Rights Issue Fraction to arrive to an average number of shares

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Sariche

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Hello,

Can anyone please help me understand why, when there is a Rights Issue, we need to apply the bonus fraction being the Market Value of shares/TERP to a number of shares to arrive to an average number of shares? I mean, I understand that we need to adjust the number of shares retrospectively to make all periods comparable, but mathematically, I do not understand why we are multiplying number of shares by a $ shares fraction to arrive to an average number of shares. I am not good at maths but I would have thought that multiplying 5 shares x $5 gives me $10 and not 10 shares.

What am I missing?

Thanks in advance
 
First, you understand that it is an ESTIMATED value per share. There can be numerous different estimates.

Second, I still do not understand what you are trying to do. Let’s first do such an estimate.

Say there are a million shares outstanding and the market value per share is $50. So the theoretical value of the company is $50 million.

And let’s say that each shareholder can buy one additional share per share held at $45 dollars per share. If all the shareholders exercise all their rights, the company will raise $45 million and will be worth $95 million and have 2 million shares outstanding, resulting in a value per share of $47.50. If no shareholder exercises any of the rights, the company will still be worth $50 million and still have 1 million shares outstanding. The value per share will still be $50. In other words, what the TERP is a guess at. what the value per share will be depending on how many of the rights are exercised.

You with me to here?

Can you rephrase your question. It sort of looks as though you are trying to use the ratio of the market price over the TERP to calculate the number of new shares issued.
 
Many thanks for your reply but it is the use of the bonus/rights fraction that confuses me. Let me try and explain a bit better: when calculating Earnings Per Share for a particular period, if during that period there has been a rights issue, say mid year, assuming it has been exercised, we need to adjust the number of existing shares prior to the rights issue in order to get to an average # of shares and calculate our EPS for the year, or two separate EPSs would need calculating within that year. What I do not undertand, is that in order to calculate that average number of shares, we apply what they call a bonus or a rights fraction to all previous periods within the year and this fraction is FV/TERP, so it a money fraction we are using here, not a # of shares fraction to arrive to an average # of shares. I will partly use your example:

At 01/01 company A had 1 million ordinary shares in issue. On 30/06 a rights issue of 1 for 1 at $45 was made and all rights were exercised. The FV of the shares immediately before the rights issue was made was $50. Say profit for the year was $ 655,000

First we need to adjust the # of shares for periods prior to the rights issue, so 1000000 x bonus fraction*

Bonus fraction = $50 / $47.5

Adjusted # of shares for the first half of the year 1000000 x 50/47.5 = 1052632

Then we need to time apportion all shares

1052632 x 6/12 = 526316
2000000 x 6/12 = 1000000

Weithed average # of shares within that year was 1,526,316.

EPS= 655000/1526316 = 43c per share

Why, when doing the adjustment for an average, are we multiplying # of shares by a $ fraction to get # of shares, that is my question.

Thanks in advance

Sara
 
This is the calculation under International Accounting Standard IAS 33 not sure if in different countries do it differently
 
OK I have found the citation. You did not initially say that this was a question about accounting and reported earnings per share. The citation does not explain why that formula was chosen. Decades ago, when I studied accounting theory, requirements were influenced by availability of objective evidence as well as economic reasoning. But I shall think about the formula in terms of math and economics and see whether I can figure out the rationale (if there is one).

The first thing to note is that the values involved are ACCOUNTING values, so called “fair value,” rather than actual market values. When I was a corporate director, I never paid any attention to the accountants’ “fair value“ calculation because it added together historical costs, estimates, legal values, and market values as of a particular date. It seemed to me to be a logical mess.
 
Many thanks for looking into it and apologies for not being clear enough since the beginning
 
Let’s start with the time adjustment.

Suppose, for the first half of the year, there were 1 million shares outstanding. On July 1, the company sold another million shares on the open market at a price of 50 dollars per share. For the second half, there were 2 million shares outstanding. It seems reasonable to calculate the average number of shares outstanding at

[math]\dfrac{1}{2} * 1000000 + \dfrac{1}{2} * 2000000 = 500000 + 1000000 = 1500000.[/math]
Suppose, however, there was a one for one stock split at opening of business on July 1. Each share in the first half of the year magically became two shares in the second half of the year. Now it seems reasonable at year end to say that there were 2 million of the current type of share outstanding for the entire year.

If there were shares issued under a rights issue, we have an intermediate situation.

What is a simple way but reasonable and consistent way to account for any situation?

What the accountants do is to weight the number of shares by “fair value“ per share and by time outstanding.

