MATH1206 by Zomak Assignments

Chapter 12
Ordinary General Annuities

Objectives
1. Compute the future value (or accumulated value) for
ordinary general annuities.
2. Compute the present value (or discounted value) for
ordinary general annuities.
3. Compute the payment for ordinary general annuities.
4. Compute the number of periods for ordinary general
annuities.
5. Compute the interest rate for ordinary general
annuities.
6. Compute future value and present value for constant-
growth annuities.

Ordinary General Annuities (1 of 4)
• 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.
• With the exception of variable rate mortgages, all
mortgages are compounded semi-annually, by law.

Definition
• “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.
Example
• Payments of $100 made at the end
of every month for five years earns
interest at 5% compounded semi-
annually.
• Note that the payments are at
the end of the month (ordinary)
and are made monthly.
• The interest conversion period
is semi-annually which is every
six months.
• The length of the payment
period is different from the
length of the conversion period.

• It’s important to understand the relationship
between the payment interval and the number of
interest conversion periods per payment interval.
• 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

Table 12.1 Some Possible Combinations of Payment Intervals
and Interest Conversion Periods (1 of 2)
Numerator: Interest
Conversion Period
Denominator: Payment
Interval
Number of Interest
Conversion Periods per
Payment Interval
monthly semi-annually
𝑐 =
12
2
= 6
monthly quarterly
𝑐 =
12
4
= 3
quarterly annually
𝑐 =
4
1
= 4
annually semi-annually
𝑐 =
1
2
= 0.5

Table 12.1 Some Possible Combinations of Payment Intervals and
Interest Conversion Periods (2 of 2)
Numerator: Interest
Conversion Period
Denominator: Payment
Interval
Number of Interest
Conversion Periods per
Payment Interval
annually quarterly
𝑐 =
1
4
= 0.25
semi-annually monthly
𝑐 =
2
12
=
1
6
quarterly monthly
𝑐 =
4
12
=
1
3
annually monthly
𝑐 =
1
12

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
𝒄 =

Ordinary General Annuities
• Computing p, allows us to treat the ordinary general
annuity problem similar to an ordinary simple annuity
problem.
• FVg = the future value (or accumulated value) of an ordinary
general annuity
• PMT = the size of the periodic payment
• n = the number of periodic payments
• c = the number of interest conversion periods per payment
interval
• i = the interest rate per interest conversion period
• p = the equivalent rate of interest per payment period

Future Value of an Ordinary
General Annuity (1 of 3)
𝐹𝑉
𝑔 = 𝑃𝑀𝑇
1 + 𝑝 𝑛 − 1
𝑝
where 𝑝 = 1 + 𝑖 𝑐 − 1
𝒄 =

• To attend school, Sam deposits $1500 at the end of
every six months for four and one-half years. What is
the accumulated value of the deposits if interest is 6%
compounded quarterly?
• 6%/4 = 1.5% = 0.015 every quarter (3 months) so there are
four quarters per year
• Payments are every 6 months or two times per year
𝒄 =
=
𝟒
𝟐
= 𝟐
𝑝 = 1 + 0.015 2
− 1 = 0.030225
• 3.0225% every six months

Interest Conversion Using the BA II
PLUS
• Given interest rate is 6% compounded quarterly.
• Payments are semi-annually.
• Convert 6% compounded quarterly to the equivalent semi-annual rate.
KEY DISPLAY KEY DISPLAY
2nd Blank ↓ NOM = 6
INCONV 0 CPT NOM = 6.045 (nominal semi-annual rate
equivalent to 6% compounded quarterly).
2nd Blank Blank Blank
CLR WORK NOM = 0 Blank Blank
6 ENTER NOM = 6 CLR 6.045
↑ C/Y = 2 (default) ÷ Blank
4 ENTER C/Y = 4 2 Blank
↑ EFF = 0 (default) = 3.0225 (periodic rate – every 6 months)
CPT EFF = 6.136355
(effective annual rate)
See previous slide for
manual computation.
Blank
↓ C/Y = 4 Blank Blank
2 ENTER C/Y = 2 Blank Blank

