In a recent post, I compared the effect of dividend yield vs. dividend growth. In that article, I did not include the effects of compounding the dividend payments back into the investment because the mathematics made my brain cell hurt. I did speculate that compounding might help the higher yield more than the lower yield since the higher initial dividends would be compounding more over the lifetime of the comparison.
So this week I’ve been spending some quality time with my brain cell to learn about the math behind compounding although it’s taken quite a bit longer than I expected. This weekend I finally found the answer I was looking for with the help of some friendly and knowledgeable people on a couple of mathematics forums. So I thought I’d share what I learned.
I’m going to try to make this explanation as gentle as possible so this is quite a drawn out affair. You may already know how to calculate compound dividend growth and be familiar with these calculations, but what I’m trying to reach here is “how long will it take my investment to reach a given value?”. And that formula isn’t so commonly found. I’ll explain each step as I go so that I can look back at it later and remember. So, are you sitting comfortably? Then let’s begin!
Calculating dividends without re-investing
I covered this in more detail in my last post; but here’s a quick recap. First, a question:
If I have $1,000 in a savings account that pays 1% interest each year, and each year I withdraw all the interest paid, how much interest will I have earned after 3 years?
Because I’m withdrawing the interest, the interest paid each year remains the same; it’s the year’s interest rate multiplied by the original investment.
Year 1: Interest = $10
Year 2: Interest = $10
Year 3: Interest = $10
Total Interest = $10 + $10 + $10 = $30
The general solution to this question is simply to calculate the interest for a given year and total it up. To describe this function in math terms, it’s written using the sigma character as shown below.
where V is the total dividends paid, R is the initial investment amount and I represents the interest rate paid in a given year k. The interest each year is summed for n years.
If the interest rate each year (I) is a constant percentage (c) then summing the value n times is the same as multiplying it by n:
And as we saw in the previous post, if the interest rate (r) increases each year (n) by a growth rate (g) then
So let’s see what happens when we re-invest the interest at the end of each year. The earlier question now becomes:
If I have $1,000 in a savings account that pays 1% interest each year, and each year I re-invest all the interest paid, how much interest will I have earned after 3 years?
Year 1: Interest = $10 ($1000 * 1%)
Year 2: Interest = $10.1 ($1010 * 1%)
Year 3: Interest = $10.2 ($1020.1 * 1%)
Total Interest = $10 + $10.1 + $10.2 = $30.3
However, since we’re re-investing the interest paid, the Future Value of the investment is changing each year:
Year 0: Initial Value = $1000
Year 1: Future Value = $1000.0 + ($1000.0 * 1%) = $1000.0 + $10.0 = $1010.0
Year 2: Future Value = $1010.0 + ($1010.0 * 1%) = $1010.0 + $10.1 = $1020.1
Year 3: Future Value = $1020.1 + ($1020.1 * 1%) = $1020.1 + $10.2 = $1030.3
So for the re-investing dividend case, it makes more sense to talk about Future Value than it does Interest paid, since the value of the investment is always changing.
Now we can write $1000 + ($1000 * 1%) as simply $1000 * (1 + 0.01). And if you look closely you’ll see that we’re actually multiplying the interest rate for each year rather than adding it like before.
Year 0: $1000
Year 1: Year 0 * (1+0.01) = $1010
Year 2: Year 1 * (1+0.01) = $1020.1
Year 3: Year 2 * (1+0.01) = $1030.2
and because multiplication is easy to combine, I can just write
Now similar to the summation symbol that we used above, there’s another symbol used in math to describe a series of multiplication. I’d never actually come across it until researching this article, and the symbol is the capital Pi letter and it’s written
where FV is the Future Value, R is the initial investment, I is the interest rate in a particular year k and n is the number of years we’re interested in.
This formula holds true regardless if I is constant or changing over time. But it’s kind of a useless formula really, you can’t just plug it into a calculator and get the answer. So you have to convert it.
When I is a constant value (c) then the solution is simple (and you should be able to predict the answer as I’ve already played this hand)
That should look familiar, especially if I say that R (the initial investment) = $1000 and the interest rate (c) = 1% or 0.01, and I want to calculate the FV for a time n = 3 years:
But what about variable yields that increase over time already?
I mentioned earlier that the formula
works both for constant and variable interest rates.
The formula for a variable interest rate I in a given year n is, as we’ve already seen
where r is the initial yield and g is the growth rate.
So putting them together we get
Once again, it’s a correct formula, but not something we can simply put into a calculator and answer. You can write it out in full and calculate it manually; for example, given r = 3%, g = 5%, R = $1000, we can calculate the Future Value at year n = 3 as
But what I really want to know is, how many years will it take my investment to reach a particular Future Value if I know the initial yield and growth rate?
