Consider the following two plans for fixed deposit with an initial principal $100,000. Plan A is to deposit the initial principal today for 1-year at a fixed deposit rate 2.5% per annum; Plan B is to first deposit the initial principal today for 6 months at a fixed deposit rate 3% (per annum), and renew the deposit at a fixed deposit rate R (per annum) for the subsequent 6 months. Assume that during the renewal, only the initial principal is deposited for the subsequent 6 months (so you take out the interest for the first 6 months and do not deposit the interest in the renewal). Using IRR analysis, for what range of R, Plan B is better than Plan A?

We are comparing two plans for fixed deposits with different structures. Plan A involves a one-time deposit for 1 year, while Plan B involves two consecutive 6-month periods with different rates. To determine when Plan B is better than Plan A, we will use Internal Rate of Return (IRR) analysis.

### Plan A:
The investment in Plan A involves depositing an initial principal $ P = 100,000 $ for 1 year at a fixed rate of $ 2.5\% $ per annum. After 1 year, the principal grows with interest compounded annually. The future value for Plan A is:

\[
\text{Future Value for Plan A} = P \times (1 + r_A) = 100,000 \times (1 + 0.025) = 100,000 \times 1.025 = 102,500.
\]

The interest earned from Plan A is:

\[
\text{Interest from Plan A} = 102,500 - 100,000 = 2,500.
\]

### Plan B:
In Plan B, the initial principal $ P = 100,000 $ is invested for 6 months at a rate of 3% per annum (which is $ 1.5\% $ for 6 months). After the first 6 months, the interest is taken out, and the initial principal is re-deposited for another 6 months at a new rate $ R $ (per annum), with no reinvestment of the earned interest. The calculations are as follows:

1. **First 6 months** at 3% per annum:
\[
\text{Interest for first 6 months} = P \times \frac{r_B}{2} = 100,000 \times \frac{0.03}{2} = 100,000 \times 0.015 = 1,500.
\]
After 6 months, the interest earned is $ 1,500 $, and the remaining principal is still $ 100,000 $.

2. **Next 6 months** at the new rate $ R $ (per annum):
Since only the principal is re-invested at the rate $ R $ for 6 months, the interest earned for this period is:
\[
\text{Interest for second 6 months} = P \times \frac{R}{2} = 100,000 \times \frac{R}{2}.
\]
Thus, the total future value for Plan B after 1 year is the principal plus the interests earned over two 6-month periods:
\[
\text{Future Value for Plan B} = 100,000 + 1,500 + 100,000 \times \frac{R}{2}.
\]
Simplifying:
\[
\text{Future Value for Plan B} = 101,500 + 50,000 \times R.
\]

The interest earned from Plan B is:
\[
\text{Interest from Plan B} = 1,500 + 50,000 \times R.
\]

### IRR Comparison:
To find when Plan B is better than Plan A, we set the condition:

\[
\text{Interest from Plan B} > \text{Interest from Plan A}.
\]

Substituting the expressions for the interests:

\[
1,500 + 50,000 \times R > 2,500.
\]

Solving for $ R $:

\[
50,000 \times R > 1,000,
\]
\[
R > \frac{1,000}{50,000} = 0.02.
\]

Thus, Plan B is better than Plan A when the renewal interest rate $ R $ is greater than 2% per annum.

### Conclusion:
For Plan B to be better than Plan A, the interest rate $ R $ for the second 6-month period must be greater than 2% per annum.

---

Would you like more details or have any questions about the analysis? Here are some related questions to consider:

1. How would the comparison change if the interest from the first 6 months were reinvested in Plan B?
2. What if the interest rates in both plans were compounded more frequently, such as quarterly?
3. How would varying the initial principal impact the results?
4. What happens to the comparison if Plan A's interest rate increases slightly to 2.6%?
5. How does the analysis change if Plan B's renewal rate $ R $ is fixed but lower than 2%?

**Tip:** The Internal Rate of Return (IRR) helps compare investments, but it's important to consider other factors such as risk and liquidity when making financial decisions.

Learn how to compare fixed deposit plans using Internal Rate of Return (IRR) analysis. This problem examines two deposit options with different interest rates, showing when Plan B is more profitable than Plan A based on IRR. Detailed formulas and interest rate calculations are provided. Suitable for advanced high school or college-level students.

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