Question

Use \(y=y_{0} e^{k t}\). If a bank offers annual interest of \(7.5 \%\) or continuous interest of \(7.25 \%\), which has a better annual yield?

Short answer

Continuous interest at 7.25% has a slightly better yield than 7.5% discrete interest.

Step-by-step solution

Express the formulas

We have two scenarios: discrete interest and continuous interest. For discrete interest compounded annually, use the formula \( y = y_0 (1 + r)^t \). For continuous interest, use \( y = y_0 e^{r \cdot t} \). Here, \( r \) is the interest rate in decimal, and \( t \) is time in years.

Calculate the yield with discrete interest

Convert the 7.5% annual discrete interest to a decimal by dividing by 100: \( r = 0.075 \). For one year \( t = 1 \), substitute in the formula: \( y = y_0 (1 + 0.075)^1 = y_0 \cdot 1.075 \). The yield is \( 7.5\% \).

Calculate the yield with continuous interest

Convert the 7.25% annual continuous interest to a decimal: \( r = 0.0725 \). For one year \( t = 1 \), substitute in the formula: \( y = y_0 e^{0.0725} \). Calculate \( e^{0.0725} \) using a calculator to find the continuous yield: \( e^{0.0725} \approx 1.07526 \).

Compare the yields

By comparing the values from the calculations, the discrete interest provides a final amount of \( 1.075 \), and the continuous interest provides approximately \( 1.07526 \). Thus, continuous interest has a higher yield than discrete interest.

Key concepts

Continuous Compounding

Continuous compounding is a concept in finance where your interest is calculated and added to the principal continuously. Unlike other methods, which compound at specific intervals (like annually or monthly), continuous compounding uses an infinite number of compounding periods per year. This is represented by the formula \[ y = y_0 e^{r imes t} \]where:

• \( y \) is the final amount after time \( t \)
• \( y_0 \) is the initial principal amount
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828
• \( r \) is the annual interest rate expressed as a decimal
• \( t \) is the time in years

By using continuous compounding, small amounts of interest are constantly being added to the principal, providing slightly higher returns over time compared to discrete compounding. This makes it a powerful tool for maximizing interest accrued.

Discrete Compounding

Discrete compounding involves interest being added to the principal at set intervals. These intervals can be annually, semi-annually, quarterly, monthly, or daily. The most basic form of discrete compounding, which occurs annually, can be calculated using\[ y = y_0 (1 + r)^t \]where:

• \( y \) is the amount after time \(t\)
• \( y_0 \) is the initial principal amount
• \( r \) is the interest rate per period as a decimal
• \( t \) is the total number of periods

Annual compounding means the interest is calculated once at the end of each year. This strategy of compounding generates interest on both the initial principal and the accumulated interest from previous periods, leading to exponential growth over time.
Discrete compounding is commonly used in savings accounts, certificates of deposit, and fixed-rate mortgages. Each time interest is compounded, the balance on which future interest is calculated increases, effectively snowballing the growth of investment.

Interest Rate Calculations

Interest rate calculations help determine how much money an investment will earn over a certain period of time. When calculating interest, the rate needs to be expressed as a decimal. For example, a 7.5% interest rate becomes 0.075. This conversion is crucial for using formulas like those for continuous and discrete compounding.
To compare interest rates effectively, one must understand the effect compounding frequency has on the yield. The more frequently interest is compounded, the higher the effective yield for an investment over similar time spans.
Interest calculations are fundamental not only for investors but for borrowers too, as they demonstrate how loan interest accumulates. Understanding these calculations allows individuals to make more informed financial decisions, weighing various investment opportunities or loan offers according to their true cost or income potential.

Annual Yield Comparison

Comparing annual yields involves analyzing the returns of different compounding methods to find the most beneficial financial option. When evaluating financial products with different compounding strategies, it's key to compare their effective annual rate (EAR) or annual percentage yield (APY).
The effective annual rate elaborates on how much interest an investment earns over a year, taking compounding into account. For continuous compounding, the formula gives a slightly higher yield compared to discrete compounding at the same nominal rate.
In practice, an investor might encounter a choice between, for instance, a discrete interest offer of 7.5% and a continuous rate of 7.25%. By calculating the final amounts one year later, it becomes apparent that continuous interest, though slightly lower in nominal terms, yields a marginally higher return. Therefore, understanding yield comparisons helps investors make informed decisions that optimize returns.