# GIC Coupon Rate Calculation

&#x20;

GIC Coupon Rate Calculation

&#x20;

We calculate the Treasury transfer coupon rate, in the case of zero coupon payment frequency, or the equivalent annualized simple rate in the case of non-zero coupon payment frequency, from which the transfer coupon rate.

&#x20;

The model assumes that the GIC holder receives deterministic payments on specified payment days, and the embedded put option is exercised by the GIC holder when the redemption value exceeds the holding value.

&#x20;

Consider a GIC specified by

·         maturity, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png),

·         set of future payment times, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png), where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png).

&#x20;

Let

&#x20;

* ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png)      be the annualized customer coupon rate,
* ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png)       be the transfer coupon rate,

·         *fc*          denote the coupon rate compounding frequency (i.e., annual, semi-annual, quarterly or monthly),

·         *fp*          be the coupon payment frequency (i.e., annual, semi-annual, quarterly or monthly),

&#x20;

We define an "equivalent simple annualized rate", which we denote EAR:

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png),                                              (1a)

&#x20;                                   ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png) .                                               (1b)

&#x20;

The payment ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png) at time *ti* is then

&#x20;

&#x20;                                   ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png),                                        (2)

&#x20;

where time interval, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png), is calculated based on the ACT/365 day counting convention. We have reviewed the generation of the payment dates, *ti*, and the respective payment amounts,![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png).

&#x20;

In the case of zero payment frequency, the 'bullet' payment at maturity is calculated as

&#x20;

&#x20;                                   ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image023.png)

&#x20;

No equivalent simple annualized rate calculation is required in this case.

&#x20;

We solve the equation,

&#x20;

&#x20;                                   ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image025.png),                               (3)

&#x20;

for the unknown Transfer Coupon Rate, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png), where ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) are the closed GIC present value and the redemption (put) option price respectively.  Here,

&#x20;

&#x20;                ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image032.png).                              (4)

&#x20;

We note that if the payment frequency,![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png), is not zero, then the spreadsheet outputs![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png), defined by eq. (1b), instead of the transfer coupon rate, *cTr*; it is the responsibility of the spreadsheet user to then calculate

&#x20;

&#x20;                                               ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png).                                     (5)

&#x20;

In the case of zero payment frequency, we directly output the transfer coupon rate (ref <https://finpricing.com/lib/FiZeroBond.html>)

&#x20;

We assume a stochastic interest rate environment, with the short interest rate dynamics governed by a HW model of the form,

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png),                                                                 (6)![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png)

&#x20;

with constant mean reversion, *a*, and volatility, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image044.png), parameters. The interest rate model is implemented as a trinomial tree.

&#x20;

We use the HW tree described above to evaluate the terms of eq. (3). It employs Brent's root finding method for solving eq. (3), which is guaranteed to converge if a root exists within the given interval.

&#x20;

One can expect monotonic behavior of the left hand side of equation (3) as a function of the transfer coupon rate, which ensures the uniqueness of the eq. (3) root. This is supported by the graph in Figure 1 that shows the difference between the left and right sides of eq. (3) for the Premium GIC specified in Section 6 below.

&#x20;

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image046.png)&#x20;

&#x20;

&#x20;

&#x20;

&#x20;

&#x20;

&#x20;

&#x20;

<br>

Figure 1. Difference between the left and right hand sides of equation (3) for the Premium GIC specified in Section 6 below. Hull-White volatility is 1.383%.

&#x20;
