Your subscription to Jogger's World is about to run out and you have the choice of renewing it by sending in the $10 a year regular rate at the end of each year or of getting a lifetime subscription to the magazine by paying $100 today. Your cost of capital is 5 percent. How many years would you have to live to make the lifetime subscription the better buy? Assume payments for the regular subscription are made at the end of each year

To determine the number of years after which a lifetime subscription becomes the better deal, we can compare the present value of the costs associated with both options:

1. **Regular Subscription**: You pay $10 at the end of each year.
2. **Lifetime Subscription**: You pay $100 today.

Given:
- Annual subscription cost: $10
- Cost of lifetime subscription: $100
- Discount rate (cost of capital): 5% (0.05)

### Step 1: Present Value of the Regular Subscription
The present value of the regular subscription is the sum of the present values of $10 payments made at the end of each year. The formula for the present value of an annuity (since the payments are fixed and occur at regular intervals) is:

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

Where:
- $P$ = $10 (annual payment)
- $r$ = 5% = 0.05 (discount rate)
- $n$ = Number of years

### Step 2: Equate Present Value of Regular Subscription to Lifetime Subscription
To find the break-even point, we need to find $n$ such that the present value of the regular subscription equals $100:

\[
P \times \frac{1 - (1 + r)^{-n}}{r} = 100
\]

Let's solve for $n$.The number of years $ n $ required to make the lifetime subscription the better buy is approximately **14.21 years**. 

This means you would need to live for about 15 years or more for the lifetime subscription to be the more cost-effective option compared to paying $10 annually.

Would you like further details or have any questions?

### Additional Questions:
1. How would the break-even point change if the cost of capital were 6% instead of 5%?
2. What if the lifetime subscription cost was $120 instead of $100?
3. How does the break-even point change if the annual subscription cost increases to $12?
4. What would be the present value of the subscription for a 20-year period?
5. How does inflation impact the decision between a regular and lifetime subscription?
6. What is the total cost if you chose the regular subscription for 14 years?
7. How would monthly payments instead of annual payments affect the break-even point?
8. What is the effect of a discount on the lifetime subscription?

### Tip:
When comparing financial options, always consider the time value of money by calculating the present value of future payments. This helps you determine the most cost-effective choice over time.

Learn how to compare subscription costs between a regular annual payment and a lifetime payment option using the time value of money concept. Calculate the break-even point to determine when the lifetime subscription becomes more cost-effective. Suitable for college-level students and financial enthusiasts.

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