Compounding Periods $1000 is invested at 12% per annum compounded
_______________ for one year.
ANNUALLY
(1 compounding period per year)
interest per compounding period
i
=
0 .12
1
=0 . 12
$1000
x 1.12
begin year
end year = $1000(1.12)
1= $ ____________
SEMI-ANNUALLY
(2 compounding periods per year)
i
=
0.12
2
=0.06
$1000
x 1.06
x 1.06
= $1000(1.06)
2= $ ____________
QUARTERLY
(4 compounding periods per year)
i
=
0 .12
4
=0 .03
$1000
x 1.03
x 1.03
x 1.03
x 1.03
= $1000(1.03)
4= $ ____________
MONTHLY
(12 compounding periods per year)
$1000 x 1.01 x 1.01 x 1.01 x 1.01 x 1.01 x 1.01 x 1.01 x 1.01 x 1.01 x
1.01 x 1.01 x 1.01
= $1000(1.01)
12= $ ____________
Compound Interest and Annuities
In all the diagrams below, arrows “above the line” represent positive cash flow (someone puts money in your pocket), and arrows “below the line” represent negative cash flow (money leaves your pocket).
Compound Interest:
1. Investment:
-you make an initial investment which grows over time. In the future you get a larger amount back
2. Loan:
-you are given an initial amount and you must pay back a larger amount in the future
Annuities:
3. Present Value:
-you are given an initial amount which you pay back over time with equal payments (eg. a mortgage)
fund)
4. Future Value:
-you make equal payments over time which are worth a lump sum in the end (eg. saving for college)
-you are paid equal payments over time, and pay back a lump sum at the end (not generally used in real life)
Using the TI-83:
APPS 1:Finance
1:TVM solver
(to solve unknown, cursor to the unknown after entering all data and use ALPHA ENTER)
N = total number of compounding periods
I% = annual interest rate as a percent
PV = present value (amount at the beginning)
PMT = payment each period
FV = future value (amount at the end)
P/Y = number of payments per year
NOTE: WATCH YOUR SIGNS. If you are loaned $20000 and your payment is $500, then
PV = 20000 and PMT = –500 (since the loan and the payments are going in opposite directions).
If you ever get the error “ERR: NO SIGN CHNG” or “DOMAIN ERROR”, it is because of your +/– signs)
EXAMPLES
1.You invest $750 at 6% per annum at a bank that pays compound interest.
How much money would be in your bank account after the 8 years if you
did not withdraw any money?
2. When you were born, your parents deposited some money into a savings
account to pay for your college or university education. The account pays
interest at a rate of 4% per year, compounded semi-annually. If you have
$20 000 on your 18
thbirthday, how much did they deposit into the account
when you were born?
3. You are saving for a trip to the 2014 World Cup in Brazil. Each month
you deposit a fixed amount of money into your savings account, which
earns 5% per year, compounded monthly. How much must you deposit each
month if you wish to have $5000 in exactly 2 years?
4. The bank lends you $200 000 to buy a house and charges you interest of
4.5% per year, compounded quarterly. To pay back the mortgage you make
equal quarterly payments of $3000. How long will it take you to pay back
the mortgage, and how much will you have paid in total back to the bank?
MCR 3U Annuities Date:
An annuity is a series of equal payments at regular interval of time. For an ordinary annuity, each payment is made at the end of each payment period. A payment period is the time between successive payments.
1. Determine the future value of 20 annual deposits of $1000 if the deposits earn 8% interest per annum, compounded annually.
2. Determine the monthly payments required to accumulate a future value of $10 000 in four years, if the payments earn 6.5% interest per annum, compounded monthly.
TVM Entries Final Answer
1. N = 8, I = 6, PV = 750, PMT = 0, FV = ?, P/Y = 1, C/Y = 1 FV = 1195.39 2. N = 36, I = 4, PV = ?, PMT = 0, FV = 20000, P/Y = 2, C/Y = 2 PV = -9804.46 3. N = 24, I = 5, PV = 0, PMT = ?, FV = 5000, P/Y = 12, C/Y = 12 PMT = -198.52 4. N = ?, I = 4.5, PV = 200000, PMT = -3000, FV = 0, P/Y = 4, C/Y = 4 N = 124, so 31 years
(FV = 45761.96) (PMT = -182.98)
3. A $10 000 loan is repaid with monthly payments of $334.54 for three years. Determine the interest rate per annum, compounded monthly.
(note: if you got “ERR: NO SIGN CHNG”, you need to set PMT = –334.54) (I% = 12.5)
4. Determine the total amount of interest earned on an annuity consisting of quarterly deposits of $1500.00 for ten years, if the annuity earns 9% interest per annum,
compounded quarterly.
($95679.26 – $60000 = $35679.26)
5. Starting in 4 months, Jeanine plans to deposit $875 on each July 31, October 31, January 31, and April 30, for 3 years, into an account. With an interest rate of 6% per year, compounded quarterly, how much will Jeanine have in her account when the last payment is made?
(FV = 11411.06)
6. Marcel wants to buy a home entertainment centre that he sees priced at $3799 plus 13% HST. He plans to buy the centre in 18 months, and he assumes that the price will stay the same. He will make a payment into an account at the end of every month for 18 months. The interest rate is 9% per annum, compounded monthly.
a) How much will each of Marcel’s payments be? b) How much interest will Marcel have earned?
