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

[Snoopy]

Here is the IRD tax document that is used to equalise earnings across reporting periods, with a worked example. This is something you need to understand if you wish to invest in bonds directly.

The following technical tax bulletin is sub-headed 'Determination G3: Yield to maturity method''

https://www.taxtechnical.ird.govt.nz/-/media/project/ir/tt/pdfs/determinations/financial-arrangements/financial-arrangements-general/determination-g3.pdf?modified=20211123023345&modified=20211123023345

I have been for a long time curious about how this is done. But I have lacked motivation to find out because frankly, up until the last few months I considered bond investment as offering a fixed interest return for an equity risk. However, I always told myself that when I could buy blue chip(ish) bonds at an interest rate starting with a 7, then I might come back to them. And lo and behold that day is here. Be warned, that as I am writing this I don't really know what I am doing. When you don't exactly know what is going on, I find it best to go back to first principles. That is what I will attempt to do in this thread. The whole purpose of this thread is to try and understand the worked example (not fully explained in the IRD document), so here goes.

We will start with someone else's opinion. Below is how SailorRob sums up the process (from the Oceania thread post 16406)

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As for the tax doc, that's just a simple DCF model where you solve for the net present value to be zero and thus determine the discount rate.

That discount rate is used to determine the income for the tax year. I wouldn't get to bothered about that, the tax is exactly the same, you just pay it on $1 of income the same way as the next $1.

It's just a simple way of calculating the income by determining a yield.

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I do note the clock has just passed 9pm, which is beyond the hour when I write my worst stuff. But Rob's explanation looks like 'accounting speak' taken out of context to me.

For a start nothing is being discounted in this IRD example (cashflows shown in bold). If you look at all the sample calculations, there is no 'time value of money' being considered here. The input cost (what the investment was bought for) is quite clear: $1,012,500 (12-03-1987). The total money received once the investment fully matures is also clear: 70k(15-05-1987)+70k(15-11-1987)+70k(15-05-1988)+70K(15-11-1988)+1,000k(15-11-1998) = $1,280,000 (total over time). The overall 'gain' made (including interest income and taxable capital gain) is $1,280,000 - $1,012,500 = $267,500. The only thing that is being changed is when the income is being recognised. Not the amount of money being recognised.

So back to 'first principles' as I promised. Quoting from the referenced technical tax bulletin:

--------------------------------

Principle

The yield to maturity method apportions the total income or expenditure under a financial arrangement so that—

(a) The amount apportioned in respect of each period between payments represents a constant annual rate R on the amount of the principal outstanding during each period; and
(b) The rate R is such that at the time the financial arrangement is issued or acquired the discounted value of the money to be given and received accumulates to zero.

The amount apportioned to each period is then allocated to income years on a daily basis, in accordance with 'Determination G1'.

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A few words of my own, setting out how I see what is being described. Buying bonds 'at a discount to face value' means that you will get more capital back than you laid out when that bond matures (should you hold it to maturity, which for the propose of this exercise I shall assume you do). This 'extra capital' you get back is subject to tax at your marginal income tax rate, whether you are classed as a 'bond trader' or not. You are also taxed on the bond interest income 'as normal'. Despite the extra capital coming to you in 'one hit', when the bond matures, the 'financial arrangement' rules require you to 'spread this future taxable capital gain out' across the remaining term of the bond.

The actual interest payments are 'set in stone' right at the beginning of when the bond was created. Thus all of the 'equalising adjustments' must be made, for tax purposes, by assuming that you are getting some of the 'extra capital back' you are due BEFORE the bond matures. Thus in the year the bond matures, the capital that is returned to you is declared to be the original capital value of the bond, EVEN THOUGH IN CASH TERMS THIS IS NOT TRUE. In reality you will get all of your bond capital back only when the bond matures, despite having had to declare some of this capital gain as income in previous tax years.

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

SNOOPY

[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/-/media/project/ir/tt/pdfs/determinations/financial-arrangements/financial-arrangements-general/determination-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.

---------------------

So putting this knowledge into practice:
.
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 EndingPrincipal OutstandingIncome in Respect of PeriodPayments 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 EndingPrincipal 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

[Nor]

You can avoid all this by being a cash basis person I believe, ie cash plus debts less than is it 1 million?
ie not cash minus debts, using absolute values in math speak.

[Snoopy]

You can avoid all this by being a cash basis person I believe, ie cash plus debts less than is it 1 million?
ie not cash minus debts, using absolute values in math speak.


