Final answer:
To determine the length of the annuity time period given the present value, annual payment, and rate of return, the Present Value of an Ordinary Annuity formula was used. After calculation, the annuity is estimated to last roughly 52 years.
Step-by-step explanation:
To find the length of the annuity time period, we can use the formula for the Present Value of an Ordinary Annuity:
PV = PMT × ¶
(1 - (1 + r)⁻¹ⁿ¹
) / r
Where:
- PV = Present Value of the annuity
- PMT = Payment per period
- r = Rate of return per period
- n = Number of periods
Rewriting the formula to solve for n gives:
n = –(ln(1 - (PV × r) / PMT) / ln(1 + r))
Substituting the given values:
n = –(ln(1 - (70000 × 0.045) / 3500) / ln(1 + 0.045))
Calculating the natural logarithms and performing the division:
n = –(ln(1 - (3150) / 3500) / ln(1.045))
n ≈ –(ln(0.1) / ln(1.045))
n ≈ –(-2.30258509299 / 0.04402060702)
n ≈ 52.34
Therefore, the annuity lasts approximately 52.34 periods. Since we're discussing annual payments, we can round to 52 years.
The length of the annuity is approximately 52 years when payments of $3,500 are made annually on a present value of $70,000 earning 4.5% annual return.