Outline

1
Outline

1. More on the use of the financial calculator and
   warnings
2. Dealing with periods other than years
3. Understanding interest rate quotes and
   conversions
4. Applications mortgages, etc.

2
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 (1r)t-1
  to get PV0.
• Note this works even if t is a fraction.

3
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 ___________________
  . (r12)

4
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-t1 cash flows in the annuity.

5
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?

6
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 0 1 1 2 2 3 3 4 4


C C
If in doubt, draw a time line.
7
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? __________________________

8
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 rates period of quotation.

9
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?

10
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.

11
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?

12
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.

13
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.

14
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 rates 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.

15
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

16
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.

17
Step 2 continuedConverting 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 Month Quarter 1 1 2 2 3 months 1 quarter 3 months 1 quarter


1 1 1.05 1.05 1.1025 1.1025 1.157625 1.157625
x 1.05
x 1.05
x 1.05
x 1.157625
18
Step 2 continuedEffective 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)31.157625
• As a formula this can be represented as
• 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.

19
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

20
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.

21
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.

22
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.

23
Step 3 additional examples

• Given an effective rate of 15.7625 per quarter,
  find the following
• The rate per six months compounded quarterly
• The rate per 2 years compounded quarterly
• The rate per month compounded quarterly
• The rate per 1.5 months compounded quarterly

24
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

25
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
• 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.

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.
26
Continuous Compounding self study (continued)

• Using the previous formula and mathematical
  limits

27
Continuous Compounding self study examples to
try

• What is the effective annual rate, given a quoted
  rate of 20 per year with
• Monthly compounding answer21.939108
• Daily compounding answer22.133586
• Compounding every hour answer22.139997
• Continuous compounding answer22.140276
• What is the rate per year compounded continuously
  if the effective annual rate is
• 10 answer9.531018
• 50 answer40.54651
• 100 answer69.31472

28
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

29
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?

30
Bond Yields

• A bonds 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?

31
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?

32
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?

33
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?

34
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.

35
Mortgage amortization schedule
E
D
C
B
A
Column
Principal outstanding at the end of the month
(after the payment)
Principal reduction with monthly payment
Monthly payment
Interest charged during the month
Principal outstanding at the beginning of the
month
Month
A - D
C - B
E0 anr
A r
 
 
100,000.00
 
 
 
 
0
99,931.11
68.89
724.71
655.82
100,000.00
1
99,861.77
69.34
724.71
655.37
99,931.11
2
99,791.97
69.80
724.71
654.91
99,861.77
3
99,721.71
70.26
724.71
654.46
99,791.97
4
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.
36
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.

37
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.