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Question 1179079: Betty received $ 500,000 from a life insurance policy to be distributed to her as an annuity certain in 10 equal annual installments with the first payment made immediately. On the day she receives her third payment, she is offered a monthly perpetuity of X in lieu of the future annual payments. The first payment will be made in exactly one month. The effective annual rate of interest is 3 %. Determine the value of X.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem step-by-step:
**1. Calculate the Annual Payment:**
* Total amount: $500,000
* Number of installments: 10
* Annual payment: $500,000 / 10 = $500,000 / 10 = $50,000
**2. Calculate the Present Value of the Remaining Annual Payments:**
* Betty receives the third payment immediately. This means there are 7 remaining payments.
* Effective annual interest rate: 3% (0.03)
* We need to find the present value of an annuity due with 7 payments.
PV = PMT * [(1 - (1 + r)^-n) / r] * (1 + r)
Where:
* PV = Present Value
* PMT = Annual Payment ($50,000)
* r = Annual Interest Rate (0.03)
* n = Number of Remaining Payments (7)
PV = 50000 * [(1 - (1.03)^-7) / 0.03] * (1.03)
PV = 50000 * [(1 - 0.81309152) / 0.03] * 1.03
PV = 50000 * [0.18690848 / 0.03] * 1.03
PV = 50000 * 6.23028267 * 1.03
PV = 50000 * 6.41719115
PV ≈ $320,859.56
**3. Calculate the Equivalent Monthly Interest Rate:**
* Effective annual interest rate: 3% (1.03)
* Monthly interest rate: (1.03)^(1/12) - 1 ≈ 0.00246627
**4. Calculate the Monthly Perpetuity Payment (X):**
* The present value of the perpetuity must equal the present value of the remaining annual payments.
PV_perpetuity = X / r_monthly
Where:
* PV_perpetuity = $320,859.56
* X = Monthly Perpetuity Payment
* r_monthly = Monthly Interest Rate (0.00246627)
X = PV_perpetuity * r_monthly
X = 320859.56 * 0.00246627
X ≈ $791.31
**Answer:**
The value of X is approximately $791.31.
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