a2【练习】会计金融基础·年金现值终值2

Financial Mathematics: Extra Practice Questions

Note: In attempting these questions, particularly the more complex ones, it may prove useful to draw a time-line to determine the magnitude and timing of cash flows before undertaking any calculations.

Question One

Calculate the future value of $1,000 invested today for a period of 20 years at an interest rate of 10% p.a. compounded daily. Show how and discuss why your answer would change if interest was compounded quarterly.

Question Two

You made a deposit in a bank account exactly 18 months ago today. You have not made any subsequent deposits, and the balance of your account is now $4,400. Calculate the value of your initial deposit given you earned an interest rate of 15% p.a. compounded semi-annually. Show how and discuss why your answer would change if interest were compounded annually.

Question Three

You have just successfully applied for a home loan. Calculate how much you are borrowing given that the terms of the loan are as follows:

?Your monthly repayments are $1,000;

?The loan is taken over 25 years; and,

?The interest rate you will pay on funds borrowed is fixed at 10% p.a.

compounded quarterly.

Show how and discuss why your answer would change if interest were compounded annually.

Question Four

Calculate the present value of an ordinary perpetuity that comprises one cash flow of $200 at the end of each year given an interest rate of 15% p.a. compounded annually. Show how and discuss why your answer would change if interest were compounded weekly.

Question Five

A man invests $500 at 15% p.a. compounded fortnightly and plans to hold this investment for 10 years. Assuming there are exactly 26 fortnights in a year, how much will he have at the end of his holding period?

Question Six

A business needs $20,000 in 2 years time to replace a piece of equipment. How much must be invested now at an interest rate of 6% p.a. compounded monthly in order to provide for this replacement?

A woman wants to provide a $10,000 university scholarship every year for 50 years. The first scholarship is to be awarded one year from now. If the university can earn a 6% p.a. compounded daily as a return on their investments, how much should the woman give now?

Question Eight

How much will $500 grow to if invested for 10 years at an interest rate of 12% p.a. compounded annually?

Question Nine

What will the following investments accumulate to if interest is compounded annually?

a. $1,000 invested at 10% p.a. for 6 years?

b. $125.47 invested at 12% p.a. for 8 years?

Question Ten

In 30 years time when I retire, I will have $4 million in my retirement fund. What is this worth in today's dollars (i.e., what is the present value) assuming an average annual interest rate of 10% p.a. compounded annually?

Question Eleven

You invest $100 for a period of 7 years, after which it has grown to $200. If interest was compounded annually, what was the average rate of interest earned?

Question Twelve

How long does it take $100 to grow to $150 if the interest rate is 10% p.a. compounded annually?

Question Thirteen

If the interest rate is 10% p.a. compounded annually, what is the present value of the following cash flows:

a. $1,000 to be received in 3 year's time?

b. $1,500 to be received in 10 year's time?

Question Fourteen

You have just been signed to a major record label and have been promised $20,000 in one year's time plus another $10,000 in two year's time. What is the value of this consideration to you today assuming that you can invest your money at 5% p.a. compounded annually?

At the end of each of the next 10 years, you will place $1,000 into an investment that returns 12% p.a. compounded annually. How much will this investment have grown to by the end of year 10?

Question Sixteen

You wish to purchase a new car, valued at $55,000. The purchase will be financed as follows. An upfront payment of $10,000 is due immediately. The balance ($45,000) will be paid off over the next four years. Repayments are due at the end of each month. The finance company quotes an interest rate of 12% p.a. compounded monthly. Calculate the monthly repayment amount.

Question Seventeen

Reconsider the previous question. You will have trouble meeting the monthly repayments calculated in the previous question. If you can talk the finance company into allowing you to pay off the balance over five years (rather than four years), by how much does this reduce your monthly repayment?

Question Eighteen

Your child will commence university in 15 year's time. You wish to put away money regularly (one deposit at the end of each year) to provide for her education, which you estimate will cost $200,000. You anticipate that the average rate of return on an investment fund will be 8% p.a. compounded annually. How much will you have to put away at the end of each of the next 15 years so that you will have the $200,000 required?

