Choice 1: Payments of $ 2600 now,
$3000 a year from now, and $3450 two years from now.
Choice 2: Three yearly payments of $ 3000 starting now.
Interest is compounded continuously.
(a) If the interest rate on savings were 4.76 %,which would you prefer?
(b) What is the interest rate that would make both choices equally lucrative?
This is not the calculus I know.
Working present value;
Option 1
PV=2600+3000/(1+i) + 3450/(1+i)^2
calculate that with i=.0476
Option 2
PV=3000+3000/(1+i) + 3000/(1+i)^2
calculate that
which is the lower PV? That is the prefered option.
1
To compare Choice 1 and Choice 2, we need to determine the present value of each choice using the formula for continuous compounding:
PV = P * e^(-rt)
Where:
PV = Present value of the cash flows
P = Future value of the cash flows
r = Interest rate
t = Time
(a) To compare the two choices with an interest rate of 4.76%, we can calculate the present value of each option.
For Choice 1:
PV = 2600 * e^(-0.0476*0) + 3000 * e^(-0.0476*1) + 3450 * e^(-0.0476*2)
For Choice 2:
PV = 3000 * e^(-0.0476*0) + 3000 * e^(-0.0476*1) + 3000 * e^(-0.0476*2)
By calculating these values, we can compare the present values to determine which choice is more preferable.
(b) To find the interest rate that would make both choices equally lucrative, we need to set the present values of both choices equal to each other and solve for the interest rate.
For example, using present value formulas for both choices, we have:
2600 * e^(-rt) + 3000 * e^(-(r*t+1)) + 3450 * e^(-(r*t+2)) = 3000 * e^(-rt) + 3000 * e^(-(r*t+1)) + 3000 * e^(-(r*t+2))
By simplifying and solving this equation, we can find the interest rate that makes both choices equally lucrative.
Please note that these calculations involve advanced mathematical operations. To get precise answers, it is recommended to use a financial calculator, spreadsheet software, or specialized financial software capable of handling continuous compounding formulas.