Yes you 'can' avoid it, but that doesn't mean you 'should' avoid it. Spreading out income more evenly over several years will likely be to your advantage for tax purposes.

More information on being a cash basis person here:
https://www.ird.govt.nz/income-tax/income-tax-for-businesses-and-organisations/types-of-business-income/interest-and-dividends/financial-arrangements-rules

SNOOPY

[mcdongle]

Thankfully i have an accountant....:)

[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. :mellow:

[Nor]

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.
SNOOPY

But ird records the buyer as the receiver of the whole interest payment even though the buyer has given the seller his share of it.

[Snoopy]

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



Period EndingPrincipal OutstandingIncome in Respect of PeriodPayments 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

[Snow Leopard]

I thought that you had realised where you had gone wrong on your first pass.
But you are totally misunderstanding this and basically nothing in your rework is correct, especially your assumptions.

Sometime tomorrow I will see if we can put you on the path to understanding.

[Bjauck]

But ird records the buyer as the receiver of the whole interest payment even though the buyer has given the seller his share of it. One can alter the IRD's prepopulated amounts, and one will need to attach a document explaining why. Perhaps the reporting system cannot cope with the complexity of the actual income tax rules for individuals. Determining what is taxable investment income has a myriad of different regimes. A shareholder's right to a Dividend does not accrue daily but bond interest does. Plus any net realised capital gain on the "financial arrangement" is taxable if the income has not been determined on an accrual basis. This is not advice.

[Snow Leopard]

[All] Bonds are Financial Arrangements but not All Financial Arrangements are Bonds:

If you do a word search on Determination G3 you will not find the word Bond or the word Interest, so amongst the many possibilities this 'Financial Arrangement' between B (the buyer) and S (the seller) could be:

A five year loan with an initial value of $1.5m with 10 semi-yearly payments of $70,000 and a final repayment of $1m that B bought off A;

A unique loan which B & S entered into where the initial value is that $1,012,500 and the repayments are as specified;

The 'Bond' that Snoopy has assumed which B bought from A.

The history of, and precise type of this arrangement is irrelevant, from 12 Mar 1987 to the end on 15 Nov 1988 they all have identical profiles and G3 shows how to correctly calculate the income and make adjustments to the outstanding principal for tax purposes for the periods between payments.

Deciding to throw the method away and calculate the linear income over the entire period of the loan, which is what in a very roundabout way has happened in this post (https://www.sharetrader.co.nz/showthread.php?12697-IRD-Determination-G3-Yield-to-Maturity-Method-for-Bonds-(worked-example)&p=1011677&viewfull=1#post1011677), is a pointless exercise.
(https://www.sharetrader.co.nz/showthread.php?12697-IRD-Determination-G3-Yield-to-Maturity-Method-for-Bonds-(worked-example)&p=1011677&viewfull=1#post1011677)
Whether such a linear approach is even permitted in some other determination or other, I do not know. But given it is simple and straight-forward I very much doubt it.

[Snoopy]

[All] Bonds are Financial Arrangements but not All Financial Arrangements are Bonds:

If you do a word search on Determination G3 you will not find the word Bond or the word Interest, so amongst the many possibilities this 'Financial Arrangement' between B (the buyer) and S (the seller) could be:

A five year loan with an initial value of $1.5m with 10 semi-yearly payments of $70,000 and a final repayment of $1m that B bought off A;

A unique loan which B & S entered into where the initial value is that $1,012,500 and the repayments are as specified;

The 'Bond' that Snoopy has assumed which B bought from A.

The history of, and precise type of this arrangement is irrelevant, from 12 Mar 1987 to the end on 15 Nov 1988 they all have identical profiles


I have hit the pause button here. All of what the Snow Leopard has written above is correct. And yes I did assume our buyer B, bought the 'bond' (if it was a bond) off an earlier buyer A. But as the Snow Leopard says, the exact history of the story of this 'bond' is irrelevant, provided we know the starting conditions, the ending conditions and the interest payments for the bond (if I am allowed to call it that) for the duration of the investment in question.



and G3 shows how to correctly calculate the income and make adjustments to the outstanding principal for tax purposes for the periods between payments.


I would argue it doesn't show the method used, as the author writes in rather ambiguous terms.
"It will be found that the constant annual rate R is 16.2308% per annum"

I went to my front door mat today, lifted it up, and didn't find anything. So if that isn't proof the IRD is not telling the full story, I don't know what is!

SNOOPY

[Snoopy]

I would argue it doesn't show the method used, as the author writes in rather ambiguous terms.
"It will be found that the constant annual rate R is 16.2308% per annum"

I went to my front door mat today, lifted it up, and didn't find anything. So if that isn't proof the IRD is not telling the full story, I don't know what is!


