I got the first two answers right. I still need help with the last two #3 and #4. Someone did help me but I did not understand and they never responded back.
Your rich uncle bequests to you a continuous, constant income stream of $6000 per year for the next 10 years. The terms of the bequest require that this income stream be paid continuously into a specific savings account that will not be available to you for 10 years. This account earns 5.2% interest, compounded continuously.
1) What is the present value of the bequest? 46786.09061
2) How much money would the bequest be worth (including all interest accrued) after 10 years? 78695.49804
3) You discover that a bank is offering 5.7% interest compounded continuously on a certificate of deposit (CD) that matures in 10 years. What is the cost of a CD at the above interest rate that would provide the same amount of money as the bequest after 10 years?
4) Because the CD earns more interest than the savings account specified in the will, you feel that you are losing out on interest. So you ask the executor of the estate to use funds from the estate to buy a CD that will be worth the same as the bequeathed income stream in 15 years. You ask her to pay you today the difference between the present value of the original bequest and the amount invested in the CD. How much should she pay you today?
To answer questions #3 and #4, we need to use the concept of present value and future value calculations.
Let's start with question #3:
To find the cost of a CD that would provide the same amount of money as the bequest after 10 years, we need to find the present value of the bequest.
The present value (PV) can be calculated using the formula:
PV = FV / e^(rt)
Where:
PV = Present Value
FV = Future Value (the amount of money the bequest would be worth after 10 years, which we calculated to be $78,695.49804)
r = Annual interest rate (5.7% in this case)
t = Time in years (10 years in this case)
e = Euler's number (approximately 2.71828)
Using this formula, we can calculate the present value (cost) of the CD.
PV = 78,695.49804 / (2.71828 ^ (0.057 * 10))
PV ≈ $51,206.35976
Therefore, the cost of a CD at the interest rate of 5.7% that would provide the same amount of money as the bequest after 10 years is approximately $51,206.35976.
Now let's move on to question #4:
To calculate how much the executor of the estate should pay you today, we need to find the difference between the present value of the original bequest and the amount invested in the CD.
The original present value of the bequest is $46,786.09061 (calculated in question #1).
The amount invested in the CD can be calculated by finding the future value (FV) of the CD after 15 years. We can use the formula for future value with continuous compounding:
FV = PV * e^(rt)
Where:
PV = Present Value ($46,786.09061)
r = Annual interest rate (5.7% in this case)
t = Time in years (15 years in this case)
e = Euler's number (approximately 2.71828)
FV = 46,786.09061 * (2.71828 ^ (0.057 * 15))
FV ≈ $97,322.22879
So, the amount invested in the CD after 15 years would be approximately $97,322.22879.
To find the difference, we subtract the original present value from the amount invested in the CD:
Difference = Amount invested in CD - Present value of the bequest
Difference = $97,322.22879 - $46,786.09061
Difference ≈ $50,536.13818
Therefore, the executor should pay you approximately $50,536.13818 today to make up for the difference in interest between the savings account and the CD over a period of 15 years.