Metro Corporation has to retire Rs.20 million of debentures each at the end of 6, 7, and 8 years from now. How much should the firm deposit in a sinking fund account annually for 5 years, in order to meet the debenture retirement need? The net interest rate earned is 10 %.

To solve this problem, we need to determine the annual deposit that Metro Corporation should make into a sinking fund in order to accumulate enough money to retire the debentures at the end of years 6, 7, and 8.

### Step 1: Determine the future value needed

Metro Corporation needs to retire debentures of Rs.20 million at the end of each of the three years: year 6, year 7, and year 8. 

- At the end of year 6: Rs.20 million
- At the end of year 7: Rs.20 million
- At the end of year 8: Rs.20 million

### Step 2: Calculate the present value of each future payment

Given that the sinking fund earns a net interest rate of 10%, we can discount each of these future payments to the end of year 5, which is when the last annual deposit will be made. 

The present value \( PV \) of a future amount \( FV \) is given by:

\[
PV = \frac{FV}{(1 + r)^n}
\]

Where:
- \( FV \) is the future value
- \( r \) is the interest rate (10% or 0.10)
- \( n \) is the number of years

1. **For the payment at the end of year 6 (1 year after the last deposit):**
   \[
   PV_6 = \frac{20,000,000}{(1 + 0.10)^1} = \frac{20,000,000}{1.10} \approx 18,181,818.18 \text{ Rs.}
   \]

2. **For the payment at the end of year 7 (2 years after the last deposit):**
   \[
   PV_7 = \frac{20,000,000}{(1 + 0.10)^2} = \frac{20,000,000}{1.21} \approx 16,528,925.62 \text{ Rs.}
   \]

3. **For the payment at the end of year 8 (3 years after the last deposit):**
   \[
   PV_8 = \frac{20,000,000}{(1 + 0.10)^3} = \frac{20,000,000}{1.331} \approx 15,032,150.77 \text{ Rs.}
   \]

### Step 3: Calculate the total present value

The total present value (amount that needs to be accumulated by the end of year 5) is:

\[
PV_{total} = PV_6 + PV_7 + PV_8 = 18,181,818.18 + 16,528,925.62 + 15,032,150.77 \approx 49,742,894.57 \text{ Rs.}
\]

### Step 4: Determine the annual sinking fund deposit

To find the annual deposit, we need to use the formula for the future value of an annuity:

\[
FV = A \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( FV = 49,742,894.57 \text{ Rs.} \) (The total present value we calculated)
- \( A \) is the annual deposit
- \( r = 0.10 \) (The interest rate)
- \( n = 5 \) (The number of years of deposits)

Rearranging the formula to solve for \( A \):

\[
A = \frac{FV}{\frac{(1 + r)^n - 1}{r}} = \frac{49,742,894.57}{\frac{(1 + 0.10)^5 - 1}{0.10}}
\]

\[
A = \frac{49,742,894.57}{\frac{1.61051 - 1}{0.10}} = \frac{49,742,894.57}{6.1051} \approx 8,145,897.02 \text{ Rs.}
\]

### Final Answer:

The firm should deposit approximately **Rs.8,145,897.02** annually for 5 years in the sinking fund account.

---

Would you like more details on any specific part of the calculation or have any questions?

### Relative Questions:
1. How does the interest rate affect the annual deposit needed for the sinking fund?
2. What would be the impact if the payments were due earlier than 6, 7, and 8 years?
3. How would the required deposit change if the interest rate was compounded semi-annually instead of annually?
4. What would happen if the firm decided to deposit different amounts each year instead of a fixed annual deposit?
5. How can the firm ensure that the sinking fund will accumulate the exact amount needed, considering potential fluctuations in interest rates?

### Tip:
Always consider the time value of money when dealing with future payments or investments. Understanding how to discount future amounts to present value is crucial in financial planning.

Learn how Metro Corporation can calculate annual sinking fund deposits to retire Rs.20 million of debentures at the end of 6, 7, and 8 years. Detailed solution using finance concepts including time value of money and sinking fund calculations.

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