In an attempt to find the mystery of what has gone on here, I have decided to take 'desperate measures'. That means reading the 'Yield to Maturity' reference document right through, starting at the beginning! Look what I found on page 1:

"The approach adopted is to define a constant annual rate R representing the yield to maturity of all the cash flows in the financial arrangement. Income derived and expenditure incurred is assumed to be compounded on the date of each payment."
"In general there is no explicit formula for a yield to maturity in terms of the cashflows."

It sounds like we have to solve an implicit equation, while selecting a suitable payment period that fits the cash flows (in this case six monthly) to 'simplify' things.

"The yield to maturity is defined as the discount rate at which the cashflows accumulate to zero."

"As part of the method, the amount of income derived or expenditure incurred to be compounded at the end of of each period is calculated as a fraction F multiplied by the principal outstanding during the period. This income derived and expenditure incurred is then added to the principal outstanding for the next period (if one exists). The final payment must equal the principal outstanding during the final period plus the income derived or expenditure incurred during that period."

Hmmm. I am starting to think the solution might be to buy all your managed fund investments through a ;fixed interest manager' and forget about all this stuff!

SNOOPY

[Snow Leopard]

....It sounds like we have to solve an implicit equation.

Iterative methods and a spreadsheet :t_up:

[Snoopy]

Iterative methods and a spreadsheet :t_up:

I have found a web reference here that might be useful
https://www.wallstreetmojo.com/yield-to-maturity-ytm-formula/

Yield to Maturity Formula

YTM considers the effective yield of the bond, which is based on compounding. The below formula focuses on calculating the approximate yield to maturity, whereas calculating the actual YTM will require trial and error by considering different rates in the current value of the bond until the price matches the actual market price of the bond. Nowadays, computer applications facilitate the easy calculation YTM of the bond.

Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the years to maturity.

SNOOPY

[Snoopy]

Here is the IRD tax document that is used to equalise earnings across reporting periods, with a worked example. This is something you need to understand if you wish to invest in bonds directly.

The following technical tax bulletin is sub-headed 'Determination G3: Yield to maturity method''

https://www.taxtechnical.ird.govt.nz/-/media/project/ir/tt/pdfs/determinations/financial-arrangements/financial-arrangements-general/determination-g3.pdf?modified=20211123023345&modified=20211123023345

IRD example (Cashflows shown in bold).

The input cost (what the investment was bought for) is quite clear: $1,012,500 (12-03-1987).

The total money received once the investment fully matures is also clear: 70k(15-05-1987)+70k(15-11-1987)+70k(15-05-1988)+70k(15-11-1988)+1,000k(15-11-1988) = $1,280,000 (total over time). The overall 'gain' made (including interest income and taxable capital gain) is $1,280,000 - $1,012,500 = $267,500.

Being changed is when the income is being recognised. Not the amount of money being recognised.




I have found a web reference here that might be useful
https://www.wallstreetmojo.com/yield-to-maturity-ytm-formula/

Yield to Maturity Formula

YTM considers the effective yield of the bond, which is based on compounding. The below formula focuses on calculating the approximate yield to maturity, whereas calculating the actual YTM will require trial and error by considering different rates in the current value of the bond until the price matches the actual market price of the bond. Nowadays, computer applications facilitate the easy calculation YTM of the bond.

Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the years to maturity.


Time to create some weekend amusement for the readers. Watching me make a fool of myself trying to calculate the 'yield to maturity' for this example bond.

When the bond is finally repaid it will be with a $1,000k (i.e. million dollar) capital repayment. It is apparent that the interest payments we know about are $70k every six months. We can use these two numbers to work out the annual 'coupon rate' C as follows:

Coupon Rate = ($70k + $70k)/ $1,000k = 14%. Annual coupon payment is $140k

There is no 'market' for the bonds in the example given. So F=P=$1,012,500.

There is one full year to maturity (year ending 30th November), plus a 'fractional year' which measured in days is:
(19+30+31+30+31+31+30+31+15)/ 365 = 0.6795

So the total number of years 'n' is: 1 + 0.6795 = 1.6795

Putting these numbers into the 'yield to maturity' formula gives me:
= [C] / [(2F/2] = C/F = $140k/ $1,0125k = 13.83%

This makes little sense to me (sigh!) I think my working has been undone because I am meant to be working out 'the market price of the bond' when there is no market.

