Melanie would like to have $25,706.30 in an ordinary annuity in 5 years. The annuity has a 10.8% annual interest rate, and she plans on making quarterly deposits. How much should she deposit per quarter?
Would this be the way I solve the question?
(1+.108/4)^5-1
[.108/4]
= .1424895016
.1424895016/( .108/4)
=5.277388946
25706.30 (5.277388946)
= 135662.14
Would 135662.14 be the answer?
Thanks!!!
No, your exponent is incorrect
i = .108/4 = .027
but n = 5(4) or 20 periods
let the payment be P
P (1.027^20 - 1)/.27 = 25,706.30
P(26.06525..) = 25,706.30
P = $986.23
Why wouldn't your answer of $135,662.14 sound totally illogical to you??
You only wanted to accumulate $25,700, how could you possible make 20 payment of over $135,000 each ??
Well, Melanie, if you want me to clown around with some math for you, let's get started!
First, you're on the right track with the formula for calculating the future value of an ordinary annuity. However, it seems like you made a small error in your calculation. Let me give you the correct breakdown:
Step 1: Calculate the interest rate per period:
10.8% annual interest rate / 4 (quarterly deposits) = 2.7% interest rate per quarter.
Step 2: Calculate the number of quarters:
5 years x 4 quarters per year = 20 quarters.
Step 3: Calculate the deposit amount per quarter:
$25,706.30 / [(1 + 2.7%)^20 - 1] = $1,169.79 (approximately).
So, the correct answer is that Melanie should deposit around $1,169.79 per quarter to reach her goal in five years. Hope that helps, and don't forget to pack some juggling balls for your annuity journey!
Yes, you are on the right track in solving the question. However, there is a small mistake in your calculation at the step where you find the present value factor of the annuity.
The correct equation to find the present value factor of the annuity is:
PV = C * [(1 - (1+r)^(-n))/(r)]
Where:
PV = Present Value (the desired amount, $25,706.30)
C = Cash Flow per period (the amount to be deposited)
r = Interest Rate per period (10.8% annual rate divided by 4 quarters = 2.7%)
n = Number of periods (5 years multiplied by 4 quarters per year = 20 quarters)
Substituting the given values into the equation:
25,706.30 = C * [(1 - (1+0.027)^(-20))/(0.027)]
Simplifying the equation, you should find:
C = 25,706.30 / [(1 - (1+0.027)^(-20))/(0.027)]
Evaluating this equation, the value of C should be approximately $1,055.77 per quarter.
So, the correct answer is that Melanie should deposit approximately $1,055.77 per quarter in order to have $25,706.30 in 5 years, given the 10.8% annual interest rate and quarterly deposits.
No, 135662.14 is not the correct answer. Let me guide you through the correct steps to solve this equation.
To determine how much Melanie should deposit per quarter, we can use the formula for the future value of an ordinary annuity.
The formula for the future value of an ordinary annuity is:
FV = P * ((1 + r/n)^(n*t) - 1) / (r/n)
where:
FV = future value of the annuity
P = periodic deposit (amount deposited per period)
r = annual interest rate (as a decimal)
n = number of compounding periods per year
t = number of years
In this case, Melanie wants to have $25,706.30 in the annuity in 5 years, the annual interest rate is 10.8%, and she plans to make quarterly deposits.
Step 1: Convert the annual interest rate to a quarterly interest rate.
r = 10.8% = 0.108
n = 4 (quarterly compounding)
Step 2: Plug in the known values into the formula and solve for P.
25706.30 = P * ((1 + 0.108/4)^(4*5) - 1) / (0.108/4)
Step 3: Simplify the equation and calculate P.
25706.30 = P * (1.027)^20 - 1) / (0.027)
25706.30 = P * (1.747422051177) / 0.027
P = 25706.30 * 0.027 / 1.747422051177
P ≈ $399.89
Therefore, Melanie should deposit approximately $399.89 per quarter to reach her goal of $25,706.30 in 5 years with an annual interest rate of 10.8%.