Find the accumulated present value of an investment over a 9 year period if there is a continuous money flow of $10,000 per year and the interest rate is 1.5% compounded continuously.

Question

Find the accumulated present value of an investment over a 9 year period if there is a continuous money flow of $10,000 per year and the interest rate is 1.5% compounded continuously.

Answer 1

This question is a bit ambiguous in that it seems to conflate present value and accumulated value. As this is a continuous annuity, I'm going to go with the present value interpretation because I've never seen a formula for accumulated value when the annuity is continuous.

The formula for present value of a continuous annuity with a payment amount of $1 is (1 - vⁿ) / δ, where v = (1 + I)^-1 and δ is the force of interest that satisfies the equation 1 + i = e^δ.

Since we're told that the interest is being compounded continuously at a 1.5% (annual?) rate, then 0.015 must be the force of interest. We still need to find the annual rate that corresponds to this force of interest. So we set up the equation 1 + i = e^0.015, and solve. The result is very close to 0.015. It is 0.0151130646. So we then plug all these values in and get

10,000·(1 - (1.0151130646)^-9) / 0.015 ≈ 84,189.39.

Answer 2

I will start with 10g. and add it each year.

fv=p(1+int/c)^(tn)

1yr 10g(1+.015/365)^(365)=10151.13

2 +10g 20455.67

3 " 30915.94

4 " 41534.29

5 " 52313.11

6 " 63254.84

7 " 74361.92

8 " 85636.86

9 " $97082.19