Question 1

If deposits of $9,900 invested at the end of every six months earn 13% compounded quarterly, how long will it take to accumulate $295,000?

Accepted Answer

The correct answer is found by using the formula for the future value of a series: $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $FV$ is the future value of the annuity, $P$ is the payment per period, $r$ is the interest rate per period, and $n$ is the total number of payments. Given that the interest is compounded quarterly but payments are made semi-annually, we adjust the annual interest rate to a per period rate by dividing by 4 (for quarterly compounding) but calculate $n$ based on semi-annual payments. The goal is to solve for $n$ when $FV = \$295,000$, $P = \$9,900$, and the annual interest rate is 13% or 0.13, leading to a per period rate of $0.13/4 = 0.0325$ for quarterly compounding. However, since the compounding frequency does not match the payment frequency, a direct application of this formula requires adjustments or iterative methods to account for the difference in compounding versus payment periods. The correct approach involves recognizing the need for an iterative solution or using financial calculators or software designed to handle such calculations, taking into account the effective interest rate per payment period and the actual compounding effect. Given the complexity of directly solving this with the provided information and without specifying the iterative steps or calculations, the key insight is understanding that the process involves calculating the time it takes for regular investments to grow to a specified amount under compound interest, which typically requires numerical methods or financial functions available in spreadsheets or financial calculators.

Question 2

Given the time-value of money of 11% compounded annually, calculate the term of an ordinary annuity of $425 monthly payments with a future value of $37,725.

Accepted Answer

The answer of Given the time-value of money of 11%...

Question 3

Moses' goal, when he retires from work in seven years, is to have $400,000 in his Retirement Fund. Assuming he achieves his goal and the fund earns 7% compounded semi-annually after he retires, Moses will, at the end of every six months, take $20,000 out of his Retirement Fund. For how long will he be able to do that before the money runs out?

Accepted Answer

To solve this, we use the formula for the present value of an annuity because Moses is withdrawing a fixed amount periodically, and the fund earns a fixed interest rate. The formula is:$$ PV = P \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] $$Where:- $PV$ is the present value or the initial amount in the retirement fund ($400,000),- $P$ is the payment amount per period ($20,000),- $r$ is the interest rate per period (7% per year compounded semi-annually gives 3.5% per period, or 0.035),- $n$ is the total number of payments.Rearranging the formula to solve for $n$:$$ n = \frac{\log(1 - \frac{PV \times r}{P})}{\log(1 + r)} $$Plugging in the values:$$ n = \frac{\log(1 - \frac{400,000 \times 0.035}{20,000})}{\log(1 + 0.035)} $$$$ n = \frac{\log(1 - 0.7)}{\log(1.035)} $$$$ n = \frac{\log(0.3)}{\log(1.035)} $$$$ n \approx 35 $$Since $n$ represents the total number of semi-annual payments, to find the number of years, we divide by 2:$$ \text{Years} = \frac{35}{2} = 17.5 $$Therefore, Moses will be able to withdraw $20,000 every six months for approximately 17.5 years before the money runs out.

Question 4

A loan of $25,000 is to be repaid by monthly payments of $400. The interest rate is 9% compounded monthly. How many months will it take for the amount owing on the debt to be reduced to less than $10,000?

Accepted Answer

The answer of A loan of $25,000 is to be...

Question 5

Vince has $35,000 to purchase an annuity that will provide him with payments of $1,000 at the end of every three months. If the funds earn 8% compounded quarterly, how long will the payments continue?

Accepted Answer

We can use the formula for the present value of an annuity to solve for the number of periods, n:

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

where PV is the present value or initial amount, PMT is the payment amount, r is the interest rate per period, and n is the number of periods.

Substituting the given values, we have:

$35,000 = $1,000 * [1 - (1+0.08/4)^(-n)] / (0.08/4)

Simplifying, we get:

35 = 1 - (1.02)^(-4n)

34 = (1.02)^(-4n)

Taking the natural logarithm of both sides, we have:

ln(34) = -4n * ln(1.02)

n = ln(34) / [-4 * ln(1.02)]

n = 14.96

Therefore, the payments will continue for approximately 15 years (rounded to the nearest whole year). Answer choice C is the best choice.

Question 6

A $350,000 loan has an interest rate of 6.2% compounded semi-annually. Determine the semi-annual payments that will reduce the balance to $80,000 after 4 years

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The answer of A $350,000 loan has an interest rate...

