Glossary

6.4 Compound Interest

2 min read•june 18, 2024

is a powerful financial concept that can make your money grow faster. It's calculated on both the initial amount and the interest earned over time, leading to . Understanding this can help you make smarter investment decisions.

The helps you figure out how much your money will grow over time. You can also use it to calculate how much you need to invest now to reach a future goal. Knowing about helps you compare different investment options.

Compound Interest

Compound interest calculations and comparisons

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  • Compound interest calculated on initial and accumulated interest from previous periods
    • Differs from only calculated on (original amount invested)
  • Compound interest formula A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}
    • AA final amount (), PP initial principal, rr annual (decimal), nn number of times interest compounded per year, tt number of years money invested
  • Higher interest rates lead to faster investment growth (5% vs 3%)
  • More frequent compounding (higher nn) results in larger final amount
    • $1,000 invested at 5% interest for 10 years
      • Compounded annually: A=1000(1+0.051)1⋅10=A = 1000(1 + \frac{0.05}{1})^{1 \cdot 10} = 1,628.89
      • Compounded monthly: A=1000(1+0.0512)12⋅10=A = 1000(1 + \frac{0.05}{12})^{12 \cdot 10} = 1,647.01

Present value for savings goals

  • formula derived from compound interest formula
    • P=A(1+rn)ntP = \frac{A}{(1 + \frac{r}{n})^{nt}}
  • Calculates amount needed to invest today to reach specific future goal
    • To have $50,000 in 20 years with 6% annual interest compounded monthly
      • P=50000(1+0.0612)12⋅20=P = \frac{50000}{(1 + \frac{0.06}{12})^{12 \cdot 20}} = 18,419.18
  • Helps determine initial investment required for retirement, college tuition, down payment on a house

Effective annual yield evaluations

  • () actual rate of return earned on investment, accounting for compounding
    • EAY formula: EAY=(1+rn)n−1EAY = (1 + \frac{r}{n})^n - 1
  • Compares investments with different compounding frequencies (monthly, quarterly, annually)
    • 5% interest compounded quarterly vs 4.9% interest compounded monthly
      • 5% compounded quarterly: EAY=(1+0.054)4−1=0.0509EAY = (1 + \frac{0.05}{4})^4 - 1 = 0.0509 or 5.09%
      • 4.9% compounded monthly: EAY=(1+0.04912)12−1=0.0501EAY = (1 + \frac{0.049}{12})^{12} - 1 = 0.0501 or 5.01%
    • 5% interest compounded quarterly yields higher return
  • Helps identify most profitable investment option based on and interest rate
  • Distinguishes between (stated rate) and effective annual yield (actual return)

Investment growth analysis tools

  • : Estimates time required to double an investment
    • Divide 72 by the interest rate to approximate years needed to double the investment
  • (CAGR): Measures average annual growth rate over time
    • Useful for comparing investments with different time periods or initial values

Key Terms to Review (24)

