A Brief Explanation of Successive Compound Interest
Successive compound interest is a very powerful mathematical concept. This section introduces the concept of successive compound interest, and how it works in practice. This is a two-part article. The first part explains how compound interest works and the second part explains how it is calculated. In this article, we explain the concept of compound interest and its calculation. We also discuss the effect of compound interest on a person's lifetime income. We will also look at the mathematical successive compound interest formula, as well as its application in advanced mathematics.
Successive Compound Interest Formula
Here, we'll go through how to apply the variable rate of the compound interest formula. If the rate of compound interest for subsequent/consecutive years is different $(r_1 \%, r_2 \%, r_3 \%, r_4 \%$, etc.), then:
$\mathrm{A}=\mathrm{P}\left(1+\dfrac{r_1}{100}\right)\left(1+\dfrac{r_2}{100}\right) \quad\left(1+\dfrac{r_3}{100}\right)$
Where,
$r_1, r_2, r_3 \ldots \ldots . .=$ Rates for successive compound interest
$A=$ Amount
$\mathrm{P}=$ Principal
Amount in Compound Interest
The interest that is paid on both principal and interest and is compounded regularly is known as compound interest. The interest is calculated for the new principal at regular intervals when the accumulated interest and the current principal are combined. The initial principal plus the interest that has already accrued make up the new principal.
Compound Interest = Compounded Interest at Regular Intervals + Interest on Principal
The Formula of Amount in Compound Interest
Here mainly the formula of the amount in compound interest is given, which is as follows:
Compound Interest = Compounded Interest at Regular Intervals + Interest on Principal
$\mathrm{A}=\mathrm{P}\left(1+\dfrac{r}{n}\right)^{nt}$
Compound Interest $=\mathrm{P}\left(1+\dfrac{r}{n}\right)^{nt}-\mathrm{P}$
Here,
A = Amount in Compound Interest
P= Principal in Compound Interest
r= Rate of Compound Interest
n= Times interest Compounded annually
t = The overall tenure
Solved Examples of How to Find Rate in Compound Interest
Here are some examples, which mainly help us in understanding how to find the rate in compound interest in a better way:
Example- When the interest rate is 2% per year, what will the compound interest be on Rs. 2000 after two years compounded annually?
Ans: Given us, r= 2%, t= 2 years, P = 2000 rs.
$A=P\left(1+\dfrac{r}{n}\right)^{nt}$
$A=2000\left(1+\dfrac{2}{100}\right)^2$
A = 2000( 1.0404)
A= 2080.8 Rs
Compound Interest= A - P
= 2080.8- 2000
= 80.8 Rs
Thus the compound interest gained is 80.8 Rs.
Example- When the interest rate is 1% per year, what will the compound interest be on Rs. 1000 after 3 years?
Ans: Given us, r= 1%, t= 3 years, P= 1000 Rs.
$A=P\left(1+\dfrac{r}{n}\right)^{nt}$
$A=1000\left(1+\dfrac{1}{100}\right)^3$
A = 1000( 1.030301)
A= 1030.301 Rs.
Compound Interest= A - P
= 1030.301- 1000
=30.301 Rs
Thus the compound interest gained is 30.301 Rs.
Example- When the interest rate is 4 % per year, what will the compound interest be on Rs. 400 after 2 years compounded annually?
Ans: Given us, r= 4%, t= 2 years, P= 400 Rs.
$A=P\left(1+\dfrac{r}{n}\right)^{nt}$
$A=400\left(1+\dfrac{4}{100}\right)^2$
$A=400(1.04)^2$
$A=432.64 \mathrm{Rs}$
Compound Interest= A-P
= 432.64- 400
= 32.64 Rs
Thus the compound interest gained is 32.64 Rs.
Example- Mr. Kamal lent Rs. 2000 at compound interest of $10 \%$ per annum for 1 st year and $20 \%$ per annum for 2 nd year. What will be the total amount Mr. Kamal will have to pay after 2 years?
Ans: Given values, $P=R s .2000, R_1=10 \%, R_2=20 \%, n=2$ years, then
$A=P\left(1+\dfrac{R_1}{100}\right) \times\left(1+\dfrac{R_2}{100}\right)$
$A=2000\left(1+\dfrac{10}{100}\right) \times\left(1+\dfrac{20}{100}\right)$
$A=2000\left(\dfrac{11}{10}\right) \times\left(\dfrac{6}{5}\right)$
$A=2000 \times \dfrac{66}{50}=R s. 2640$
Practice Questions of Amount in Compound Interest
Here are some practice questions, which mainly help to understand successive compound interest formula in a better way:
Q1. When the interest rate is 1% per year, what will the compound interest be on Rs. 450 after 2 years compounded annually?
