# Solving an Actuarial Problem: Investing $10,000 for Annual Scholarships • ToxicBug In summary, the conversation discusses a problem involving an investment made in 1970 with an annual interest rate of 5%. The goal is to determine when the first payment of$2000 can be made and what smaller payment can be made one year earlier while still allowing for the annual scholarships of $2000. The conversation also touches on finding the duration of a fund with a continuous 4% growth rate and continuous withdrawals of$2300 per year.
ToxicBug
This is a problem on an assignment for my actuarial class.
A sum of 10,000 was invested on September 1, 1970 at an effective annual interest rate of 5% in order to provide an annual scholarship of 2000 every September 1 forever, starting as soon as possible. In what year will the first payment of 2000 be made? What smaller payment could be made one year earlier while still permitting the annual scholarships of 2000 thereafter? Assume that interest is credited every August 31.

First of all I found how much time it would take for the investment to reach the present value of the perpetuity:

10000(1 + i)^n = 2000/i
10000(1 + 0.05)^n = 2000/0.05

n = ln(4)/ln(1.05)
n = 28.41339817

Then for the second part I did this:

X + 10000(1.05)^(28.41339817 - 1) = 2000/0.05
X = 1904.7618

But the answer in the back of the book is 1161.36

Anyone know what is my mistake?

do you mean $$10000(1+i)^{t}$$?

ToxicBug said:
Then for the second part I did this:

X + 10000(1.05)^(28.41339817 - 1) = 2000/0.05
X = 1904.7618

But the answer in the back of the book is 1161.36

Anyone know what is my mistake?

The scholarship payments aren't starting at year 28.41339... Year 29 is the first year you have enough to sustain the perpetuity, the first payment is at year 30 though. The excess payment would be at year 29.

Brilliant, thanks!

Another question if you don't mind, I would like to get a hint on what I'm supposed to do:
There is $40,000 in a fund which is accumulating at 4% per annum convertible continuously. If money is withdrawn continuously at a rate of$2300 per year, how long will the fund last?

$$P = P_{0}e^{rt}$$

Tried that, didn't work.

Any ideas?

## 2. How much should I expect to earn from this investment?

The amount you can expect to earn from this investment depends on several factors such as the performance of the market, the types of investments you choose, and the fees associated with managing your investments. It's important to have realistic expectations and to regularly review and adjust your investment strategy to maximize your returns.

## 3. How can I ensure the safety of the invested funds?

To ensure the safety of your invested funds, it's important to do thorough research on the investment options available. Look for investments with a track record of stable returns and low risk. It's also important to regularly monitor and review your investments to make sure they are still meeting your safety criteria.

## 4. How can I make sure the funds will be available for annual scholarships?

To ensure the funds are available for annual scholarships, it's important to have a long-term investment strategy and to regularly review and adjust it. This will help you to maintain a steady stream of returns and ensure the funds are available for the scholarships each year.

## 5. Are there any tax implications for investing in scholarships?

The tax implications of investing in scholarships may vary depending on the type of investments you choose. It's important to consult with a tax professional to understand any potential tax implications and to ensure you are following all tax laws and regulations.

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