Problema Solution
To fund a down payment on a home, you decide to invest $13,000 into an
account earning 3.5% interest compounded daily.Find a function which gives the amount in the account A, after t years. If you need $25,000 in 15 years to make a down payment, will you have enough in your
account at that time? If not, how much should be invested now to guarantee that you do?
Answer provided by our tutors
A = P(1 + i)^n
where and
A = is the future (maturity) value;
P = $13,000 is the principal;
r = 3.5% = 0.035 is the annual interest rate;
m = 365 is the number of compounding periods per year, compounded daily;
t = is the number of years;
n =m*t = 365t is the number of compounding periods;
i = r/m = 0.035/365 is the interest rate per period.
A(t) = 13000(1 + 0.035/365)^(365t)
A(t) = 13000*1.0351^t
In 15 years that is for t = 15 we have
A(15) = 13000*1.0351^15
A(15) = $ 21.811.1 < 25.0000 meaning that there will not be enough money in the account.
How much should be invested now to guarantee that A(15) >= 25000 that is for what P we will have A(15) >= 25000
A(15) <= P*1.0351^15
25000 <= P*1.0351^15
P >= 25000/(1.0351^15)
P >= $14.900.7
to guarantee that after 15 years there will be $25.000 in the account i must invest at least $14.900.7.