Mortgage Transfer Coupon Rate Calculation

A model is presented for calculating the transfer coupon rate for forward starting, closed Canadian commercial mortgage certificates.

Mortgage Transfer Coupon Rate Calculation

With respect to a closed commercial mortgage certificate, the transfer coupon rate is defined as an annualized, monthly compounded interest rate, such that the fair value of the closed mortgage certificate, from Treasury’s point of view, is par.

A model is presented for calculating the transfer coupon rate for forward starting, closed Canadian commercial mortgage certificates.

Let n denotes the mortgage term in months, and and denote the forward start date and the maturity date of the mortgage, respectively. Furthermore, let denote dates that are obtained by rolling back the maturity date of the mortgage monthly until the forward start date of the mortgage. If the maturity date is the end of the month, then each corresponds to the end of the month date. For example, if the mortgage term is 60 months and the maturity date is April 30, 2009, then is April 30, 2009, is March 31, 2009, is February 28, 2009, and, finally,is May 31, 2004.

The payment dates, are given by the following business day rule. The payment date, is equal to if is a weekday. On the other hand, if is either a Saturday or a Sunday, then is set equal to the following Monday, provided that it falls in the same month; otherwise, is set equal to the Friday immediately preceding

See the schematic below for a pictorial description of the valuation date, forward start date and payment dates. Here, denotes the valuation date; moreover, .

Valuation date

Forward start date Payment dates

5.2 Discount Factors Generation for Key Terms

We use the previously vetted par-bond bootstrapper, employed by the previously vetted mortgage pricing spreadsheet, Mtg.xls, to generate discount factors based on Treasury rates (see https://finpricing.com/lib/IrCurveIntroduction.html) for key terms. This spreadsheet takes a set of terms and corresponding rates as input parameters. Table 5.2.1 below specifies the set of input terms used in our testing.

The output is a two-dimensional array, where is the discount factor associated with the key term The output key terms range from one month to 30 years, that is,

yr., yr., yr., yr., yr., yr.,

And

, for

Table 5.2.1 – Input Terms

Term (Yr.) Term (Yr.)

Table 6.1. 2

Table 6.1. 3

corresponding rates with 0.250 4 corresponding rates with terms less than one year 0.500 5 terms longer than one year

are expressed as 0.750 7 are expressed as an

annualized, simple 1.000 10 annualized semi-annually interest rates 12 compounded rates

15

20

30

5.3 Calculating Discount Factors for Payment Dates

Let

, for

Moreover, let j be an integer such that where and are key terms described in Section 5.2. We obtain the discount factor for the term by log-linear interpolation,

where and are the discount factors associated with the terms and

5.4 Monthly Principal Balance Generation

Let MP denotes the monthly mortgage payment. Then

Where

· is the principal at the forward start time,

· C is the annualized monthly compounded coupon, and

· N is the number of amortization months of the mortgage.

We calculate the month end balance, as follows,

5.5 Transfer Coupon Rate Calculation

We solve

for the unknown, r, that is,

(1)

Next, we show that . Observe that

(2)

and assume that

for (3)

From (2) and (3),

which leads to the conclusion.