is because the prepayment amounts must be reinvested even though interest rates tend to be low when prepayments occur. Analysts must take the prepayment option into consideration when valuing mortgage portfolios.
• This chapter discusses how mortgage payments can be calculated and then explains how pools of mortgages are formed to create tradeable investment vehicles. It also outlines how prepayments can be modeled to calculate the value of an investment in a mortgage pool.

Calculating Monthly Payments

• The compounding frequency with which a mortgage interest rate is expressed does not always match the frequency of payments.

To calculate the monthly interest payments on a fixed-rate mortgage, it is necessary to first convert the quoted rate to a rate with monthly compounding. For example, if a mortgage rate in Canada is quoted as 4% with semi-annual compounding, the rate with monthly compounding is

In the United States, mortgage rates are quoted with monthly compounding and thus such a calculation is not necessary.

• Monthly payment can be calculated using the simple logic of discounting. Consider a 30-year mortgage where the fixed rate is 6% with monthly compounding. The monthly payments (constant each month for 30 × 12 = 360 months) must totally amortize the mortgage. This means that the monthly payments must provide the lender with a return of principal as well as interest at 0.5% (= 6%/12) per month on the outstanding principal.

Let the USD payment made at the end of each month be X. Since the present value of the summed payments (outflows) must be equal to the amount borrowed (PV of inflows).

Payments of USD 1,498.88 per month therefore fully amortize (i.e., repay) borrowings of USD 250,000 over 30 years. In general, the relationship between the amount borrowed A, the interest rate R (compounded monthly), and the monthly payment X is

where T years is the life of the mortgage.

Calculating Monthly Payments – Amortization Tables

• An amortization table shows the monthly principal and interest payments on a mortgage (assuming no prepayment of principal). At the beginning of the mortgage, most of the monthly payment is interest. Toward the end of the mortgage, most of the monthly payment is principal.

Example

Consider the same 30-year mortgage example where the fixed rate is 6% with monthly compounding. The monthly payments (constant each month for 30 × 12 = 360 months) must totally amortize the mortgage. This means that the monthly payments must provide the lender with a return of principal as well as interest at 0.5% (= 6%/12) per month on the outstanding principal.

Using the TVM buttons in the Financial calculator, the monthly payment comes out to be the same USD 1,498.88.

Month	End of Month Interest Payment	End of Month Principal Payment	Balance at end of month
0	–	–	250,000.00
1	1,250.00	248.88	249,751.12
2	1,248.76	250.12	249,501.00
3	1,247.51	251.37	249,249.63
4	1,245.25	252.63	248,997.00
……	……	……	……
356	36.92	1,461.96	5,921.30
357	29.61	1,469.27	4,452.03
358	22.96	1,476.62	2,975.42
359	14.88	1,484.00	1,491.42
360	7.46	1,491.42	0.00

This figure shows the way in which the principal declines with maturity for the previous table.

Calculating Monthly Payments – Outstanding Principal

• As an alternative to using an amortization table, analysts can calculate the outstanding principal by discounting the remaining cash flows. For example, when there are ten years (i.e., 120 months) remaining in the life of the mortgage, the outstanding USD principal (assuming there have been no prepayments) is

Mortgage Pools

• Mortgage portfolios (known as mortgage pools) can be created for investment purposes. The mortgages in a pool are usually similar in terms of loan type, interest rate, and origination date.
• The weighted-average coupon (WAC) is the weighted-average interest rate on the mortgages in the pool, with the weight assigned to each mortgage being proportional to its outstanding principal. The weighted-average maturity (WAM) is similarly calculated as the weighted-average of the number of months to maturity, with the weight assigned to each mortgage being proportional to its outstanding principal.

For example, suppose a pool consists only of a USD 200,000 mortgage with an interest rate (referred to as a coupon) of 4% and a USD 400,000 mortgage with an interest rate of 5%, and have maturities of 340 and 280 months (respectively).

In general:

where

n is the number of mortgages in the pool,

is the coupon of the ith mortgage,

is the remaining life of the ith mortgage, and

where

is the remaining principal of the ith mortgage.

