ch.6 Flashcards
You are buying a previously owned car today at a price of $6,890. You are paying $500 down in cash and financing the balance for 36 months at 7.9%. What is the amount of each loan payment?
199.94
A perpetuity differs from an annuity because:
Perpetuity payment never cease
You have $2,500 that you want to use to open a savings account. You have found five different accounts that are acceptable to you. All you have to do now is determine which account you want to use such that you can earn the highest rate of interest possible. Which account should you use based upon the annual percentage rates quoted by each bank?
Account A: 3.75%, compounded annually
Account B: 3.70%, compounded monthly
Account C: 3.70%, compounded semi-annually
Account D: 3.65%, compounded continuously
Account E: 3.66%, compounded quarterly
Account B
The effective annual rate is defined as the rate which:
would apply if interest were compounded annually.
The interest rate expressed as if it were compounded once per year is called the _____ rate.
effective annual
Today, you are retiring. You have a total of $413,926 in your retirement savings and have the funds invested such that you expect to earn an average of 3%, compounded monthly, on this money throughout your retirement years. You want to withdraw $2,500 at the beginning of every month, starting today. How long will it be until you run out of money?
213.29
Your credit card company charges you 1.35% per month. What is the annual percentage rate on your account?
16.2%
You are considering two perpetuities which are identical in every way, except that perpetuity A will begin making annual payments of $P to you two years from today while the first $P payment for perpetuity B will occur one year from today. It must be true that the present value of perpetuity:
B, exceeds that of A by the PV of $P for one year.
Today, you signed loan papers agreeing to borrow $4,954.85 at 9% compounded monthly. The loan payment is $143.84 a month. How many loan payments must you make before the loan is paid in full?
40.00
If you are investing money, you should prefer an ______ and if you are borrowing money you should prefer an _____.
annuity due, ordinary annuity
You currently have $7,000 in a bank account earning 8% interest. You think you will be able to deposit an additional $4,000 at the end of each of the next three years.
How much will you have in three years?
Total value in 3 years = $8,817.98 + $4,665.60 + $4,320 + $4,000 = $21,803.58
(6.1 slides)
Suppose you invest $500 in a mutual fund today and $600 in one year.
If the fund pays 9% annually, how much will you have in two years?
FV = $500 × (1.09)2 + $600 × (1.09) = $1,248.05
Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years.
How much will be in the account in five years if the interest rate is 8%?
FV = $100 × (1.08)4 + $300 × (1.08)2 = $136.05 + $349.92 = 485.9
You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the year after, and $800 at the end of the following year. You can earn 12% on similar investments.
How much is this investment worth today?
Year 1 CF: $200/(1.12)1 = $178.57
Year 2 CF: $400/(1.12)2 = $318.88
Year 3 CF: $600/(1.12)3 = $427.07
Year 4 CF: $800/(1.12)4 = $508.41
=> Total PV = $178.57 + $318.88 + $427.07 + $508.41 = $1,432.93
Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years.
If you require a 15% return on investments of this risk, should you take the investment?
PV = $40/(1.15)1 = $34.78
PV = $75/(1.15)2 = $56.71
=> PV = $34.78 + $56.71= $91.49 < $100
You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years.
How much would you be willing to invest today if you desire an interest rate of 12%?
𝑃𝑉=$25,000/(1+0.12)^40 +$25,000/(1+0.12)^41 +$25,000/(1+0.12)^42 +$25,000/(1+0.12)^43 +$25,000/(1+0.12)^44 =$268.67+$239.88+$214.18+$191.23+$170.74=$1,084.71
Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7%.
What is the value of the cash flows at year 5?
What is the value of the cash flows today?
What is the value of the cash flows at year 3?
Value at year 5 = 131.08 + 245.01 + 228.98 + 321 + 300 = 1226.07
Present value today = 93.46 + 174.69 + 163.26 + 228.87 + 213.90 = 874.18 (difference due to rounding)
Value at year 3 = 114.49 + 214 + 200 + 280.37 + 262.03 = 1070.89
What is annuity
- finite series of equal payments that occur at regular intervals
Ordinary annuity
If the first payment occurs at the end of the period
Annuity due
If the first payment occurs at the beginning of the period
After carefully going over your budget, you have determined that you can afford to pay $632 per month towards a new sports car. Your bank will lend to you at 1% per month for 48 months.
How much can you borrow?
After carefully going over your budget, you have determined that you can afford to pay $632 per month towards a new sports car. Your bank will lend to you at 1% per month for 48 months.
How much can you borrow?
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years.
If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
PV = $333,333.33 × [1 – 1/1.0530]/0.05 = $5,124,150.29
You know the payment amount for a loan and you want to know how much was borrowed.
