Outline
I.
More on the use of the financial
calculator and warnings
II. Dealing with periods other than
years
III. Understanding interest rate quotes
and conversions
IV. Applications – mortgages, etc.
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© 2002 David A. Stangeland
I. Warnings for annuities and
perpetuities
 Remember the PV formulas given for
annuities and perpetuities always
discount the cash flows to exactly
one period before the first cash
flow.
 If the cash flows begin at period t, then
you must divide the PV from our
formula by (1+r)t-1 to get PV0.
 Note: this works even if t is a fraction.
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© 2002 David A. Stangeland
Example

A retirement annuity of 30 annual payments (each
payment is $50,000) begins 20 years from today.
The value of that annuity 20 years from today is
__________________. The value of that annuity
19 years from today is ___________________.
The value of that annuity today is
___________________. (r=12%)
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© 2002 David A. Stangeland
Be careful of the number of annuity payments
 Count the number of payments in an annuity. If the
first payment is in period 1 and the last is in period 2,
there are obviously 2 payments. How many payments
are there if the 1st payment is in period 12 and the
last payment is in period 21 (answer is 10 – use your
fingers). How about if the 1st payment is now (period
0) and the last payment is in period 15 (answer is 16
payments).
 If the first cash flow is at period t and the last cash
flow is at period T, then there are T-t+1 cash flows in
the annuity.
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© 2002 David A. Stangeland
Example

Five years from now Mary will deposit $1,000 into
a savings fund for her daughter Margaret. Each
year she will make an additional $1,000 deposit.
The last deposit will be twenty years from now.
How much will accumulate into the savings fund by
the time the last deposit is made?
___________________________

What is the present value of the cash flows today?
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© 2002 David A. Stangeland
Be careful of the wording of when a
cash flow occurs
 A cash flow occurs at the
end of the third period.
 A cash flow occurs at
time period three.
 A cash flow occurs at the
beginning of the fourth
period.
 Each of the above
statements refers to the
same point in time!
0
1
2
3
4
C
If in doubt, draw a time line.
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© 2002 David A. Stangeland
Example
 What is the value at the end of the 12th
year of $100,000 that is invested at the
beginning of the 5th year?
__________________________
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© 2002 David A. Stangeland
II. Dealing with periods other than years
 Definition: Effective interest rates are
returns with interest compounded once
over the period of quotation. Examples:
 10% per year compounded yearly
 0.5% per month compounded monthly
 PV and FV Calculations for a single cash
flow
 As long as you have an effective interest rate
there is only one thing to ensure: set the
number of periods for PV or FV calculation in the
same units as the effective rate’s period of
quotation.
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© 2002 David A. Stangeland
Examples
 You expect to receive $50,000 in 90 days. What is the
PV if your relevant opportunity cost of capital is an
effective rate of 6% per year?
 Note if the you are told it is an effective rate of 6%
per year, then this implies 6% per year compounded
yearly.
 You have just invested $100,000 and expect your
return to be 4% per quarter compounded quarterly.
How much do you expect to accumulate after 5 years?
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© 2002 David A. Stangeland
Annuities and perpetuities
 The annuity and perpetuity formulae require the rate
used to be an effective rate and, in particular, the
effective rate must be quoted over the same time
period as the time between cash flows. In effect:
 If cash flows are yearly, use an effective rate per
year
 If cash flows are monthly, use an effective rate per
month
 If cash flows are every 14 days, use an effective rate
per 14 days
 If cash flows are daily, use an effective rate per day
 If cash flows are every 5 years, use an effective rate
per 5 years.
 Etc.
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© 2002 David A. Stangeland
Example
 You are obtaining a car loan from your bank and the loan
will be repaid in 5 years of monthly payments beginning
in one month. The amount borrowed is $20,000. Given
the rate that the bank quoted, you have determined the
effective monthly interest rate to be 0.75%. What are
your monthly payments?
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© 2002 David A. Stangeland
III. Understanding Interest Rate
Quotes and Conversions
 The TVM formulae we have used all require
rates that are effective. Unfortunately, rates
are rarely quoted in a way that we can
input, as is, into our TVM formulae or
calculator functions.
 Thus we must be competent in converting
between the rates that are quoted to us
and the equivalent rates that are necessary
for our calculations.
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© 2002 David A. Stangeland
Interest Rate Conversions
Step 1: finding the implied effective rate
 Identify how the rate is quoted and, if not an effective
rate, convert into the implied effective rate. Examples:
 10% per year compounded yearly
 This rate is already effective, so there is nothing to do
for step 1.
 60% per year compounded monthly
 This rate is not effective, but it implies – by definition –
an effective rate of 5% per month
 Note: the quoted rate of 60% per year with monthly
compounding is compounded 12 times per the
quotation period of one year. Thus the implied effective
rate is 60% ÷ 12 = 5% and this implied effective rate is
over a period of one year ÷ 12 = one month.
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© 2002 David A. Stangeland
Step 1: finding the implied effective rate
 In words, step 1 can be described as follows:
 Take both the quoted rate and its quotation
period and divide by the compounding
frequency to get the implied effective rate
and the implied effective rate’s quotation
period.
 The quoted rate of 60% per year with monthly
compounding is compounded 12 times per the
quotation period of one year. Thus the implied
effective rate is 60% ÷ 12 = 5% and this implied
effective rate is over a period of one year ÷ 12 = one
month.
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© 2002 David A. Stangeland
Step 1: additional examples to find
the implied effective rate
 16% per year compounded quarterly
 9% per year compounded semi-annually
 11% per year compounded bi-yearly (every
two years)
 100% per decade compounded every 10
years
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© 2002 David A. Stangeland
Step 2: Converting to the desired
effective rate
 Example: if you are doing loan
calculations with quarterly payments,
then the annuity formula requires an
effective rate per quarter.
 Once we have done step 1, if our
implied effective rate is not our
desired effective rate, then we need
to convert to our desired effective
rate.
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© 2002 David A. Stangeland
Step 2: continued
Converting between equivalent effective rates
 Use the example of 60% per year compounded monthly
and the implied effective rate of 5% per month . . . we
need an effective rate per quarter. Consider how $1
grows after 3 months . . .
Month:
Quarter:
$1
x 1.05
1
2
3 months
1 quarter
$1.05
$1.1025
$1.157625
x 1.05
x 1.05
x 1.157625
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© 2002 David A. Stangeland
Step 2: continued
Effective to effective conversion