• In addition to WAC and WAM, mortgage pool statistics of interest are as follows:

a) The average loan balance is the total current outstanding principal of the mortgages in the pool divided by the number of mortgages.

b) The pool’s factor is the total current outstanding pool principal as a percentage of the original pool principal. A pool’s factor declines with time because part of each monthly payment by the borrower is a repayment of principal. It also declines as a result of prepayments.

c) The weighted-average FICO score (FICO is short for Fair Isaac Corporation) is a FICO score that measures the creditworthiness of the borrower. It can range from 300 to 850, with a score above 650 being considered acceptable by many lenders. The weights are given by the previous equation, but with usually set equal to the original principal of the i^th mortgage.

d) The weighted-average loan-to-value ratio (LTV) for a mortgage is the principal amount of the loan divided by the assessed value of the mortgaged property at the time of the loan. The weights are given by the previous equation, but with usually set equal to the original principal of the mortgage.

e) The geographical distribution of the loans describes the location of the properties being financed by the mortgages.

Mortgage Pools – SMM And CPR

• Prepayments are monitored in mortgage pools. In any given month, some mortgages may totally prepay and some may curtail (i.e., partially prepay). The single monthly mortality rate (SMM) is the percentage of the outstanding principal that was prepaid during a given month. This does not include scheduled payments of principal, such as those shown in the earlier table, The constant prepayment rate (also known as the conditional prepayment rate) is the annualized SMM.

Example – If a remaining mortgage balance is $250 million with a next month SMM of 0.55% and the scheduled principal payment for that month is $5 million, the prepayment for the month can be calculated as

• Conditional Prepayment Rate (CPR) is an annualized SMM rate which assumes that SMM is constant throughout the year and some amount of prepayment is bound to happen apart from scheduled principal repayment
  • Best predictor of CPR is past prepayment rates
  • It’s called conditional since it’s conditional on the remaining mortgage balance

The CPR is calculated as:

Mortgage Pools – SMM And CPR And PSA

• PSA prepayment benchmark is expressed as the monthly series of CPRs. PSA benchmark assumes that prepayment is low for a new mortgages, and it speeds up until 30 months, remaining constant thereafter.
• 100 PSA (or 100% PSA) assumes CPR of 0.2% for the first month and thereafter increases by 0.2% per month for the next 30 months
  • If t < 30 months, CPR = 6% × (month/30)
  • If t > 30 months, CPR = 6%
• Different prepayment characteristics are quoted as a percentage of PSA (e.g. 180% PSA has a higher pre-payment rate where CPR is 1.8 times more than the CPR in 100 PSA). The percent PSA increases as the yield decreases, which further increases prepayment risk (low interest rate).

• Compute the prepayment for the 17th and 33rd month on a 250 PSA mortgage loans of $500 m and scheduled repayment of $17.5 m

Mortgage Backed Securities (MBSs)

• In the United States, there are three agencies that buy mortgages from banks and create mortgage polls. The agencies are:
  1. Government National Mortgage Association, referred to as Ginnie Mae (GNMA);
  2. Federal National Mortgage Association, referred to as Fannie Mae (FNMA); and
  3. Federal Home Loan Mortgage Corporation, referred to as Freddie Mac (FHLMC).

GNMA is a government agency, whereas FNMA and FHLMC are private companies known as government-sponsored enterprises. FNMA and FHLMC loans are not explicitly guaranteed by the U.S. government, but most market participants consider that there is an implicit guarantee.

• The three agencies enable banks to make long-term loans to home buyers without keeping the loans on their books. The funds banks receive from selling their mortgages can then be used to make new loans to home buyers. The mortgages bought by these agencies must satisfy certain criteria in areas such as size and credit quality. Once the mortgages are purchased, cash flows from the mortgage pools are used to create mortgage-backed securities that are then sold to investors. The three agencies guarantee their mortgages for a fee so that pool investors have protection against mortgage defaults. However, there is no protection for prepayment risk.

• The securities created by the agencies are known as agency mortgage-backed securities (agency MBSs). Non-agency MBSs are issued by private corporations (typically financial institutions) and are not guaranteed by government-sponsored institutions.
• The simplest agency MBSs are pass-throughs, with all investors in a pool receiving the same return. Investors get their share of the cash flows from the mortgages in the pool minus the agency’s fees for guaranteeing and servicing the mortgages. The pools are structured so that they offer returns in 50-basis point increments (e.g., 3%, 3.5%, and 4%). Although the returns are known as coupons, they are different from the coupons provided by Treasury bonds and corporate bonds. MBS coupons are made at the end of each month (rather than semi-annually) and the payments are a blend of interest and principal on the underlying mortgages.
• Pass-throughs are characterized by their issuers, their coupons, and their maturities. For example, one pass-through security could be an investment in a GNMA 30-year 4% pool. The “30-year” descriptor refers to the original lives of the mortgages in the pool (rather than their current lives). Pass-throughs have prepayment risk.

