## Request for help in understanding relationship between distributions, share value, and returns

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Topic Author
theostrich
Posts: 1
Joined: Mon Aug 12, 2019 2:06 pm

### Request for help in understanding relationship between distributions, share value, and returns

I've been reading this fantastic forum for about a year. Thank you to all who contribute. I've been learning so much.

I'm really stuck on a few concepts. I've read many posts related to distributions, share value, and returns in the forum, but something is not making sense to me.

Can someone please help fill in or correct what I am missing?

To simplify, assume I have a single share of a fund in a tax-deferred account. The fund does not distribute dividends or capital gains. I don't buy or sell any shares and hold it for 2 years. Research on the fund says that the annual return is 5%.

Here is my understanding/questions:
1. When we talk about an annual "return" we are talking about the change in value of the initially invested money, not the value of the thing you own (the share).

So after two years (assuming the same return rate for the second year) the returns would be:
YR1 total = initial investment + (initial investment)5%.
YR2 total = YR1 total + YR1 total(5%).

This describes returns on the previous years returns or "compounding".

2. If I only have one share (and no distrubutions), did the value of the share increase by 5%? If I still hold only a single share for the second year and only the value of the share increases by 5% doesn't that mean the total "return" is 10%? Where are the "returns on returns"? Is that not happening in this case?

3. I removed dividends from the example to simplify. You don't get anything that you didn't already have when a fund distributes a dividend. The NAV goes down to compensate for the distribution, leaving you with the same investment value (if you reinvest the dividend). But you now have more shares. The math (from a previous BH post) shows that it doesn't matter, you haven't gained anything. But it seems like you have gained the potential for faster future growth. Is this not true because the funds ability to grow is now decreased because of the distribution? This seems to be widely misunderstood - as so many people on this forum and others tout the "benefits" of dividends.

4. I know that dividends are not "free" and the reported "return" number for a fund includes reinvested distributions, and does not directly correlate to the change in NAV. But I don't understand how "returns on returns" happen or why this doesn't make sense to me.

Thanks in advance.

dbr
Posts: 29798
Joined: Sun Mar 04, 2007 9:50 am

### Re: Request for help in understanding relationship between distributions, share value, and returns

theostrich wrote:
Tue Aug 13, 2019 1:17 pm
I've been reading this fantastic forum for about a year. Thank you to all who contribute. I've been learning so much.

I'm really stuck on a few concepts. I've read many posts related to distributions, share value, and returns in the forum, but something is not making sense to me.

Can someone please help fill in or correct what I am missing?

To simplify, assume I have a single share of a fund in a tax-deferred account. The fund does not distribute dividends or capital gains. I don't buy or sell any shares and hold it for 2 years. Research on the fund says that the annual return is 5%.

Here is my understanding/questions:
1. When we talk about an annual "return" we are talking about the change in value of the initially invested money, not the value of the thing you own (the share).

So after two years (assuming the same return rate for the second year) the returns would be:
YR1 total = initial investment + (initial investment)5%.
YR2 total = YR1 total + YR1 total(5%).

This describes returns on the previous years returns or "compounding".

This is correct. There may be a confusing statement in your reference to the "initially invested money." As you have already correctly stated the 5% applies repeatedly to the investment value at the beginning of each year, which would not be the "initially invested money" at the beginning of the first year. Just delete that whole idea and then everything is fine.

2. If I only have one share (and no distrubutions), did the value of the share increase by 5%? If I still hold only a single share for the second year and only the value of the share increases by 5% doesn't that mean the total "return" is 10%? Where are the "returns on returns"? Is that not happening in this case?

At the end of the first year the value of the share (which is also the value of your investment) will have been increased by 5%. At the end of the second year that value will be increased by another 5% for a total of 10% plus 5% of 5% or 10.25% The GAIN over two years is 10.25% but return is always a rate relative to time so that remains 5%/year and it is compounded.

3. I removed dividends from the example to simplify. You don't get anything that you didn't already have when a fund distributes a dividend. The NAV goes down to compensate for the distribution, leaving you with the same investment value (if you reinvest the dividend). But you now have more shares. The math (from a previous BH post) shows that it doesn't matter, you haven't gained anything. But it seems like you have gained the potential for faster future growth. Is this not true because the funds ability to grow is now decreased because of the distribution? This seems to be widely misunderstood - as so many people on this forum and others tout the "benefits" of dividends.

If there are distributions and they are reinvested you have more shares each of which is worth less relative to what they would have been if the return rate doesn't change. The end result is you just continue to apply the math you are already doing using the dollar value of the investment and avoid dealing with the number of shares and the value of a share.

4. I know that dividends are not "free" and the reported "return" number for a fund includes reinvested distributions, and does not directly correlate to the change in NAV. But I don't understand how "returns on returns" happen or why this doesn't make sense to me.

The formula for returns on returns works like this:

start of account invest A0
end of first year at return of r account value A1 is A1 = A0 + r * A0 = (1 + r) * A0
end of second year account value A2 is A2 = (1 + r) * A1 = (1 + r)^2 * A0
. . .
end of year n account value An is An = (1 + r)^n * A0

The interesting thing is, for example at end of year 2 (1 + r)^2 = 1 +2r + r^2

1 is the original amount
2r is two years return on the original amount
r^2 is the return in the second year on the return earned in the first year

At the end of the 3rd year (1+r)^3 = 1 + 3r + 3r^2 + r^3

It is left as an exercise to untangle what is return on the original investment, what is return on the return in the first year, what is return on the return in the second year.

Thanks in advance.