Choice 1: Payments of $ 2600 now, $ 3200 a year from now, and $ 3880 two years from now.
Choice 2: Three yearly payments of $ 3200 starting now.
Modification: Interest is compounded continuously instead of annually.
(a) If the interest rate on savings were 4.86 %, which would you prefer?
(Type in 1 for Choice 1, or 2 for Choice 2.)
(b) What is the interest rate that would make both choices equally lucrative?
a.) choice 1
To answer these questions, we will calculate the present value of both choices and compare them.
(a) To find the present value of Choice 1 with continuous compounding, we will use the formula:
PV = A / e^(rt)
Where:
PV = Present Value
A = Future Value of the payment
e = Euler's Number (approximately 2.71828)
r = Interest rate (in decimal)
t = Time in years
For Choice 1, the payments are $2600 now, $3200 in one year, and $3880 in two years.
PV1 = 2600 / e^(0.0486*0) + 3200 / e^(0.0486*1) + 3880 / e^(0.0486*2)
Calculating this expression, we find PV1 ≈ $8880.14
For Choice 2, the annual payments are $3200 each. We will calculate the present value using the formula:
PV2 = Payment / r [1 - (1 + r)^(-n)]
Where:
Payment = Annual Payment
r = Interest rate (in decimal)
n = Number of payments
PV2 = 3200 / (0.0486) [1 - (1 + 0.0486)^(-3)]
Calculating this expression, we find PV2 ≈ $8563.68
Since the present value of Choice 1 (PV1) is greater than the present value of Choice 2 (PV2), if the interest rate is 4.86%, you should prefer Choice 1.
Therefore, the answer to (a) is 1 for Choice 1.
(b) To find the interest rate that would make both choices equally lucrative, we can set the two present values equal to each other and solve for the interest rate.
2600 / e^(r*0) + 3200 / e^(r*1) + 3880 / e^(r*2) = 3200 / r [1 - (1 + r)^(-3)]
This equation is not easily solvable algebraically. You would need to use numerical methods such as approximation techniques or a solver tool to find the interest rate that satisfies this equation.
Therefore, we cannot provide a specific interest rate that would make both choices equally lucrative without further calculations.
To determine which choice is preferable in scenario (a), we can compare the present values of each choice using the continuous compound interest formula:
PV = A / e^(rt)
Where:
PV is the present value
A is the future value
e is the base of the natural logarithm (~2.71828)
r is the interest rate
t is the time in years
For Choice 1:
PV1 = 2600 / e^(0.0486 * 0) + 3200 / e^(0.0486 * 1) + 3880 / e^(0.0486 * 2)
For Choice 2:
PV2 = 3200 / e^(0.0486 * 0) + 3200 / e^(0.0486 * 1) + 3200 / e^(0.0486 * 2)
Calculating these values will give us the present value of each choice.
For scenario (b), we need to find the interest rate that makes both choices equally lucrative. We can set up an equation by equating the present values of the two choices:
2600 / e^(rt) + 3200 / e^(rt) + 3880 / e^(rt) = 3200 / e^(rt) + 3200 / e^(rt) + 3200 / e^(rt)
Simplifying the equation will give us the desired interest rate.
Let's calculate the values for both scenarios (a) and (b):
(a) Calculation:
PV1 = 2600 / e^(0.0486 * 0) + 3200 / e^(0.0486 * 1) + 3880 / e^(0.0486 * 2)
PV2 = 3200 / e^(0.0486 * 0) + 3200 / e^(0.0486 * 1) + 3200 / e^(0.0486 * 2)
Compare PV1 and PV2 to determine which choice is preferable.
(b) Calculation:
Set up the equation:
2600 / e^(rt) + 3200 / e^(rt) + 3880 / e^(rt) = 3200 / e^(rt) + 3200 / e^(rt) + 3200 / e^(rt)
Solve for r to find the interest rate that makes both choices equally lucrative.
Perform the necessary calculations to obtain the values for both scenarios (a) and (b).
Assuming choices are payment options, i.e. we will be paying the said amounts.
Total amount of payment for each choice is $9600.
Assuming we have $9600 in the bank, we calculate what would we have accumulated in interest at the end of two years.
Choice 1:
3800 for 2 years +
3200 for 1 year.
Interest at annual rate 4.86%
=3800*1.0486²-3800+3200*0.0486
=533.86
Choice 2:
3200 for 2 years + 3200 for 1 year
Interest at continuous rate of 4.86%
=3200(e^(.0486*2))-3200+3200(e^(.0486))-3200
=486.02
So the first choice is more profitable.