• Pass-through agency securities trade as specified pools and to-be-announced (TBAs).
• In the specified pools market, buyers and sellers agree to trade a certain amount of a specified pool at a specified price. The characteristics of a given pool affect the price of a trade. For example, pools having high loan-balances are associated with higher prepayments, and trade for lower prices.
• In the TBA market, buyers and sellers agree on:
  • Issuer (e.g., FNMA);
  • Maturity (original maturity) of the mortgages (e.g., 30 years);
  • Coupon (e.g., 4.5%);
  • Price per USD 100 of par value (e.g., USD 104.50);
  • Par value (e.g., USD 100 million); and
  • Settlement month (e.g., August).

The TBA market is a forward market and is more liquid than the specified pools market. Based on a pool allocation process, the actual pools are not disclosed to the seller until two days before settlement.

Dollar Roll

• A trade known as a dollar roll involves selling a TBA for one settlement month and buying a similar TBA for the following settlement month. For example, a trader could sell a USD 100 million 30-year FNMA pool with a 4.5% coupon for August settlement and buy a USD 100 million 30-year FNMA pool with a 4.5% coupon for September settlement.
• A dollar roll is similar (in some respects) to a repo transaction (i.e., where one party sells securities to another party and agrees to buy them back at a future time for a slightly higher price). But there are two important differences:
  1. The securities purchased in the second month may not be the same as the securities provided in first month. The party on the other side of the transaction can sell back the same securities, but it may also deliver securities with worse prepayment properties.
  2. No interest is added to the price at which the securities are repurchased. The dollar roll transaction involves the initiating party losing one month of interest payments from a pool with the specified coupon, while the party on the other side gains one month of interest.
• If the following terms are defined –
  

  A – The price at which the pool is sold during the first month (including accrued interest),

  B – The price at which the pool is purchased during the second month (including accrued interest),

  C – The interest earned on the proceeds of the sale for one month, and

  D – The coupon and the principal repayment that would have been received on the pool sold during the first month.

  The value of the roll is calculated as:

  A – B + C – D

  For example, suppose that a USD 1 million par value of a 4.5% pool is sold for USD 102.50 in March and repurchased for USD 102.00 in April. It is assumed that the payment date is the twelfth of the month for both months. This means that the accrued interest in total for both transactions.

  It follows that A = 1025000 + 1500 = 1026500 and B = 1020000 + 1500 = 1021500

  Now, it is assumed that the proceeds of the sale in the first month can be invested a 0.1% for the month so that C = 1026.5. In calculating D, it is assumed that if the pool had been not been sold, interest and principal payments on the pool during the month of the roll would have amounted to 0.45% of the par value. This means that D = 4500. In this case, the value of the roll (USD) is

  1026500 – 1021500 + 1026.5 – 4500 = 1526.5

  • The agency securities discussed till now have been pass-throughs. Other products will be discussed from the next slide.

Other Agency Products

• In a collateralized mortgage obligation (CMO), classes of securities that bear different amounts of prepayment risk are created. These classes are referred to as tranches.
• As a simple example, suppose that there are tranches A, B, and C with the following properties:
  • Tranche A investors finance 30% of the MBS principal,
  • Tranche B investors finance 50% of the MBS principal, and
  • Tranche C investors finance the remaining 20% of the MBS principal.

Cash flows to the tranches are defined so that each tranche gets interest on its outstanding principal. However, there are special rules for principal payments (both scheduled and prepayments). Payments are first channelled to tranche A. When the principal of tranche A has been fully repaid, principal payments are then channelled to tranche B. When the principal of tranche B has been fully repaid, all remaining principal payments are channelled to tranche C.

In this example, most of the prepayment risk is borne by tranche A and very little is borne by tranche C. However, the distribution prepayment risk can be adjusted by changing the percentage of the pool financed by the different tranches.

• Two other agency securities are interest-only securities (IOs) and principal-only securities (POs). These are also called stripped MBSs. As their names imply, all the interest payments from a mortgage pool go to the IOs, while all the principal payments go to the POs. Both IOs and POs are risky instruments.
• As prepayments increase, a PO becomes more valuable because cash flows are received earlier than expected. By contrast, IOs become less valuable because fewer interest payments are made overall. As prepayments decrease, the reverse happens.