Annual compounding: Annual compounding refers to the process of calculating interest on an investment or loan, where the interest is added to the principal balance once a year. This means that each year, the interest earned in the previous period is included in the principal for calculating future interest, leading to exponential growth of the investment over time. The effectiveness of annual compounding lies in its ability to leverage the power of compound interest, which results in a higher amount of total interest earned compared to simple interest methods.
Certificate of deposit: A certificate of deposit (CD) is a financial product offered by banks and credit unions that allows individuals to deposit a fixed amount of money for a specified period of time, earning interest at a higher rate compared to traditional savings accounts. CDs are considered low-risk investments and are ideal for savers looking to grow their money over time without the volatility associated with stocks or other investment vehicles.
Compound annual growth rate: The compound annual growth rate (CAGR) is a measure used to calculate the mean annual growth rate of an investment over a specified period of time, assuming the profits are reinvested at the end of each period. CAGR is significant because it reflects the actual growth of an investment, providing a more accurate picture than simple average growth rates, especially when dealing with fluctuating returns. This metric is particularly useful in finance and investing for comparing the historical performance of various investments or for forecasting future growth.
Compound interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods, allowing investments to grow at a faster rate over time. This concept connects to various financial topics, including how investments can appreciate more significantly compared to simple interest, the role of exponents in calculating growth, and the importance of percentages in determining returns. Understanding this concept is crucial for making informed decisions about savings, investing, and managing loans or mortgages.
Compound interest formula: The compound interest formula is a mathematical expression used to calculate the amount of money that will be accumulated over time when interest is applied to both the initial principal and the interest that has been added to it. This formula highlights the power of compounding, where interest earns interest, leading to exponential growth of investments or savings over a period. Understanding this concept is essential for evaluating investment opportunities and making informed financial decisions.
Compounding frequency: Compounding frequency refers to the number of times interest is calculated and added to the principal balance of an investment or loan during a specified period, typically expressed on an annual basis. This concept is crucial in understanding how compound interest works, as more frequent compounding leads to a higher effective interest rate, which can significantly increase the amount earned or owed over time. The choice of compounding frequency can vary, with common options being annually, semi-annually, quarterly, monthly, or daily.
Compounding Period: A compounding period is the interval of time at which interest is calculated and added to the principal balance of an investment or loan. It is a critical concept in understanding how compound interest works, as the more frequently interest is compounded, the more total interest accumulates over time. Different financial products can have varying compounding periods, which can significantly affect the growth of an investment or the cost of a loan.
Continuous compounding: Continuous compounding is a method of calculating interest where the frequency of compounding is theoretically infinite. Instead of compounding at regular intervals, continuous compounding allows interest to accumulate at every possible moment, leading to a higher overall return on investment. This concept is particularly relevant in finance, as it highlights the power of exponential growth in investments and savings.
EAY: EAY stands for Effective Annual Yield, which measures the actual annual return on an investment or savings account after accounting for compounding over a year. It's important because it provides a clearer picture of the true earning potential of an investment compared to nominal rates, especially when interest is compounded more frequently than annually. Understanding EAY helps in comparing different investment options that may have varying compounding frequencies.
Effective annual yield: Effective annual yield (EAY) is the real annual return on an investment, taking into account the effect of compounding interest. It provides a more accurate measure of financial performance than nominal interest rates.
Effective Annual Yield: Effective annual yield (EAY) is the actual annual rate of return on an investment, taking into account the effects of compounding interest over a year. It reflects how much an investment grows in value after accounting for compounding periods within that year, making it a crucial measure for comparing different financial products with varying compounding frequencies.
Exponential growth: Exponential growth is a process where the quantity of something increases at a rate proportional to its current value, resulting in a rapid increase over time. This type of growth can be modeled mathematically using exponents, where the function grows faster as it continues. It has significant implications across various fields, including finance, environmental science, and healthcare, highlighting how quickly a population or investment can grow under ideal conditions.
Fixed interest rate: A fixed interest rate is an interest rate on a loan that remains constant throughout the term of the loan. It does not fluctuate with market conditions, providing predictable payments for borrowers.
Future Value: Future value refers to the amount of money an investment will grow to over a period of time at a given interest rate. This concept is essential for understanding how savings and investments can increase in value due to interest, and it plays a significant role in making financial decisions. Future value helps individuals plan for financial goals by estimating how much money they will have in the future based on current investments, whether it's through simple interest, compound interest, or savings methods.
Interest rate: An interest rate is the percentage at which interest is charged or paid for the use of money, typically expressed as an annual percentage. This rate affects how much money you can earn from savings or how much extra you have to pay on borrowed funds. Understanding interest rates is crucial because they influence loan affordability, savings growth, and overall financial planning.
Logarithms: Logarithms are the mathematical operations that help to solve for the exponent in equations involving exponential growth or decay. They essentially tell us how many times we need to multiply a base number to get another number. In the context of finance, logarithms play a crucial role in understanding compound interest, particularly when determining how long it takes for an investment to grow to a certain value.
Nominal interest rate: The nominal interest rate is the percentage increase in money that a borrower pays to a lender, without adjusting for inflation. It reflects the basic cost of borrowing or the return on investment without taking into account how inflation affects purchasing power. This rate is crucial for understanding how investments grow over time, especially in contexts like compound interest, where the compounding effect can significantly amplify returns.
Present Value: Present value refers to the current worth of a sum of money that is to be received or paid in the future, discounted back to today's value at a specific interest rate. This concept is essential for understanding how money can grow over time, whether through simple interest, compound interest, savings methods, loans, or when considering the costs and benefits of renting versus owning a home. By calculating present value, individuals can make informed financial decisions based on the time value of money.
Principal: The principal is the original sum of money borrowed in a loan or invested, before interest or earnings. It is the base amount on which interest is calculated.
Principal: The principal refers to the initial amount of money that is either deposited or borrowed, which serves as the basis for calculating interest in various financial contexts. Understanding the principal is crucial, as it directly impacts how much interest will be earned or paid over time, influencing savings strategies and loan repayment plans.
Rule of 72: The Rule of 72 is a simple formula used to estimate the number of years required to double an investment based on a fixed annual rate of return. By dividing 72 by the annual interest rate, investors can quickly gauge how long it will take for their money to grow, making it a valuable tool for understanding compound interest and effective savings strategies. This rule can also help individuals weigh the potential returns of renting versus homeownership by considering how quickly investments can grow over time.
Savings account: A savings account is a deposit account held at a financial institution that allows individuals to store money while earning interest on the balance. It provides a safe way to save funds for short-term goals or emergencies, and the interest earned can vary based on the institution and type of account. This type of account typically has lower liquidity compared to checking accounts, as it may have restrictions on the number of withdrawals.
Simple interest: Simple interest is a method of calculating the interest charge on a loan or investment based on the original principal amount, without compounding. This means that the interest is calculated only on the principal and does not take into account any interest that has previously accrued. It is essential for understanding financial transactions, comparing different saving methods, and grasping the concept of compound interest.
Time horizon: A time horizon is the period over which an investment is expected to grow or be held before the funds are needed. It plays a crucial role in determining investment strategies and the type of interest calculations applied, particularly in compound interest scenarios. Understanding the time horizon helps investors balance risk and reward, as longer horizons can accommodate more volatile investments that may offer higher returns.
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