In the previous example, 5% per month is equivalent to
15.7625% per 3 months (or quarter year). This result is due to
the fact that (1+.05)3=1.157625
As a formula this can be represented as
Ld
Ld
Lg
Lg
g
d
d
g
(1  r )


 (1  r )
or
r  (1  r )
1
where rg is the given effective rate, rd is the desired effective
rate.
Lg is the quotation period of the given rate and Ld is the
quotation period of the desired rate, thus Ld/Lg is the length of
the desired quotation period in terms of the given quotation
period.
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© 2002 David A. Stangeland
Step 2: additional examples to find
the desired effective rate
 9% per year compounded semi-annually; from step one
this gives us 4.5% per six months (effective rate).
 Suppose we desire an equivalent effective rate per
month
 Suppose we desire an equivalent effective rate per year
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© 2002 David A. Stangeland
Step 3?
 For the purpose of doing TVM calculations,
generally we are ready after doing steps 1
and 2 as we have obtained our desired
effective rate and can now use it in the TVM
formulae.
 Unfortunately, there are some
circumstances when we desire a final rate
quoted in a manner that is not effective –
here a third step is necessary.
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© 2002 David A. Stangeland
Step 3: finding the final quoted rate

Identify how the final rate is to be quoted and, if not an
effective rate, convert from the desired effective rate
(determined in step 2) into the desired quoted rate. Examples:
 Desired rate is to be quoted as a rate per quarter
compounded quarterly


This rate is already effective and was determined in step 2
(where, using a previous example, we calculated 15.7625%
per quarter), so there is nothing to do for step 3.
Desired rate is to be quoted as a rate per year compounded
quarterly


