2. Learning Objectives
• Compute the future value (or accumulated value) for
ordinary general annuities
• Compute the present value (or discounted value) for
ordinary general annuities
• Compute the payment for ordinary general annuities
• Compute the number of periods for ordinary general
annuities
• Compute the interest rate for ordinary general
annuities
• Compute future value and present value for constant-
growth annuities
3. Ordinary General Annuity
• When the length of the interest conversion period
is different from the length of the payment interval,
these annuities are called general annuities
• In Canada, home mortgages are usually
compounded semi-annually and payments are
often made on a monthly, semi-monthly, or weekly
basis
• The quoted interest rates are not always quoted so
4. Ordinary General Annuity
• “Ordinary” means that the payments are made or
received at the end of each payment interval
• “General” means that the payment interval and
interest conversion intervals are different
5. Ordinary General Annuity
• There is a relationship between payment interval
and interest conversion period
• The number of interest conversion periods per
payment interval, designated by the letter c, can be
determined from the following ratio:
𝒄 =
𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
6. Typical Conversion Periods
• If the interest conversion period is longer than the
payment period, each payment interval contains
only a fraction of one conversion period
• In this case, c has a fractional value
• If the interest conversion period is shorter than the
payment period, each payment interval contains
more than one conversion period
• In this case, c has a value greater than 1
7. Equivalent Rate of
Interest Per Payment Period “p”
• To change the given rate of interest to the
equivalent interest per payment period “p” of the
problem, use the following formula:
𝑝 = 1 + 𝑖 𝑐
− 1
8. Practical Solutions
• For general annuities we will use the calculator to
set up the payment and interest conversion periods
• We set c/y as different than p/y (note on calculators set
p/y first then c/y)
• After that solving problems is similar to ordinary
simple annuities
9. Practice Questions - FV
• Q1. Tristian saves $100 at the end of each week
and deposits the money in an account paying 4%
compounded monthly
• A. How much will she accumulate in 8 years?
• B. How much of the accumulated amount is interest?
• Q2. At the end of each quarter, Bruce Machinery
pays $7000 into an account to fund a charity event.
If interest earned on the account is 0.95%
compounded annually, how much would the
account balance be in five years?
10. Practice Questions - PV
• Q3. The law firm of Blitzen and Donner borrowed
money to purchase equipment. The loan is repaid by
making payments of $824.37 at the end of every three
months over nine years. If interest is 4.3%
compounded annually, what was the original loan
balance?
• Q4. Peter Puck, a popular hockey player, has been
offered a two year salary deal. He can either accept
$2,200,000 now or accept monthly amounts of
$120,000 payable at the end of each month. If money
can be invested at 5.7% compounded quarterly, which
option is the better option for Peter and by how much?
11. Practice Questions - Payments
• Q5. How much must be deposited at the end of
each quarter for 12 years to accumulate to $14,000
at 8% compounded monthly?
• Q6. Jordyn bought a car priced at $11,300 for 15%
down and equal monthly payments for four years.
If interest is 8% compounded semi-annually, what is
the size of the monthly payment?
12. Practice Questions – Number of
Payments
• Q7. For how many years will Prasad make payments
on the $48,000 he borrowed to start his machine
shop if he makes payments of $9400 at the end of
every three months and interest is 8.08%
compounded semi-annually?
• Q8. Mr. Musk accumulated $220,000 in an RRSP. He
converted the RRSP into a RRIF and started to
withdraw $7500 at the end of every three months
from the fund. If interest is 6.75% compounded
monthly, for how long can Mr. Musk make
withdrawals?
13. Practice Questions - Interest
• Q9. A car valued at $11,400 can be purchased for
10% down and monthly payments of $286.21 for
three and a half years. What is the nominal rate of
interest compounded annually?
• Q10. A vacation property valued at $350,000 was
bought for 15 payments of $32,000 due at the end
of every 6 months. What nominal annual rate of
interest compounded monthly was charged? What
was the effective interest rate?
