Mathematics of Finance with a Calculator

1
With a Financial Calculator
Mathematics of Finance
Presentation
2
Compound Value
Parameters

• Interest rate (i)
• Amount that is invested, present value (PV)
• Time money remains invested (n)
• Future value of the investment in n years (FVn)
• Periodic equal payment (or deposit) (PMT)

3
Compound Value
Future Value of a Lump Sum (one time payment)

• Value at some time in the future of an investment
• Interest compounds earn interest on interest in
  later years.
• Future value in one year is present value plus
  the interest that is earned over the year.

4
Compound Value
Future Value of a Lump Sum (one time payment)

• In General

FVn PV(1 i)n
5
Compound Value
Present Value of a Lump Sum (one time payment)

• Value today of an amount to be received or paid
  in the future.

Example Expect to receive 100 in eight years.
If can invest at 10, what is it worth today?
6
Compound Value
Present Value of a Lump Sum (one time payment)

• Value today of an amount to be received or paid
  in the future.

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
0 1 2 3 4 5
6 7 8
?
100
7
Compound Value
Present Value of a Lump Sum (one time payment)

• Value today of an amount to be received or paid
  in the future.

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
0 1 2 3 4 5
6 7 8
?
100
8
Financial Calculator
Setting Display

• Should show at least 2 decimal places on dollar
  amounts and 4 decimal places on percentages

HP10B Calculator
9
Financial Calculator
Setting Display

• Should show at least 2 decimal places on dollar
  amounts and 4 decimal places on percentages

1
HP10B Calculator
10
Financial Calculator
Setting Display

• Should show at least 2 decimal places on dollar
  amounts and 4 decimal places on percentages

1
2
HP10B Calculator
11
Financial Calculator
Setting Display

• Should show at least 2 decimal places on dollar
  amounts and 4 decimal places on percentages

3
1
2
HP10B Calculator
12
Financial Calculator
Clearing Memory

• Financial calculators contain a number of memory
  registers. These registers should be cleared to
  prevent carry-over errors.

HP10B Calculator
13
Financial Calculator
Clearing Memory

• Financial calculators contain a number of memory
  registers. These registers should be cleared to
  prevent carry-over errors.

1
HP10B Calculator
14
Financial Calculator
Clearing Memory

• Financial calculators contain a number of memory
  registers. These registers should be cleared to
  prevent carry-over errors.

2
1
HP10B Calculator
15
Financial Calculator
Setting Compounding Frequency

• Compounding should be set to annual, i.e. P/YR1,
  not the factory setting of 12.

HP10B Calculator
16
Financial Calculator
Setting Compounding Frequency

• Compounding should be set to annual, i.e. P/YR1,
  not the factory setting of 12.

1
HP10B Calculator
17
Financial Calculator
Setting Compounding Frequency

• Compounding should be set to annual, i.e. P/YR1,
  not the factory setting of 12.

1
2
HP10B Calculator
18
Financial Calculator
Setting Compounding Frequency

• Compounding should be set to annual, i.e. P/YR1,
  not the factory setting of 12.

3
1
2
HP10B Calculator
19
Financial Calculator
Setting Compounding Frequency

• Compounding should be set to annual, i.e. P/YR1,
  not the factory setting of 12.
• To check setting CLEAR the calculator (holding
  down the CLEAR ALL key)

2
1
HP10B Calculator
20
Financial Calculator Solution
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
Using Formula
100 (1.1)8
46.65
PV
21
Financial Calculator Solution
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
8.0000
N
I/YR
PV
PMT
FV
8
22
Financial Calculator Solution
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
10.000
N
I/YR
PV
PMT
FV
Enter the Interest Rate as a WHOLE
8 10
23
Financial Calculator Solution
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
100.0000
N
I/YR
PV
PMT
FV
100
8 10
24
Financial Calculator Solution
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
- 46.65
N
I/YR
PV
PMT
FV
100
8 10 ?
25
Compound Value
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
Additional Calculator Notes
- 46.65
Can change any or all parameters without
reentering others
100
8 10
26
Compound Value
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
Additional Calculator Notes
5.0000
Can change any or all parameters without
reentering others
N
I/YR
PV
PMT
FV
Change Interest rate to 5
100
8 10
5
27
Compound Value
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
Additional Calculator Notes
- 67.68
Can change any or all parameters without
reentering others
N
I/YR
PV
PMT
FV
Change Interest rate to 5
100
8 10 ?
5
28
Compound Value
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
Additional Calculator Notes
Can check the number entered in each memory
location using the recall (RCL) key.
RCL
29
Compound Value
Present Value of a Lump Sum (one time payment)