Question 7

How many monthly withdrawals of $1,400 will an investment of $75,000 sustain if the first withdrawal is made 13 months from now and the money earns 8.4% compounded monthly?

Accepted Answer

The answer of How many monthly withdrawals of $1,400 will...

Question 8

For a long time Sally has been investing $7,000 at the end of every six months and today the value of her savings has reached $2,500,000. She has made wise investment decisions and has earned a return of 16.4% compounded semi-annually. How long ago (rounded up to the nearest whole year) did she start this savings plan?

Accepted Answer

The answer of For a long time Sally has been...

Question 9

You have $45,000 earning 9.6% compounded monthly. For how long will you be able to withdraw $5,000 at the end of every year?

Accepted Answer

The answer of You have $45,000 earning 9.6% compounded monthly....

Question 10

A loan of $16,525 at 10.8% compounded monthly is to be paid off by equal monthly payments of $650. How long will it take to pay off the loan?

Accepted Answer

First, we need to find the total interest charged on the loan:
I = P*r*t
I = 16525*0.108*t
Since the interest is compounded monthly, we need to convert the annual interest rate to a monthly rate:
r = 0.108/12 = 0.009
Now, we can use the formula for the present value of an annuity to find the time needed to pay off the loan:
PMT = P*r/(1-(1+r)^-n)
Where PMT is the monthly payment, P is the principal, r is the monthly interest rate, and n is the number of months. We want to solve for n.
Plugging in the numbers, we get:
650 = 16525*0.009/(1-(1+0.009)^-n)
Solving for n using a calculator or trial and error gives us n ≈ 29 months. Therefore, the answer is B.

Question 11

Bosley has $750,000 which is earning 7.2% compounded monthly. At the end of every month for the next 10 years he plans to withdraw $5,000. After that he wants to increase his equal monthly withdrawals to the largest amount possible for 15 more years. What is that amount?

Accepted Answer

Using the formula for monthly compound interest, we can calculate the value of Bosley's investment after 10 years:

PV = $750,000
i = 7.2%/12 = 0.6% per month
n = 10 years x 12 months/year = 120 months

FV = PV x (1 + i)^n
FV = $750,000 x (1 + 0.006)^120
FV = $1,681,351.14

Now, we need to calculate the present value of Bosley's retirement payments, using the same interest rate and number of periods, but with a negative payment of $5,000 at the end of each month:

PV = $0 (since we are solving for the payment amount)
i = 0.6% per month
n = 15 years x 12 months/year = 180 months
FV = $1,681,351.14 (from previous calculation)
PMT = -$5,000

PV = PMT x [(1 + i)^n - 1]/i - FV x (1 + i)^-n
0 = PMT x [(1 + 0.006)^180 - 1]/0.006 - $1,681,351.14 x (1 + 0.006)^-180

Solving for PMT, we get:

PMT = ($1,681,351.14 x 0.006) / [(1 + 0.006)^180 - 1]
PMT = $6,029.34

Therefore, Bosley can withdraw $6,029.34 per month for the last 15 years of his retirement. The answer is C, $6,029.

Question 12

How many monthly payments of $919 will it take to pay off a mortgage loan if a $121,500 loan at 7% compounded semi-annually?

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Question 13

A $12,000 loan is repaid by semi-annual payments of $1,500 each. Interest on the loan is 10% compounded semi-annually. How long will it take to pay off the loan?

Accepted Answer

The loan is repaid semi-annually, meaning twice a year. To find the number of payments, we use the formula for the present value of an annuity: $PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)$, where PV is the present value (loan amount), PMT is the payment amount, r is the interest rate per period, and n is the total number of payments. Rearranging to solve for n, we get $n = \frac{\log(1 - \frac{PV \times r}{PMT})}{\log(1 + r)}$. Plugging in the values: $PV = 12000$, $PMT = 1500$, $r = 0.05$ (since 10% annual interest compounded semi-annually gives a 5% semi-annual rate), we can solve for n. However, without needing to solve it explicitly, knowing that $1,500 is paid semi-annually means it will take 8 payments (4 years) to reach $12,000 without interest. Considering the interest, it will take more than 4 years but less than 11 years (the next possible whole number of years with semi-annual payments). The only option fitting this description is 5.5 years, implying 11 payments.

Question 14

A loan of $47,000 calls for payments of $2,700 at the end of every three months until the debt is settled. At an interest rate of 14% compounded semi-annually, how many payments will be made?

