Compound Interest – Multiplying Your Money
Perhaps One of the Most Important Functions You Will Ever Use
Arguably, one of the most useful (and valuable) exponential functions in existence is the following.
The value \(P\) in the formula above is often called the Principal Amount (or Value). The annual rate \(R\) is often called an account or investment’s APY (Annual Percentage Yield).
The above formula is certainly a (transformed) exponential function that requires an input number of years \(n\) to compute the account’s value. In problems involving this formula, the rate and number of compoundings per year are often known in advance, allowing us to compute the base \(\left(1+\frac{r}{k}\right)\). The exponent is calculated by taking the input value of \(n\) and multiplying it by the number of compoundings. It may not look like a transformed exponential function as it stands, but it should once we plug in values for \(k,P\) and \(r\). See the quick example below.
Let’s first set up the compound interest function. From the given situation, we first need to determine:
- The rate \(r\) as a decimal
- \(k\), the number of compoundings per year
- \(P\), the principal amount of the investment
Teasing this info out of the problem we have:
- \(r=0.04\)
- which comes from writing 4% as a decimal. Note that 0.4 is %40 as a decimal. Be careful
- \(k=4\)
- compounded quarterly, so 4 compoundings per year
- \(P=1250\)
- Your principal, or initial/starting amount.
Using this information to build our compound interest formula gives us
$$\begin{align}Value(n)&=P\left(1+\frac{r}{k}\right)^{k\cdot n}\\&=1250\left(1+\frac{0.04}{4}\right)^{4\cdot n}\\&=1250(1.01)^{4n}\end{align}$$
Now all that’s left is to plug in different values for \(n\), which represents the number of years we allow the investment to grow.
$$Value(1)=1250(1.01)^{4\cdot 1}=\$1300.76$$
$$Value(2)=1250(1.01)^{4\cdot 2}=\$1353.57$$
$$Value(5)=1250(1.01)^{4\cdot 5}=\$1525.24$$
$$Value(10)=1250(1.01)^{4\cdot 10}=\$1861.08$$
The number of compoundings (for variable \(k\)) represents the number of times the account accrues interest throughout the year. Some accounts compound daily; some compound biannually; some compound only once per year. More compoundings per year allow the investment to build on itself more quickly.
Suppose you have $100 to invest at a rate of growth of 10% per year. If the account compounds only twice per year (i.e. k=2), then after one year we will have
$$\begin{align}Value(1)&=100\left(1+\frac{0.10}{2}\right)^{2\cdot 1}\\&=100\cdot(1.05)^{2}\\&=\$110.25\end{align}$$
Now if instead the account compounds daily (i.e. 365 times per year), then after one year we will have
$$\begin{align}100\left(1+\frac{0.10}{365}\right)^{365\cdot 1}&=100(1.000273926)^{365}\\&=110.5156\end{align}$$
While the difference between the results above may seem negligible (only 25 cents different), imagine you were investing $10,000 or $100,000. The difference between the two would be $25 and $250, respectively. Still might not seem like much of a difference, but having $25 or $250 is better than not having it if you ask me!
$$Value(2)= 1000\left(1+\frac{0.03}{365}\right)^{365\cdot 2} =\$1061.83$$
$$Value(5)= 1000\left(1+\frac{0.03}{365}\right)^{365\cdot 5} =\$1161.83$$
$$Value(10)= 1000\left(1+\frac{0.03}{365}\right)^{365\cdot 10} =\$1349.84$$
First, we need to gather all the pieces we need to construct the compound interest function:
- Number of compoundings: \(k=365\) (days per year)
- Yearly rate as a decimal: \(r=0.03\) (3% as a decimal)
- Principal amount invested: \(P=1000\)
From this info, our compound interest formula is given as:
$$\begin{align}Value(n)&=P\left(1+\frac{r}{k}\right)^{k\cdot n}\\&=1000\left(1+\frac{0.03}{365}\right)^{365n}\\&=1000(1.0000821917808)^{365n}\end{align}$$
Now, plugging in the different number of years gives us the corresponding value of the account:
$$Value(2)= 1000( 1.0000821917808 )^{365\cdot 2} =\$1061.83$$
$$Value(5)= 1000( 1.0000821917808 )^{365\cdot 5} =\$1161.83$$
$$Value(10)= 1000( 1.0000821917808 )^{365\cdot 10} =\$1349.84$$
$$Value(1)= 500\left(1+\frac{0.08}{52}\right)^{52\cdot 1} =\$541.61$$
$$Value(2)= 500\left(1+\frac{0.08}{52}\right)^{52\cdot 2} =\$586.68$$
$$Value(5)= 500\left(1+\frac{0.08}{52}\right)^{52\cdot 5} =\$745.67$$
First, we need to gather all the pieces we need to construct the compound interest function:
- Number of compoundings: \(k=52\) (weeks per year)
- Yearly rate as a decimal: \(r=0.08\) (8% as a decimal)
- Principal amount invested: \(P=500\)
Note that all other information and assumptions given in the problem are there to simplify the situation so that the compound interest formula can be used… and to prevent nasty emails from accountants and economists. The real world of investing is more messy than these problems admit. Anyhoo…
From this info, our compound interest formula is given as:
$$\begin{align}Value(n)&=P\left(1+\frac{r}{k}\right)^{k\cdot n}\\&=500\left(1+\frac{0.08}{52}\right)^{52n}\\&=500(1.0015384)^{52n}\end{align}$$
Now, plugging in the different number of years gives us the corresponding value of the account:
$$Value(1)= 500(1.0015384)^{52\cdot 1} =\$541.61$$
$$Value(2)= 500(1.0015384)^{52\cdot 2} =\$586.68$$
$$Value(5)= 500(1.0015384)^{52\cdot 5} =\$745.67$$
$$Value(1)= 6\left(1+\frac{0.062}{365}\right)^{365\cdot 1} =\$6.38$$
$$Value(2)= 6\left(1+\frac{0.062}{365}\right) ^{365\cdot 2} =\$6.79$$
$$Value(4)= 6 \left(1+\frac{0.062}{365}\right) ^{365\cdot 4} =\$7.68$$
First, we need to gather all the pieces we need to construct the compound interest function:
- Number of compoundings: \(k=365\) (365 per year)
- Yearly rate as a decimal: \(r=0.062\) (6.2% as a decimal)
- Principal amount invested: \(P=6\)
From this info, our compound interest formula is given as:
$$\begin{align}Value(n)&=P\left(1+\frac{r}{k}\right)^{k\cdot n}\\&=6\left(1+\frac{0.062}{365}\right)^{365n}\\&=6(1.00016986)^{365\cdot n}\end{align}$$
Now, plugging in the different number of years gives us the corresponding value of the account:
$$Value(1)= 6(1.00016986)^{365\cdot 1} =\$6.38$$
$$Value(2)= 6(1.00016986)^{365\cdot 2} =\$6.79$$
$$Value(4)= 6(1.00016986)^{365\cdot 4} =\$7.68$$
Moral of the story: everything you buy on loan and wait to repay is going to cost a LOT more in the future. Might be best to make your own lattes at home or in your dorm. Its easy, fun, tastes 1000x better, and it costs 50 cents per cup after you buy your supplies, not $7.68 per cup. Not to mention, you can show off your barista skills to your friends!