Do you compute a present value or a future value?
You want to receive $5,000 per month in retirement. If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement?
PV = 5000[1 – 1 / 1.0075300] / .0075 = 595,808
Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8%/12 = 0.66667% per month).
If you take a 4-year loan, what is your monthly payment?
$20,000 = C × [1 – 1/1.006666748]/0.0066667
=> C = $488.26
You ran a little short on your February vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5% per month.
How long will you need to pay off the $1,000
$1,000 = $20 × (1 – 1/1.015t)/0.015
0.75 = 1 – 1/1.015t
1/1.015t = 0.25
1/0.25 = 1.015t
t = ln(1/.25)/ln(1.015) = 93.111 months or 7.76 years
Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42.
How long before you pay off the loan?
=> t = ln(1.157624287)/ln(1.05) = 3 years
You want to receive $5000 per month for the next 5 years.
How much would you need to deposit today if you can earn 0.75% per month?
PV = 5000(1 – 1 / 1.007560) / .0075 = 240,867
Suppose you begin saving for your retirement by depositing $2000 per year in an RRSP.
If the interest rate is 7.5%, how much will you have in 40 years?
FV = $2,000 × (1.07540 – 1)/0.075 = $454,513.04
You are saving for a new house and you put $10,000 per year in an account paying 8% compounded annually. The first payment is made today.
How much will you have at the end of 3 years?
FV = $10,000 × [(1.083 – 1)/0.08] × (1.08) = $35,061.12
(slide 34)
The Home Bank of Canada want to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter.
What dividend would the Home Bank have to offer if its preferred stock is going to sell?
First, find the required return for the comparable issue:
40 = 1/r
=> r = 0.025 or 2.5% per quarter
Then, using the required return found above, find the dividend for new preferred issue:
$100 = C/0.025
=> C = $2.50 per quarter
Hoffstein Corporation is expected to pay a dividend of $3 per share next year. Investors anticipate that the annual dividend will rise by 6% per year forever. The required rate of return is 11%.
What is the price of the stock today?
PV=$3.00/(0.11−0.06) =$60.00
Gilles Lebouder has just been offered a job at $50,000 a year. He anticipates his salary will increase by 5% a year until his retirement in 40 years.
Given an interest rate of 8%, what is the present value of his lifetime salary?
PV=$50,000/(0.08−0.05) x[1−(1.05/1.08)^40 ] =$1,126,571
You want to have $1 million to use for retirement in 35 years.
If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? What if the first payment is made today?
155.50
slide 41
You are considering preferred stock that pays a quarterly dividend of $1.50.
If your desired return is 3% per quarter, how much would you be willing to pay?
50
What is an effective annual rate
This is the actual rate paid (or received) after accounting for compounding that occurs during the year.
If you want to compare two alternative investments with different compounding periods, you need to compute the EAR for both investments and then compare the EAR’s
Annual Percentage rate
This is the annual rate that is quoted by law
By definition APR = period rate times the number of periods per year
Period rate = APR/number of periods per year
Suppose you can earn 1% per month on $1 invested today.
What is the APR?
How much are you effectively earning?
APR: 1% × 12 = 12%
Earning: FV = $1 × (1.01)^12 = 1.1268
=> Rate = (1.1268 – 1)/1 = 0.1268 or 12.68%
You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding.
Which account should you use?
First account:
EAR = (1 + 0.0525/365)365 – 1 = 0.0539 or 5.39%
Second account:
EAR = (1 + 0.053/2)2 – 1 = 0.0537 or 5.37%
You should choose the first account (the account that compounds daily), because you are earning a higher effective interest rate.
Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis
APR=12 × [(1+0.12)^(1⁄12)−1]=0.1138655152 or 11.39%
Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding.
What is your monthly payment?
Monthly rate = 0.169/12 =.0140833333
Number of months = 2 × 12 = 24
3500 = C × [1 - (1/1.0140833333)24]/0.0140833333
=> C = 172.88
slide 55
Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding.
How much will you have in the account in 35 years?
Monthly rate = 0.09/12 = 0.0075
Number of months = 35 × 12 = 420
=> FV = $50 × [1.0075420 – 1]/0.0075 = $147,089.22
You need $15,000 in 3 years for a new car.
If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?
Daily rate =0.055/365 = 0.00015068493
Number of days = 3 × 365 = 1095
=> PV = $15,000/(1.00015068493)1095= $12,718.56
What is the effective annual rate of 7% compounded continuously?
EAR = e0.07 – 1 = 0.0725 or 7.25%
WHat is the reward to risk ratio
The risk-reward ratio is a mathematical calculation used by investors to measure the expected gains of a given investment against the risk of loss.
