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

[Snoopy]

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

https://www.taxtechnical.ird.govt.nz...20211123023345

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.

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

[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/finan...ld-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).
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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.


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