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

[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, is a pointless exercise.

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

[Snoopy]

Iterative methods and a spreadsheet

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

[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