How much money should be deposited today in an account that earns 5% compounded monthly so that it will accumulate to $13,000 in three years

Answer 1

A = P(1 + r/n)^nt where r = rate of interest, n= number of compounding periods annually, t = years, P = original amount invested. A = Amount at the end of t years

r=5% = .05, n=12, t=3, A=$13,000

13,000= P(1.+.05/12)^12(3)

13,000 = P(1.0041666...)^36

P= 13,000/(1.0041666...)^36

P = 13,000/1.161471961

P= 11192.69

$11,192.69 will grow to $13,000 in 3 years, compounded monthly at 5%

For continuously compounded interest the formula is

A = Pe^rt

13000=Pe^.05(3)

13,000/P = e^.15

ln(13,000/P) = .15

ln13000 -lnP = .15

lnP = ln13000 -.15

lnP = 9.322704636

P =$11,189.20= $3.49 less than needed at monthly compounding

For just annual compounding

13000=P(1.05)^(3)

P =13000/(1.05)^3

P = 13000/1.157625

P=$11,229.89 needed