# Mortgage Transfer Coupon Rate Calculation

 

 

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 ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png)and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png)denote the forward start date and the maturity date of the mortgage, respectively. Furthermore, let ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) 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 ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png)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 ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) is April 30, 2009, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png) is March 31, 2009, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png) is February 28, 2009, and, finally,![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png)is May 31, 2004.

 

The payment dates, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png) are given by the following business day rule. The ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png)payment date, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png) is equal to ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image024.png) if ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png) is a weekday. On the other hand, if ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png) is either a Saturday or a Sunday, then ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) is set equal to the following Monday, provided that it falls in the same month; otherwise, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) is set equal to the Friday immediately preceding ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image032.png) 

 

See the schematic below for a pictorial description of the valuation date, forward start date and payment dates. Here, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png) denotes the valuation date; moreover, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png).

 

 

 

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image039.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image041.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image043.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image044.png)                    ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png)                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png)               ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image046.png)               ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png)                                         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image050.png)             ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png)

Valuation date                      &#x20;

&#x20;                            Forward start date                                Payment dates

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5.2 Discount Factors Generation for Key Terms

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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.

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The output is a two-dimensional array, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png)where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image054.png)is the discount factor associated with the key term ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image056.png) The output key terms range from one month to 30 years, that is,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image058.png)yr., ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image060.png) yr., ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image062.png) yr., ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image064.png) yr., ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image066.png) yr., ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image068.png) yr.,

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And

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image070.png), for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image072.png)

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**Table 5.2.1 – Input Terms**

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image075.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png)                                                  Term (Yr.)               Term (Yr.)                 &#x20;

Table 6.1.                                  2

Table 6.1.                                  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image077.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image078.png)3

&#x20;           corresponding rates with            0.250                             4             corresponding rates with             terms less than one year                              0.500                            5              terms longer than one year

&#x20;    are expressed as                         0.750                            7              are expressed as an

&#x20;    annualized, simple                     1.000                          10              annualized semi-annually interest rates                                                                                      12              compounded rates

&#x20;                                                                                          15

&#x20;                                                                                          20

&#x20;                                                                                          30

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5.3 Calculating Discount Factors for Payment Dates

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Let

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image080.png)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image082.png),     for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image084.png)

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Moreover, let *j* be an integer such that ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png) where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image088.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image090.png) are key terms described in Section 5.2. We obtain the discount factor ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image092.png) for the term ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image094.png) by log-linear interpolation,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image096.png)

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where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image098.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image100.png)are the discount factors associated with the terms ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image088.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image103.png)

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5.4 Monthly Principal Balance Generation

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Let *MP* denotes the monthly mortgage payment. Then

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image105.png)

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Where

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·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image107.png)is the principal at the forward start time, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image109.png)&#x20;

·         *C* is the annualized monthly compounded coupon, and

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

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We calculate the ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png) ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png) month end balance, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png) as follows,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image114.png)

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5.5 Transfer Coupon Rate Calculation

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We solve

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image116.png)

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for the unknown, *r*, that is,

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&#x20;                                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image118.png)                                              (1)

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Next, we show that ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image120.png). Observe that

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&#x20;                                                          ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image122.png)                                                       (2)

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and assume that

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&#x20;                                                      ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image124.png) for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image126.png)                                                (3)

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From (2) and (3),

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image128.png)

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which leads to the conclusion.

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