• Number of payments n = 4.5 × 2 = 9
𝐹𝑉
𝑔 = 1500
1 + 0.030225 9 − 1
0.030225
𝐹𝑉
𝑔 = 1500
1.030225 9 − 1
0.030225
𝐹𝑉
𝑔 = 1500 10.168425 = $15,252.64

Calculator Solution (1 of 2)
Manually
CE/C Blank ) 1.0168425
1500 Blank = 15,252.63701
× Blank Blank Blank
( Blank Blank Blank
1.030225 Blank Note: You could also compute the terms
inside the square brackets first and then
multiply by 1500 – this eliminates the need
for brackets on the calculator.
Blank
yx Blank Blank Blank
9 Blank Blank Blank
−1 1.03073416 Blank Blank
÷ 0.03073406 Blank Blank
0.030225 Blank
𝐹𝑉
𝑔 = 1500
1.030225 9
− 1
0.030225
Blank

TVM Financial functions
2nd Blank 6* Blank
CLR TVM 0 I/Y* I/Y = 6*
2nd Blank 0 Blank
P/Y Blank PV PV = 0
2 ENTER P/Y = 2 −1500 Blank
↓ Blank PMT PMT = −1500
4 ENTER C/Y = 4 CPT Blank
2nd Blank FV FV = 15,252.64
QUIT 0 * Enter the nominal rate given as C/Y (set to
4 for quarterly). There is no need for a
conversion.
Blank
9 Blank Blank Blank
N N = 9 Blank Blank

Ordinary General Annuities—
Finding the Present Value PV (1 of 3)
𝑃𝑉
𝑔 = 𝑃𝑀𝑇
1 − 1 + 𝑝 −𝑛
𝑝
𝑝 = 1 + 𝑖 𝑐 − 1

• A 30-year mortgage on a condominium requires
payments of $915.60 at the end of each month. If
interest is 5.5% compounded semi-annually, what was
the mortgage principal?
• i = 5.5  2 = 2.75% = 0.0275 every 6 month period (twice per
year).
• Payments are monthly which means twelve payments per
year.
𝒄 =
=
𝟐
𝟏𝟐
=
𝟏
𝟔
𝑝 = 1 + 0.0275
1
6 − 1 = 1.0275
1
6 − 1 = 0.00453168
• 0.453168% per month (5.438018% compounded
monthly)

Interest Conversion Using the BA II
PLUS
• Given interest rate is 5.5% compounded semi-annually.
• Payments are monthly.
• Convert 5.5% compounded semi-annually to the equivalent monthly rate.
2nd Blank ↓ NOM = 5.5
INCONV 0 CPT NOM = 5.438018062 (nominal monthly rate
equivalent to 5.5% compounded semi-
annually).
CLR WORK NOM = 0 Blank Blank
5.5 ENTER NOM = 5.5 CLR 5.438018062
↑ C/Y = 2 (default) ÷ Blank
2 ENTER C/Y = 2 12 Blank
↑ EFF = 0 (default) = 0.453168172 (periodic rate – every month)
CPT EFF = 5.575625
(effective annual rate)
See previous slide for
manual computation.
Blank
↓ C/Y = 2 Blank Blank
12 ENTER C/Y = 12 Blank Blank

• Number of payments n = 12 × 30 = 360
𝑃𝑉
𝑔 = 915.60
1 − 1 + 0.00453168 −360
0.00453168
𝑃𝑉
𝑔 = 915.60
1 − 1.00453168172 −360
0.00453168172
𝑃𝑉
𝑔 = 915.60 177.334432 = $162,367.41

