My brain cell has been occupied recently about comparing yield and growth. I mean what’s better – 1% yield with 10% growth or 2% yield with 5% growth? Does it even matter? Armed with some vague memories of math classes in my dim and distant past, and some astute web searches, I did some investigating to see if I could find out.
Using a cute analogy to distract you from the math that’s coming up, if you compare dividend income to hamsters running in a wheel then the hamsters’ speed represents the current yield, and the hamsters’ ability to run even faster is the dividend growth. The total distance run by the hamsters in a given amount of time represents the total amount of dividends paid out.
If you wanted to judge a race between two hamsters that start the race running at different speeds and have different acceleration, how would you determine a winner? You’d probably simply set a finish line out in the distance and declare the first hamster to reach it the winner. Or else you could declare the winner to be the first hamster to reach an arbitrary speed, say 6 mph.
I’m going to start with the first to reach 6 mph analogy since this is simpler and we’ll cover the basic growth formula along the way. In our case, we want to be able to calculate the future dividend yield of a stock if it continued from a current yield and increased by the same percentage each year.
So given an initial annual dividend yield (y) and a projected annual dividend growth rate (g), the future yield (F) in a number of years (t) is calculated using the formula
Future Yield = Current Yield x (1 + Dividend Growth)^Number of Years
F = y(1+g)^t
You can see from the graph that despite a 50% higher growth rate, the 2% stock couldn’t meet the yield of the 3% stock in a 10 year period. In a longer time period obviously it would, but there’s also less chance that high growth rates could be maintained.
But my stock yields never reach 7% or higher and I’ve held them for years!
Well technically they do, but it’s a matter of perspective. If you went back and calculated the yield on cost of a dividend growth stock that you’ve held for a long time, you’ll find that the yield of your original investment amount is much higher than the current yield.
Let’s say I bought shares in LMT back on 3/9/2009. I paid $1000 and bought 17.10 shares for $58.475 each. The dividend at the time was $0.57 per share and back then the yield was 3.90%. Today, LMT’s share price is $158.08 and my $1,000 investment has increased to $2,703. With the current dividend per share of $1.33, the current yield is 3.37%.
However, I only actually spent $1000 and bought the shares at $58.475, not $158.08. So my yield on cost is actually 4 x $1.33 / $58.475 which is 9.1% – quite a bit different from the current 3.37%.
So when I say future or target yield in 10 year’s time, I’m really referring to what the ‘yield on cost’ would be in 10 year’s time. You’ll not actually see the 10% yield due to stock price changes. I’m just using the calculation of future yield to compare performance between two investments.
How to determine the first stock to reach a given target yield
One way to compare performance between two different
hamsters stocks is to set an arbitrary target yield and calculate how many years it would take a given dividend yield and growth rate to reach that level. So for example, how long will it take a stock to reach a 10% yield?
This approach is the basis of an article that I came across when thinking about this topic and it’s referenced as 10 x 10 investing. The article compares the time needed to reach a final target yield of 10%.
I wanted to know how to generate the numbers in the table in that article and create my own tables if necessary. Here’s how you do that.
The formula for future yield we used above calculates the target yield (F) for a given value of time (t). This time we want the opposite – we want to calculate ‘t’ using a given value for ‘F’, say 10%. This means we have to solve the equation for t and the steps are:
F = y(1+g)^t
F/y = (1+g)^t
log(F/y) = t.log(1+g)
log(F/y) / log(1+g) = t
or, writing that out in full
time (t) = log (Target Yield / Current Yield) / log(1+Dividend Growth)
where Target Yield = 10% (0.01) or whatever target value you want.
I’ve plotted some example numbers for your viewing pleasure below
If you follow the 2% line, you can see from the chart that it’ll take nearly 55 years to reach 10% yield, and that even at 15% growth, it’ll take about 12 years to reach a 10% yield.
The 3% stock is better obviously and will reach a 10% yield in 10 years with a growth rate of 13% or higher.
Here’s a calculator I made showing the 10 x 10 table in more detail. I’ve tailored it towards typical dividend yields with smaller percentages. Yield and growth values required to meet the given target yield are highlighted in green. You should also be able to change the target 10% and target 10 year timeframe values yourself using a dropdown list at the top of the table and the table will update accordingly.
Okay let’s take a short break and make a couple of assumptions before we continue. Sadly I don’t have any refreshments to offer.
Effect of Inflation
To keep things simple, I’m not including inflation in these calculations. If you are interested in the effects of inflation, Kristi wrote a good article on the time value of money on her blog at MoneyIsYourFriend.
I’m also not including re-investing of dividends back into the stock as part of these calculations. I’ve not done the calculations but my intuition tells me this would favor current yield rather than growth since the higher dividend payments at the beginning of the investing period would compound more.
In any case, I don’t re-invest dividends immediately back into the same stock so I don’t have to worry about this case. I simply use dividend income to invest more money than I would have otherwise been able to afford.
