Question 1

Your aunt has $500,000 invested at 5.5%, and she now wants to retire.She wants to withdraw $45,000 at the beginning of each year, beginning immediately.When she makes her last withdrawal (at the beginning of a year), she also wants to have enough left in the account so that you can make a final withdrawal of $50,000 at the end of that year (her last withdrawal is at the beginning of the year, your withdrawal is at the end of that same year).What is the maximum number of $45,000 withdrawals that she can make and still have enough in the account so that you can make a $50,000 withdrawal at the end of the year of her last withdrawal? (Hint: If your solution for N is not an integer, round down to the nearest whole number.)

Accepted Answer

We can use the formula for the present value of an annuity to solve this problem:
Present Value = Payment x ((1 - (1 + interest rate)^-n) / interest rate)
Where payment is the withdrawal amount, interest rate is the annual interest rate (5.5%), and n is the number of withdrawals.

We can first calculate the present value of the $50,000 withdrawal at the end of the final year:
PV = $50,000 / (1 + 0.055) = $47,393.36

Next, we can use trial and error to find the number of withdrawals that will result in a remaining balance of at least $47,393.36 after the last $45,000 withdrawal. Starting with 15 withdrawals, we get:
PV = $45,000 x ((1 - (1 + 0.055)^-15) / 0.055) = $525,772.43
Remaining balance = $525,772.43 x 1.055 - $45,000 = $484,419.51
This is more than the required $47,393.36, so we can try with 14 withdrawals:
PV = $45,000 x ((1 - (1 + 0.055)^-14) / 0.055) = $480,529.69
Remaining balance = $480,529.69 x 1.055 - $45,000 = $436,101.66
This is less than the required $47,393.36, so we need to try with more than 14 withdrawals. Continuing in this way, we find that 15 withdrawals result in a remaining balance of $484,419.51, which is sufficient for the final $50,000 withdrawal. Therefore, the maximum number of $45,000 withdrawals that she can make is 15.

Question 2

Your sister's pet supplies business obtained a 30-year amortized mortgage loan for $250,000 at a nominal annual rate of 7.0%, with 360 end-of-month payments.The firm can deduct the interest paid for tax purposes.What will the interest tax deduction be for for the first year of the loan? (Assume she took out the loan on January 1.)

Accepted Answer

The answer of Your sister's pet supplies business obtained a...

Question 3

Your aunt wants to retire and has $375,000.She expects to live for another 25 years and to earn 7.5% on her invested funds.How much could she withdraw at the end of each of the next 25 years and end up with zero in the account?

Accepted Answer

The answer of Your aunt wants to retire and has...

Question 4

Suppose you borrowed $15,000 at a rate of 8.5% and must repay it in 5 equal installments at the end of each of the next 5 years.By how much would you reduce the amount you owe in the first year?

Accepted Answer

To solve this problem, we need to use the formula for the present value of an annuity:

PV = PMT * (1 - 1 / (1 + r)^n) / r

where PV is the present value of the annuity (i.e. the amount borrowed), PMT is the equal annual payments, r is the annual interest rate, and n is the number of periods.

In this case, PV = $15,000, PMT = PV / n = $15,000 / 5 = $3,000, r = 8.5%, and n = 5.

Plugging these values into the formula, we get:

PV = $3,000 * (1 - 1 / (1 + 0.085)^5) / 0.085 = $12,655.09

This means that the borrower will have to pay a total interest of $15,000 - $12,655.09 = $2,344.91 over the next 5 years. Since the payments are equal, we can divide this amount by 5 to find the interest paid in the first year:

Interest in Year 1 = $2,344.91 / 5 = $468.98

Therefore, the amount the borrower would reduce the amount owed in the first year is:

Reduction in Year 1 = PMT - Interest in Year 1 = $3,000 - $468.98 = $2,531.49

Therefore, the best choice is B.

Question 5

Which of the following statements is CORRECT?

Accepted Answer

Time lines can be constructed where some of the payments constitute an annuity but others are unequal and thus are not part of the annuity. This is because time lines can be used to visualize any set of cash flows, whether they are part of an annuity or not. Time lines can be used to show the timing and magnitude of each cash flow, regardless of whether they occur annually, quarterly, or at any other frequency. They can also be used to show the effects of interest and compounding on the cash flows over time.