This rate is not effective, but 15.7625% per quarter (from step
2) implies a desired quoted rate per year compounded
quarterly of 63.05%
Note: the desired quoted rate is quoted per year with quarterly
compounding; i.e., compounded 4 times per the quotation
period of one year. Thus the desired quoted rate is 15.7625%
per quarter x 4 = 63.05% quoted over one year (= 4 x one
quarter of a year) compounded quarterly.
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© 2002 David A. Stangeland
Step 3: finding the final quoted rate
 In words, step 3 can be described as follows:
 Take both the implied effective rate and its
quotation period and multiply by the
compounding frequency of the desired final
quoted rate. This results in the desired final
quoted rate and its quotation period.
 In our example, the desired quoted rate is a rate
per year compounded quarterly. Therefore the
compounding frequency is 4. We multiply
15.7625% per quarter by 4 to get 63.05% per
year compounded quarterly.
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© 2002 David A. Stangeland
Step 3: additional examples
 Given an effective rate of 15.7625%
per quarter, find the following:
 The rate per
quarterly
 The rate per
quarterly
 The rate per
quarterly
 The rate per
quarterly
six months compounded
2 years compounded
month compounded
1.5 months compounded
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© 2002 David A. Stangeland
Interest rate conversions:
additional examples
 Bank of Montreal is offering car loans at 8% per
year compounded monthly. You manage Catfish
Credit Union where rates are quoted as “per year
compounded semiannually”. What is the most you
could quote to remain competitive with Bank of
Montreal?
 Step 1:
 Note: since your final quoted rate will be
compounded semiannually, you would like to (in
step 2) convert the B of M rate into an effective
rate per 6 months. So step 2 depends on the
desired outcome in step 3!
 Step 3
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© 2002 David A. Stangeland
Interest rate conversions – continuous
compounding – self study
 Consider steps 1 and 2 combined together in a formula
to convert a quoted rate per period compounded m
times into an effective rate over the same quotation
period …
m
 rquoted 
1 
  1  reffective
m 


Do not use this formula.
Use the 3-step method
shown in the prior slides as
that method works generally
and this formula only works
in one special situation.
Note: this formula only handles steps 1 and 2 when the final effective rate has
the same quotation period as the initial quoted rate. This formula is not
recommended as it does NOT work in most situations and is only shown because
of the derivation that follows.
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© 2002 David A. Stangeland
Continuous Compounding – self
study (continued)
Using the previous formula and mathematical limits …
As m  , 1  reffective  e q u o ted
r
As m  , rquoted is said to be the
continuous ly compounded rate of interest
To convert in the other direction ...
from reffective per period to rper period with continuouscompounding ,
rper period with continuouscompounding  ln(1  reffective per period )
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© 2002 David A. Stangeland
Continuous Compounding – self
study – examples to try
 What is the effective annual rate, given a
quoted rate of 20% per year with…




Monthly compounding
Daily compounding
Compounding every hour
Continuous compounding
answer=21.939108%
answer=22.133586%
answer=22.139997%
answer=22.140276%
 What is the rate per year compounded
continuously if the effective annual rate is …
 10%
 50%
 100%
answer=9.531018%
answer=40.54651%
answer=69.31472%
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© 2002 David A. Stangeland
IV. Applications of TVM
Quotations on mortgages
Quotations on bonds
Quotations on credit cards
Quotations on personal loans and car
loans
 Mortgage and loan amortizations




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© 2002 David A. Stangeland
Canadian Mortgage Quotes
 Canadian mortgages are quoted as rates per year
compounded semiannually. In this course, unless
otherwise noted, assume all mortgage quotes are quoted
in the above manner.

(Note, some of the text problems may not make this
assumption, but all class assignments and exams will make
this assumption unless otherwise noted).
 Normally a constant series of monthly payments is
required to repay the mortgage. What interest rate is
required to do TVM calculations for the mortgage if the
quoted rate is 6%?
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© 2002 David A. Stangeland
Bond Yields
 A bond’s yield is essentially the IRR of the bond and is
quoted as a rate per year compounded semiannually. In
this course, unless otherwise noted, assume bond yields
are quoted in the above manner.

