1 Mortgages and Mortgage Valuation Based on Blackwell, Griffiths and Winters (Chapter 6) and other material

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Mortgages and Mortgage Valuation

Based on Blackwell, Griffiths and Winters (Chapter 6) and other material

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Mortgage Terminology and Institutional Features

A mortgage loan is divided into two parts.

1. Loan agreement: the borrower, or mortgagee, agrees to receive a fixed amount of money at origination and repay it through a series of payments to the lender, or mortgagor, over a fixed period of time.

The loan agreement includes all the terms of the loan, such as number of payments, the interest rates and fees.

2. Mortgage: the mortgagee pledges a piece of real property as collateral for the loan.

The mortgage places a lien on the property. The lien requires paying off the loan when the property is sold and allows the lender to seize the property in the case of a loan default.

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Mortgage Terminology and Institutional Features

Scheduled Principal and Interest Payments

The most common type of mortgage loan is called a fixed-rate mortgage because the interest rate on the loan is fixed for the life of the loan.

One feature of a fixed-rate mortgage is the monthly loan payment (P&I payment) is the same every month.

The monthly mortgage loan payment is known as a P&I payment because the payment includes the interest due (I) on the loan and an amount that reduces the principal (P).

A P&I payment fixed for the life of the loan can be calculated using the present value of an annuity formula.

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The formula for the monthly P&I payment

The formula for the monthly P&I payment is

,

1

11

1

r

r

FPICF

tT

tt F = principal amount

r = monthly loan rateT = initial length of loan in monthst = time remaining on the loan

5

Example of P&I payment calculation

Inputs:

30-year fixed-rate mortgage

Initial principal of $150,000

Annual coupon rate of 6%

Monthly payments

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Example of P&I Payment CalculationContinued

3258.899

005.0

005.1

11

000,150

360

1

PICF

In Excel:=PMT(6%/12,30*12,-150000,0,0)

$899.33

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Calculating the P&I Payment on a Fixed-Rate Mortgage during the Life of the Mortgage

Inputs

15-year fixed-rate mortgage that is 4 years old with current principal of $141,163,

Annual coupon rate of 5.40%,

Monthly payments.

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Calculating the P&I Payment on a Fixed-Rate Mortgage during the Life of the Mortgage

6292.420,1

0045.0

0045.1

11

163,141

48180

49

PICF

In Excel:=PMT(5.4%/12,11*12,-141163,0,0)$1,420.63

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Mortgage Terminology and Institutional FeaturesSecured Lending and Default

Residential mortgages are quite safe loans because people are generally careful to protect their homes. However, because a house is a major investment for most people, mortgage loans have a number of features that protect the lender.

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Protecting the Lender

1. Buyer provides part of the purchase price of the house which is called the down payment. With a down payment, the value of the loan is less than the value of the collateral, which is referred to as over-collateralization.

2. Part of a monthly payments applies toward principal reduction. Thus, the loan to value (LTV) ratio declines over the life of the loan which provides increasing protection for the lender.

2. Borrowers are required to maintain adequate insurance on the house to cover fires and natural disasters. Typically, monthly amounts to cover insurance are added to the monthly P&I payment. This is referred to as escrow.

3. Government claims take priority over the lenders claims, so lenders typically require that monthly payments include an escrow amount for property taxes.

4. Private mortgage insurance (PMI) is an additional protection for the lender. Lenders often require PMI when the buyer wants to provide a down payment of less than 20% of the purchase price.

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Prepayment Options

Prepayment OptionMortgages have an embedded prepayment option. This option is similar to a call option on a corporate bonds.

Prepayment option allows the borrower to pay off the mortgage at any time and without penalty. This means that the borrower can retire the mortgage by repaying the remaining principal balance plus any interest that has accrued since the last payment was made.

The most common reason for prepayment is the sale of the house.

Another common reason to prepay a mortgage loan is that market interest rates have declined, thus enabling borrowers to refinance their loan at lower rates.

However, borrowers are also allowed to prepay small amounts by paying more than required monthly payment. This is referred to as a curtailment.

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Business Aspects of Mortgages

Major aspect to mortgage lending.1. Origination: Making the

decision to accept or reject the loan and then with the accept decisions making the loan.

2. Servicing: Accepting payments over the life of a mortgage, pursuing late payments, and handling foreclosure when necessary.

3. Investment: Owning the loan (holding in a portfolio) to generate a return to cover interest to depositors and returns to shareholders.

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Intermediation and Distintermediation

Historically, all three aspects of the mortgage business for an individual loan were handled by one financial institution. This is referred to as financial intermediation.

Financial intermediaries stand between savers and borrowers, writing different contracts with each side.

Disintermediation allows a financial institution to participate in only one or two aspects of the mortgage loan process. This allows for specialization by institutions.