Accepted Answer

First, we need to find the semi-annual interest rate. We have an annual interest rate of 14%, so the semi-annual rate is 7%.

Next, we need to find the number of periods. Each payment is made at the end of every three months, which means there are 4 payments per year. Let's call the number of periods "n." We can use the formula for the future value of an annuity to solve for n:

FV = PMT * [(1 + r)^n - 1] / r

Where: 
FV = future value of the loan (which is $47,000) 
PMT = payment per period ($2,700) 
r = interest rate per period (7% / 2 = 3.5% since it's compounded semi-annually)

Plugging in the values:

$47,000 = $2,700 * [(1 + 0.035)^n - 1] / 0.035

Simplifying:

[(1.035)^n - 1] / 0.035 = 17.41

(1.035)^n - 1 = 0.60935

(1.035)^n = 1.60935

Taking the natural log of both sides:

n * ln(1.035) = ln(1.60935)

n = ln(1.60935) / ln(1.035)

n ≈ 26.95

Since we can't have a fractional number of payments, we round up to 27. Therefore, the answer is D) 27.

Question 15

MaryAnne is taking out a loan of $70,000 at 8% compounded semi-annually. Calculate the monthly payments that will reduce her balance owing to $30,000 in five years.

Accepted Answer

Firstly, we need to calculate the semi-annual interest rate:
i = 8% / 2 = 4%

In five years, there will be 2 * 5 = 10 semi-annual periods.
Let the monthly payment be P. Then, at the end of the five-year period, the remaining balance will be:

30000 = P * [(1 + i)^10 - 1] / (i / 12)

Solving for P, we get:
P = 1005.15

Rounded to the nearest dollar, the monthly payment is $1,005, which is answer choice B.

Question 16

Five years before her daughter was expected to enter college, Frieda deposited $15,000 in an education fund. Her daughter will make 48 equal quarterly withdrawals from the fund starting on the date she enters college. If the education fund earns 10% compounded quarterly, what will be the maximum quarterly withdrawal?

Accepted Answer

The answer of Five years before her daughter was expected...

Question 17

Martina's RRSP earns 12% compounded monthly and her goal is to accumulate $1,000,000 by making deposits of $4,100 at the end of every year. How many years will it take to reach her goal?

Accepted Answer

We can use the formula for future value of an annuity to solve this problem.
FV = PMT[((1+r)^n-1)/r]
where FV is the future value of the annuity, PMT is the regular payment, r is the interest rate per period, and n is the number of periods.
In this case, PMT = $4,100, r = 0.12/12 = 0.01 (since the interest rate is compounded monthly), and we want to find n such that FV = $1,000,000.
$1,000,000 = $4,100[((1+0.01)^n-1)/0.01]
Simplifying this equation, we get:
244.81 = (1.01)^n - 1
Taking the logarithm of both sides, we get:
log(244.81) = n*log(1.01)
n = log(244.81)/log(1.01) = 28.75
Therefore, it will take approximately 29 years to reach Martina's goal. Answer choice B is the best answer.

Question 18

If $2,000 were invested at the end of every three months, how long would it take to accumulate $475,000 at 10% compounded quarterly? (rounded up to the nearest whole year)

Accepted Answer

The answer of If $2,000 were invested at the end...

Question 19

Clarence has been investing $1,000 at the end of every year at 11% compounded annually. With today's contribution his investment has grown to $44,500. How many years ago did he make the first contribution?

Accepted Answer

We can use the formula for future value of an annuity to solve this problem:
FV = A[(1 + r)^n - 1]/r
where FV is the future value, A is the annuity payment, r is the annual interest rate, and n is the number of periods.

Plugging in the given values, we get:
44,500 = 1,000[(1 + 0.11)^n - 1]/0.11

Simplifying this equation, we get:
(1.11^n - 1) = 44.5
1.11^n = 45.5
n = log(45.5)/log(1.11)
n = 16.85

Therefore, Clarence made his first contribution 16.85 years ago. Rounded to the nearest year, this is 17 years ago. So the answer is C.

Question 20

Marvin's goal is to accumulate $1,000,000 in his retirement plan by making contributions of $250 at the end of each month. He expects to earn 10.8% compounded monthly. How long will it take Marv to reach his goal? (rounded up to the nearest whole year)

Accepted Answer

The answer of Marvin's goal is to accumulate $1,000,000 in...

Quiz 11: Annuities: Periodic Payment, Number of Payments, and Interest Rate