Non – Agency MBSs

• Non-agency MBSs are those that are not issued by GNMA, FNMA, or FHLMC. In a typical non-agency securitization, a mortgage portfolio is sold by a bank to a special purpose vehicle (SPV), which in turn passes the cash flows to the various securities it creates. In this case, there is usually no guarantee protecting investors against defaults. Indeed, default risk (rather than prepayment risk) is the major risk being taken by investors. Typically, the SPV creates several tranches that are subject to different amounts of default risk and promised returns. This securitization model played a key role in the 2007–2008 crisis.

Modeling Prepayment Behavior

• Calculating the cost of the prepayment option to investors is more complicated than valuing an interest rate option. This is because prepayment behavior depends on more than just interest rates.
• There are generally four reasons for prepayments:

1. Refinancing
2. Turnover
3. Defaults
4. Curtailments

Modeling Prepayment Behavior – Refinancing

• Refinancing arises when a borrower prepays a mortgage in order to refinance the underlying property.
  • The most likely reason for this is a decline in interest rates. By refinancing in such an environment, the borrower can reduce his or her monthly payments.
  • Another reason is improvement in the borrower’s credit rating so that he or she is able to obtain a lower rate even when interest rates have not changed.
  • Another reason for refinancing can be that the value of the property has increased so that a higher loan can be negotiated. (This is referred to as cash-out refinancing.)

The extent to which refinancing is likely to occur is measured by the incentive function.

A simple incentive function I for a pool could be

I = WAC – R

where

WAC is the weighted-average coupon, and R is the current mortgage rate available to borrowers. The incentive function is then the amount by which refinancing allows borrowers to reduce their interest rates.

A slightly more elaborate incentive function is

I = (WAC – R) × ALS × A – K

where

ALS is the average loan size,

K is the estimated cost of refinancing a loan, and

A is an annuity factor giving the present value of one dollar of payments per year for a period equal to the weighted-average maturity.

This incentive function reflects the amount by which refinancing allows borrowers to reduce the total present value of their remaining payments. It reflects the empirical evidence that the prepayment rate increases as the average loan size increases.

For a given incentive function I, the annualized prepayment rate is sometimes modeled as:

where

a, b, and c are parameters estimated from empirical data. It is illustrated in the next figure assuming a = 4, b = 0.02, and c = 25 with interest rates measured in basis points.

This is an S-shaped function. The shape of the CPR in this figure is broadly consistent with prepayment experience. It shows that refinancing creates very little prepayment when mortgage rates have increased (so that WAC-R is negative). As mortgage rates decrease, the prepayment rate increases quite fast and then levels off.

The parameters a, b, and c may depend on the economic environment. For example, increases in housing prices make prepayments more likely. Another phenomenon that may affect the parameters is termed burnout. When a mortgage pool has been in existence for some time and interest rates have declined, the mortgage holders most likely to refinance (e.g., those who are financially sophisticated, have a good credit rating, or high loan balances) will tend to have already done so. The remaining mortgage holders are (on average) less likely to refinance. The burnout phenomenon shows that a prepayment function can be path dependent; prepayments on a mortgage pool that is a few years’ old can depend on where interest rates were in the past as well as where they are today.

Modeling Prepayment Behavior – Turnover

• Turnover prepayments arise when a borrower sells the house. Turnover is higher in the summer months than the winter months. It is also lower early in the life of a mortgage because homeowners usually do not relocate immediately after taking out a mortgage.
• The prepayment model developed by the Public Securities Association (PSA) for analyzing American MBSs has already been discussed earlier.
• The turnover rate is liable to depend on the geographical location of the properties and on the average age of the mortgage holders. It does not depend on interest rates, except to the extent that a homeowner may be less inclined to move if he or she is paying below-market rates on his or her mortgage. (A reluctance to move when the mortgage rate is low is referred to as the lock-in effect.)

Modeling Prepayment Behavior – Defaults

• When a mortgage holder defaults and the mortgage is part of an agency pool, the agency pays the outstanding balance on the mortgage. This is treated as a prepayment and therefore defaults are relevant to the calculation of prepayments despite agency guarantees. Defaults added considerably to the prepayment experience of mortgage pools during the 2007–2008 crisis. Models use average FICO scores, LTVs, and the history of housing price movements to predict the default component of prepayments.

