After having a good year Business Cat decides it is time to

After having a good year, Business Cat decides it is time to expand his company, and decides he would like to save up enough money to do this after 5 years. His financial advisor determines that he will need 1.2 million dollars to expand. If Business Cat finds an account that accrues 7.8% interest, compounded quarterly, what should his initial investment be, in millions of dollars? Round to four decimal places. If Business Cat can increase his initial investment by $1,000 from his investment in part a, how much sooner will he get to expand?

Solution

The formula for annual compound interest is A = P (1 + r/n) ^nt:

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

PART: (a)

here, r/n = 7.8/100, since it is compounded quarterly(given) and rate = interest percent/100

n = 4, t=5 and A = 1.2

hence, P = A*1/(1+r/n)^nt  = 1.2/(1+ 7.8/100)^4*5 = 1.2/ 4.4913

= 0.2672 million dollars

so, initial investment sholud be 0.2672 million dollars or 267183.2209 dollars

PART: (b)

now, P = 267183.2209 + 1000 = 268183.2209 and we need to calculate t.

A is same as before and so is the rate, putting respective values in the above equation

1.2* 10⁶  = 268183.2209*(1+7.8/100)^4_*t

=> (1.078)^4*t = 1.2*10⁶ / 268183.2209

=>   (1.078)^{4*t =} 4.4745

=> 4*t = log_1.0784.475

=> t = 1/4*(log(4.475)/log(1.078))

=> t = 4.98 years

so, business cat can get to expand 0.02(5-4.98) years sooner i.e. (0.02*365) = 7 days sooner