SNOOPY

[Snoopy]

I have found a web reference here that might be useful
https://www.wallstreetmojo.com/yield-to-maturity-ytm-formula/

Yield to Maturity Formula

YTM considers the effective yield of the bond, which is based on compounding. The below formula focuses on calculating the approximate yield to maturity, whereas calculating the actual YTM will require trial and error by considering different rates in the current value of the bond until the price matches the actual market price of the bond. Nowadays, computer applications facilitate the easy calculation YTM of the bond.

Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the years to maturity.


I think there is a problem with the above formula. It looks 'dimensionally inconsistent' to me. Specifically, what I mean here is that the units measured on each side of any equals sign must be the same. The 'yield to maturity' answer we seek, on the left side of the equation, is a 'percentage figure' which is a dimensionless term.

By contrast on the right hand side of the equation we have 'C', a 'coupon rate' which is also a percentage figure and a dimensionless term (so far so good). But '[F-P]' is measured in 'dollars' and n is measured in 'years'. So the result of (F-P)/n is measured in units of 'dollars per year'. Now I guess you could argue that if you put some money in an interest bearing account at a bank (say $100) at an interest rate of say 5%, then you would earn 5 'dollars per year'. So at a stretch you could say that 'C' and '(F-P)/N' are 'kind of compatible'.

Yet if you accept my above (doubtful) conclusion, you then have to divide a 'percentage figure' by an amount in dollars ( [F+P]/2 ).

So you end up with a percentage number on the LHS of the equation equalling a percentage number divided by a dollar amount on the RHS of the equation. This doesn't make sense. What am I missing?

SNOOPY

[Snow Leopard]

...What am I missing?

SNOOPY

Snoopy.

C is the Coupon (an amount of dollars) not the Coupon Rate ( a fraction or percentage or...)

See example 1 on the page you reference.

The formula is also not precise but approximate and you should refer to this masterful post (https://www.sharetrader.co.nz/showthread.php?12697-IRD-Determination-G3-Yield-to-Maturity-Method-for-Bonds-(worked-example)&p=1012098&viewfull=1#post1012098)

[Snoopy]

IRD example (Cashflows shown in bold).

The input cost (what the investment was bought for) is quite clear: $1,012,500 (12-03-1987).

The total money received once the investment fully matures is also clear: 70k(15-05-1987)+70k(15-11-1987)+70k(15-05-1988)+70k(15-11-1988)+1,000k(15-11-1988) = $1,280,000 (total over time). The overall 'gain' made (including interest income and taxable capital gain) is $1,280,000 - $1,012,500 = $267,500.

Being changed is when the income is being recognised. Not the amount of money being recognised.





Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the years to maturity.




Time to create some weekend amusement for the readers. Watching me make a fool of myself trying to calculate the 'yield to maturity' for this example bond.

When the bond is finally repaid it will be with a $1,000k (i.e. million dollar) capital repayment. It is apparent that the interest payments we know about are $70k every six months. We can use these two numbers to work out the annual 'coupon rate' C as follows:

Coupon Rate = ($70k + $70k)/ $1,000k = 14%. Annual coupon payment is $140k

There is no 'market' for the bonds in the example given. So F=P=$1,012,500.

There is one full year to maturity (year ending 30th November), plus a 'fractional year' which measured in days is:
(19+30+31+30+31+31+30+31+15)/ 365 = 0.6795

So the total number of years 'n' is: 1 + 0.6795 = 1.6795

Putting these numbers into the 'yield to maturity' formula gives me:
= [C] / [(2F/2] = C/F = $140k/ $1,0125k = 13.83%

This makes little sense to me (sigh!) I think my working has been undone because I am meant to be working out 'the market price of the bond' when there is no market.

Thanks to the Snow Leopard for pointing out that the annual coupon 'C' is actually a 'dollar value', not a percentage.

Furthermore I have decided that 'F' and 'P' are probably not equal. The face value of the bond 'F' originally is likely the face value that will be returned to the bondholder, when that bond expires. OTOH the price 'P' paid might well be considered a 'market price'.

So we can substitute these numbers into our 'Yield to Maturity' Formula as follows:

= [C + (F-P)/n] / [(F+P)/2]
=[($70k+$70k) + ($1,000k-$1,012,5k)/1.6795] / [($1,000k+$1,012.5k)/2]
= ($140k-$7.443k) / $1,006.25k
= 13.173%

This is of course, not equal to the constant annual rate of R being 16.2308% per annum that 'will be found' as laid out in the in the IRD 'Yield to maturity' determination G3. Hmmmm?!?

SNOOPY

[Snoopy]

Snoopy.

