Problema Solution
Determine the time necessary for P dollars to double when it is invested at interest rate r compounded annually, monthly, daily, and continuously. (Round your answers to two decimal places.) The rate is 10%
Answer provided by our tutors
Let
P = is the principal
r = 0.10 the annual interest rate of 10%
t = the time in years
A = 2P future value
- Annually:
A = P(1 + i)^n
m = 1 is the number of compounding periods per year
n = m*t = t is the number of compounding periods
i = r/m = 0.10/1 = 0.10 is the interest rate per period
2P = P(1 + 0.10)^t
1.10^t = 2
........
........
t = 7.27 years
- Monthly:
A = P(1 + i)^n
m = 12 is the number of compounding periods per year
n = m*t = 12t is the number of compounding periods
i = r/m = 0.10/12 = 0.1/12 is the interest rate per period
2P = P(1 + 0.1/12)^(12t)
(1 + 0.1/12)^(12t) = 2
12t*log(1.00833) = log(2)
........
........
t = 6.96 years
(d) Daily:
A = P(1 + i)^n
m = 365 is the number of compounding periods per year
n = m*t = 365*t is the number of compounding periods
i = r/m = 0.10/365 = 0.1/365 is the interest rate per period
2P = P(1 + 0.1/365)^(365t)
(1 + 0.1/365)^(365t) = 2
365t*log(1.00027 = log(2)
........
........
t = 7.03 years
(e) Continuously:
A = P*e^(r*t), where e = 2.71828 is Napier's constant
2P = P*e^(0.10*t)
e^(0.10*t) = 2
0.10t = ln(2)
........
........
t = 6.93 years