Manually
CE/C Blank = 177.334432
1.00453168172* Blank × Blank
yx Blank 915.60 Blank
360 Blank = 162,367.41*
+ | − −360 Blank
= 0.196377 Note: You could also compute the terms inside the
square brackets first and then multiply by 1500 – this
eliminates need for brackets on the calculator.
Blank
+ | − −0.196377 Blank Blank
+ Blank Blank Blank
1 Blank Blank Blank
÷ 0.8036231 Blank Blank
0.00453168172* Blank
𝑃𝑉
𝑔 = 915.60
1 − 1.00453168 −360
0.00453168
Blank

2nd Blank 5.5* Blank
CLR TVM 0 I/Y* I/Y = 5.5*
2nd Blank 0 Blank
P/Y Blank FV FV = 0
12 ENTER P/Y = 12 −915.60 Blank
↓ Blank PMT PMT = −915.60
2 ENTER C/Y = 2 CPT Blank
2nd Blank PV PV = 162,367.41
QUIT 0 * Enter the nominal rate given as C/Y has been set to 2 for semi-
annually. There is no need for a conversion.
* Using the Time value of money (TVM) keys the solution is
computed with 9 decimal places for the rate.
Blank
360 Blank Blank Blank
N N = 360 Blank Blank

Finding the Periodic Payment PMT
• When the future value of a general annuity is
known
𝑃𝑀𝑇 =
𝐹𝑉
𝑔𝑝
1 + 𝑝 𝑛 − 1

Ordinary General Annuities—Finding
the Periodic Payment PMT when FV
Known
• What payment made at
the end of each year for
18 years will amount to
$16,000 at 4.2%
compounded monthly?
𝑃𝑀𝑇 =
𝐹𝑉
𝑔𝑝
1 + 𝑝 𝑛 − 1
• p = (1 + 0.0035)12/1 −1 =
0.042818
• n = 18
𝑃𝑀𝑇 =
16000 × 0.042818
1.042818 18 − 1
= $607.92

Manually
CE/C Blank 1 Blank
16000 Blank ) 1.126930957
× Blank = 607.92
0.042818007 Blank Blank Blank
= 685.088112 Blank Blank
÷ Blank
𝑃𝑀𝑇 =
16000 × 0.042818007
1.042818007 18 − 1
= $607.92
Blank
( Blank Blank Blank
yx Blank Blank Blank
− Blank Blank Blank

2nd Blank I/Y* I/Y = 4.2*
CLR TVM 0 0 Blank
2nd Blank PV PV = 0
P/Y Blank 16000 Blank
1 ENTER P/Y = 1 FV FV = 16000
↓ Blank CPT Blank
12 ENTER C/Y = 12 PMT PV = 607.92
QUIT 0 * Enter the nominal rate given because C/Y has been set to 12
for semi-annually. There is no need for a conversion.
* Using the time value of money (TVM) keys, the solution is
Blank
4.2* Blank Blank Blank

the Periodic Payment PMT (3 of 4)
When the PV of an annuity is known

the Periodic Payment PMT (4 of 4)
• When the present value of a general annuity is
known;
𝑃𝑀𝑇 =
𝑃𝑉
𝑔𝑝
1 − 1 + 𝑝 −𝑛

the Periodic Payment PMT when FV Known
• What payment is
required at the end of
each month for five
years to repay a loan of
$6000 at 7%
compounded semi-
annually?
𝑃𝑀𝑇 =
𝑃𝑉
𝑔𝑝
1 − 1 + 𝑝 −𝑛
• p = (1 + 0.035)2/12 −1 =
0.005750039
• n = 12 × 5 = 60
𝑃𝑀𝑇
=
6000 × 0.005750039
1 − 1.005750039 −60
= $118.52

Manually
CE/C Blank yx Blank
6000 Blank −60 Blank
× Blank ) 0.708918835
0.005750039 Blank ) 0.291081165
= 34.500234 = 118.52
÷ Blank
𝑃𝑀𝑇 =
6000 × 0.005750039
1 − 1.005750039 −60
= $118.52
Blank
( Blank Blank Blank
1 Blank Blank Blank
− Blank Blank Blank
( Blank Blank Blank

