Your grandmother bought an annuity from Rock Solid Life Insurance Company for $200,000 when she retired. In exchange for the $200,000, Rock Solid will pay her $25,000 per year until she dies. The interest rate is 5%. How long must she live after the day she retired to come out ahead (that is, to get more in value than what she paid in)?

To determine how long your grandmother must live after retirement to come out ahead, we need to figure out when the total value of the payments she receives from the annuity exceeds the original $200,000 she paid. Since the annuity includes an interest rate of 5%, we need to account for that in our calculations.

### Key Information:
- Initial amount paid for the annuity: \$200,000
- Annual payment: \$25,000
- Interest rate: 5% (which means the annuity value grows at 5% per year)

### Approach:

To solve this, we’ll determine how long it takes for the present value of the payments to exceed the \$200,000 investment. We'll use the concept of **present value of an annuity**, which adjusts future payments for the time value of money (5% interest in this case).

The formula for the present value of an annuity is:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $ PV $ is the present value (in this case \$200,000),
- $ P $ is the annual payment (\$25,000),
- $ r $ is the interest rate per period (5% or 0.05),
- $ n $ is the number of years (what we're solving for).

We can rearrange this equation to solve for $ n $. I'll do the calculations next.Your grandmother must live at least **10.47 years** after retiring to receive more in value than what she originally paid for the annuity, considering the 5% interest rate. This is approximately 10 years and 6 months.

Would you like further details or clarification?

Here are five related questions you may find interesting:
1. What would happen if the interest rate was higher or lower?
2. How would the outcome change if the annual payment was increased or decreased?
3. What is the total amount she would receive after 15 or 20 years?
4. How does the concept of present value apply to other financial products?
5. How could inflation affect the real value of these payments over time?

**Tip:** Understanding the time value of money is crucial in financial decisions, as it shows how future payments are worth less in today's terms due to inflation or interest rates.

Determine how long someone must live to come out ahead on a $200,000 annuity with a 5% interest rate and annual payments of $25,000. Learn how to use the Present Value of Annuity formula and account for interest rates in this real-world financial problem.

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