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

[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

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...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

[Snow Leopard]

Iterative methods and a spreadsheet

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

Xcel-1.png

Xcel-2.png

Xcel-3.png

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/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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