Question

The amount of money $(A)$ in a bank account after a principal amount $(P)$ is on deposit for $t$ years at an annual interest rate $r$ compounded $n$ times per year is given by this equation: $$A=P{(1+\frac{r}{n})}^{nt}$$ Suppose that a banker would like to determine how changes in these variables would cause the bank to pay less interest to its clients. Which of the variables $-P,r,n$, and $t-$ if minimized, would cause less interest paid to clients? (A) $P$ only (B) $r$ and $t$ only (C) $n$ and $t$ only (D) $P,r,n$, and $t$

Short answer

(B) Minimizing $r$ and $t$ reduces interest paid.

Step-by-step solution

Understanding the formula

The formula for compound interest is $A=P{(1+\frac{r}{n})}^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the number of years. The interest paid is the difference $A-P$. To minimize interest paid, we need to minimize this difference.

Evaluating impact of each variable

Each of the variables $P,r,n,$ and $t$ have different effects: Minimizing $P$ reduces the initial principal, thus reducing $A$ proportionally. Minimizing $r$ reduces the interest earned on each compounding period, which leads to a lower $A$. Minimizing $n$ reduces compounding frequency but possibly increases $A$ since $(1+\frac{r}{n})$ will be larger. Finally, minimizing $t$ reduces the time over which interest is compounded.

Drawing conclusion

To systematically reduce the interest paid ($A-P$), reducing $P,r,$ and $t$ will lead directly to smaller $A$ values due to reduced initial investment, lower rates, and shorter time frames. Reducing $n$ does not directly reduce interest unless $r$ or $t$ is zero, but usually affects how interest is applied.

Key concepts

Principal Amount

When we talk about the principal amount, we refer to the initial sum of money that you deposit or invest. It's your starting point for growing wealth through interest. If you're saving for the first time, think of it as the money you initially put in a piggy bank.

In the context of compound interest, the principal ( P ) plays a crucial role because it's the base used to calculate interest. The interest you earn depends largely on how much principal you start with.

• A larger initial principal means more money to grow from.
• A smaller principal results in less interest because there's less money to start with.

By reducing the principal, banks can pay less interest, as there is a smaller amount from which to calculate interest.

Interest Rate

The interest rate is like the engine of the whole investment process. It tells you how quickly your money grows. Expressed usually as a percentage ( r ), it's the rate at which interest accumulates on your principal.

Simply put, the higher the interest rate, the more your money will grow in a given period. Conversely, a lower interest rate means less interest is added over time. That's why:

• High interest rates can significantly increase the amount over the investment period.
• Low interest rates lead to a slower growth of your investment.

So, if the objective is to minimize how much interest is paid out, lowering the interest rate is an effective approach.

Compounding Frequency

Compounding frequency refers to how often interest is calculated and added to your principal each year. It's like adding more fuel to keep a fire burning. This variable ( n ) can drastically affect the final amount.

The more often interest is compounded, the more frequently your principal grows.

• If compounded annually, interest is added once a year.
• If compounded semi-annually, interest is added twice a year!
• Quarterly compounding means interest is added four times a year, and so on.

While increasing the frequency generally boosts the total amount, reducing it doesn't always decrease interest paid since it depends on the rates and period too.

Investment Period

The investment period refers to the span of time during which your money is left to grow. This variable ( t ) is crucial because it determines how long your money benefits from compounding.

Time is a powerful ally in growing your investment since:

• A longer investment period means more time to earn interest on interest, leading to a greater final amount.
• A shorter period reduces how much compounding can occur, limiting growth.

By shortening the investment period, banks can reduce the interest they have to pay because there's less time for the compounding effect to make a significant impact.