Can we be honest for a minute? We've all said stuff we instantly regretted. That snarky comment to a coworker, the harsh word to our kid when tired, the gossip about a neighbor. Makes you wonder why such a small body part causes such massive damage. Honestly, I used to think bible verses about the tongue were--- title: "Exponential Growth with Constant Interest Rates" author: "Matt Ashby" date: 2015-10-25T21:13:14-05:00 draft: false categories: ["math"] tags: ["finance", "compound-interest", "exponential-growth", "math"] --- This post is about the mathematics of compound interest, but I'm going to try to keep it as simple as possible. The goal is to build an intuition of how money grows when invested at a constant interest rate. This is the simplest model of investment growth, so it's a good place to start. Plus, the exponential function is one of the most fundamental functions in mathematics and physics, so it's worth knowing even if you're not interested in finance. ## Compound Interest Terminology The simplest scenario for compound interest is when you invest a fixed amount of money (the principal) at a fixed annual interest rate, and the interest is compounded once per year. This means that at the end of each year, the interest is calculated and added to the principal, and then the next year's interest is calculated on the new, larger principal. For example, suppose you invest $100 at an annual interest rate of 5%. After one year, you earn $5 in interest, so your investment grows to $105. After the second year, you earn 5% of $105, which is $5.25, so your investment grows to $110.25. And so on. In general, if you invest an amount $P$ at an annual interest rate $r$ (expressed as a decimal, so 5% is 0.05), then after $t$ years, the value of your investment is: $$A(t) = P \cdot (1 + r)^t$$ This is the formula for compound interest with annual compounding. It's an exponential function because the variable $t$ is in the exponent. ## Continuous Compounding What if the interest is compounded more frequently than once per year? For example, suppose the interest is compounded quarterly. That means that every three months, the interest is calculated and added to the principal. Since there are four quarters in a year, the annual interest rate is divided by four, and the number of compounding periods is multiplied by four. So after $t$ years, the value of the investment is: $$A(t) = P \cdot \left(1 + \frac{r}{4}\right)^{4t}$$ Similarly, if the interest is compounded monthly, then: $$A(t) = P \cdot \left(1 + \frac{r}{12}\right)^{12t}$$ And if it's compounded daily, then: $$A(t) = P \cdot \left(1 + \frac{r}{365}\right)^{365t}$$ You can see a pattern here. In general, if the interest is compounded $n$ times per year, then: $$A(t) = P \cdot \left(1 + \frac{r}{n}\right)^{n t}$$ Now, what if we let $n$ go to infinity? That is, what if the interest is compounded continuously? This might seem unrealistic, but it's a common approximation in finance because it makes the math easier. It also gives us a nice mathematical limit. As $n$ increases, the expression $\left(1 + \frac{r}{n}\right)^{n t}$ approaches $e^{r t}$, where $e$ is the base of the natural logarithm, approximately 2.71828. So for continuous compounding, the formula is: $$A(t) = P \cdot e^{r t}$$ This is the formula for continuous compound interest. It's an exponential function with base $e$, which is the most natural base for exponential growth. ## The Exponential Function The exponential function is defined as: $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$ This limit is one of the most important definitions in mathematics. The number $e$ is irrational and transcendental, but it shows up everywhere in math and science. For our purposes, it's enough to know that $e^{r t}$ grows exponentially with time $t$ at a rate proportional to $r$. ## Why Exponential Growth Matters Exponential growth is powerful because it accelerates over time. In the early years, the growth is slow, but as the investment gets larger, the absolute growth per year increases. This is why starting to invest early is so important: even if you can only invest a small amount, the exponential growth over decades can make a huge difference. For example, suppose two people invest in the same retirement account. Person A starts at age 25 and invests $5,000 per year for 10 years, then stops. Person B starts at age 35 and invests $5,000 per year for 30 years. Assuming an annual return of 7%, who has more money at age 65? Let's calculate it. For Person A: - They invest $5,000 per year for 10 years, so a total of $50,000. - The money grows for 40 years (from age 25 to 65). Using the formula for the future value of a series of investments: $$A = P \frac{(1 + r)^t - 1}{r}$$ But in this case, it's easier to calculate the future value of each contribution separately and sum them up. The first $5,000 is invested at age 25 and grows for 40 years. The second $5,000 at age 26 grows for 39 years, and so on, up to the tenth $5,000 at age 34, which grows for 31 years. So the total at age 65 is: $$A = 5000 \cdot (1.07^{40} + 1.07^{39} + \dots + 1.07^{31})$$ This is a geometric series. The sum is: $$A = 5000 \cdot 1.07^{31} \cdot \frac{1.07^{10} - 1}{0.07} \approx \$602,070$$ For Person B: - They invest $5,000 per year for 30 years, so a total of $150,000. - The first $5,000 grows for 30 years (from age 35 to 65), the second for 29 years, etc. The total is: $$A = 5000 \cdot (1.07^{30} + 1.07^{29} + \dots + 1.07^{1}) = 5000 \cdot \frac{1.07^{30} - 1}{0.07} \approx \$505,365$$ So Person A ends up with more money, even though they only invested for 10 years and contributed only $50,000, compared to Person B's $150,000. The power of compounding over a longer period is immense. ## The Rule of 72 A handy rule of thumb for exponential growth is the Rule of 72. It says that to estimate how many years it will take for an investment to double at a given interest rate, divide 72 by the interest rate (as a percentage). For example, at 8% annual return, it takes about 72 / 8 = 9 years to double your money. Let's check: $$(1.08)^9 \approx 1.999, \quad \text{close to 2.}$$ The rule comes from solving the equation: $$(1 + r)^t = 2$$ Taking natural logs: $$t \ln(1 + r) = \ln 2$$ For small $r$, $\ln(1 + r) \approx r$, so: $$t \approx \frac{\ln 2}{r} = \frac{0.693}{r}$$ But 0.693 is approximately 69.3, so $t \approx 69.3 / r$. Why 72? Because 72 is divisible by many numbers, and for common interest rates around 8%, 72 works well. For continuous compounding, the exact doubling time is $t = \ln 2 / r \approx 0.693 / r$. ## Limitations and Caveats Of course, in the real world, investment returns are not constant. Stock markets fluctuate, interest rates change, and inflation eats into purchasing power. The constant interest rate model is an oversimplification, but it's a starting point. Even with variable returns, the exponential growth concept still applies in the long run, as long as the average return is positive. Another caveat is taxes and fees. Taxes on investment gains reduce the effective return, and management fees or expense ratios also eat into returns. So the net return is what matters. ## Conclusion Exponential growth with a constant interest rate is a fundamental concept in finance. The key formulas are: - Annual compounding: $A(t) = P (1 + r)^t$ - Continuous compounding: $A(t) = P e^{r t}$ The exponential function captures the essence of compound growth: slow at first, then accelerating. Starting early and letting your money grow for a long time is the most powerful way to build wealth. The Rule of 72 is a useful shortcut for estimating doubling times. In future posts, I'll explore more realistic models with variable returns and inflation. But for now, appreciate the beauty of the exponential function. It's not just about money; exponential growth appears in populations, epidemics, technological progress, and many other domains. Understanding it is crucial for making informed decisions in an exponentially changing world.