C is the Coupon (an amount of dollars) not the Coupon Rate ( a fraction or percentage or...)

See example 1 on the page you reference.


Thanks



The formula is also not precise but approximate and you should refer to this masterful post (https://www.sharetrader.co.nz/showthread.php?12697-IRD-Determination-G3-Yield-to-Maturity-Method-for-Bonds-(worked-example)&p=1012098&viewfull=1#post1012098)


An iterative solution is the way to go when you have an equation that cannot be solved algebraically, but has the variable you wish to find on both the RHS and LHS of the equation. But that doesn't seem to apply here. Because the variable you want to find is on the LHS of the equation and all the other variables are known and on the RHS , as far as I can tell.

SNOOPY

[Snoopy]

Time to create some weekend amusement for the readers. Watching me make a fool of myself trying to calculate the 'yield to maturity' for this example bond.

When the bond is finally repaid it will be with a $1,000k (i.e. million dollar) capital repayment. It is apparent that the interest payments we know about are $70k every six months. We can use these two numbers to work out the annual 'coupon rate' C as follows:

Coupon Rate = ($70k + $70k)/ $1,000k = 14%. Annual coupon payment is $140k

There is no 'market' for the bonds in the example given. So F=P=$1,012,500.

There is one full year to maturity (year ending 30th November), plus a 'fractional year' which measured in days is:
(19+30+31+30+31+31+30+31+15)/ 365 = 0.6795

So the total number of years 'n' is: 1 + 0.6795 = 1.6795


There is another way of looking at this problem. I can work out the 'period return' by looking at returns every six months (rather than annually) , because payment is received every six months. Then once I get the six month return number, I can multiply the two adjacent six month period returns to get the annual return.

Under this approach we have four return periods, not two.

i/ The coupon for each period becomes C=$70k, not $140k.
ii/ The length of each full return period in days becomes 365/2= 182.5 days.
iii/ The first return period is the short one from 12th March to 15th May. In terms of days that equates to 19+30+15= 64. In terms of a 'fraction of the return period' this amounts to: 64/182.5= 0.3507. This means the total number of return periods over which our investment runs pans out to: n= 3+0.3507 = 3.3507.
iv/ The face value of the bond F=$1,000k AND
v/ the market price of the bond P=$1,012.5k remain unchanged.

Now let's put these new numbers into our return formula and see what comes out:

Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the 'investment periods' to maturity.

YTM = [ $70k + ($1,000k-$1,012.5k)/3.3507 ] / [ ($1,000k + $1,012.5k)/2 ] = $66.27k /$1,006.25k = 6.584%

This 'yield to maturity' is over six months. To get the return over twelve months, we have to multiply two six month period returns together:

1.06584 x 1.06584 = 1.13601, which equates to an annual rate of 13.60%

That is a little bit higher than the 13.17% I calculated in post 20, due to the compounding effect across investing periods more closely reflecting actual cashflow timing.

But it isn't the 16.2308% figure we were looking for. Hmmmmm.

SNOOPY

[Snoopy]

[All] Bonds are Financial Arrangements but not All Financial Arrangements are Bonds:

If you do a word search on Determination G3 you will not find the word Bond or the word Interest, so amongst the many possibilities this 'Financial Arrangement' between B (the buyer) and S (the seller) could be:

A five year loan with an initial value of $1.5m with 10 semi-yearly payments of $70,000 and a final repayment of $1m that B bought off A;

A unique loan which B & S entered into where the initial value is that $1,012,500 and the repayments are as specified;

The 'Bond' that Snoopy has assumed which B bought from A.

The history of, and precise type of this arrangement is irrelevant, from 12 Mar 1987 to the end on 15 Nov 1988 they all have identical profiles and G3 shows how to correctly calculate the income and make adjustments to the outstanding principal for tax purposes for the periods between payments.


The above post from the Snow Leopard has got me thinking




i/ The (annual) coupon for each period becomes C= $140k.
iv/ The face value of the bond F=$1,000k AND

v/ the market price of the bond P=$1,012.5k remain unchanged.


To get the return over twelve months, equates to an annual rate of 13.60%

That is a little bit higher than the 13.17% I calculated in post 20, due to the compounding effect across investing periods more closely reflecting actual cashflow timing.

But it isn't the 16.2308% figure we were looking for. Hmmmmm.


There has been a particular fundamental flaw in my calculations of the 'Yield to Maturity' (YTM) figure to date. My above calculation is from a base level of assuming a $1m 'bond' trades over an unknown period before it is ultimately repaid on 15-11-1988 for $1m ($1,000k).