• Previous Example

Example Expect to receive 100 in EIGHT years.
If can invest at 10, what is it worth today?
?
100
Additional Calculator Notes
8.0000
Can check the number entered in each memory
location using the recall (RCL) key.
N
I/YR
PV
PMT
FV
Check setting for years
RCL
30
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

31
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
32
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
33
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
or FVn PV(1 i)n
34
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
or FVn PV(1 i)n
2
35
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
When Entering inflows and outflows of cash, enter
as follows (-) cash outflow () cash inflow
or FVn PV(1 i)n
36
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
When Entering inflows and outflows of cash, enter
as follows (-) cash outflow () cash inflow
200.00
or FVn PV(1 i)n
N
I/YR
PV
PMT
FV
2
-200
37
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
230.00
or FVn PV(1 i)n
N
I/YR
PV
PMT
FV
2
-200
230
38
Compound Value
Solve for other parameters (I/YR)

• Given any three of the following PV, FV, i and
  n, the fourth can be computed.

Example A 200 investment has grown to 230 over
two years. What is the ANNUAL return on this
investment?
200
230
7.24
or FVn PV(1 i)n
N
I/YR
PV
PMT
FV
2
-200
230
?
39
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
40
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
0 1 N
300
500
41
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
0 1 N
300
500
42
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
0 1 N
300
500
300.00
N
I/YR
PV
PMT
FV
-300
43
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
0 1 N
300
500
500.00
N
I/YR
PV
PMT
FV
-300
500
44
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
0 1 N
300
500
6.00
N
I/YR
PV
PMT
FV
-300
500
6
45
Compound Value
Solve for other parameters (N)

• Given any three of the following PV, FV, i and
  n, the forth can be computed.

Example How long will it take for a 300
investment to grow to 500 if 6 annual interest
is earned?
0 1 N
300
500
8.77
N
I/YR
PV
PMT
FV
?
-300
500
6
46
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.

47
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.

Example Deposit 1,000 at 10 nominal annual
interest rate. How much will you have at end of 1
year?
ANNUAL COMPOUNDING
1,000
1,000(1.1)
1,100
SEMI-ANNUAL COMPOUNDING
1,000
48
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.

Example Deposit 1,000 at 10 nominal annual
interest rate. How much will you have at end of 1
year?
ANNUAL COMPOUNDING
1,000
1,000(1.1)
1,100
Earn 10/25 each compounding period
SEMI-ANNUAL COMPOUNDING
1,000
1,000(1.05)
1,050
49
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.

Example Deposit 1,000 at 10 nominal annual
interest rate. How much will you have at end of 1
year?
ANNUAL COMPOUNDING
1,000
1,000(1.1)
1,100
Earn 10/25 each compounding period
SEMI-ANNUAL COMPOUNDING
1,000
1,000(1.05)
1,050
1,050(1.05)
1,102.50
50
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.
• When compounding more than once a year, must
  adjust formula

m of compounding periods in a year
i m
FVn PV(1 )mn
51
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.
• When compounding more than once a year, must
  adjust formula

m of compounding periods in a year
i m
FVn PV(1 )mn
Example Deposit 1,800 at 8 nominal annual
interest rate, compounded quarterly. How much
will you have at end of 3 years?
52
Financial Calculator Solutions
Setting Compounding Frequency

• Calculator makes adjustments for differing
  compounding periods based on the setting of P/YR
• For Quarterly compounding set P/YR 4

xP/YR
3
1
2
HP10B Calculator
53
Financial Calculator Solutions
Setting Compounding Frequency

• Calculator makes adjustments for differing
  compounding periods based on the setting of P/YR
• For Quarterly compounding set P/YR 4
• I/YR/YR is automatically adjusted by the P/YR
  setting.

xP/YR
HP10B Calculator
54
Financial Calculator Solutions
Setting Compounding Frequency

• Calculator makes adjustments for differing
  compounding periods based on the setting of P/YR
• For Quarterly compounding set P/YR 4
• I/YR/YR is automatically adjusted by the P/YR
  setting.
• To adjust N by P/YR enter the number of years on
  the xP/YR key.