How to calculate the “fair value per share“ weight? Calculate the fair value of the company immediately before the transaction and divide by the number of shares outstanding immediately before the transaction. That gives a before-value in dollars per share, and lets us compute the fair value of the firm at that instant. Then calculate the fair value of the company immediately after the transaction and divide by the number of shares outstanding immediately after the transaction. That gives an after-value in dollars per share. Now take the ratio of the before-value over the after-value. Dollars per share in the numerator and in the denominator cancel out, and we are left with a pure number to get our weight.

Let’s see how this works out in the extreme cases.

We had a million shares and sold a million more at a price of $50 per share. What do the accountants say is fair value in this case? They say that we have objective evidence that the value of each share before the transaction occurred is at least 50 dollars per share. So we estimate the before-value at $50 per share, giving an estimated fair value for the firm of $50,000,000 before the transaction. But the transaction added $50,000,000 to the assets of the firm without any increase in liabilities. So the fair value per share after the transaction is

[math]\dfrac{50,000,000 + 50,000,000}{1,000,000+ 1,000,000} = \dfrac{100,000,000}{2,000,000} = 50 \text { dollars per share}[/math]
But the ratio of 50 / 50 is 1. So

[math]\dfrac{1}{2} * 1 * 1,000,000 + \dfrac{1}{2} * 2,000,000 = 500,000 + 1,000,000 = 1,500,000.[/math]
This gives us the common-sense result shown above.

Now let’s think about the stock split example: we went from a million to two million shares with an old stock price of 50 per share. The new shares were issued for nothing or zero. The value of the underlying firm has not changed. The only objective evidence of the fair value per share before the split is the market value per share before the split of $50. So the fair value per share after the split is

[math]\dfrac{50,000,000 + 0}{1,000,000 + 1,000,000} = \dfrac{50,000,000}{2,000,000} = 25 \text { dollars per share}[/math]
But the ratio of 50 over 25 is 2.

[math]\dfrac{1}{2} * 2 * 1,000,000 + \dfrac{1}{2} * 2,000,000 = 1,000,000 + 1,000,000 = 2,000,000.[/math]
Again, this gives us the common-sense result shown above. What the weighting does is to count shares before the transaction in terms of relative fair value to shares after the transaction.

Let’s see what we are doing in common sense terms. We know the market price per share before the transaction. So the fair value of the firm before the transaction is simply the price per share times the shares outstanding. The fair value of the firm after the transaction is the sum of the fair value of the firm before the transaction and the cash realized from the transaction. To get the new fair value per share, we simply divide the fair value of the firm after the transaction by the number of shares outstanding after the transaction. We compute our fair-value weight as the ratio of the before value per share over the after value per share. It is all a very common sense way to make a reasonable adjustment in a consistent way. Let’s apply it to the rights case.

We have a million shares outstanding on June 30 with a market price per share. Under a rights offering, an extra million shares are issued at a price of $45 per share. The old value of the firm was $50 million. The new value is
$50 million + $45 million = $95 million. There are now 2 million shares outstanding. That gives a new value per share of $47.50.

We say the shares outstanding before July 1 count as 50/47.50 as many shares as the shares after June 30.

It is just expressed in fancy words.
 
Many thanks for your explanation, but I am still having trouble understanding

1/2 * 1000000 *50/47.5 = 526316

526316 * $47.50 is the equivalent of having 500000 * $50? So $ 25 million.

But what does the $1.05 (50/47.5) represent exactly?
 
Would it be translated into something like '$25 million would buy me 526316 shares @ $47.5? So there is an increment of 1.05 in the number of shares compared to the amount of shares I can buy @ $50?
 
First, you understand, I hope, 50/47.50 does not exactly equal 1.05. The 1.05 is an approximation, and not a good approximation for many purposes.

Second, saying $1.05 is just a mistake.

[math]\dfrac{50 \text { dollars}}{47.50 \text { dollars}} = \dfrac{50 \cancel { \text { dollars}}}{47.50 \cancel { \text { dollars}}} = \dfrac{50}{47.50} \approx 1.05.[/math]
The dollars cancel. 50/47.50 is just a number.

Third, do you fully understand why, when we sold additional shares for cash, we effectively adjusted the number of shares only for time?

Do you fully understand why, when we split the shares, we effectively adjusted the number of shares only for fair value?

This is really my most important question. If the reasons for those adjustments are hazy, the rights example will be meaningless.
 
Sorry but my maths knowledge is very basic, so 50/47.50 is a ratio, not a fraction, so what does this ratio mean/represent? why are we putting 50 in the numerator and 47.50 in the denominator? Would it be that each one of my old shares at fair value prior to the rights issue count for approx 1.05 of the new shares at the theoretical price? Is that what you mean by what the weighting does is to count shares before the transaction in terms of relative fair value to shares after the transaction?

