Interest Rate Conversions

1
Supplement to Chapter 4
Interest Rate Conversions

• Definitions
• Length of Compounding period (n)
• How often (in years) is interest computed
  (calculated)?
• e.g. if interest is compounded
• ...annually -gt n 1
• semi-annually -gt n 1/2
• every decade -gt n 10
• Effective Annual Interest Rate (EAR)
• Actual percentage interest (simple
  compounded) charged
• during a year
• Stated (nominal) Annual Interest Rate (rn)
• Only simple percentage interest charged
  during a year
• (What the bank usually quotes)

2
Interest Rate Conversions
3
(No Transcript)
4
(No Transcript)
5
(No Transcript)
6
Interest Rate Conversions in your HP 10B
Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on example
2 (ii))
Yellow
C C ALL
When finished - dont forget to set your
calculator to 1P_Yr
15
Key in EAR
PV EFF
Yellow
Key in number of compounding periods per year
(1/n)
2
Yellow
PMT P/YR
Compute stated (nominal) annual rate compounded
s.a.
Display should show 14.47610590
I/YR NOM
Yellow
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
Canadian Mortgages

• What is special about Canadian Mortgages
• Canadian banks quote the annual interest
  compounded semi-annually for mortgages, although
  interest is calculated (compounded) every month
• The terms of the mortgage are usually
  renegotiated during the term of the mortgage. For
  example, the interest of my 25-year mortgage will
  be negotiated 5 years after the initiation of my
  mortgage.

12
Canadian Mortgages
Q. You have negotiated a 25-year, 100,000
mortgage at a rate of 7.4 per year
compounded semi-annually with the
Toronto-Dominion Bank. To answer most
mortgage questions, we first have to convert the
quoted (stated) annual interest compounded
semi-annually, to the actual interest rate
charged each month. That is, we need to calculate
the Effective monthly Period rate, EPR1/12. We
have r1/2 7.4. By equation (3)
EPR1/20.50.0740.037. Thus, using equation
(1), we get Equation (4) gives us EPR1/12
13
Calculating the Effective Monthly Mortgage Rate
Key in stated (nominal) annual rate compounded
semiannually
7.4
Yellow
I/YR NOM
Key in 2 compounding periods per year
2
Yellow
PMT P/YR
Display should show 7.5369
Compute EAR
Yellow
PV EFF
Key in 12 compounding periods per year
12
Yellow
PMT P/YR
Compute stated (nominal) annual rate compounded
monthly
Display should show 7.28843047
Yellow
I/YR NOM
Display should show 0.607369
Divide by 12 to get the Effective Monthly Rate

12

Store this result for future calculations
RCL STO
Yellow
1
14
We are now ready to solve mortgage problems Q-a
What is the monthly payment on the above
mortgage? A-a We have PV0 100,000,
EPR1/120.607369, and T2512300
months. Since a mortgage is an annuity with
equal monthly payments, we use the present value
of annuity formula Solving for PMT, we
get the monthly payment on the mortgage PMT
725.28
15
Mortgage Payments in your HP 10B Calculator
Set your calculator to 1 P/YR Input data
(based on above PV example)
Key in 1 compounding periods per year
1
Yellow
PMT P/YR
Recall Effective Monthly Rate
Display should show 0.607369
RCL
1
Key in this rate
I/YR
0.607369
Key in number of monthly periods
N
300
/-
PV
Key in PV
100,000
Compute monthly PMT
Display should show 725.28464244
PMT
16
Q-b How much of the first three mortgage payment,
goes toward principal and interest? A-b In
general, to calculate the interest portion of
each monthly payment, use EPR1/12(Balance of
Principal at the Beginning of Month) The
principal portion of each monthly payment is
given by PMT - Interest Payment For the first
three payments
17
Q-c Assuming that the mortgage rate remains at
7.4 for the remaining time of the mortgage,
what is the total amount of interest paid during
the 25-year period? A-c The total amount of
interest to be paid is given by Total
Payments - Total Principal Payments Since after
25 years the entire principal will be paid, the
value total principal payments is 100,000. With
300 monthly payments of 725.28 each, we get a
total amount of interest to be paid Total
Payments - Total Principal Payments
(300725.28) - 100,000 217,584 -
100,000 117,584
18
Q-d Assuming that the mortgage rate remains at
7.4 for the remaining time of the mortgage,
after you have paid two-thirds of your monthly
payments, what is the amount still remaining to
be paid on the mortgage? A-d Two-thirds of your
monthly payments will be paid right after the
200th payment. The remaining value of the
mortgage at that time is given by the present
value of the remaining 100 monthly payments. We
have EPR1/120.607369, PMT725.28 and T100
months. We use the present value of annuity
formula to get
19
Continuous Compounding

• EAR of continuous compounding er - 1
• The TVM Relationship for continuous compounding
• FVtT PVte (rT)
• where r is the stated annual continuously
• compounded interest rate
• Note r ln(1EAR)

20
Q. Your bank offers you a certificate which pays
you 100 at the end of each of the following
three years, carrying a stated annual
continuously compounded interests of 8. What is
the present value of this security? A. A Time
Line To find the PV, we use the present
value of annuity formula with an EAR of EAR e
r - 1 e 0.08 - 1 0.08328707 8.328707.
Time
0
1
2
3
r 8 c.c.
Cashflow
100
100
100