IRD Determination G3: Yield to Maturity Method for Bonds (worked example)

[Snoopy]

The next step is to find out what the 'Determination G1' referred to is.

I have not been able to find 'Determination G1' for good reason. It was replaced by document 'Determination G1A' as linked to below:

https://www.taxtechnical.ird.govt.nz...nation-g1a.pdf

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2. This determination requires that income derived or expenditure incurred in respect of a period shall be apportioned on a straight line basis among the income years in which the period falls, according to the number of days in the period calculated on either a 365 or a 360 day basis (we won't go into the 360 day year option here).

It assumes that income and expenditure accrue at a flat dollar rate over each day in the period.

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So putting this knowledge into practice:
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Our 'example bond' was bought on 12th March 1987 and the first interest payment on that bond to our new bond owner was made on 15th May 1987. That adds up to a period of 19+30+15= 64days between 'bond purchase' and 'first payment'. There are three half year periods subsequent to that over which interest is earned, before the bond finally matures on 15th November 1988. On that date, the final interest payment is made and the original capital is returned.

This means the total time over which our investors 'extra capital ' is returned is deemed to be: 64 + 1.5x 365= 612 days

We have already calculated the 'total income' that our bondholder is to receive for the whole investment period is to be $267,500 (refer post 1).

This amounts to $267,500/612= $437.09 per day.

The next step is to find out how this information relates to the tabulated information in 'Determination G3'.

SNOOPY

[Snoopy]

The next step is to find out how this information relates to the tabulated information in 'Determination G3'.

Below is the first table, exactly as it appears in Determination G3.

Period Ending	Principal Outstanding	Income in Respect of Period	Payments Received at End of Period
15-05-1987	$1,012,500	$28,815	$70,000
15-11-1987	$971,315	$78,826	$70,000
15-05-1988	$980,141	$79,542	$70,000
15-11-1988	$989,638	$80,317	$1,070,000
Total		$267,500	$1,280,000

The periods are split into six monthly intervals, which is a strange thing to do as tax years always have twelve months. Nevertheless I will go with the numbers 'as presented' and see what comes out.

The first period, ending 15-05-1987, represents 64 days (refer post 2) of a half year period over which $70k of interest would be paid to the bondholder, if they had held those bonds for 6 months. But since they only held those bonds for 64days, their share of that interest would be: $70,000 x [64/(365x0.5)]= $24,547.95

The total capital gain held by our investor over the whole investment period was:

($1,280,000 - 4x$70.000) - $1,012,500 = -$12,500 (i.e. a loss)
This represents a loss of -$12,500/[64+1.5x365] = -$20.44 per day

Yet for income tax purposes, we must add (adding a negative number is subtraction) to this $24,547.95, the portion of 'extra capital' our bondholder will get back that has been apportioned to this period: 64x-$20.44=-$1,308.16. So taxable earnings for the initial period reduce to: $24,547.95-$1,308.16=$23,239.79

Now we move on to the three six monthly income periods. Interest income for each of those is $70,000. But once again there is a 'capital adjustment' to be made for each period of: $20.44 x (365/2)= $3,730.30. So taxable earnings for the full six month period add to: $70,000-$3,730.30= $66,269.70

As previously discussed (post 1), all of these adjustments are 'capital' adjustments that change the book value of the capital of the bond remaining for income tax purposes (column 2 in the above table). Using my 'first principles' analysis, I would rewrite the reallocated income table that I have replicated above, as below:

Period Ending	Principal Outstanding (1)	Adjusted Gross Income in Respect of Period	Already Income Incorporated Taxable Capital Loss	Payments Received by End of Period
Day of Opening Investment	$1,012,500
15-05-1987	$1,011,192	$23,240	($1,308m)	$70,000
Income transfer to previous bond owner (2)				($45,452)
15-11-1987	$1,007,462	$66,270	($3,730)	$70,000
15-05-1988	$1,003,731	$66,270	($3,731)	$70,000
15-11-1988	$1,000,000	$66,270	($3,731)	$1,070,000
After Investment matures	$0
Resultant Capital Loss	$12,500
Total		$222,050	($12,500)	$1,234,548

Notes

1/ Capital adjustments period to period are as follows:
@15-05-1987: $1,012,500-$1,308=$1,011,192
@15-11-1987: $1,011,192-$3,730=$1,007,462
@15-05-1988: $1,007,462-$3,730=$1,003,731
@15-11-1988: $1,003,731-$3,730= $1,000,000

2/ Transfer required of income paid to you that belonged to the Previous Bond Owner:$ $70,000-$24,548=$45,452

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There is a obvious juxtaposition between my table above and the IRD table at the head of this post. That being that my table incorporates a 'capital loss', with declared income being lower than the coupon rate income received, whereas the IRD table is showing income higher than the coupon rate received (i.e. our bondholder has received a 'capital gain'). That IRD determination was made on 13th May 1987. So either I am the first person in the 36 years since publication of that document that has picked up the IRD's error, or I have made a mistake ;-P.

Actually I think I have made a mistake. With shares, where if you sell a share ex-dividend, but the registration process is delayed such that you are delivered the dividend in error, you are required to forward on that dividend to the new owner. With bonds it doesn't work like that. I believe that in a bond, interest due is priced in as though it was being accrued daily. So when you sell a bond coming up to the interest due date, you automatically retain your share of the upcoming interest payment. Or more pertinently, the buyer of the bond does not acquire the portion of the interest that was due to you the seller. That means there is no separate 'refund' transaction as shown in my version of the table. And if there is no refund, that means once all transactions have been tallied up, I make a capital gain, not a capital loss. So I will now rework my own version of the table, given these revised assumptions.

