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?
I got the first two answers. I just need help with the last two #3 and #4
#3. if the cost is P, then the amount after x years will be P * 1.057^x
#3 gives you the present value P of the CD. S just subtract.
I disagree with the answer to 1) and 2)
To use the standard annuity formulas, the periods of payment and interest compounding MUST correspond, e.g. you can't have quarterly payments with an interest rate compounded semi-annually.
In this question, the payment period is annual, but the compounding is continuous. So we have to find the equivalent rate of i compounded annually which is equivalent of 5.2% compounded continuously.
(1 + i)^1 = e^(.052)
1 + i = 1.053375....
i = .053375..... (I stored that in my calculator's memory)
1)
so now:
PV = 6000( 1 - 1.053375....^-10)/.053375..
= $45,580.19
2) quick way:
Amount = 45,560.19 e^(10*.052) = $77,667.15
or , using the annuity formula with the equivalent rate:
amount = 6000(1.053375..^10 - 1)/.053375... = 77,667.15
btw, what did you do to get your answers ?
I apologize but I am still confused on #3 and #4 mate.
#1 I did integral symbol 10^ on the top and 0 on the bottom 6000e^(-0.052t).
#2 I did the same equation but the 0.052 is positive. Both of the answers are right though.
To answer questions #3 and #4, we need to use the concept of present value in finance. Present value allows us to compare the value of cash flows that occur at different points in time.
Let's break down each question:
3) To find the cost of a CD that would provide the same amount of money as the bequest after 10 years at an interest rate of 5.7%, compounded continuously, we need to determine the present value (PV) of the bequest. The formula for calculating the present value of a continuous cash flow is:
PV = CF / e^(r*t)
Where:
PV = Present value
CF = Cash flow
r = Interest rate
t = Time
In this case:
CF = $6,000 (annual income stream)
r = 0.057 (5.7% interest rate)
t = 10 years
Using the formula, we can calculate the present value:
PV = $6,000 / e^(0.057*10)
PV ≈ $3,857.76
Therefore, the cost of a CD with a 5.7% interest rate that would provide the same amount of money as the bequest after 10 years is approximately $3,857.76.
4) In this question, we need to determine how much the executor should pay you today to make up for the lost interest due to investing in the lower-interest savings account. This is calculated by finding the difference between the present value of the original bequest and the amount invested in the CD.
First, let's calculate the present value of the original bequest using the given interest rate of 5.2% compounded continuously:
PV = $6,000 / e^(0.052*10)
PV ≈ $4,786.54
Now, we need to calculate the present value of the CD investment that will be worth the same as the bequeathed income stream in 15 years. Using the 5.7% interest rate:
PV = X / e^(0.057*15)
We don't know the value of X yet, but we can set it equal to the present value of the original bequest and solve for X:
$4,786.54 = X / e^(0.057*15)
Solving for X:
X ≈ $9,616.61
Therefore, the executor should pay you today the difference between the present value of the original bequest ($4,786.54) and the amount invested in the CD ($9,616.61), which is approximately $4,830.07