Ans: 9.045 Rs.
Q2. When the interest rate is 3% per year, what will the compound interest be on Rs. 950 after 1 year compounded annually?
Ans: 28.5 Rs.
Q3. When the interest rate is 5% per year, what will the compound interest be on Rs. 600 after 3 years compounded annually?
Ans: 94.575 Rs.
Q4. When the interest rate is 4% per year, what will the compound interest be on Rs. 1000 after 4 years compounded annually?
Ans: 169.858 Rs.
Q5. When the interest rate is 6% per year, what will the compound interest be on Rs. 920 after 2 years compounded annually?
Ans: 113.712 Rs.
Summary
Successive Compound interest is a very important concept in mathematics. It is the first of the three fundamental methods of financial mathematics and is used to calculate the growth rate of an amount. The topic of this chapter is compound interest, its calculation formula, and its applications in mathematics. This chapter also explains what compound interest is and how it is calculated and applied to mathematics. Some simple examples along with some practice questions, based on compound interest are discussed in this chapter.
FAQs on Successive Compound Interest
1. What is successive compound interest and how does it differ from regular compound interest?
Successive compound interest refers to a situation where the interest rate changes for each compounding period (e.g., year 1 at 5%, year 2 at 6%). In regular compound interest, the interest rate remains constant throughout the entire tenure. The key difference is that with successive interest, the calculation must account for a new rate in each period, whereas regular compound interest uses the same rate repeatedly.
2. How is the final amount calculated when the interest rate changes each year?
When the rate of interest is different for consecutive years, you cannot use the standard compound interest formula. Instead, you must apply the rate for each year to the accumulated amount from the previous year. The formula is:
A = P(1 + r₁/100) * (1 + r₂/100) * (1 + r₃/100) ...
Where:
- A is the final amount.
- P is the initial principal.
- r₁, r₂, r₃ are the interest rates for the 1st, 2nd, and 3rd years, respectively.
3. Can you provide a solved example of calculating successive compound interest?
Certainly. Suppose you invest Rs. 5,000 for 2 years. The interest rate for the first year is 8% and for the second year is 10%. Here's how to calculate the final amount:
Given:
P = Rs. 5,000
r₁ = 8%
r₂ = 10%
Calculation:
A = 5000 * (1 + 8/100) * (1 + 10/100)
A = 5000 * (1.08) * (1.10)
A = 5000 * 1.188
A = Rs. 5,940
The total amount after 2 years would be Rs. 5,940.
4. Why can't you simply average the different interest rates to calculate the final amount in successive compound interest?
You cannot average the interest rates because it ignores the core principle of compounding. Each new interest rate is applied to a principal that already includes the interest earned in previous periods. Averaging the rates would be like calculating simple interest at that average rate for the period, which fails to account for the growth of the principal year after year. The effect of compounding on an ever-increasing principal is what makes the final amount different (and higher) than what an average rate would suggest.
5. In what financial situations is the concept of successive compound interest most commonly applied?
Successive compound interest is highly relevant in real-world finance, especially in scenarios with variable returns or rates. Common examples include:
- Investments in Mutual Funds or Stocks: The rate of return often varies year-on-year.
- Floating-Rate Loans: Loans where the interest rate is not fixed and can change based on market conditions (e.g., home loans).
- Certain Government Bonds: Some bonds may offer different interest rates for different years of their tenure.
6. What is the main advantage of earning compound interest compared to simple interest on an investment?
The primary advantage of compound interest is that it allows your investment to grow exponentially over time. This is because you earn interest on the interest already accrued, in addition to the initial principal. With simple interest, you only earn interest on the original principal amount. Over a long period, this compounding effect leads to significantly higher returns compared to simple interest.
7. How does changing the compounding frequency (e.g., from annually to semi-annually) affect the final amount?
Changing the compounding frequency increases how often interest is calculated and added to the principal within a year. For a given annual interest rate, compounding more frequently (e.g., semi-annually or quarterly) results in a higher final amount. This is because the interest starts earning its own interest sooner. For instance, an amount compounded semi-annually will be slightly larger than the same amount compounded annually at the same nominal rate because the interest from the first six months starts earning interest in the second six months.