Question 6

Which of the following statements is CORRECT?

Accepted Answer

Time lines can be constructed to deal with situations where some of the cash flows occur annually but others occur quarterly. Time lines are a useful tool for visualizing complex problems prior to doing actual calculations, and can be constructed for both ordinary annuities (where payments occur at the end of the periods) and annuities due (where payments occur at the beginning of the periods). However, time lines can include both annuity payments and other cash flows that are not part of the annuity, such as lump sum payments or uneven periodic payments.

Question 7

Your uncle just won the weekly lottery, receiving $375,000, which he invested at a 7.5% annual rate.He now has decided to retire, and he wants to withdraw $35,000 at the end of each year, starting at the end of this year.What is the maximum number of whole payments that can be withdrawn before the account is exhausted, i.e., before the account balance would become negative? (Hint: Round down to the nearest whole number.)

Accepted Answer

The answer of Your uncle just won the weekly lottery,...

Question 8

Which of the following statements is CORRECT?

Accepted Answer

Time lines are commonly used tools in finance to visualize cash flows over time. They facilitate understanding complex problems and help in making appropriate decisions. They can accommodate annuity payments as well as uneven cash flows, and need not be restricted to annually occurring cash flows. The last statement about payments at the beginning of periods is incorrect as time lines can be constructed for such annuities.

Question 9

Your Aunt Elsa has $500,000 invested at 6.5%, and she plans to retire.She wants to withdraw $40,000 at the beginning of each year, starting immediately.What is the maximum number of whole payments that can be withdrawn before the account is exhausted, i.e., before the account balance would become negative? (Hint: Round down to the nearest whole number.)

Accepted Answer

The answer of Your Aunt Elsa has $500,000 invested at...

Question 10

You plan to borrow $35,000 at a 7.5% annual interest rate.The terms require you to amortize the loan with 7 equal end-of-year payments.How much interest would you be paying in Year 2?

Accepted Answer

First, we need to calculate the annual payment using the amortization formula:
PMT = r * PV / (1 - (1 + r)^(-n))
where
PMT = annual payment
r = annual interest rate
PV = present value of the loan
n = number of payments

Plugging in the given values:
PMT = 0.075 * 35,000 / (1 - (1 + 0.075)^(-7))
PMT = $6,188.01

In the second year, we need to calculate the interest portion of the second payment. To do this, we first need to find the principal portion of the first payment, which we can do using the formula:
Principal = PMT - r * PV
where
Principal = principal portion of the payment
PMT = payment amount
r = annual interest rate
PV = present value of the loan

Plugging in the given values:
Principal = $6,188.01 - 0.075 * $35,000
Principal = $3,758.51

Since the loan is being amortized, the principal portion of each payment will increase and the interest portion will decrease. In the second year, the principal portion will be higher and the interest portion will be lower than in the first year. To find the interest portion of the second payment, we need to subtract the principal portion of the first payment from the total interest paid over the first two payments:

Total interest paid over first two payments = r * PV + r * (PV - Principal)
Total interest paid over first two payments = 0.075 * $35,000 + 0.075 * ($35,000 - $3,758.51)
Total interest paid over first two payments = $5,371.49

Interest portion of second payment = Total interest paid over first two payments - Interest portion of first payment
Interest portion of second payment = $5,371.49 - ($6,188.01 - $3,758.51)
Interest portion of second payment = $2,326.27

Therefore, the answer is D) $2,326.27.

Question 11

Suppose you borrowed $15,000 at a rate of 8.5% and must repay it in 5 equal installments at the end of each of the next 5 years.How much would you still owe at the end of the first year, after you have made the first payment?

Accepted Answer

The answer of Suppose you borrowed $15,000 at a rate...

Question 12

Your aunt wants to retire and has $375,000.She expects to live for another 25 years, and she also expects to earn 7.5% on her invested funds.How much could she withdraw at the beginning of each of the next 25 years and end up with zero in the account?