An important part of disintermediation is the securitization of mortgages. Securitization allows funds to flow efficiently from investors to borrowers and has resulted in lower mortgage rates that more closely track other market interest rates.

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Types of Residential Mortgages

Most residential mortgages are one of three types:

(1) fixed-rate mortgage,

(2) adjustable rate mortgage (ARM), and

(3) balloon mortgage.

We will also briefly discuss an interest-only mortgage.

A fixed-rate mortgage is a mortgage loan with a fixed-coupon rate for the entire life of the loan. The loan is scheduled to repay the full amount of the principal using the monthly P&I payment.

Having a fixed rate allows us to calculate the remaining principal balance at any point in time.

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Formula to calculate remaining principal balance

,

1

11

r

rPICFF

tT

t

F = remaining principal amount at time tPICF = monthly P&I paymentr = monthly coupon rateT= original life of loan in months

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Calculating the remaining principal balance

Inputs

30-year fixed-rate mortgage,

6% annual coupon rate, and

monthly P&I payment = $899.33.

What is remaining balance after one year?

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Calculating the remaining principal balance

68.158,148$

005.0

005.1

11

33.89912360

12

F

See accompanying amortization table for excel approach

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Types of Residential Mortgages

Two important points about the formula.

1. The formula assumes that all payments are made exactly when scheduled.

2. The monthly P&I payment amount is a rounded number which creates some error in the calculation when compared to a calculator or spreadsheet.

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Adjustable-Rate Mortgage

An adjustable-rate mortgage (ARM) has a coupon rate that changes periodically over the life of the mortgage loan.

The coupon rate changes reflect changes in the level of market interest rates.

Having an adjustable coupon rate means the monthly P&I payments are not the same across the life of the loan.

Since the interest rate on an ARM changes, the loan contract must specify three additioinal items:

1. Interest rate index for the base rate on the loan,

2. A fixed spread over the index that reflects borrower specific risk, and

3. The repricing frequency ( how often the interest rate adjusts).

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Adjustable-Rate Mortgage

The most common frequency for adjusting the interest rate on an ARM is annual.

When the annual rate adjustment is made, a new P&I payment must be calculated. The payment is calculated assumimg that the new interest rate will hold over the remaining life of the loan.

Hybrid ARMs.

An ARM allows rates to adjust from the beginning of the loan based on the repricing frequency.

In a hybrid ARMs, the interest rate is fixed for a pre-specified time period.

Hybrid ARMs are quoted based on their fixed rate period and the repricing frequency.

For example a 5/1 ARM starts with a 5-year fixed rate period followed by annual interest rate adjustments for the remainder of the life of the loan.

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ARMs and Risk from Changing Interest Rates

One concern for a borrower in an ARM is that interest rates could increase to the point where the borrower could not make the scheduled payment. This is bad for both the borrower and the lender.

Most ARMs have interest rate caps and floors that limit the amount by which the interest rate can change.

ARMs have both annual and lifetime caps and floors. It is common to have an annual limit of 1% to 2% with a lifetime limit of 5% to 6%.

Teaser Rates are artificially low starting interest rates used to attract borrowers.

When the initial rate is a teaser rate, the loan rate adjusts frequently so the lender can get up to market interest rates quickly.

A balloon mortgage has a fixed rate and monthly P&I payments based on a 30-year life. However, it requires that the entire remaining principal to be repaid at some point in time before the end of 30 years.

For example, a 5-year balloon mortgage acts like a 30-year fixed rate mortgage for five years, but at the end of five years all the remaining principal is due in a lump-sum payment

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Basics of Mortgage Valuation

In theory, a single fixed-rate mortgage can be valued in a manner similar to a fixed coupon bond.

To do this, we would need the coupon rate and the discount rate that reflects the default risk of the individual borrower.

Because almost all mortgages prepay, an estimate of when the mortgage will be prepaid is needed to determine the cash flows the lender expects to receives.

Prepayments occur for financial reasons (decline in market interest rates) and liquidity reasons (moving for new job), so we need estimates for both.

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Basics of Mortgage ValuationContinued

It is clear that the process for valuing a fixed-rate mortgage becomes very complicated.

In addition, much of the information is specific to one borrower and does not generalize well.

Accordingly, analysts focus on valuing portfolios of similar mortgages.

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Valuing a Mortgage Portfolio

Valuing a mortgage portfolio uses borrower averages instead of specifics for individual borrowers. • It is much easier to predict the average behavior of a

group than specific behavior of an individual.The first step to mortgage portfolio valuation is to create a

portfolio of similar mortgages.

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Use the adjusted cash flow (ACF) approach to value a portfolio of mortgages.

The ACF approach uses monthly expected cash flows for the portfolio that reflect prepayments and defaults and then discounts the expected cash flows using an appropriate risk-adjusted discount rate.

The process begins by estimating the scheduled monthly P&I payments for the portfolio.