I understand that we have to time apportion the number of shares before and after the rights issue. Obviously I do not fully understand why we are applying the bonus fraction retrospectively.
 
It would help if I knew whether you were taking a course (and if so in what) or if you are merely a curious investor.

Mathematically, a ratio is a fraction, but we tend to think of a fraction as part of a whole, and that is not what is going on here. We are talking about relative values before and after a date. I still think the easiest way to follow what the accountants are thinking is to consider a stock split rather than a rights issue.

Lets consider the following example. A company earned 30 million dollars from January 1 through December 31 and had 1 million shares outstanding from January 1 through November 30, but it split the stock in the ratio of 4 shares for 1 on December 1. At the end of the year, 4 million shares are outstanding.

On average, how many shares were outstanding during the year.

[math]\dfrac{1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 4}{12} \text { million} = 1.25 \text { million.}[/math]
And this is what the time adjustment factor does more simply.

[math]\dfrac{11}{12} * 1 + \dfrac{1}{12} * 4 = 1.25.[/math]
Let’s calculate earnings per share using what is the actual average number of shares outstanding through out the year.

[math]\dfrac{30 \text { million dollars}}{1.25 \text { million shares}} = 24 \text { dollars per share.}[/math]
It is February of the following year. You are thinking of buying a share of stock in this company. Let’s say its current market price is 120 dollars a share. You see that last year it earned 24 dollars a share. You say this stock is selling at five times earning, what a bargain! But in fact, the proportionate claim of each share now outstanding in those earnings is

[math]\dfrac{30 \text { million dollars }}{4 \text { million shares}} = 7.50 \text { dollars per share.}[/math]
In terms of the shares now outstanding, it is selling at sixteen times earnings. There is a huge difference in meaning between five times earnings and 16 times earnings.

The 24 dollars per share that we calculated using the actual average number of shares outstanding during the year is highly deceptive. Why? Because each share before the split represented a one millionth interest in the company, and each share after the split represented a one quarter of a millionth interest in the company. We were adding apples and oranges when we added number of pre-split shares and number of post-split shares.

We adjust the number of pre-split shares retrospectively so that each represents the same interest in the company as does a post-split share. We do this by multiplying by 4. Each pre-split share was worth 4 of the post-split shares. Our new computation is

[math]\dfrac{4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4 * 1 + 4}{12} \text { million}\\ = 4 \text { million.}[/math]
Now we calculate earnings per share as

[math]\dfrac{30 \text { million dollars}}{4 \text { million shares}} = 7.50 \text { dollars per share.}[/math]
And that is resonable.

The case of a stock rights issue raises the same type of problem but in much more complex form because the rights issue not only changes the number of shares outstanding but simultaneously changes the value of the company. The formula is a simple but consistent method to calculate the retrospective adjustment to the NUMBER of shares in a way that considers both the change in value of the company and the change in the the number of shares resulting from a single transaction.

So YES, (going back to the original example), the adjustment factor of 50/47.50 represents a reasonable and objective quantitative estimate of the fact that “each one of [the] old shares at fair value prior to the rights issue counts for about 1.05 of the new shares.”

You seem to have the concept. The reason for making a retrospective adjustment is to make the earnings per share reflect current shares, not outdated shares. It is important to remember that accountants make estimates. They want such estimates to be consistent, reasonable, and based on objective evidence; they do not worry about economic perfection.

Have I answered your question or do you still want to understand the logic of the formula?
 
You have answered all my questions, now I can see it clearly. Thanks so much, your explanations have been very helpful.

I am indeed sturying ACCA, Financial Reporting. I am doing it on my own and then whenever I feel ready I sit the exam, so I usually get the information from books and from websites which offer free tuition. The tutors there told me to just learn how it is done, but I want to understand what I am doing, it makes sense to me. Otherwise, why studying?

Thanks again
 
At least 50 years ago, I took a course in accounting theory. I was amazed at how many students in the class had no interest whatsoever in why accounting was done in different ways in different countries or what the advantages and disadvantages were with different methods. Most of the students were interested only in how to apply what was then standard practice in the U.S.

I am not surprised that you were given no rationale at an accounting site.
 
I see. I cannot do that. I cannot retain anything that I do not understand and I am not finding much help amongst accountants, maybe because they do not know the rationale behind these methods but I want to think it is because they just cannot be bothered to give so much information for free.

Thanks again for your explanations. They have been of great help. I will add them to my study notes.
 
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