Calculating Annuity Due and Sinking Fund Surplus

Calculating the Future Value of an Annuity Due

Step 1: Determine the Variables

The problem provides the following details:

• Annual payment: Rs. 200. Therefore, the half-yearly payment (Pmt) is:
  
  Rs. 200 / 2 = Rs. 100
  Rs. 200 / 2 = Rs. 100
• Annual interest rate (r): 4% or 0.04. Since the interest is compounded half-yearly, the interest rate per period (i) is:
  
  0.04 / 2 = 0.02
  0.04 / 2 = 0.02
• Term: 20 years. Payments are made half-yearly, so the total number of periods (n) is:
  
  20 × 2 = 40
  20 × 2 = 40
• The annuity type is an annuity due, meaning payments are made at the beginning of each period.

Step 2: Apply the Future Value Formula

The formula for the Future Value (FV) of an annuity due is given by:

FV = Pmt × [((1 + i)^n - 1) / i] × (1 + i)

FV = Pmt × [((1 + i)^n - 1) / i] × (1 + i)

Substitute the values into the formula:

FV = 100 × [((1 + 0.02)^40 - 1) / 0.02] × (1 + 0.02)

FV = 100 × [((1 + 0.02)^40 - 1) / 0.02] × (1 + 0.02)

Step 3: Calculate the Future Value

First, calculate (1.02)^40:

(1.02)^40 ≈ 2.20804

(1.02)^40 ≈ 2.20804

Now, complete the calculation:

FV = 100 × [2.20804 - 1] / 0.02 × (1.02)

FV = 100 × [2.20804 - 1] / 0.02 × (1.02)

FV = 100 × (1.20804 / 0.02) × 1.02

FV = 100 × (1.20804 / 0.02) × 1.02

FV = 100 × 60.402 × 1.02

FV = 100 × 60.402 × 1.02

FV = 6040.2 × 1.02

FV = 6040.2 × 1.02

FV ≈ 6161.00

FV ≈ 6161.00

Answer: The amount of the annuity due is Rs. 6161.00.

Calculating Sinking Fund Surplus

Step 1: Calculate the Future Value of the Sinking Fund

The future value (FV) of an ordinary annuity is calculated using the formula:

FV = PMT × [((1 + i)^n - 1) / i]

FV = PMT × [((1 + i)^n - 1) / i]

Where:

• 
  PMT = Annual payment (Rs. 5,000)
  PMT = Rs. 5,000
• 
  i = Interest rate per period (0.05)
  i = 0.05
• 
  n = Number of periods (10 years)
  n = 10

Plugging in the values:

FV = 5000 × [((1 + 0.05)^10 - 1) / 0.05]

FV = 5000 × [((1 + 0.05)^10 - 1) / 0.05]

FV = 5000 × [(1.05^10 - 1) / 0.05]

FV = 5000 × [(1.05^10 - 1) / 0.05]

FV = 5000 × [(1.62889462677 - 1) / 0.05]

FV = 5000 × [(1.62889462677 - 1) / 0.05]

FV = 5000 × (0.62889462677 / 0.05)

FV = 5000 × (0.62889462677 / 0.05)

FV = 5000 × 12.5778925354

FV = 5000 × 12.5778925354

FV ≈ Rs. 62,889.46

FV ≈ Rs. 62,889.46

Step 2: Calculate the Surplus

The surplus is the difference between the accumulated fund's future value and the debenture's redemption value.

Surplus = Rs. 62,889.46 - Rs. 60,000

Surplus = Rs. 62,889.46 - Rs. 60,000

Surplus ≈ Rs. 2,889.46

Surplus ≈ Rs. 2,889.46

Answer: The surplus after redeeming the debenture will be Rs. 2,889.46.