We do know for sure what the coupon payment 'C' for this bond is: Two payments of $70k per year, amounting to $140k per year. If I assume those payments are based around the ultimate capital to be repaid of $1,000k, this means our 'bond' has a 'Coupon Rate' of:

$140k/$1,000k = 14%

However, this annual 14% coupon rate figure cannot be correct. Why is that?

We are told the 'yield to maturity' of this financial arrangement is 16.2308%. In order to get a yield to maturity greater than the coupon rate, an investor must buy that bond at a discount. Yet, we were told the bond was bought at a total price of $1,012.5k. Paying $1,012.5k for a $1,000k 'bond' is doing the opposite - paying a premium. Yet we know that the price paid for the bond, $1,025.5k is right. So it must be that the bond issue price was greater than the base capital figure that I used, greater than $1,012.5k in fact. And that means that my assumption that the bond was issued at its redemption price of $1,000k must be wrong.

This in turn means that the Snow Leopard conjecture that what we are looking at is:
"A five year loan with an initial value of $1.5m with 10 semi-yearly payments of $70,000 and a final repayment of $1m that B bought off A"

could be the right answer, - or is a least on the right track.

SNOOPY

[Snoopy]

Yield to Maturity Formula = [C + (F-P)/n] / [(F+P)/2]

Where,

C is the Coupon.
F is the Face Value of the bond.
P is the current market price.
n will be the 'investment periods' to maturity.



Following my revelation in post 23, I can now return to the above formula, and solve for F, the face value of the bond (because we know the value of all the other numbers in the equation.

Looking again at this calculation on an annual basis:

C is the Coupon = $140k.
F is the Face Value of the bond (to be determined).
n is the 'investment periods' to maturity. 1.6795 years (post 17)
and the YTM we are told is 16.2308%.

So putting the above numbers into the YTM formula we get:

0.162308 = [$140k + (F-$1,012.5)/1.6795] / [(F+$1,012.5)/2]

Now we need to guess a value of F that will make the above equation true. How about we guess a bond face value of $1,059.82?
Put that number in for 'F' and we get the YTM number we are seeking! What an incredible stroke of luck to guess the number right first time ;-) !

SNOOPY

[Snoopy]

So putting the above numbers into the YTM formula we get:

0.162308 = [$140k + (F-$1,012.5)/1.6795] / [(F+$1,012.5)/2]

Now we need to guess a value of F that will make the above equation true. How about we guess a bond face value of $1,059.82?
Put that number in for 'F' and we get the YTM number we are seeking! What an incredible stroke of luck to guess the number right first time ;-) !



For those that have not figured this out, there was no luck involved at all in my guess, because this is an exercise in algebra. We are given the equation below to solve for F

Y = [C + (F-P)/n] / [(F+P)/2]

<=> [(F+P)/2]Y = [C + (F-P)/n]
<=> (F+P)Y = [2C + 2(F-P)/n]
<=> n(F+P)Y = 2nC + 2(F-P)
<=> nFY+nPY = 2nC + 2F-2P
<=> 2P+nPY = 2nC + 2F-nFY
<=> F(2-nY) = P(2+nY) - 2nC
<=> F = [P(2+nY) - 2nC]/(2-nY)

Stick in the numbers you know of the RHS of the equation and you get 'F', the 'capital value' of the bond.

SNOOPY

[Snoopy]

F = [P(2+nY) - 2nC]/(2-n)

Stick in the numbers you know of the RHS of the equation and you get 'F', the 'capital value' of the bond.



I think it is worth going through this equation again, this time using a six month investment period for reference.

C is the Coupon = $70k.
F is the Face Value of the bond (to be determined)
P is the current market price of the bond: $1,012.5k
n is the 'investment periods' to maturity. = 3,3507 sets of six months (post 21)
Y, the yield to maturity we are told is 16.2308% for the year, implying a multiplication factor of 1.162308. For six months the multiplication factor must be the square root of that figure.: 1.078104. This is equivalent to a yield to maturity of 7.8104% compounding for every six months.

F = [P(2+nY) - 2nC]/(2-nY)
= [ $1,012.5k(2 + 3.3507x0.078104) - 2 x 3.3507 x $70k ] / (2-3.3507 x 0.078104)
= [ $2,290.0k - $469.1k ] / 1.7383
= $1,047.52k

This is slightly different result to the annual figure calculation that I did before (post 24), because in this calculation the coupon is generated more frequently (every six months, not summed over a year). That in turn means you need slightly less discount on your bond purchase price to create the same coupon income stream per unit of investment time.

