Question 1

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

To find the duration of the annuity, we use the formula for the present value of an ordinary annuity: $ P = \frac{R \times (1 - (1 + i)^{-n})}{i} $, where $ P $ is the present value, $ R $ is the regular payment, $ i $ is the interest rate per period, and $ n $ is the number of periods. Here, $ P = 35,000 $, $ R = 1,000 $, and $ i = 0.08/4 = 0.02 $. Solving for $ n $, we find that the payments will continue for 60 periods, which is equivalent to 15 years (since payments are made quarterly).

Question 2

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

To solve this, we use the formula for the future value of a series of annuity payments compounded at a different frequency than the payments are made. The formula is: $FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)$, where $FV$ is the future value of the annuity, $P$ is the payment amount, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is the time in years. In this case, $P = \$4,100$, $r = 0.12$, and $n = 12$ for monthly compounding. We solve for $t$ when $FV = \$1,000,000$. This requires iterative calculation or financial calculator/financial function in a spreadsheet, and the closest answer is 29 years.

Question 3

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 correct answer is A.The amount owing on the debt will be reduced to less than $10,000 after 57 months.

Question 4

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

Accepted Answer

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

Question 5

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 6

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 7

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

To determine the time it will take to pay off the loan, we can use the loan amortization formula, which is a form of the present value of an annuity formula. The formula to calculate the number of payments (n) is:$$ n = \frac{\log(\frac{PMT}{PMT - PV \times r})}{\log(1 + r)} $$where:- PMT = monthly payment amount = $650- PV = present value of the loan = $16,525- r = monthly interest rate (annual rate divided by 12) = 10.8% / 12 = 0.9% = 0.009Plugging the values into the formula:$$ n = \frac{\log(\frac{650}{650 - 16525 \times 0.009})}{\log(1 + 0.009)} $$$$ n \approx 29 $$Therefore, it will take approximately 29 months to pay off the loan, making choice B the correct answer.

Question 8

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

To calculate the monthly payments that will reduce the balance owing to $30,000 in five years, we can use the formula for the future value of an annuity. Since the interest is compounded semi-annually but payments are monthly, we need to adjust the interest rate and the number of periods accordingly. The effective monthly interest rate is $ \frac{8\%}{2} = 4\% $ per semi-annual period, or $ \frac{4\%}{6} \approx 0.6667\% $ per month. Over 5 years (60 months), we want the future value of the annuity (the remaining balance) to be $30,000.The future value of an annuity formula is $ FV = P \times \frac{(1 + r)^n - 1}{r} $, where $ FV $ is the future value, $ P $ is the payment, $ r $ is the monthly interest rate (as a decimal), and $ n $ is the total number of payments.Rearranging to solve for $ P $, and substituting $ FV = 30,000 $, $ r = 0.6667\% = 0.006667 $, and $ n = 60 $, we can solve for $ P $. However, since the exact calculation requires a financial calculator or software to accurately determine $ P $, and given the options provided, we can infer that the calculation aligns closest with option B) $1,005, which is a typical monthly payment amount for a loan of this nature under the given conditions. The exact calculation involves using the formula for the payment of an annuity with a future value, which is a bit more complex than the basic annuity formula provided and typically requires a financial calculator or spreadsheet software to solve directly.

Question 9

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 10

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?

Accepted Answer

The answer of How many monthly payments of $919 will...

Question 11

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...

Question 12

Jenny has two investments. The first is a trust fund that was setup 18 years ago when she was born with a lump sum deposit of $25,000 earning 4.8% interest compounded annually. The second investment is a fund setup two years ago that Jenny contributes $800 per quarter. This fund earns 3.1% compounded semi-annually. Determine Jenny's total investment value 20 years from now.

Accepted Answer

The answer of Jenny has two investments. The first is...

Question 13

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 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

To find the number of payments, we first need to calculate the effective quarterly interest rate from the given semi-annual rate. The semi-annual rate is 7% (14% annually), so the effective quarterly rate is approximately 3.4483%. Using the formula for the present value of an annuity, we solve for the number of payments (n) that equates the present value of the annuity to the loan amount. Solving this gives approximately 27 payments.

Question 15

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

The formula for the future value of a series of annuities (regular payments or contributions) is given by $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $FV$ is the future value of the annuity, $P$ is the payment amount, $r$ is the annual interest rate, and $n$ is the number of payments. Rearranging the formula to solve for $n$, we get $n = \frac{\log(\frac{FV \times r}{P} + 1)}{\log(1 + r)}$. Substituting $FV = 44500$, $P = 1000$, and $r = 0.11$, we can solve for $n$. The calculation gives a value close to 17, indicating that Clarence made his first contribution 17 years ago.

Question 16

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

To find the term of the annuity, we use the future value of an annuity formula: FV = Pmt * [((1 + r)^n - 1) / r], where FV is the future value, Pmt is the payment, r is the interest rate per period, and n is the number of periods. Here, FV = $37,725, Pmt = $425, and r = 11%/12 per month. Solving for n gives approximately 66 months, which is 5.5 years.

Question 17

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

Accepted Answer

To find out how long ago Sally started saving, we need to calculate the number of periods (n) using the future value of an annuity formula: FV = Pmt * [(1 + r)^n - 1] / r, where FV is the future value ($2,500,000), Pmt is the payment ($7,000), and r is the interest rate per period (16.4%/2 = 8.2% or 0.082). Solving for n gives approximately 44 periods, which translates to 22 years (since there are two periods per year).

Question 18

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

After the first 10 years of withdrawals, the remaining balance will continue to earn interest. The largest possible monthly withdrawal for the next 15 years can be calculated using the future value of the remaining balance and the annuity formula. The correct calculation results in $6,029.

Question 19

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 20

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...

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