Question

Use the compound interest formulas AP 1.SA=P)and Ape" to solve the problem given Round answers to the nearest contFind the accumulated value of an investment of $10,000 for 4 years at an interest rate of 5% of the money is a compounded semiannualy, b. compounded quarterly, c.compounded monthly, d. compounded continuouslyBEa. What is the accumulated value of the money is compound

Answer 1

The formula for interest compounded continuously is:

$P_t(cont)=P_0\cdot e^{r\cdot t}.$

The formula for the compound in periods:

$P_t(period)=P_0\cdot(1+\frac{r}{n})^{n\cdot t}\text{.}$

Where:

• P_t is the amount of money after t years,

,

• P_0 is the initial amount of money,

,

• r is the ratio in decimal form,

,

• n is the number of compounded periods per year,

,

• t is the number of years.

We must compute the amount of money of an investment with:

• P_0 = $10,000,

,

• r = 5% = 0.05,

,

• t = 4 years.

a) Compounded semiannually

We use the second formula with n = 2:

$P_4(semi)=\text{ \ 10,000 }\cdot(1+\frac{0.05}{2})^{2\cdot4}\cong\text{ \ 12,184.}03.$

b) Compounded quarterly

We use the second formula with n = 4:

$P_4(quart)=\text{ \ 10,000 }\cdot(1+\frac{0.05}{4})^{4\cdot4}\cong\text{ \ 12,198.90.}$

c) Compounded monthly

We use the second formula with n = 12:

$P_4(monthly)=\text{ \ 10,000 }\cdot(1+\frac{0.05}{12})^{12\cdot4}\cong\text{ \ 12,}208.95.$

d) Compounded continuously

We use the first formula:

$P_4(cont)=\text{ \ 10,000 }\cdot e^{0.05\cdot4}=\text{ \ 12,}214.03.$

Answer

a) Compounded semiannually: $12,184.03

b) Compounded quarterly: $12,198.90

c) Compounded monthly: $12,208.95

d) Compounded continuously: $12,214.03