Calculating the Present Value of a Deferred Annuity
In this article, we will explore how to calculate the present value of a deferred annuity. A deferred annuity is a financial product that pays a series of payments beginning at a future date, after an initial deferral period. We will break down the calculation into two main steps: calculating the present value of the annuity payments at the time they begin, and discounting that value back to the present purchase time. Let's dive into the details.
Step 1: Calculate the Present Value of the Annuity Payments
A deferred annuity in this case pays $5000 per quarter for 10 years starting 5 years from now. The annual interest rate is 6%, compounded quarterly. First, we need to consider the components of the calculation:
Quarterly interest rate: Since the interest is compounded quarterly, the quarterly interest rate is: r r frac{6}{4} 0.015
The total number of payment quarters is: r n 10 , text{years} times 4 , text{quarters/year} 40 , text{quarters}
The present value PV of the annuity payments at the time they begin can be calculated using the present value of an annuity formula: r RV frac{left(1 - r^{-n}right)}{r}
Substituting the values: r RV frac{left(1 - left(0.015^{-40}right)right)}{0.015}
Calculating r 1 - 0.015^{-40} approx 0.5337
Now substituting back into the equation: r RV 5000 times left(frac{1 - 0.5337}{0.015}right) approx 5000 times 31.0867 approx 155433.5
Step 2: Discount the Present Value Back to Today
Now, we need to discount this present value back to the present time, 5 years before the annuity starts. The present value PV0 can be calculated using the formula: r {mi PV_0 PV times left(1 - r^{-t}right)}
Where:
( t 5 , text{years} times 4 , text{quarters/year} 20 , text{quarters} )Substituting the values: r {mi PV_0 155433.5 times left(1 - 0.015^{-20}right)}
Calculating r 1 - 0.015^{-20} approx 0.7435
Now substituting back into the equation: r {mi PV_0 155433.5 times 0.7435 approx 115556.6}
Conclusion
The present value of the deferred annuity is approximately $115556.60. This comprehensive approach ensures a clear understanding of how to calculate the present value of a deferred annuity, providing valuable insights for financial planning and decision-making.