Question

A) what is the accumulated value of the money is compounded semi annual?B) what is the accumulated value of the money as compounded quarterly?C) what is the accumulated value if the money is compounded monthly? D) what is the accumulated value if the money is compounded continuously?

Answer 1

We need to find the accumulated value A of an investment of P =$10000 for t = 6 years at an interest rate of r = 4% = 0.04.

When the investment is compounded n times a year, A is given by:

`A=P\left(1+(r)/(n)\right)^(nt)`

When it is compounded continuously, we need to use the limit of the above formula for n going to infinity. Then, A is given by:

`A=Pe^(rt)`

Part a

When the investment is compounded semiannually, n = 2. Then, we have:

`A=10000\left(1+(0.04)/(2)\right)^(2\cdot6)=10000\left(1.02\right)^(12)\cong12682.42`

Answer (in dollars): 12682.42

Part b

When the investment is compounded quarterly, n = 4. Then, we have:

`A=10000\left(1+(0.04)/(4)\right)^(4\cdot6)=10000\left(1.01\right)^(24)\cong12697.35`

Answer (in dollars): 12697.35

Part c

When the investment is compounded monthly, n = 12. Then, we have:

`A=10000\left(1+(0.04)/(12)\right)^(12\cdot6)=10000\left(1+(0.01)/(3)\right)^(72)\cong12707.42`

Answer (in dollars): 12707.42

Part d

When the investment is compounded continuously, we have:

`A=10000(e^(0.04\cdot6))=10000(e^(0.24))=12712.49`

Answer (in dollars): 12712.49