Juan and Rachel Burpo plan to buy a time-share in six years of $16,860. In order to have adequate funds to do so, the Burpo want to make a deposit to their money market fund today. Assume that they will be able to earn an investment rate of 5.75%, compounded annually. How much will Juan and Rachel need to deposit today to achieve their goal? (Round off to the nearest dollar.)
How do you compute the compounded annually?
P = Po(1+r)^n.
P = $16,860.
r = 5.75%/100 = 0.0575
n = 1Comp./yr. * 6yrs. = 6 Compounding periods.
Po = ?.
12,055.22
To compute compound interest annually, you can use the formula:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the Burpos want to buy the time-share in 6 years, the investment rate is 5.75% (or 0.0575 as a decimal), compounded annually.
Using the formula:
A = P(1 + r/n)^(nt)
The future value A is the target amount they want to achieve, which is $16,860. The principal P (the initial deposit) is what we need to find.
So, we rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Plugging in the values:
P = $16,860 / (1 + 0.0575/1)^(1*6)
P = $16,860 / (1 + 0.0575)^6
P = $16,860 / (1.0575)^6
P ≈ $12,006
Therefore, Juan and Rachel need to deposit approximately $12,006 today to achieve their goal of buying the time-share in 6 years.
To compute compound interest annually, you can use the following formula:
A = P (1 + r/n)^(n*t)
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the principal amount (P) is what the Burpos need to deposit today, and they want to achieve a future value (A) of $16,860 in six years. The annual interest rate (r) is 5.75% (0.0575 as a decimal), and the interest is compounded annually (n = 1). The number of years (t) is 6.
Now, let's substitute these values into the formula and solve for P:
16,860 = P (1 + 0.0575/1)^(1*6)
Simplifying the equation further:
16,860 = P (1 + 0.0575)^6
Now, we raise the expression (1 + 0.0575) to the power of 6:
16,860 = P (1.0575)^6
Calculating (1.0575)^6, you get approximately 1.3748:
16,860 = P * 1.3748
Now divide both sides of the equation by 1.3748 to isolate P:
P ≈ 16,860 / 1.3748
P ≈ 12,259.36
So, Juan and Rachel need to deposit approximately $12,259 today to achieve their goal of $16,860 in six years, considering an investment rate of 5.75% compounded annually.