2nd Blank I/Y* I/Y = 7*
CLR TVM 0 6000 Blank
2nd Blank PV PV = 6000
P/Y Blank 0 Blank
12 ENTER P/Y = 12 FV FV = 0
↓ Blank CPT Blank
2 ENTER C/Y = 2 PMT PV = 118.52
QUIT 0 * Enter the nominal rate as given because C/Y has been set to 2 for
semi-annually. There is no need for a conversion.
* Using the time value of money (TVM) keys, the solution is
Blank
7* Blank Blank Blank

Finding the Term n
• When the future value of a general annuity is
known;
𝑛 =
𝑙𝑛
𝐹𝑉
𝑔𝑝
𝑃𝑀𝑇
+ 1
𝑙𝑛 1 + 𝑝

the Term n when the FV is Known
• For how long must
contributions of $2000
be made at the end of
each year to accumulate
to $100,000 at 6%
compounded quarterly?
𝑛 =
𝑙𝑛
𝐹𝑉
𝑔𝑝
𝑃𝑀𝑇
+ 1
𝑙𝑛 1 + 𝑝
• p = (1 + 0.015)4/1 − 1 =
0.061363551effective
• Can also use ICONV
function.
𝑛 =
𝑙𝑛
100000 × 0.061363551
2000 + 1
𝑙𝑛 1.061363551
= 23.561549
• 23.56 years

Manually
CE/C Blank ÷ Blank
100000 Blank 1.061363551 Blank
× Blank LN 0.05955445
0.061363551 Blank = 23.561549*
÷ Blank * years Blank
2000 Blank
𝑛 =
𝑙𝑛
100000 × 0.061363551
2000
+ 1
𝑙𝑛 1.061363551
Blank
+ 3.06817755 Blank Blank
1 Blank Blank Blank
= 4.06817755 Blank Blank
LN 1.403195123 Blank Blank

2nd Blank PV PV = 0
CLR TVM 0 −2000 Blank
2nd Blank PMT PMT = −2000
1 ENTER P/Y = 1 FV FV = 100000
↓ Blank CPT Blank
4 ENTER C/Y = 4 N N = 23.561549*
QUIT 0 *years Blank
6 Blank Blank Blank
I/Y I/Y = 6 Blank Blank
0 Blank Blank Blank

Finding the Term n
• When the present value of a general annuity is
known;
𝑛 =
𝑙𝑛 1 −
𝑃𝑉
𝑔𝑝
𝑃𝑀𝑇
−𝑙𝑛 1 + 𝑝

Finding the Periodic Rate of
Interest i (1 of 2)
• Using preprogrammed financial calculators
• Determining i without a financial calculator is extremely
time consuming.
• When the future value or present value, the
periodic payment (PMT), and the term n of a
general annuity are known, retrieve the value of I/Y
by pressing CPT I/Y.

Finding the Periodic Rate of
Interest i (2 of 2)
• Victoria saved $416
every six months for
eight years. What
nominal rate of
interest
compounded
annually is earned if
the savings account
amounts to $7,720
in eight years?
2nd Blank PV PV = 0
CLR TVM 0 −416 Blank
2nd Blank PMT PMT = −416
2 ENTER P/Y = 2 FV FV = 7720
↓ Blank CPT Blank
1 ENTER C/Y = 1 I/Y I/Y = 3.9247*
QUIT 0 *compounded
annually
(effective)
Blank
0 Blank Blank Blank

Constant-Growth Annuities (1 of 12)
• Constant-growth annuities differ from fixed-
payment-size annuities in that the periodic
payments change (usually grow) at a constant rate.
• The assumption of constant growth is often used in
sales forecasting and long-term financial planning.
• It is consistent with the indexing of pensions or
annuities.