SNOOPY

[Snoopy]

When all else fails, go back to the original data.

https://www.taxtechnical.ird.govt.nz/-/media/project/ir/tt/pdfs/determinations/financial-arrangements/financial-arrangements-general/determination-g3.pdf?modified=20211123023345&modified=20211123023345



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



Period EndingPrincipal OutstandingIncome in Respect of PeriodPayments 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.

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 capital loss)

But the IRD table is showing income higher than the coupon rate received (i.e. our bondholder has received a 'capital gain').

(Note: That IRD determination example was made and published way back on 13th May 1987).


There is something very strange about the above table (taken straight from the referenced IRD document I might add).

"How can one table show both a capital loss and a capital gain for the same set of cashflows?"
This is the question I want answered that I do not understand!

It is quite clear that when we take out the 4x$70k 'coupon payments' our investor has made a capital loss over the investment period of $12,500 (at the cash transactional level).

Yet we also know that the 'deemed constant rate of return. on this investment of 16.2308% is greater than the coupon rate (based on the bond redemption price) of 14%. The only possible way to explain this 'interest rate differential' is that our investor bought the bond at a discount and will make a capital gain. Indeed those last three deemed 'income in respect of period' payments being greater than the coupon rate of $70k would suggest this.

This table above is showing us that the IRD is requiring us, as the new bond holder, to declare a 'capital gain' over an 18month+ period over which we make a net cash loss on our investment. WTF?

At times like this I am tempted to lie on top of my doghouse and just go "woof, woof." Much easier than playing this bond investment game.

Yet there must be more to this than what our table is showing at first glance. So some of my own thought processes, as I try to get a handle on what is going on, will follow.

SNOOPY

[Snoopy]

There must be more to this than what our table is showing at first glance. So some of my own thought processes, as I try to get a handle on what is going on, will follow.


That first $70k coupon payment was 'not fully earned' by our investor, because our investor bought into the bond part way through that first shown investment period (on 12-03-1987, which was 64 days before the ex-coupon pay date). Put another way, our investor was only entitled to part of that 'investment coupon payment'. Rather than have our investor refund part of that payment to the previous owner of that bond, 'the bond market' deals with this situation in a different way to the sharemarket.

IF this was a share poised to pay a $70k dividend on the ex-dividend date, THEN as a 'share owner', one would expect the market value of one's shares to drop by $70k after the share ex-dividend date (all other factors in the market being equal). But this isn't what happens on the bond market. Instead, the market prices a bond in such a way that any 'coupon payment' is, in this case, spread across each of the days in the half year between actual coupon payments occurring, on an equal basis. Thus the 'refund' that 'our bond owner' was due to make to 'the previous bond owner' was incorporated into the capital price our new bond owner paid for the bond on market. In this way, the 'income loss', required to be paid back to the former bond owner, by our new bond investor, has been 'adjusted for' by our new bond investor taking an equivalent 'capital loss'. And the way our new investor takes this 'capital loss' on market, is to pay a 'capital premium' above the value for the underlying capital asset.

Thus from an IRD tax 'currently owned bond investor perspective', both 'capital gains and/or losses' and 'coupon payments' get mixed up into one 'payment pot'. How much did our bond investor 'earn' during that 64 day portion of the half year ownership period during which our bond investor made their bond purchase? It is all based on a constant annual rate R which we are told is 16.2308% (this number is too difficult for you plebs reading the IRD documentation to work out. So just accept what we, the IRD, tell you., O.K.?)

All right, I admit the IRD did not actually state that bit in brackets. But that is my take on what was implied. I am calling BS on this 'patronizing position' as espoused by the IRD. In order to calculate this 'constant annual return rate' you need to know the opening capital value of each bond when it was first set up. I had naively assumed that this must have been the same as the capital value paid back at maturity. But this assumption must be wrong (see my post 23). Since we are not given the opening capital value position of the bond, we do not know how much capital was 'lost' at repayment time. And if we don't know how much capital was lost, that means we do not have the information needed to calculate the constant annual return rate 'R' over any time period. Thus we are forced to accept the constant annual return rate figure of 16.2308% figure is 'true', with no way to verify that figure.

Maybe I am reading this whole thing the wrong way. But I think this apparent lack of disclosure by the IRD on this matter in what purports to be an 'explanatory determination' is disgraceful for a worked example that is there to show 'we plebs' how interest income should be distributed over multiple time periods.

From the referenced IRD document:
"In general there is no explicit formula for a yield to maturity in terms of the cashflows."

