Comparing Investment Schemes with Different Compounding Frequencies

The question asks us to determine which investment scheme is most beneficial for Karuna if she invests a principal amount P for one year at an annual interest rate of 28%, with varying compounding frequencies. The benefit of an investment scheme is usually measured by the final amount received at the end of the investment period. A higher final amount means a more beneficial scheme.

Understanding Compound Interest and Compounding Frequency

Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. The compounding frequency is how many times per year the interest is calculated and added to the principal. The more frequently interest is compounded, the faster the balance grows, leading to a higher final amount for the same annual rate and investment period.

The formula for compound interest is:

Where:

• \( A \) is the final amount after \( t \) years.
• \( P \) is the principal amount.
• \( r \) is the annual interest rate (as a decimal).
• \( n \) is the number of times interest is compounded per year.
• \( t \) is the time the money is invested for, in years.

In this problem, the principal \( P \) is the same for all schemes, the annual rate \( r \) is 28% or 0.28, and the time \( t \) is 1 year. The difference lies only in the compounding frequency \( n \).

Analyzing Each Investment Scheme

Scheme I: Compounded Quarterly

In this scheme, interest is compounded every quarter. There are 4 quarters in a year.

• Annual interest rate, \( r = 0.28 \).
• Time period, \( t = 1 \) year.
• Compounding frequency, \( n = 4 \) (quarterly).
• Interest rate per period, \( \frac{r}{n} = \frac{0.28}{4} = 0.07 \).
• Total number of periods, \( nt = 4 \times 1 = 4 \).

The final amount for Scheme I is:

Let's calculate \( (1.07)^4 \):

So, \( A_I \approx P \times 1.3108 \).

Scheme II: Compounded Half-Yearly

In this scheme, interest is compounded every half year. There are 2 half-years in a year.

• Annual interest rate, \( r = 0.28 \).
• Time period, \( t = 1 \) year.
• Compounding frequency, \( n = 2 \) (half-yearly).
• Interest rate per period, \( \frac{r}{n} = \frac{0.28}{2} = 0.14 \).
• Total number of periods, \( nt = 2 \times 1 = 2 \).

The final amount for Scheme II is:

Let's calculate \( (1.14)^2 \):

So, \( A_{II} = P \times 1.2996 \).

Scheme III: Compounded Yearly

In this scheme, interest is compounded every year. There is 1 year in a year.

• Annual interest rate, \( r = 0.28 \).
• Time period, \( t = 1 \) year.
• Compounding frequency, \( n = 1 \) (yearly).
• Interest rate per period, \( \frac{r}{n} = \frac{0.28}{1} = 0.28 \).
• Total number of periods, \( nt = 1 \times 1 = 1 \).

The final amount for Scheme III is:

So, \( A_{III} = P \times 1.28 \).

Comparing the Outcomes

To find the most beneficial scheme, we compare the final amounts \( A_I, A_{II}, \) and \( A_{III} \). Since the principal \( P \) is the same and positive, we can compare the factors by which \( P \) is multiplied:

• Scheme I: \( 1.3108 \)
• Scheme II: \( 1.2996 \)
• Scheme III: \( 1.28 \)

Comparing these values:

This shows that \( A_I > A_{II} > A_{III} \). The final amount is highest for Scheme I, where the interest is compounded quarterly.

Conclusion: Most Beneficial Scheme

The scheme that results in the highest final amount for the same principal, rate, and time period is the most beneficial for the investor. Our analysis shows that Scheme I, with quarterly compounding, yields the highest final amount compared to half-yearly or yearly compounding for the given rate and period. Therefore, Scheme I is the most beneficial scheme for Karuna.

Revision Table: Investment Schemes Summary

Additional Information: Effective Annual Rate (EAR)

The benefit of higher compounding frequency can also be understood through the Effective Annual Rate (EAR). The EAR is the actual rate of interest earned in one year, taking into account the effect of compounding. The formula for EAR is:

Let's calculate the EAR for each scheme:

• Scheme I (Quarterly): \( EAR_I = (1 + 0.07)^4 - 1 \approx 1.3108 - 1 = 0.3108 \) or 31.08%.
• Scheme II (Half-Yearly): \( EAR_{II} = (1 + 0.14)^2 - 1 = 1.2996 - 1 = 0.2996 \) or 29.96%.
• Scheme III (Yearly): \( EAR_{III} = (1 + 0.28)^1 - 1 = 1.28 - 1 = 0.28 \) or 28%.

Comparing the EARs, \( 31.08\% > 29.96\% > 28\% \). The scheme with the highest EAR is the most beneficial, which is Scheme I. This confirms our earlier finding. A higher compounding frequency leads to a higher Effective Annual Rate, thus yielding a greater return on the investment.