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When calculating compound interest often a constant interest rate is assumed. However, when applying this to dividend stocks for example, the dividend yield changes every year. This made me think about how variations in interest rate impact the value of your investment.

In a simplified version we can calculate compound interest in the following way:

$finalvalue= initial* interest^n$

where n is the number of years.

Let us assume our initial investment to be 1 and the number of years to be 3. Then we create 2 scenarios:

  1. The interest rate is 5% each year giving us: $1.05^3 = 1.157625$
  2. The interest rates for the three years are 4%, 5%, 6% which on average is equal to 5% each year. This gives us: $1.04*1.05*1.06= 1.15752$

Doing a couple of these examples seems to suggest that the closer to the average the interest rates are, so the more consistent dividend is paid out, the better the result for your investment.

Therefore I was wondering whether this is actually true and if so, how this can be explained mathematically in order to answer my main question which is how compound growth is essentially being affected by variations in interest.

Just van der Veeken
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  • Yes, notice how 1.041.06 = (1.05)^2 - (0.01)^2, so 1.05^3 > 1.041.05*1.06. In fact, you can set up an AM/GM if they are not equally spaced, and get to the same result – Gabe Aug 18 '19 at 13:22

1 Answers1

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This is a consequence of the so-called arithmetic mean-geometric mean inequality ("AM-GM" for short). It says that for any positive numbers $x_1,x_2,\cdots,x_n$, we have $$ \frac{x_1+x_2+\cdots+x_n}n\geq \sqrt[n]{x_1x_2\cdots x_n} $$ with equality iff all the $x_i$ are equal. Raise both sides to the $n$th power if you'd like.

Considering $n$ and the value of the left-hand side are fixed for the cases you're comparing, this result tells you exactly that your conjecture holds true in general.

Arthur
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