Once the expected P&I payment is determined, the next step is to determine expected prepayments.

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Experts estimate the expected prepayments on an annual basis by the the Cumulative Prepayment Rate (CPR).

Mortgage cash flows are monthly, so the CPR is converted to a monthly rate referred to as the Single Monthly Mortality (SMM).

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The formula for the SMM is: Assuming that the prepayment cash flow occurs after the monthly P&I payment, the prepayment cash flow cash flow is calculate as follows

.)1(1 12/1CPRSMM

Once the SMM is calculated, the expected prepayment cash flow for the portfolio can be calculated.

11 tttt FrPICFFSMMPCF

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Default and Recovery

The final piece of the puzzle for the expected portfolio cash flow is to estimates defaults and recoveries.

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For valuing mortgage portfolios, default means the buyer stops making payments and the lender will not receive any more cash flows until the property is seized and sold.

The frequency of default will depend on many things: the state of the economy, housing prices, unemployment rates, the loan-to-value ratios of mortgages in the portfolio, and so on.

Recall that mortgage defaults have been historically relatively rare because the borrower protects this major investment and typically can sell the property for more than the remaining loan principal balance.

The estimated default rate is referred to as the annual cumulative default rate, which we label as AD.

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Again, annual rates must be adjusted to monthly rates to fit the time of mortgage cash flows. So the monthly default rate is

Then the expected default amount (DA) is a percentage of the remaining principal balance and is calculated as follows:

.)1(1 12/1ADMD 11 tttt FrPICFFMDDA

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However, investors do not lose the total amount of the defaulted loan because the property is sold and the proceeds are used to cover foreclosure costs and repay the loan.

The loan repayment from selling the property is a cash flow to the mortgage portfolio investor.

The loan repayment from selling foreclosed property is calculated as a fixed percentage of the defaulted amount referred to as the recovery rate (rr).

The recovery rate is converted into an expected cash flow as follows:

.tt DArrRCF

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The formula for the expected monthly cashflow is

tttt RCFPCFPICFCFE ][

then for our portfolio the expected cash flow for the first month is

.46.522,1052.47026.362,768.689,2][ 1 CFE

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Calculating the Rest of

the Expected Cash Flows

We calculated the first expected cash flow from the portfolio. Now we have to repeat the process for the remainder of the life of the portfolio.

In calculating the remaining expected cash flows, there are two important points to remember.

1. As each month passes, the WAM must decrease by 1.

2. Each month the remaining principal amount must be calculated using the prepayment and default/recovery cash flows.

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The new principal balance is calculated as follows:

.)(1 tttttt DAPCFFrPICFFF

Notice that full default amount (DA) is subtracted. This is done because the full amount of defaults will no longer contribute to month P&I payments.

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The process is repeated for each month in the life of the portfolio.

Clearly, the best way to do this is in the spreadsheet since spreadsheets were created to handle repetitive calculations with numbers that carry forward.

Once we have all the expected future monthly cash flows for the mortgage portfolio, we need to discount the cash flows back to the beginning of the life of the portfolio to determine the portfolio’s value.

The ACF method uses expected cash flows.

Expected cash flows take into account all expected prepayments and defaults. This means that the expected cash flows are theoretically stripped of prepayment risk and default risk.

Therefore, the appropriate discount rate should be for a security that has no prepayment risk or default risk.

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U.S. Treasury securities have no prepayment risk or default risk so a Treasury yield is a reasonable discount rate.

Is a single discount rate across the life of the portfolio or the theoretical spot rate curve for each month?

With all the estimation of the cash flow calculation process, the precision of the theoretical spot rate curve is not needed.

Most analysts suggest using a single discount rate and suggest using the yield from the Treasury security that best tracks mortgage rate. A popular choice is the yield on a 10-year Treasury note.

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Convexity

Convexity was introduced in the bond chapter to improve on the duration-based price change model.

Convexity would serve the same purpose for mortgages, but in mortgages the same difficulties exist for calculating convexity as for calculating duration.

Accordingly, we will not attempt to calculate convexity for mortgage portfolios. Instead, we will discuss the insight available from mortgage portfolio convexity.

Mortgage portfolios have negative convexity while bonds have positive convexity.

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Mortgage Price Characteristics (cont.)

Figure 6-10: Prepayments and Negative Convexity

85%

90%

95%

100%

105%

110%

5.0% 6.0% 7.0% 8.0% 9.0%

discount rate

PV

(%

of

par

)

changing prepayment rate fixed prepayment rate

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Mortgage Price Characteristics

Negative convexity means that a duration-based price change model will always overstate the value of the mortgage portfolio because it will understate the loss in value from a change in interest rates.

Documents

1 Mortgages and Mortgage Valuation Based on Blackwell, Griffiths and Winters (Chapter 6) and other material