To answer that question, we need to transform the formula to a more usable form with more math.
Please make the math stop!
Time for a short break…
Calculating Future Value
OK so let’s continue and figure out how to simplify this formula. Logarithmic functions have an interesting property where
Now think of our multiplication formula
Then we can use this to say that
And another useful property of logarithmic functions is that
so we can now say
which means the original FV formula now becomes
Let’s check that formula out with some manual calculations
Using our tried and tested values of R=1000, r=0.03, g = 0.05 and n = 3 which previously gave us:
Dang! We’ve successfully turned a fairly simple multiplication formula into a complicated bunch of logs and exponent expressions only to get the same answer. But wait! There’s more…
300 Year Old Math to the Rescue!
In 1715, a mathematician named Brook Taylor published what’s now known as the Taylor Series. The theory was actually developed by Scottish mathematician called James Gregory, and somehow without the aid of Wikipedia, Google and the Internet, he proved that
where the series continues to Infinity (and beyond!)
The formula means that ln(1+x) is approximately equal to the series – the more terms you use, the more accurate the answer. But for our simple case we can just say
or in our case
Which means our FV formula now becomes
We’ve seen this summation formula before way back when we calculated the non-compounding case. But here it is again
So we now have
Let’s check it out to our original calculated values, using the same R = 1000, r = 0.03, g = 0.05 that you know and love
So it’s pretty close although not exact. Since this whole exercise is an estimation to begin with I’m not going to fuss about it as I’ll be comparing all investments with the same amount of error; so relative to each other it should be sufficient.
Showing some values
The following graph compares two investments, one at 3% yield with 10% growth and another at 2% yield with 20% growth.
Not the most exciting chart to look at perhaps, but it’s interesting in that these are the same values I used in the non-compounded calculations. When dividends weren’t reinvested, the 2% investment caught up to the 3% yield in just over 7 years; here it takes a little more than 9 years. So re-investing dividends did delay the faster growth investment from catching up; although once it did catch up it grew faster.
Are we there yet??
No. But I did want to take a short break to add a reminder against using the formulas in this post to comparing the future value of investments where dividends are re-invested with investments where dividends aren’t re-invested. If you re-invest dividends and I don’t, then your portfolio will be more valuable because your capital is being increased. However your total value may or may not be higher than mine depending on what I do with my dividend payments. The comparison is comparing apples and oranges.
Determining the time taken to reach a given Future Value
The main point of deriving a function for FV was to allow us to solve it for n so we can calculate the time taken to reach a future value at a given yield and dividend growth when re-investing dividends. To do this, we have to solve the equation in terms of n.
We start by putting a target value for FV into the formula. In this case, we’ll put FV = G*R where R is the original investment amount and G represents the target percentage that we’re trying to reach; e.g. G = 2 means 2 x R or double the original investment. Note that a value of G less than 1 is meaningless – the formula will never reach a value lower than the original investment; only a higher one. So keep G >= 1 in the following.
So we have
Taking logs on both sides
Taking logs again
Let’s take it for a spin, with G = 1.099 which we hope should give us the FV of $1099 in about 3 years’ time given r = 0.03 and g = 0.05:
Not bad, eh?
Comparison of two investments
The following chart shows two different investments, one has a 4% yield at 5% growth; the other has 2% yield at 20% growth.
Despite the latter investment growing at 4 times the higher yield investment, it still takes over 9 years to catch the low growth investment.
Effect of reinvesting dividends on growth vs yield
Finally here’s an update calculator which uses the formula to compare the effects of dividend growth vs. yield. If you’re interested in the effects of reinvested dividends you can use this table.
Just watch out for the value of Target Percentage – 100% means the time taken for the investment to reach 100% of your original investment (which is where you start at), so the results will be 0. If you want to calculate the time taken to increase the investment by 10%, you’ll have to enter 110 into the calculator. A value less than 100 is not allowed as the formula can’t reach a lower future value than it starts with.
You can see from the initial value of the calculator that the results are skewed towards the higher yield, more so than in the previous non-reinvested calculator.
And that’s it!
Well that’s all for now on this subject; I’ve learned a lot of math that I didn’t know at the outset and it’s been quite entertaining for me really. At a future date I may look at the effects of a decreasing yield since most companies can’t keep a constant growth rate, but I’ll save that for another day!
PS. No cats, hamsters or any furry animal was harmed in the writing of this post.
Quote of the day
You can do anything if you set your mind to it. Look out for kids, help them dream and be inspired. We teach calculus in schools, but I believe the most important formula is courage plus dreams equals success.