Calculating the total dividends paid by a future date
So an alternative way to measure performance of two stocks is to determine the first one to cross a finish line. In our case, this would mean “how long would this investment take to generate $100 dollars of dividends”. And we can make that more general by talking percentages, e.g. how long would this investment take to generate dividend payments equal to 100% of the original investment (or whatever target percentage we want to use here).
Before we can answer that question however, we need to know how to calculate the amount of dividends paid (D) in a time period when the dividend rate is always increasing.
If the dividend interest rate was constant, and since we’re not re-investing dividends, this calculation would be a snap:
Total Dividend % = Current Yield x Number of years
E.g. if my investment earns a constant 3% then after three years I’ve earned 3% x 3 = 9% of my original investment. But we have to work harder for the answer since our dividends are increasing each year.
The above formula still works for us, it’s just that the current yield isn’t constant. But we know what the future dividend yield in a given year since we’ve already done that previously using the formula F = y(1+g)^t. So what we want to do is simply add the dividend yield for each year in the time period we’re interested in.
If this doesn’t make sense to you then perhaps the following example may help. I have a $200 investment which has 3% yield in the first year and 3.5% yield in the second year. I’m not re-investing, so in the first year I earn $6 (3% of $200) and in the second year I earn $7 (3.5% of $200). So at the end of the second year, I’ve earned a total of $13 and if I make that a percentage of my investment, I get $13 / $200 which is 6.5%, i.e. 3% + 3.5%.
In math there’s a special symbol for summing things called Sigma or Σ and you’d write the formula as
Total Investment % = Σy(1+g)^t
You’d also add to that equation the fact that you’re interested in summing all positive values of t starting from 0. Now to use that formula, you need to translate it into something you can actually use.
I don’t know how to prove the following – many people smarter than I can do so – but there are standard summation formulas that you can use. It just so happens that if you want to sum “x = a^t” for each positive value of t from 0, you can actually use the formula “total = (a^(t+1)) – 1) / (a – 1)” and you can plug in whatever value of t you’re interested in and get the total. In our case you can look at (1+g) as being the same as ‘a’ in that standard formula.
So determining the equation goes something like this, where we’re calculating the total dividend payments (V) as a percentage of our investment:
V = Σy(1+g)^t
V = y[((1+g)^(t+1) – 1) / (1+g) -1]
V = y[((1+g)^(t+1) – 1) / g]
V = y((1+g)^(t+1) – 1)/g
It’s not the most readable formula but let’s check it out with a couple of values. Assuming the yield is 2%, the growth is 5%, what is the total dividend payment in the first year when t = 0?
V = 0.02((1+0.05)^(0+1) -1)/0.05 = 0.02((1.05)^1 -1)/0.05 = 0.02(0.05)/0.05 = 0.02 = 2%
and in the second year when t = 1?
V = 0.02((1+0.05)^(1+1) -1)/0.05 = 0.02((1.05)^2-1)/0.05 = 0.02(0.1025)/0.05 = 0.041 = 4.1%
If you check the future yield using the original formula, you’ll see that the first year gives 2% and the second year gives 2.1% – if you add those together you get 4.1% so the formula is good.
Here are some examples I plotted; with a growth rate twice as large, it takes a 2% yield 8 years to pay the same total percentage of investment as a 3% yield.
Calculating the time needed to pay a particular percentage of investment
One more step to go and then we’re done, I promise!
So now we can calculate how much of the original investment a given yield and growth rate will return over a time period that we specify. What we need to do now is calculate how much time a particular yield / growth combination will take to pay a specified percentage of our investment. This is similar to what we did before for the yield; we need to solve our new formula for t again.
V = y((1+g)^(t+1) – 1)/g
Vg = y((1+g)^(t+1) – 1)
(Vg/y) = (1+g)^(t+1) – 1
(Vg/y) + 1 = (1+g)^(t+1)
log((Vg/y) + 1) = (t+1)log(1+g)
log((Vg/y)+1) / log(1+g) = t+1
t = (log((Vg/y)+1) / log(1+g) ) – 1
Here are some results graphically for the same yield / growth combinations as above.
The results match the previous chart and show that the breakeven point is 8 years down the road, after that time the high growth investment wins easily.
Here’s another calculator that allows you to play with different percentage returns and see the required yield and growth rate to meet those expectations.
So yield vs growth – who wins? It depends on your goals, objectives and investing timeframe.
With high yields there are risks that the dividends can’t be continued long term; but he same can be said with high growth rates. Is a bird in the hand (yield) worth two in the bush (growth)? That’s for you to decide and evaluate but I hope that this article has given you some food for thought rather than indigestion!
PS. No hamsters, real or imaginary, were harmed in the writing of this post!
Quote of the day
I don’t know why I should have to learn Algebra… I’m never likely to go there.