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

and

## Dividends, Re-invested

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

as

where

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

Year(n) | Calculatd | Estimated |

1 | 1030 | 1030 |

2 | 1062 | 1063 |

3 | 1098 | 1099 |

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.

PPS. Many thanks to the kind people who answered my posts at MathHelpForum and math.stackexchange.com when I was lost in my way.

## 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.

Or……you can use the future value button on you business calculator like I do ðŸ™‚

Hi Astute!

Lol! Yes that’s true and I probably would if I had one. Though I like a challenge and the math was fun! I’m weird like that.

Best wishes!

-DL

I think the interest after each year would be $12, but using $10 makes the example easier to understand. Seems like you put a lot of work into this. I like using bankrate savings calculator. lol

Hi DividendMongrel,

Yes the savings calculators are great but they don’t tend to include the effect of dividend increases; they assume the interest rate is constant. I wasn’t looking at the impact of adding additional money to the investment which is why the initial interest is $10 – I was only looking at how to compare the earning potential of two different dividend paying stocks with different estimated future dividend growth.

Thanks for stopping by!

Best wishes,

-DL

I added the calculator table to the Resources page for easier access.

That’s a lot of math. Thanks for doing the hard work! Have you thought about taking future stock price into consideration too? Earnings for the lower yield might be higher given the lower payout ratio.

Cheers!

Hey Henry,

Yes, there was a bit more math than I wanted and I did spell it out quite verbosely (and even added my homegrown warning sign!) ; but on the other hand I’ve now learned the LaTex language quite well!

That’s a good point although I have no way to predict a stock price; the stock price is constant in all these examples. But certainly it should be interesting to look at the effect of capital growth on the yield and the relation between capital growth and current yield. This has come up in a number of posts I’ve read recently.

The lower yield investment’s higher earnings from a lower payout ratio should be somewhat baked into the estimation in the sense that a low yield, high growth company likely has a lower P/O and so is able to maintain the high dividend growth so in the long term it’ll earn more.

The other comparison I’m thinking about is to simulate the effect of a decreasing growth rate; e.g. assume the initial growth rate halves every 10 years or so; this should be more likely than a company continually growing at a fixed growth rate for a long period. Although I expect that will further lead to high yield low growth stocks as being better in the long term.

Thanks for stopping by – I appreciate your insight!

Best wishes!

-DL

Tough part is deciding which way to go. I guess I lean toward higher current yield with a lower (hopefully average) growth rate. That is real vs. future estimated growth rates. On the other hand there are allot of companies in the US Dividend Champions spreadsheet that have 15%+ dividend growth rates for 10 years or more.

Hi DFG,

Yes me too; I aim quite low so I’d be quite happy with just a percent or two over inflation as far as growth is concerned if the yield is good and the dividend payments are stable.

I think ultimately I’m trying to answer the question – are low dividend yield stocks with high growth worth purchasing now or should I just wait until they’re paying higher dividends and invest in something else in the meantime? Currently I won’t purchase any new stock if it’s under a 2% yield but is this a good rule?

And that leads to another question: if a stock doesn’t increase its dividend very much (e.g. less than its capital growth so that the yield drops) – at what point is it worth selling the stock and moving into another higher current yield, or is it? I don’t mean chasing high yield (4+%) in this case but more moving from (say) a 1% to a 2-3% yield – my current guidelines promote a buy and hold approach in this case, but is that a good rule too?

So lots to think about still as I try to question and challenge my current investing rules!

Thank you for your sharing your thoughts!

Best wishes,

-DL

I hold a combination of mutual funds, non-dividend paying stocks and dividend paying stocks in my taxable portfolio. (my portfolio yield is around 1%). I may be different than many on this site in that I am not looking for income out of my portfolio right now. The main thing that concerns me with Dividends is taxes. If I structure my entire portfolio for Dividend income, say 4-5% I am hit with a 31-36% tax rate on that as opposed to 15-20% on STCG and LTCG. Are you guys doing this in IRA’s or other tax-advantaged accounts?

Hi Astute,

In my case, the dividend stocks and mutual funds I’m using for current income to reach Financial Independence are held in taxable accounts. Aside from some bond funds which are taxed at my marginal tax rate of 28%, all stock and stock fund dividends that I cover in this blog are Qualified and taxed at 15% tax. I realize that this is by no means tax efficient but I don’t mind paying my unfair – I mean – ‘fair share’.

I view my income portfolio as separate from my retirement portfolio which is held in a combination of Roth IRA and 401(k) accounts and it consists of low cost index stock / bond funds plus a little mix of TIPS/REIT funds. I don’t write much about my retirement portfolio since it’s so boring.

This all comes from my personal view that Financial Independence is different than Retirement. For many other blogs I read they are the same thing. I know that money is fungible and that I could do things differently to minimize taxes but I want ready access to my current income portfolio without penalties. I may or may not chose to Retire when I reach Financial Independence, but I’m focused more about reaching that point first via both investing and controlling costs.

Anyway that’s my $0.02 on the method behind my madness. Thanks for stopping by!

Best wishes,

-DL

Man that took me back to my math classes. Taylor series, summation loops, log functions… I’d be interested to see a comparison of the FV numbers with something along the lines of a 5-6% yield with 4-5% growth to see how many more years you get out of it. It’s a pretty delicate balance when you’re building your portfolio but I try to stick with the 2.5-4% yield with 6-10% growth. I think those are fairly reasonable assumptions and can deliver the best of both worlds as far as current income and future growth.

Also, I know it won’t be completely accurate but I think you could use the assumption that the share price increases at the same rate as the dividend/earnings. In the long run it should be pretty close to the same.

Hi PIP,

Thanks for the suggestion – I’ll look into that. I think what you’re looking for is to show the future value of an investment where, along with re-investing of growing dividends, the stock price increases also by the same growth rate? In other words to see the capital gains / total return aspect that I’ve not shown in my current table.

Let me know if I misunderstood, otherwise I’ll give that some thought. Thanks for stopping by!

Best wishes,

-DL

Hi PIP,

Yes it was a history trip for me too! I tend to stay in the yield range that you mention; although I’m wondering if I should adjust my minimum yield which from 2% to 2.5% which is the reason I’ve been comparing the effects of yield vs. growth.

Best wishes!

-DL

DL,

I just had a flashback to a method of least squares example from college calculus. ðŸ™‚ Now I remember why I switched majors from engineering to business. In all seriousness, Brent @ Allaboutinterest recently compared investing in high yielding T vs low yielding V. I personally like dripping the high yielders as they can create a lot of future income by consistently increasing share count over time. MO, T, and BP are a few great stocks to drip for a 20 year period.

MDP

Hi Dividend Pipeline,

Yes I can honestly say that I’ve not had to use Taylor’s series in my professional career (engineering) or personal life since I left university until now. But I do find it amazing that people back in the 16/17th century were able to figure this stuff out!

I really appreciate the heads up on the article at AllAboutInterest.com – I believe you’re referring to Dividend Growth vs. Dividend Yield which is a great read, so thank you for pointing that out!

I definitely agree with you – I think dripping the high yielders would tend to give better results on average, since although the high growth stock will eventually win, it does come with a lot more risk in maintaining that high growth over the long term. It’s a case of the hare vs. the tortoise in this respect. If the hare keeps running it’ll win, but if it stops for a nap then the tortoise might just plod on by!

I hold T in my portfolio but not MO and BP so I’ll be considering those for my watchlist; thank you!

Best wishes,

-DL