The formula for the amount A in an investment account with a nominal interest rate r at any time t is given by A(t) = a(e)rt, where a is the

The formula for the amount A in an investment account with a nominal interest rate r at any time t is given by A(t) = a(e)^rt, where a is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time t can be calculated with the formula I(t) = e^rt − 1.

Solutions

Expert Solution

Consider the formula for nominal interest as following:

A(t) = a(e)rt

The annual percentage is determined using the formula:

I(t) = ert - 1

Simplify the above equation as follows:

I(t) + 1 = ert – 1 + 1

I(t) + 1 = ert

Now take natural log on both sides of the above equation,

ln{I(t) + 1} = ln(ert)

rt = ln{I(t) + 1}

r = ln{I(t) + 1}/t

Therefore, the percentage of interest is:

r = ln{I(t) + 1}/t.

Therefore, the percentage of interest is:

r = ln{I(t) + 1}/t.

Nabeel answered 1 year ago

Next > < Previous

Related Solutions

An initial investment amount P, an annual interest rate r, and a time t are given....

An initial investment amount P, an annual interest rate r, and a time t are given. Find the future value of the investment when the interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. Then find (e) the doubling time T for the given interest rate. Round to the nearest cent or nearest tenth of a year as needed. P= $750, r=2.03%, t=11 yr

$1000 is deposited into an account at time t=0. The nominal interest rate is 8% compounded...

$1000 is deposited into an account at time t=0. The nominal interest rate is 8% compounded semiannually during the first year. An additional $118.40 is deposited into the account at time t=1 and the fund grows at a force of interest 100 2 t  t  during the second year. 1) Calculate the semiannual effective interest rate during the first year. 2) Calculate the account value at time t=1 before the additional $118.40 deposit. 3) Calculate the total account...

Problem 1. Calculate the nominal rate of interest given the following (you must show the formula):...

Problem 1. Calculate the nominal rate of interest given the following (you must show the formula): The real rate of interest = 2% Expected Inflation = 4% The Risk Free rate = 6% All risk factors = 3%

Use properties of rational exponents to solve the compound interest formula for the interest rate, r. For the following exercises, use the compound interest formula, A(t) = P(1 + r/n)nt.

For the following exercises, use the compound interest formula, A(t) = P(1 + r/n)nt.Use properties of rational exponents to solve the compound interest formula for the interest rate, r.

The amount of carbon-14 present in a tree is given by the formula C(t)=C0e−.00012t, where C0...

The amount of carbon-14 present in a tree is given by the formula C(t)=C0e−.00012t, where C0 is the amount of carbon-14 present in it at the time of its death and t is the number of years that have elapsed since it died. If the plant has 15% of its original carbon-14, how old is it?

4. For any given interest rate, the quantity of investment which is optimal for a firm...

4. For any given interest rate, the quantity of investment which is optimal for a firm will be the one where ________=_________ 5. We normally assume that production functions exhibit __________________ returns to scale and _____________________ marginal product of all inputs. 6. If there is a technological advancement which makes firms more productive, it will cause a(n) ___________________ in the _________________ for loanable funds.

Given a nominal interest rate of 6.5% compounded semi-annually, compute: a) The equivalent nominal rate of...

Given a nominal interest rate of 6.5% compounded semi-annually, compute: a) The equivalent nominal rate of discount compounded quarterly b) The equivalent constant force of interest

For a given investment amount, more proceeds require a lower interest rate. is this true or false?

Find T(t), N(t), aT, and aN at the given time t for the space curve r(t)....

Find T(t), N(t), aT, and aN at the given time t for the space curve r(t). [Hint: Find a(t), T(t), aT, and aN. Solve for N in the equation a(t)=aTT+aNN. (If an answer is undefined, enter UNDEFINED.) Function Time r(t)=9ti-tj+(t^2)k t=-1 T(-1)= N(-1)= aT= aN=

The balance in an investment account years t after the account is opened is given by...

The balance in an investment account years t after the account is opened is given by 7500(1.065t). Compute the average rate of change for 2.5 ≤ t ≤ 3.5. (Round your answer to two decimal places.) Δ balance/ Δt=?

Subjects