Question Nineteen

Reconsider the previous question. As opposed to putting money away regularly to accumulate $200,000, you decide to make a once-off investment now. How much will you have to invest today to have the required $200,000 in 15 year's time? Question Twenty

Your generous uncle decides to endow his alma mater with sufficient monies to fund a scholarship of $5,000 per year in perpetuity. If the school can earn a return of 8% p.a. compounded annually on the endowment, how much does he need to donate as a lump sum today? In providing an answer, assume that the first scholarship payment will be made one year from now.

You have won a lottery and will receive $10,000 at the end of each year in perpetuity. If we assume an interest rate of 10% p.a. compounded annually, what this infinite series of payments worth in today's dollars?

Aside: As an interesting exercise to prove the formula for the present value of a perpetuity, you might try preparing a spreadsheet. Listing the payments you will receive (maybe go out to 200 years), discount each $10,000 payment to present value using the interest rate and the relevant number of years, and add up the present values. You'll see that a payment made 200 years from now is effectively worthless in today's dollars.

Question Twenty-Two

What is the present value of $500 to be received in 5 year's time if the interest rate is 8% p.a. compounded quarterly?

Question Twenty-Three

What will $550 amount to in four year's time at a nominal interest rate of 12% p.a. if interest is:

a. Compounded annually?

b. Compounded monthly?

c. Paid on daily balances (assume that the bank ignores the extra day in leap

years).

Question Twenty-Four

Different banks offer different interest rates. Which bank gives the greatest return?

?Commonstealth 15% p.a. compounded annually;

?Eastpac 14.75% p.a. compounded quarterly;

?Metaway 14.675% p.a. compounded semi-monthly; and,

?ANX 14.5% p.a. compounded monthly.

Question Twenty-Five

You decide to start saving for a vacation to the Whitsunday Islands, leaving on New Year's day (1st January). You will invest $100 on the first day of each month (starting today 1st March), with the final investment on 1st December. Assuming you earn interest at a rate of 12% p.a. compounded monthly, how much will you have to spend when you withdraw all invested funds on New Year’s day?

Question Twenty-Six

It is the first day of January 2003. Starting from the first day of the year 2006 you will deposit $5,000 into a bank. You will continue to deposit this amount into the bank every New Year's day up to and including New Year’s day 2010. On New Year's day 2011, instead of depositing any money, you will instead withdraw all of your deposited funds and accumulated interest. Assuming an interest rate of 15% p.a. compounded annually, how much will you withdraw from your account?

You are considering the purchase of a home for $300,000. You have available a deposit of $50,000. The bank will lend you the balance ($250,000) at 6% p.a. over a period of 20 years. Interest is compounded monthly.

a.Calculate your regular monthly repayment.

b.Five years later, you have made 60 repayments. What is the payout figure on

the loan? That is, how much do you still owe the bank?

c.Assume that, just after your 60th payment, the interest rate rises to 9% p.a. (still

compounded monthly). Of course, this means your monthly payment must rise if you are to pay the loan off over 20 years in total (there are now 15 years to go). Calculate your revised monthly loan repayment.

Aside: A spreadsheet is ideal for checking and proving these calculations.

Question Twenty-Eight

At the end of each of the next four years, you will receive a payment of $1,000. The interest rate is 10% p.a. compounded annually.

a.Equate this series of cash flows to a single cash flow received today. That is,

calculate the present value of the 4-payment annuity.

b.Equate this series of cash flows to a single cash flow received at the end of

year four. That is, calculate the future value of the annuity.

c.Take your answer to (a). Assume that you invest this amount for four years.

How much will it grow to?

d.Take your answer to (b). If this was a once-off payment to be received at the

end of four years, what is its present value.

e.Equate the original series of cash flows to a single cash flow received after two

years.

f.Take your answer to (e) and discount it back to present value.

Question Twenty-Nine

Consider the following series of cash flows. Today is time 0. You receive nothing for the first two years. At the end of years 3 and 4, you receive $2,000. At the end of years 5 and 6, you receive $5,000. The interest rate is 6% p.a. compounded annually. Calculate the present value of this series of payments (there are several different ways of approaching this question - all giving the correct answer).