SNOOPY

[Snow Leopard]

....or I have made a mistake ;-P.

Actually I think I have made a mistake....

If it helps I can confirm that you made a mistake.

....I believe that in a bond interest due is priced in as though it was being accrued daily. So when you sell a bond coming up to the interest due date, you automatically retain your share of the upcoming interest payment. Or more pertinently, the buyer of the bond does not acquire the portion of the interest that was due to you the seller.....

I believe I understand what you are saying here, and so yes the 'pricing' allows for accumulated interest.

example: $10,000 of 4% bonds trading at 4.00% 1/4 of a year after the last payment will cost $10,100

You may proceed.

[Snoopy]

Below is the first table, exactly as it appears in Determination G3.

Period Ending	Principal Outstanding	Income in Respect of Period	Payments Received at End of Period
15-05-1987	$1,012,500	$28,815	$70,000
15-11-1987	$971,315	$78,826	$70,000
15-05-1988	$980,141	$79,542	$70,000
15-11-1988	$989,638	$80,317	$1,070,000
Total		$267,500	$1,280,000

The table is modelled using six month timing blocks. After each block 'times through', a $70k interest payment is made. But because the bond acquisition was made half way through the first period, not all of that first $70k interest payment will accrue to our bond buyer, and now bond owner.

I had assumed (during my first attempt) the purchase price of $1,012,500 did not extinguish the requirement for the bond buyer to refund the seller of the bond their interest due ($45,452.05). However, I now believe it does. I can model this by not recording the bond purchase price as $1,012,500, but rather as: $1,012,500-$45,452=$967,048




The first period, ending 15-05-1987, represents 64 days (refer post 2) of a half year period over which $70k of interest would be paid to the bondholder - if they had held those bonds for 6 months. But since 'our bondholder' only held those bonds for 64days, their share of that interest would be just: $70,000 x [64/(365x0.5)]= $24,547.95. That leaves the interest balance ($70,000-$24547.95= $45,452.05) to be refunded to the previous bondholder. (In practice this refund was not a retrospective transaction, as the interest accrued is being priced into the bond in the 'market price' already, on a daily basis).

The total capital gain only (all taxable of course) on these bonds held by our investor over the whole investment period covered in the table was therefore:
($1,280,000 - 4x$70.000) - $967,048 = +$32,952
This represents a capital gain of $32,952/[64+1.5x365]days = $53.89 per day

For income tax purposes, we must add to the 'share of interest' of $24,547.95, the portion of 'extra capital' (capital gain) our bondholder will get back that has been apportioned to this period: 64x$53.89=$3,448.96. So total taxable earnings (income and capital gain) for the initial period now add to: $24,547.95+$3,448.96=$27,996.91





Now we move on to the three six monthly income periods. Interest income for each of those is $70,000. But once again there is a 'capital adjustment' to be made for each period of: $53.89 x (365/2)= $9,834.93. So taxable earnings for the full six month period add to: $70,000+$9,834.93= $79,834.93

As previously discussed (post 1), all of the 'capital adjustments' change the book value of the capital of the bond remaining for income tax purposes (column 2 in the above table). Using my 'first principles' analysis, I will now rewrite the reallocated income table that I have replicated above, as below:

Period Ending	Principal Outstanding (for tax accounting purposes) (1)	Adjusted Gross Income - including capital gains - in Respect of Period	Already Income Incorporated Taxable Capital Gain	Payments Received by End of Period
Day of Opening Investment (2)	$967,048
15-05-1987	$963,599	$27,997	$3,449	$70,000
15-11-1987	$953,764	$79,835	$9,835	$70,000
15-05-1988	$943,929	$79,835	$9,835	$70,000
15-11-1988	$934,094	$79,835	$9,835	$1,070,000
After Investment matures	.$0
Total		$267,502	$32,994	$1,280,000

Notes

1/ Capital adjustments period to period are as follows:
@15-05-1987: $967,048-$3,449=$963,599
@15-11-1987: $963,599-$9,835=$953,764
@15-05-1988: $953,764-$9,835=$943,929
@15-11-1988: $943,929-$9,835= $934,094

2/ Transfer required of income paid to 'our investor' that belonged to the Previous Bond Owner:$ $70,000-$24,548=$45,452. Therefore, the equivalent starting capital was: $1,012,500-$45,452 = $967,048


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The above table looks pretty close to the one published in the IRD Determination D3 now. The 'adjusted income' is $2 more than the reference table. But that $2 is simply a 'rounding error'. If we add back to the 'principal outstanding' my pre-investment 'interest owed' adjustment of $45,452 we get: $934,094+ $45,452 = $979,546. That is about 10 grand smaller than the $989,638 declared in the reference table.

As Snow Leopard has noted (post 12), I have calculated a 'linear income over the entire period of the loan', which is not what the authors of the reference table have done. But it is in accord with IRD Determination G1A which states:

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G1A 4 December 1989

2. This determination requires that income derived or expenditure incurred in respect of a period shall be apportioned on a straight line basis among the income years in which the period falls, according to the number of days in the period calculated on either a 365 or a 360 day basis.

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A 'straight line basis' means 'linear income' in my books.

SNOOPY