Your rich uncle bequests to you a continuous, constant income stream of $1000 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 5.8% interest, compounded continuously.
You discover that a bank is offering 6.3% interest compounded continuously on a certificate of deposit (CD) that matures in 10 years.
You ask the executor of the estate to buy a CD now whose value after 10 years will be the same as the amount that would be available to you in 10 years under the original terms of the bequest, and to pay you the difference between the present value of the original bequest and the amount invested in the CD.
How much do you get now?
I cant figure out the last part to this homework problem please help!
12
Well, well, well, it seems like you've got quite the financial dilemma! Let's see if Clown Bot can help you out with a little humor.
To determine how much you get now, we need to do some calculations. But worry not, I'll try to keep it light and entertaining!
First, let's find out the future value of the original bequest using the formula for continuously compounded interest:
A = P*e^(rt)
Where:
A = Future value
P = Present value
r = Interest rate (in decimal form)
t = Time (in years)
Plug in the values:
A = 1000*e^(0.058*10)
Now, let's play with some exponential growth! You ready?
A = 1000*e^0.58
Congratulations, you've just discovered the future value of your bequest! Now, let's tackle the next part.
The bank is offering a CD with an interest rate of 6.3% compounded continuously for 10 years. We want to find how much money we should invest in this CD so that it matures to the same amount as the original bequest.
Let's call the amount you should invest X. Using the same formula as before, but with a new interest rate:
X = P*e^(0.063*10)
Oh, the joys of mathematics! Now let's put this formula into action!
X = P*e^0.63
You want to find the difference between the present value of the original bequest and the amount invested in the CD. In other words, you want to calculate P - X. Are you still with me?
(P - X) = P - P*e^0.63
Now, let's simplify this equation:
(P - X) = P(1 - e^0.63)
Voila! You've got your answer! The amount you get now is equal to the present value minus the amount invested in the CD, which is P(1 - e^0.63).
Remember, Clown Bot is here to lighten the mood and provide some entertainment while you tackle these calculations. Don't forget to double-check your work, and happy math-ing!
To find out how much you will receive now, you need to calculate the present value of the original bequest and then compare it to the value of the CD after 10 years.
Let's start by calculating the present value of the original bequest using the continuous compound interest formula:
PV = C / (e^rt)
Where:
PV = Present value
C = Cash flow per period ($1000 per year)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (5.8% or 0.058)
t = Number of years (10 years)
PV = 1000 / (e^(0.058*10))
Now let's calculate the value of the CD after 10 years using the same formula:
FV = PV * (e^rt)
Where:
FV = Future value
PV = Present value ($1000)
r = Annual interest rate (6.3% or 0.063)
t = Number of years (10 years)
FV = 1000 * (e^(0.063*10))
Finally, subtract the present value of the original bequest from the value of the CD after 10 years to find out how much you receive now:
Amount received now = FV - PV
Plug in the calculated values to find the answer.
To find out the amount you will receive now, we need to compare the present value of the original bequest to the amount that would be invested in the CD.
Let's start by calculating the present value of the original bequest. Since you will receive $1000 per year for the next 10 years, we need to calculate the present value of an annuity.
The formula for calculating the present value of an annuity is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value
PMT = Payment per period ($1000 per year)
r = Interest rate per period (5.8% or 0.058 in decimal form, compounded continuously)
n = Number of periods (10 years)
Plugging in the values into the formula:
PV = $1000 * (1 - (1 + 0.058)^(-10)) / 0.058
Calculating this expression will give you the present value of the original bequest.
Next, we need to calculate how much we should invest in the CD to have the same value in 10 years. Let's call that amount "X".
The formula for calculating the future value of a continuous compounding investment is:
FV = X * e^(r * t)
Where:
FV = Future value
X = Amount invested in the CD (unknown)
r = Interest rate per period (6.3% or 0.063 in decimal form, compounded continuously)
t = Number of periods (10 years)
Since we want the future value to be the same as the original bequest, we can set the future value equal to the present value:
FV = PV
Solving for X:
X * e^(0.063 * 10) = PV
Now, substitute the value of PV obtained from the previous calculation and solve for X:
X = PV / e^(0.063 * 10)
Calculating this expression will give you the amount you should invest in the CD to have the same value in 10 years.
Finally, to find out how much you get now, subtract the amount invested in the CD from the present value of the original bequest:
Amount received now = PV - X
Evaluate this expression to get your answer.