Accepted Answer

To calculate the annual withdrawal amount, we can use the formula for the present value of an annuity:

Annual Withdrawal Amount = (Present Value * Interest Rate) / (1 - (1 + Interest Rate)^(-Number of Years))

Plugging in the given values, we get:

Annual Withdrawal Amount = (375,000 * 0.075) / (1 - (1 + 0.075)^(-25)) = $31,294.42

Therefore, the aunt could withdraw $31,294.42 at the beginning of each of the next 25 years and end up with zero in the account.

Question 13

Your uncle has $300,000 invested at 7.5%, and he now wants to retire.He wants to withdraw $35,000 at the end of each year, beginning at the end of this year.He also wants to have $25,000 left to give you when he ceases to withdraw funds from the account.What is the maximum number of $35,000 withdrawals that he can make and still have at least $25,000 left in the account? (Hint: If your solution for N is not an integer, round down to the nearest whole number.)

Accepted Answer

We can use the present value of an annuity formula to determine how much money will be left in the account after the 13th withdrawal of $35,000.

PV = $\frac{PMT}{i} \cdot \left[1 - \frac{1}{(1+i)^n}\right]$

Where PV is the present value (the amount left in the account after the 13th withdrawal), PMT is the withdrawal amount ($35,000), i is the interest rate (7.5%), and n is the number of withdrawals.

Plugging in the values, we get:

PV = $\frac{35000}{0.075} \cdot \left[1 - \frac{1}{(1+0.075)^{13}}\right]$

PV = $342,507.68$

So your uncle will have $342,507.68 left in his account after the 13th withdrawal. Since he wants to have at least $25,000 left, he can make up to 12 more withdrawals of $35,000. Therefore, the maximum number of $35,000 withdrawals he can make is 13.

Question 14

Your bank offers to lend you $100,000 at an 8.5% annual interest rate to start your new business.The terms require you to amortize the loan with 10 equal end-of-year payments.How much interest would you be paying in Year 2?

Accepted Answer

In Year 2, the interest payment is part of the second annual payment. To find this, we first need to calculate the annual payment amount using the formula for an amortizing loan, which is A = P * [r(1+r)^n] / [(1+r)^n - 1], where A is the annual payment, P is the principal ($100,000), r is the annual interest rate (8.5% or 0.085), and n is the number of payments (10). After calculating A, we determine the interest portion of the second payment by calculating the remaining balance after the first payment and applying the interest rate. The correct calculation shows that the interest in Year 2 is $7,927.

Question 15

Suppose you earned a $275,000 bonus this year and invested it at 8.25% per year.How much could you withdraw at the end of each of the next 20 years?

Accepted Answer

The answer of Suppose you earned a $275,000 bonus this...

Question 16

Which of the following statements is CORRECT?

Accepted Answer

Time lines can be constructed for annuities where the payments occur at either the beginning or the end of the periods. Option A is incorrect because cash flows can be uneven amounts. Option B is incorrect because time lines can be constructed for any periodicity of cash flows. Option C is incorrect because time lines are useful for visualizing problems before doing calculations. Option D is incorrect because time lines can be constructed for any periodicity of cash flows.

Question 17

Suppose you inherited $275,000 and invested it at 8.25% per year.How much could you withdraw at the beginning of each of the next 20 years?

Accepted Answer

The answer of Suppose you inherited $275,000 and invested it...

Question 18

Suppose you just won the state lottery, and you have a choice between receiving $2,550,000 today or a 20-year annuity of $250,000, with the first payment coming one year from today.What rate of return is built into the annuity? Disregard taxes.

Accepted Answer

To calculate the rate of return built into the annuity, we can use the formula:
PMT = C x (1 - 1 / (1 + r)^n ) / r
where PMT = annuity payment, C = original amount, n = number of years, and r = rate of return.
Substituting the given values, we get:
$250,000 = $2,550,000 x (1 - 1 / (1 + r)^20 ) / r
We can solve for r using Excel or trial and error. The answer is approximately 7.49%. Therefore, choosing the annuity would yield a better return than taking the lump sum today, assuming a constant rate of return over 20 years. However, other factors such as personal preferences, inflation, and investment opportunities should also be considered.

Question 19

Your business has just taken out a 1-year installment loan for $72,500 at a nominal rate of 11.0% but with equal end-of-month payments.What percentage of the 2nd monthly payment will go toward the repayment of principal?