So what is this then?
https://www.wallstreetmojo.com/yield...y-ytm-formula/

YTM = [C + (F-P)/n] / [(F+P)/2]

SNOOPY

[Snow Leopard]

Iterative methods and a spreadsheet :t_up:

PM me an email address and I will send this Excel Spreadsheet:

14740

14741

14742

Molehill >>> Mountain.

[Snoopy]

From the referenced IRD document:
"In general there is no explicit formula for a yield to maturity in terms of the cashflows."

So what is this then?
https://www.wallstreetmojo.com/yield...y-ytm-formula/

Y = YTM = [C + (F-P)/n] / [(F+P)/2]


OK, I have found another reference on this topic which is more in line with what I was expecting to find.
https://www.omnicalculator.com/finance/yield-to-maturity

Bond Price= ∑(k=1,n) [ C/ (1+Y)^k ]

Where 'C' is the coupon (as before), and k is the 'investment period' for a whole series of cashflows, starting at 1 and ending at n, and lastly Y is the 'yield to maturity' (as before).
​
This is a 'summation calculation of the cashflows' formula that I was expecting. Under certain circumstances, there are ways of summing these 'sigma' calculations with a simplified formula. I suspect the YTM calculation in the quoted text above might be that, for the special case where 'C' is the same for all investment periods. But I should get the same answer 'either way' . So let's see.......

The 'bond price' in the equation in this post refers to the price the investor must pay to acquire the bond, which = $1,012.5k in this case.


​
,

[Snoopy]

PM me an email address and I will send this Excel Spreadsheet:


Thanks for the offer Snow Leopard, but those 3 screen shots you gave allowed me to create the spreadsheet myself. Breaking down the spreadsheet into an underlying form, what we have here is a series of four sequential equations covering the four sequential investment time periods. The job of the spreadsheet is to select an unknown 'yield to maturity' figure to make the whole sequence of equations sum to zero. The four sequential equations from the spreadsheet being:

$989,683.38 (1 + Y/2) - $1,070,000 = 0
$980,141.15 (1 + Y/2) - $70,000 = $989,683.38
$971,315.18 (1 + Y/2) - $70,000 = $980,141.15
$1,012,500.00 (1 + 64Y/365) -$70,000 = $971,315.18

It strikes me that those intermediate principal figures always connect one investment period with a subsequent one. Their only purpose is a connective function . That means you don't need to know the exact figures they are. And the above equations can equally well be represented as below:

$A (1 + Y/2) - $1,070k = 0
$B (1 + Y/2) - $70k = $A
$C (1 + Y/2) - $70k = $B
$1,012.5k (1 + 64Y/365) -$70k = $C

Looking at the $A amount in the first equation, we can eliminate $A by substituting from the second equation.
($B (1 + Y/2) - $70k) (1 + Y/2) - $1,070k = 0

Likewise we can eliminate $B by using the third equation
(($C (1 + Y/2) - $70k) (1 + Y/2) - $70k) (1 + Y/2) - $1,070k = 0

Lastly we can eliminate $C by using the fourth equation
[{($1,012.5k (1 + 64Y/365) -$70k) (1 + Y/2) - $70k} (1 + Y/2) - $70k] (1 + Y/2) - $1,070k = 0

The above equation looks complicated. But taking out the dollar values in the calculation and looking at the above equation representing 5 'sequential events' (numbered sequentially 1,2,3,4,5, the fifth being the final repayment of the bond), the equation takes on the simpler form below.
[{(1)2}3](4) - (5) = 0

That might make it easier to see what is going on for some.


Observations

i/ The above 'combined equation' represents four periods of investment 'sequentially cascaded together' over four distinct time periods.
ii/ The return on the first period (innermost bracket (1) ) gets multiplied by the returns in the subsequent period (2), and so on.
iii/ The negative numbers in the above equation represent the money being 'pulled out' by our investor. Because this money is removed, it does not get 'multiplied up' to produce downstream earnings in subsequent periods, which is all as you might expect.
iv/ The 'total of the money put in' is less than the 'total of the money pulled out'. We can explain this by considering the original investment ($1,012.5k) as being multiplied by a multiplier containing a 'constant yield factor Y' (scaled when any constituent time period is shortened of course), with the retained earnings of such an investment subsequently 'multiplied forwards'. This yield factor Y is unknown and must be 'solved' to complete the equation.

Personally I find the above 'combined equation', that I have highlighted in bold, better communicates what is going on in this valuation process than the spreadsheet. But I guess other peoples' mileage may vary?

SNOOPY