# Seller Swap Model

&#x20;

Seller Swap Valuation

&#x20;

One party sells mortgage pools on its balance sheet and pays the bond interest by entering into a pay-fixed swap with CHT, and receives the interest from MBS pool sold to CHT. This is a seller swap. The fixed leg is semi annual, and the float leg, MBS coupons, is monthly. In addition, the MBS sold to the trust generates principal cash flows. CHT buys new-pooled mortgages from the party with this principal flows every month until the maturity of the swap.

&#x20;

CHT sold bullet bonds of total notional *N* that pays *C*% semi annual coupon for 60 months to investors for *P* dollars. Using these proceedings, CHT bought a mortgage pool, whose dollar price was *P*, from the party and entered into a “seller swap” with the party. CHT will buy new mortgage pools from the principal payments generated from the mortgage pools that already bought from the party

&#x20;

The party agreed to pay CHT the coupon of the bond, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png)dollars every six months for 5 years. On the floating side, CHT agreed to pay, monthly, all the coupons generated by the mortgage pools it bought from the party for 60 months.

&#x20;

Let ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) denotes the first day of month *i,* for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) assuming ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png) is the inception of the swap and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) denotes the coupon the party receive at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png) for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png) Furthermore, let ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png)denote the time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png) dollar price and unit price of one dollar of notional, respectively of the mortgage pool, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png)sold to CHT at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image024.png) Moreover, assume the valuation date *t* belongs to ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png), where *m* is an integer between 1 and 60.  Then

&#x20;

The fair value of the Seller-Swap                                  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png)

&#x20;

where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) is time *t* price of a zero coupon bond that matures at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image032.png) is the price of one dollar notional of the bond at inception.

&#x20;

The time *t* price of the fixed leg, which includes the notional at maturity, is time *t* price of the bond less present value of the notional, *N*.

We show that the time *t* value of the float- leg is equal to the sum of time *t* dollar prices of the mortgages the party sold to CHT less expected discounted value of mortgages at the maturity of the swap.

&#x20;

Let ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png) denotes the time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) price and notional of all the mortgages the party sold to CHT up to and including time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png) respectively.

&#x20;

At inception, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png), the party sold one MBS, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png) to CHT.

&#x20;

Evolution of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png)

&#x20;

time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png) dollar Price of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png):![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image044.png) measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png)

Coupon for the first month: ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image046.png) measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png)

&#x20;

Then ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image048.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image050.png)

&#x20;

Principal payment at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png): ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image054.png) measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png)

time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png)dollar Price of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png): ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image056.png) measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png)

Coupon for the second month: ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image058.png) measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png)

&#x20;

At time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png) the party received ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image060.png) and sold another MBS, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image062.png)to CHT. i.e.,&#x20;

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image064.png)

&#x20;

where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image066.png)is the time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png) price of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image068.png) Additionally, we note that

&#x20;

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

&#x20;

where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image074.png)is the time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png)price of  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png) per one dollar of notional. Table 1 describes the evolution of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png) during the second month.

&#x20;

## Table 1

&#x20;

\\

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png)                                  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png)

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image078.png)time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png) price                                            ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image080.png)                                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image066.png)

&#x20;             Coupon for month 2                                  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image082.png)                                        ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image084.png)

&#x20;             Prepay at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png)                                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image088.png)                                        ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image090.png)

&#x20;             time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png) price                                             ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image092.png)                                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image094.png)

&#x20;

Here, we observe that the first two rows, *time* ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png) *price* and *Coupon for month 2*, are measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image052.png), and the last two rows, *Prepay at time* ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png)and *time* ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png) *price* respectively, are measurable at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png). The Coupon ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image096.png) the party received at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image098.png) in dollars is given by

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image100.png)

&#x20;

At time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png), i.e., beginning of the third month, the party received ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image102.png) amount of cash and sold another MBS, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image104.png) to CHT. We note that

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image106.png)     and     ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image108.png)

&#x20;

where  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image110.png) is the time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png) price of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image114.png) is the time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png) price of  ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png) per one dollar of notional.

&#x20;

Table 2 describes the evolution of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png) ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png) during the third month.

&#x20;

Table 2

&#x20;

\\

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image116.png)                                                           ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png)                   ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image076.png)                ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png)

Price at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png)                                        ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image118.png)                         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image120.png)                      ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image110.png)

Coupon for month 3                                 ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image123.png)                         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image125.png)                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image127.png)

Prepay at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image129.png)                                     ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image131.png)                         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image133.png)                      ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image135.png)

Price at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image129.png)                                       ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image137.png)                         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image139.png)                      ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image141.png)

&#x20;

Here, we note that the dollar price of ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image112.png)at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image086.png), ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image143.png)is equal to ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image102.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image146.png) the Coupon the party received at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image148.png) is given by

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image150.png)

&#x20;

References:

<https://finpricing.com/lib/IrCurve.html>

&#x20;

&#x20;