Accepted Answer

The percentage of the 2nd monthly payment that goes toward the repayment of principal increases with each payment because as the outstanding loan balance decreases, the interest portion of each payment decreases, and the principal portion increases. To calculate this, first, find the monthly interest rate, which is 11.0% / 12 = 0.917%. Then, use the loan formula to find the monthly payment amount. The formula for the monthly payment (PMT) on an installment loan is PMT = P[r(1+r)^n] / [(1+r)^n - 1], where P is the loan amount, r is the monthly interest rate as a decimal, and n is the total number of payments. Substituting the given values (P = $72,500, r = 0.00917, n = 12), we find the monthly payment. Then, calculate the interest for the first month (P*r), subtract this from the monthly payment to find the principal paid in the first month, and calculate the new balance. For the second month, calculate the interest on the new balance, subtract this interest from the monthly payment to find the principal portion of the second payment, and then calculate the percentage of the second payment that goes towards the principal. This calculation shows that the percentage of the 2nd monthly payment that goes toward the repayment of principal is highest among the given options, which aligns with option E, 90.45%.

Question 20

You were left $100,000 in a trust fund set up by your grandfather.The fund pays 6.5% interest.You must spend the money on your college education, and you must withdraw the money in 4 equal installments, beginning immediately.How much could you withdraw today and at the beginning of each of the next 3 years and end up with zero in the account?

Accepted Answer

To find out how much can be withdrawn each year, we can use the formula for the present value of an annuity:
PV = PMT x [(1- (1/(1+r)^n))/r]
where PV is the present value, PMT is the payment per period, r is the interest rate per period, and n is the number of periods.

In this case, we have:
PV = $100,000
r = 6.5%/4 = 1.625% (since we are withdrawing the money in 4 equal installments)
n = 4

Substituting these values and solving for PMT, we get:
PMT = PV / [(1- (1/(1+r)^n))/r] = $27,408.82

Therefore, we can withdraw $27,408.82 at the beginning of each year for the next 4 years and end up with zero in the account. However, since we need to withdraw the money in 4 equal installments beginning immediately, we can only withdraw $27,408.82/4 = $6,852.21 today and at the beginning of each of the next 3 years.

Therefore, the answer is C.

Quiz 4: Time Value of Money

Your aunt wants to retire and has $375,000.She expects to live for another 25 years and to earn 7.5% on her invested funds.How much could she withdraw at the end of each of the next 25 years and end up with zero in the account?

Suppose you borrowed $15,000 at a rate of 8.5% and must repay it in 5 equal installments at the end of each of the next 5 years.By how much would you reduce the amount you owe in the first year?

Which of the following statements is CORRECT?

Which of the following statements is CORRECT?

Which of the following statements is CORRECT?

You plan to borrow $35,000 at a 7.5% annual interest rate.The terms require you to amortize the loan with 7 equal end-of-year payments.How much interest would you be paying in Year 2?

Suppose you borrowed $15,000 at a rate of 8.5% and must repay it in 5 equal installments at the end of each of the next 5 years.How much would you still owe at the end of the first year, after you have made the first payment?

Your aunt wants to retire and has $375,000.She expects to live for another 25 years, and she also expects to earn 7.5% on her invested funds.How much could she withdraw at the beginning of each of the next 25 years and end up with zero in the account?

Your bank offers to lend you $100,000 at an 8.5% annual interest rate to start your new business.The terms require you to amortize the loan with 10 equal end-of-year payments.How much interest would you be paying in Year 2?

Suppose you earned a $275,000 bonus this year and invested it at 8.25% per year.How much could you withdraw at the end of each of the next 20 years?

Which of the following statements is CORRECT?

Suppose you inherited $275,000 and invested it at 8.25% per year.How much could you withdraw at the beginning of each of the next 20 years?

Suppose you just won the state lottery, and you have a choice between receiving $2,550,000 today or a 20-year annuity of $250,000, with the first payment coming one year from today.What rate of return is built into the annuity? Disregard taxes.

Your business has just taken out a 1-year installment loan for $72,500 at a nominal rate of 11.0% but with equal end-of-month payments.What percentage of the 2nd monthly payment will go toward the repayment of principal?