(Note, some of the text problems may not make this assumption,
but all class assignments and exams will make this assumption
unless otherwise noted).
 Most corporate and government bonds have constant
semiannual coupon payments and a lump sum terminal
payment. What interest rate is required to do TVM
calculations on the bond coupons if the yield is quoted as
8%?
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© 2002 David A. Stangeland
Credit Cards
 CIBC Visa quotes the annual interest rate of 19.50% and
the daily interest rate of 0.05342%. How are the two
rates quoted? What is the effective rate per year charged
by CIBC Visa?
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© 2002 David A. Stangeland
Personal Loans and Car Loans
 Most banks quote the interest rates on personal
loans and car loans as rates per year compounded
monthly.
 Since personal loans and car loans generally
require equal monthly payments, what interest rate
would be used in TVM formulae if the quoted rate
was 12%?
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© 2002 David A. Stangeland
Mortgage and loan amortizations






A mortgage contract specifies the quoted rate and the
amortization period for the payments. The amortization
period is often longer than the duration of the contract.
Thus we must determine the payments, the amount of
interest and principal paid each month, and the outstanding
principal at the end of the contract.
Example: You have just negotiated a 5 year mortgage on
$100,000 amortized over 30 years at a rate of 8%.
What are the monthly payments?
What are the principal and interest payments each month
for the first 3 months of payments?
How much will be left at the end of the 5 year contract?
If the mortgage terms do not change over then entire
amortization period, how much interest and principal
reduction result from the 300th payment?
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© 2002 David A. Stangeland
Mortgage example
 First determine the relevant effective rate for TVM
calculations.
 Next determine the monthly payment.
 Now utilize the table on the next page to understand
how a mortgage amortization schedule works.
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© 2002 David A. Stangeland
Mortgage amortization schedule
Column:
A
B
C
D
Month
Principal outstanding at the beginning
of the month
Interest charged during the
month
Monthly payment
Principal reduction with
monthly payment
= A • r%
= E0 ÷ anr%
=C-B
$655.82
$655.37
$654.91
$654.46
$724.71
$724.71
$724.71
$724.71
$68.89
$69.34
$69.80
$70.26
0
1
2
3
4
$100,000.00
$99,931.11
$99,861.77
$99,791.97
E
Principal outstanding at the end
of the month (after the
payment)
=A-D
$100,000.00
$99,931.11
$99,861.77
$99,791.97
$99,721.71
Note: as time goes by, the principal outstanding is reduced and therefore the interest charge per
month drops. This results in more of the monthly payment going toward principal reduction as
time elapses. The way the annuity payments are calculated, the last payment will have just
enough principal reduction to repay the remaining principal owed and then the loan will be
repaid.
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© 2002 David A. Stangeland
Mortgage continued
 How much will be left at the end of the 5-year contract?

After 5 years of payments (60 payments) there are 300
payments remaining in the amortization. The principal
remaining outstanding is just the present value of the
remaining payments.
 How much interest and principal reduction result from
the 300th payment?

When the 299th payment is made, there are 61 payments
remaining. The PV of the remaining 61 payments is the
principal outstanding at the beginning of the 300th period
and this can be used to calculate the interest charge which
can then be used to calculate the principal reduction.
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© 2002 David A. Stangeland
Summary and conclusions





Cash flows that occur in different time periods cannot be added
together unless they are brought to one common time period.
We usually use PV to do this and sometimes FV.
PV and FV calculations were done for single cash flows,
constant annuities and perpetuities and growing annuities and
perpetuities. In addition, we used the concepts of NPV and
IRR.
For annuities and perpetuities, we must ensure the discount
rate is effective and quoted over a period the same as the time
period between cash flows.
TVM principles are useful for analyzing consumption and
investment decisions. TVM principles are also useful for
working with loan and mortgage amortizations.
If you understand TVM principles, you do not need to blindly
rely on another party to determine value or interest costs. You
know what factors affect these and you can determine
reasonable numbers for yourself.
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© 2002 David A. Stangeland