Modeling Prepayment Behavior – Curtailments

• Curtailments are partial prepayments.
  

  These tend to occur when loans are relatively old and balances are relatively low. Prepayments from curtailments can rise as high as 5% when the loans in a mortgage pool have only one or two years to maturity.

Valuation of an MBS Pool

• The first step in valuing an MBS pool is to develop a prepayment model. Two variables that future prepayments may depend on are as follows:

1. Level of interest rates: The burnout phenomenon shows that the complete pattern of interest rates because mortgage origination is relevant to determining prepayments in a given month.
2. Housing prices: The history of housing prices since the mortgage origination may also be relevant. Sharp increases in housing prices may lead mortgage holders to refinance (i.e., use cash-out refinancing). Sharp decreases in housing prices may lead to defaults. Both are liable to increase prepayments.

In addition, there are several relevant parameters describing the mortgage pool. The following are examples:

1. The prepayment rate tends to increase as the average loan size increases.
2. The geographical distribution of the loans may affect the model used to project housing prices and expected turnover.
3. Average FICO scores and average LTVs affect default predictions.
4. Average loan age affects curtailment estimates.

• Once a prepayment model has been specified, a technique known as Monte Carlo simulation is used to value an MBS. This involves the following steps:

1. Randomly sample from probability distributions to determine a hypothetical month-by-month path for risk-free interest rates and housing prices. The interest rate path will be accompanied by estimates of the spread of the mortgage rate over the risk-free rate. The housing price path may depend on interest rates and will reflect the geographical distribution of the mortgage holders.
2. For each month, determine prepayment rates using the specified prepayment model, the path for interest rates, and housing prices up to that month, as well as relevant parameters describing the mortgage pool.
3. Use the prepayment rates to calculate month-by-month cash flows from the MBS.
4. Starting at the end of the life of the MBS, discount cash flows month-by-month back to today. The discount rate for a month is the risk-free interest rate sampled for that month.
5. Repeat steps 1 to 4 many times.
6. Calculate the value of the MBS pool as the average of the calculated present values.

• The advantage of Monte Carlo simulation is that it can consider what is referred to as path dependence. This means that a given month’s prepayments can depend on the history of interest rates and the history of housing prices, as well as on their current levels. Other analytical tools, such as the use of trees, cannot easily accommodate this feature.

Option Adjusted Spread

• The option-adjusted spread (OAS) is the excess of the expected return provided by a fixed-income instrument over the risk-free return adjusted to account for embedded options. The return on an MBS is adjusted for prepayment options as follows

OAS = Expected MBS Return – Return on Treasury Instruments

The earlier procedure involving Monte Carlo simulation for determining the value of an MBS can be adjusted to determine the OAS provided by the MBS. The procedure is as follows:

1. Make an initial estimate of the OAS.
2. Carry out a Monte Carlo simulation as described in the previous section but using discount rates equal to the Treasury rate plus the current estimate of the OAS.
3. Compare the price obtained with the market price.
4. If the market price is higher than the simulated price, the OAS estimate is reduced. If the market price is lower than the simulated price, the OAS estimate is increased.
5. Continue changing the OAS estimate until the simulated price equals the market price.

• The method of successive bisection can be used to create a workable algorithm. First, initial high and low OAS estimates are produced. This can be done by reducing OAS until the simulated price is higher than the market price and then increasing OAS until the simulated price is lower than the market price. The average of the high and low prices is then used in the simulation. If it proves to be too high, it becomes the new high OAS. If it proves to be too low, it becomes the new low OAS. The procedure is then repeated. Note that the ranges between the high and low OAS are halved on each iteration.
• OAS can be useful in determining the relative valuation of different MBS pools. For example, an MBS with an OAS of 80 basis points (i.e., with a return of 0.8% over the Treasury rate) should be a better buy than one that provides an OAS of 40 basis points (i.e., with a return of 40 basis points over the Treasury rate). Of course, the OAS calculated depends on the extent to which the underlying model correctly accounts for prepayments. If the model is incorrect or has not been calibrated properly, the results cannot be relied upon. When an analyst finds a high-OAS pool, he or she should look for institutional or technical reasons why that pool might trade differently from the rest of the market. It may be that the analyst can find an assumption in the model being used that leads to the high OAS. He or she should then critically examine the validity of that assumption.