This exercise concerns a single investment of $10,000. Find the continuous interest rate per year, yielding a future value of$20,000 in the given time period. 5 years

Find the future value of an income stream of $2000 per year, deposited into an account paying 2% interest per year, compounded continuously, over a 15-year period.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 25,000$ is invested for $30$ years with an APR of $3.7 \%$ and quarterly compounding.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 25,000$ is invested for $5$ years with an APR of $3 \%$ and daily compounding.

Demetrios opens an account with an initial investment of $2000. The annual interest rate is 5%. (a) If the interest is compounded continuously and Demetrios makes an additional$1000 deposit every year, what will be the balance at the end of 10 years? (b) If the interest is compounded quarterly (four times per year) and $250 is deposited at the end of each compounding period, what will be the balance after 10 years? (c) What happens if the interest is compounded daily?

Suppose that $P$ dollars is invested at an annual interest rate of $r \times 100 \%$. If the accumulated interest is credited to the account at the end of the year, then the interest is said to be compounded annually; if it is credited at the end of each 6-month period, then it is said to be compounded semiannually; and if it is credited at the end of each 3-month period, then it is said to be compounded quarterly. The more frequently the interest is compounded, the better it is for the investor since more of the interest is itself earning interest.

One can imagine interest to be compounded each day, each hour, each minute, and so forth. Carried to the limit one can conceive of interest compounded at each instant of time; this is called continuous compounding. Thus, from part (a), the value $A$ of $P$ dollars after $t$ years when invested at an annual rate of $r \times 100 \%$, compounded continuously, is

$$A=\lim _{n \rightarrow+\infty} P\left(1+\frac{r}{n}\right)^{n t}$$

Use the fact that $\lim _{x \rightarrow 0}(1+x)^{1 / x}=e$ to prove that $A=P e^{r t}$.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 5000$ is invested for $10$ years with an APR of $2 \%$ and quarterly compounding.

This exercise concerns a single investment of $10,000. Find the continuous interest rate per year, yielding a future value of$20,000 in the given time period. 60 years

A series of equal quarterly payments of $\$ 1,000$ extends over a period of five years. What is the present worth of this quarterly-payment series at $9.75 \%$ interest compounded continuously?

A new bond you are considering buying has a face value of $1,000, an interest rate of 5 percent, and a maturity date eight years away. The bond is currently selling at 104. What is the current yield? What amount of interest would you earn in that time period?

Suppose that P dollars is invested at an annual interest rate of

$$r \times 100 \%$$

. If the accumulated interest is credited to the account at the end of the year, then the interest is said to be compounded annually; if it is credited at the end of each 6-month period, then it is said to be compounded semiannually; and if it is credited at the end of each 3-month period, then it is said to be compounded quarterly. The more frequently the interest is compounded, the better it is for the investor since more of the interest is itself earning interest. One can imagine interest to be compounded each day, each hour, each minute, and so forth. Carried to the each hour, each minute, and so forth. Carried to the instant of time; this is called continuous compounding. Thus the value A of P dollars after t years when invested at an annual rate of

$$r \times 100 \%$$

, compounded continuously, is

$$A = \lim _ { n \rightarrow + \infty } P \left( 1 + \frac { r } { n } \right) ^ { n t }$$

. Use the fact that

$$\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 1 / x } = e$$

to prove that

$$A = P e ^ { r t }$$

.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 4000$ is invested for $5$ years with an APR of $3 \%$ and daily compounding.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 10,000$ is invested for $5$ years with an APR of $2.75 \%$ and monthly compounding.

Simple interest is the amount of money the borrower pays based on the amount borrowed (the principal) for a given period of time (months or years) It is calculated this way: $I = p r t .$ If a person borrows $20,000 ( p ) to buy a car, pays 6.95% interest ( r ), and takes 5 years ( t ) to repay the loan, how much will the borrower pay in simple interest?