Question

Suppose that you deposit your money in a bank that pays interest at a rate of 18% per year.

How long will it take for your money to triple if the interest is

1. compounded weekly? (1year= 52 weeks)
2. compounded continuously?
3. compounded quarterly?

Answer 1

Annual Interest Rate,i = 18%

Future Value, FV = PV*(1+i/n)^n*t

Where PV is the present value of deposit

i is the annual interest rate

n is the number of compounding per year

t is the deposit period

Implies,

(1+i/n)^n*t = FV/PV -----------(1)

Applying natural logarithm to both sides of equation (1)

n*t*ln(1+i/n) = ln(FV/PV)

Since money triples FV = 3PV

n*t*ln(1+i/n) = ln(3PV/PV)

n*t*ln(1+i/n) = ln(3)

t = ln(3)/(n*ln(1+i/n)) ---------------(2)

1. Weekly compounding

For weekly compounding, n = 52

i = 18%

Applying the above values in equation (2)

t = ln(3)/(52*ln(1+18%/52)) = ln(3)/(52*ln(1.003461538)) = 1.098612289/(52*0.003455561127)

= 1.098612289 / 0.1796891786 = 6.113959 years

2. Continuous compounding

For continuous compounding

FV = PV*e^i*t

e^i*t = FV/PV

Since money triples FV = 3PV

e^i*t = 3PV/PV = 3

Applying natural logarithm to both sides of equation

ln(e^i*t)= ln(3)

i*t*ln(e) = ln(3)

i*t = ln(3)

t = ln(3)/i

i = 18%

t = ln(3)/18% = 1.098612289 / 18% = 6.103402 years

3. Quarterly compounding

For quarterly compounding, n = 4

i = 18%

Applying the above values in equation (2)

t = ln(3)/(4*ln(1+18%/4)) = ln(3)/(4*ln(1.045)) = 1.098612289/(4*0.04401688542)

= 1.098612289 / 0.1760675417 = 6.239721 years