• In general, if the first payment is represented by PMT
and the constant rate of growth by k, the constant
growth factor for the annuity payments is (1 + k).
• The size of the successive payments is as follows:
• 1st payment = PMT
• 2nd payment = PMT(1 + k)
• 3rd payment = PMT(1 + k)2
• 4th payment = PMT(1 + k)3
↓ ↓ ↓ ↓
↓ ↓ ↓ ↓
• 10th payment = PMT(1 + k)9
• SIZE OF THE nth PAYMENT = PMT(1 + k)n – 1

• PMT is the size of the first payment.
• k is the periodic compounding rate.
• n is the number of payments.

• Sum of the Periodic Constant-Growth Payments is
equal to
𝑃𝑀𝑇
1 + 𝑘 𝑛 − 1
𝑘

• Future value of an ordinary simple constant-growth
annuity:
𝐹𝑉 = 𝑃𝑀𝑇
1 + 𝑖 𝑛 − 1 + 𝑘 𝑛
𝑖 − 𝑘
• PMT = the size of the first annuity payment
• i = the interest rate per conversion period
• k = the constant growth rate of the annuity payments
• n = the number of conversion periods

• Seven deposits increasing at a constant rate of 1.5%
are made at the end of each of six successive years.
The size of the first deposit is $5,000 and the fund
earns interest at 6% compounded annually.
1. What is the size of the last deposit?
2. How much was deposited in total?
3. What is the accumulated value of the deposits?
4. What is the interest earned by the deposits?

• What is the size of the last deposit?
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 𝑃𝑀𝑇 1 + 𝑘 𝑛−1
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1 + 0.015 7−1
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ
𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1.015 6
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1.015 6
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = $𝟓, 𝟒𝟔𝟕. 𝟐𝟐

• How much was deposited in total?
𝑃𝑀𝑇
1+𝑘 𝑛−1
𝑘
k = the constant growth rate of the annuity payments
5,000
1 + 0.015 7
− 1
0.015
= $𝟑𝟔, 𝟔𝟏𝟒. 𝟗𝟕

• What is the accumulated value of the deposits?
𝐹𝑉 = 𝑃𝑀𝑇
1 + 𝑖 𝑛 − 1 + 𝑘 𝑛
𝑖 − 𝑘
𝐹𝑉 = 5,000
1 + 0.06 7 − 1 + 0.015 7
0.06 − 0.015
𝐹𝑉 = 5,000
1.06 7 − 1.015 7
0.045
= $𝟒𝟑, 𝟕𝟓𝟑. 𝟗𝟑

• What is the interest earned by the deposits?
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 43,753.93 − 36,614.97
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $𝟕, 𝟏𝟑𝟖. 𝟗𝟔

• Present value of an ordinary simple constant-
growth annuity;
𝑃𝑉 = 𝑃𝑀𝑇
1 − 1 + 𝑘 𝑛
1 + 𝑖 −𝑛
𝑖 − 𝑘
• PMT = the size of the first annuity payment
• i = the interest rate per conversion period
• k = the constant growth rate of the annuity payments
• n = the number of conversion periods

• When the constant-growth rate of the annuity
payments (k) and the periodic interest rate (i) are
the same, use
𝐹𝑉 = 𝑛 𝑃𝑀𝑇 1 + 𝑖 𝑛−1
𝑃𝑉 = 𝑛 𝑃𝑀𝑇 1 + 𝑖 −1

Summary
In an ordinary general annuity, the payment interval and
interest conversion interval do not coincide (General).
It is necessary to calculate an equivalent rate of interest p
so that the payment interval and interest conversion
interval do coincide.
Once an equivalent rate of interest is computed, the FV,
PV, PMT, and n can be calculated in the same manner as
ordinary simple annuities.
Constant-growth (ordinary simple) annuities have a first
payment PMT and then subsequent payments increase
by a constant amount k.

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