( a) $223.64, b) $267.35)
7. Michael wants to make a lump sum investment so that he would receive $4000 every 6 months for 5 years. How much money should he invest now at 7% per year, compounded semi-annually?
(PV = -33266.42)
8. To provide an annual scholarship for 25 years, a donation of $50 000 is invested in an account for a scholarship that will start at the end of the first year. If the money is invested at 5.5% per annum, compounded annually, how much is the yearly scholarship?
(PMT = 3727.47)
Solutions to Example Problems
1. Determine the future value of 20 annual deposits of $1000 if the deposits earn 8% interest per annum, compounded annually.
N = 20; I% = 8; PV = 0; PMT = -1000; FV = 45761.96; P/Y = 1; C/Y = 1
2. Determine the monthly payments required to accumulate a future value of $10 000 in four years, if the payments earn 6.5% interest per annum, compounded monthly.
N = 48; I% = 6.5/12; PV = 0; PMT = -182.98; FV = 10000; P/Y = 1; C/Y = 1
3. A $10 000 loan is repaid with monthly payments of $334.54 for three years. Determine the interest rate per annum, compounded monthly.
N = 36; I% = 12.5/12; PV = 10000; PMT = -334.54; FV = 0; P/Y = 1; C/Y = 1
4. Determine the total amount of interest earned on an annuity consisting of quarterly deposits of $1500.00 for ten years, if the annuity earns 9% interest per annum, compounded quarterly.
N = 40; I% = 9/4; PV = 0; PMT = -1500; FV = 95679.26; P/Y = 1; C/Y = 1
Interest earned = 95679.26 – 40(1500) = $35679.26
5. Starting in 4 months, Jeanine plans to deposit $875 on each July 31, October 31, January 31, and April 30, for 3 years, into an account. With an interest rate of 6%, compounded quarterly, how much will Jeanine have in her account when the last payment is made?
N = 12; I% = 6/4; PV = 0; PMT = -875.00; FV = 11411.06; P/Y = 1; C/Y =1
6. Marcel wants to buy a home entertainment centre that he sees priced at $3799 plus GST and PST. He plans to buy the centre in 18 months, and he assumes that the price will stay the same. He will make a payment into an account at the end of every month for 18 months. The interest rate is 9% per annum, compounded monthly.
a) How much will each of Marcel’s payments be? b) How much interest will Marcel have earned?
N = 18; I% = 9/12; PV = 0.00; PMT = -225.63; FV = 4330.86; P/Y = 1; C/Y = 1
Interest earned = 4330.86 – 18(225.63) = $269.52
7. Michael wants to make a lump sum investment so that he would receive $4000 every 6 months for 5 years, with the first payment to start in 6 months. How much money should he invest now at 7%, compounded semi-annually?
N = 10; I% = 7/2; PV = -33266.42; PMT = 4000; FV = 0; P/Y = 1; C/Y = 1
8. To provide an annual scholarship for 25 years, a donation of $50 000 is invested in an account for a scholarship that will start a year after the investment is made. If the money is invested at 5.5% per annum, compounded annually, how much is each scholarship?
Changing Conditions of an Annuity - Practice
Use the TVM Solver to solve each problem.
1. A student begins saving for college by making regular monthly payments of $200.00 into an account that earns 5% per annum interest, compounded monthly.
a) Determine the value of the annuity after 4 years.
b) If the amount of the payments was changed to $300.00, what would the future value after four years be?
c) Determine the amount of additional interest earned using $200.00 monthly payments with an interest rate of 8% per annum instead of 5% per annum.
2. a) Determine the monthly payments required to accumulate a future value of $10000.00 in four years, if the payments earn 6.5% interest per annum, compounded monthly.
b) What would the required weekly payments be to accumulate the same amount of money in the same amount of time? (Interest rate is 6.5% per annum, compounded weekly.)
3. A $20 000 car loan is charged 3.9% per annum interest, compounded quarterly.
a) Determine the quarterly payments needed to pay the loan off in five years.
b) How much faster would the loan be paid off using the same payments, if the interest rate was lowered to 1.9%?
c) How much in interest charges could be saved (compared to part a.) by making weekly payments of $100.00, if interest is charged at 3.9%, compounded weekly.
MCR3U – Finance – In the interest of a challenge…
CONSIDER USING A GRAPHING CALCULATOR FOR SOME/ALL QUESTIONS
1. You invest $5000 at 8%/a, compounded monthly. How long will it take for you to have $9000?
2. You want to have $6000 in 7 years. You currently have $4000. At what interest rate, compounded semi-annually, must you
invest?
3. You invest some money with interest compounded weekly. After 5 years you have $4500 and after 9 years you have
$7800. What is the interest rate, and how much did you initially invest?
4. You are going to invest some money at 6.5%/a, compounded quarterly. You want to be able to withdraw $3000 in 4 years,
and then to have another $5000 5 years after that. How much should you invest now?
5. You invest $1000 at an interest rate compounded annually. After 3 years the interest rate goes up by 1%. 5 years after that
your initial investment has doubled. What was the original rate of interest?
6. You invest $5000 for 5 years, with interest compounded annually. After those 5 years you withdraw $1000, and leave the
remainder invested for another 5 years. After the entire 10 years you have $8000. What was the interest rate on your
investment?
7. You borrow $3500 at 6%/a interest, compounded annually. You immediately take that $3500 at loan it to your friend at
7.5%/a, compounded monthly.
a) How much profit will you make after 3 years?
b) How long will it take you to make a profit of $1000?
8.
