Which savings account will earn you the least money?

Question: Which savings account will earn you the least money?

a. One that compounds interest daily.
b. One that earns simple interest daily.
c. One that compounds interest monthly.
d. One that earns simple interest monthly.

Answer: b. One that earns simple interest daily.

Brief Explanations:

Option b: One that earns simple interest daily.
This is the correct answer. Simple interest is calculated only on the principal amount, regardless of how frequently the interest is calculated. Therefore, whether interest is calculated daily or monthly under a simple interest scheme does not affect the total amount of interest earned. Consequently, a simple interest account will earn less money compared to compound interest accounts, which calculate interest on both the principal and the accumulated interest.

Why the Other Options Are Incorrect:

• Option a: One that compounds interest daily.
  Incorrect. Compound interest accounts earn interest on both the principal and the previously accumulated interest. Daily compounding means interest is calculated and added to the principal every day, resulting in higher overall earnings compared to less frequent compounding or simple interest.

• Option c: One that compounds interest monthly.
  Incorrect. Although monthly compounding is less frequent than daily compounding, compound interest still yields more than simple interest because interest is calculated on the accumulated interest over time.

• Option d: One that earns simple interest monthly.
  Incorrect. Similar to option b, this account uses simple interest, but the frequency of interest calculation (monthly) does not increase the total interest earned. However, since simple interest is inherently less profitable than compound interest, this option also earns less than the compound interest accounts.

Extended Knowledge:

Understanding Interest Types and Compounding Frequencies

1. Simple Interest vs. Compound Interest

• Simple Interest:

  • Definition: Interest is calculated only on the principal amount.
  • Formula: $$\text{Simple Interest} = P \times r \times t$$ Where:
    • $P$ = Principal amount
    • $r$ = Annual interest rate
    • $t$ = Time in years
  • Implication: The interest earned remains consistent each period, as it does not account for previously earned interest.

• Compound Interest:

  • Definition: Interest is calculated on both the principal and the accumulated interest from previous periods.
  • Formula: $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ Where:
    • $A$ = Amount of money accumulated after n years, including interest.
    • $P$ = Principal amount
    • $r$ = Annual interest rate
    • $n$ = Number of times interest is compounded per year
    • $t$ = Time the money is invested for in years
  • Implication: The more frequently interest is compounded, the more interest is earned over time due to the "interest on interest" effect.

2. Impact of Compounding Frequency

• Daily Compounding:

  • Interest is calculated and added to the principal every day.
  • Results in higher overall earnings compared to monthly, quarterly, or annual compounding.

• Monthly Compounding:

  • Interest is calculated and added to the principal once a month.
  • Yields more than quarterly or annual compounding but less than daily compounding.

• Annual Compounding:

  • Interest is calculated and added to the principal once a year.
  • Yields the least interest among common compounding frequencies.

3. Practical Example: Comparing Interest Earnings

Assume a principal of $1,000, an annual interest rate of 5%, and a time period of 1 year.

• Simple Interest:

  $$\text{Interest} = 1000 \times 0.05 \times 1 = \$50$$
  • Total Amount: $1,050

• Compound Interest (Daily):

  $$A = 1000 \left(1 + \frac{0.05}{365}\right)^{365 \times 1} \approx \$1,051.27$$
  • Interest Earned: $51.27

• Compound Interest (Monthly):

  $$A = 1000 \left(1 + \frac{0.05}{12}\right)^{12 \times 1} \approx \$1,051.16$$
  • Interest Earned: $51.16

• Compound Interest (Annually):

  $$A = 1000 \left(1 + 0.05\right)^1 = \$1,050$$
  • Interest Earned: $50

Observation:

• Compound interest, even when compounded annually, yields the same as simple interest.
• More frequent compounding (daily vs. monthly) results in slightly higher earnings.
• Simple interest and compound interest with annual compounding yield the same amount.

Conclusion

To maximize earnings from a savings account, choosing an account that offers compound interest, especially with more frequent compounding (daily or monthly), is beneficial. Simple interest accounts, regardless of the frequency of interest calculation, do not take advantage of the "interest on interest" effect and thus earn less over time. Among the options provided, a simple interest account with daily calculations (Option b) earns the least money.