3
2
1
HP10B Calculator
55
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.
• When compounding more than once a year, must
  adjust formula

m of compounding periods in a year
i m
FVn PV(1 )mn
Example Deposit 1,800 at 8 nominal annual
interest rate, compounded quarterly. How much
will you have at end of 3 years?
P/Yr 4
Enter Years using Shift xP/YR combination
P/YR
xP/YR
3
56
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.
• When compounding more than once a year, must
  adjust formula

m of compounding periods in a year
i m
FVn PV(1 )mn
Example Deposit 1,800 at 8 nominal annual
interest rate, compounded quarterly. How much
will you have at end of 3 years?
P/Yr 4
P/YR
xP/YR
3
8
57
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.
• When compounding more than once a year, must
  adjust formula

m of compounding periods in a year
i m
FVn PV(1 )mn
Example Deposit 1,800 at 8 nominal annual
interest rate, compounded quarterly. How much
will you have at end of 3 years?
58
Non-Annual Compounding

• All equations and calculator solutions thus far
  have assumed compounding occurs ONCE a year.
• When compounding more than once a year, must
  adjust formula

m of compounding periods in a year
i m
FVn PV(1 )mn
Example Deposit 1,800 at 8 nominal annual
interest rate, compounded quarterly. How much
will you have at end of 3 years?
P/Yr 4
2,282.84
P/YR
xP/YR
N
I/YR
PV
PMT
FV
3
8
-1800
?
59
Financial Calculator Solutions
Automatic
Alternative Settings
P/Yr 4
2,282.84

• Calculator make compounding adjustments
  automatically based on P/YR setting.

P/YR
xP/YR
N
I/YR
PV
PMT
FV
3
8
-1800
?
60
Financial Calculator Solutions
Automatic
Alternative Settings
P/Yr 4
2,282.84

• Calculator make compounding adjustments
  automatically based on P/YR setting.
• You can keep P/YR1 and make the adjustments to N
  and I/YR manually.
• Advantage should never need to change P/YR,
  therefore fewer errors on later problems.
• If change P/YR, always change back to 1 P/YR
  after doing problem.

P/YR
xP/YR
N
I/YR
PV
PMT
FV
3
8
-1800
?
Manual
P/Yr 1
2,282.84
P/YR
N
I/YR
PV
PMT
FV
12
2
-1800
?
61
Future Value of an Annuity

• Annuity- string of deposits with constant value
  and fixed interval.

0 1 2 3
0
100
100
100
Compute FV3
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
62
Future Value of an Annuity

• Annuity- string of deposits with constant value
  and fixed interval.

0 1 2 3
0
100
100
100
Compute FV3
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
3.00
N
I/YR
PV
PMT
FV
3
63
Future Value of an Annuity

• Annuity- string of deposits with constant value
  and fixed interval.

0 1 2 3
0
100
100
100
Compute FV3
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
8.00
N
I/YR
PV
PMT
FV
3
8
64
Future Value of an Annuity

• Annuity- string of deposits with constant value
  and fixed interval.

0 1 2 3
0
100
100
100
Compute FV3
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
100.00
N
I/YR
PV
PMT
FV
3
8
-100
65
Future Value of an Annuity

• Annuity- string of deposits with constant value
  and fixed interval.

0 1 2 3
0
100
100
100
Compute FV3
How much would this account have in it at the end
of 3 years if interest were earned at a rate of
8 annually?
324.64
NOTE PV 0 since the cashflow in time period 0
0
N
I/YR
PV
PMT
FV
3
8
-100
?
66
Future Value of an Annuity
Example

• Susan is able to save 980/yr for retirement. She
  makes these deposits at the end of each year. If
  she invests her savings at 12 compounded
  annually, how much will she have upon retirement
  in 45 years?

67
Future Value of an Annuity
Example

• Susan is able to save 980/yr for retirement. She
  makes these deposits at the end of each year. If
  she invests her savings at 12 compounded
  annually, how much will she have upon retirement
  in 45 years?

0 1 2 3
44 45
980
980
980
980
980
68
Future Value of an Annuity
Example

• Susan is able to save 980/yr for retirement. She
  makes these deposits at the end of each year. If
  she invests her savings at 12 compounded
  annually, how much will she have upon retirement
  in 45 years?

0 1 2 3
44 45
980
980
980
980
980
69
Future Value of an Annuity
Example 1a

• Susan will make equal quarterly payments totaling
  980/yr for retirement. She makes these deposits
  at the end of each quarter. If she invests her
  savings at 12 compounded quarterly, how much
  will she have upon retirement in 45 years?

0 1 2 45
P/Yr 1
245
70
Present Value of an Annuity

• How much would the following cash flows be worth
  to you today if you could earn 8 on your
  deposits?

0 1 2 3
0
100
100
100
71
Present Value of an Annuity

• How much would the following cash flows be worth
  to you today if you could earn 8 on your
  deposits?

0 1 2 3
0
100
100
100
100/(1.08)
92.60
100 / (1.08)2
85.73
100 / (1.08)3
79.38
257.71
257.71
N
I/YR
PV
PMT
FV
3 8 ? -100
72
Present Value of an Annuity
Loan Amortization

• Borrow 1,000 today, how much would the annual
  payments be if you are required to repay in two
  years and the interest rate is 10?

73
Present Value of an Annuity
Example 1a

• Bob borrows 5,000 from his children to purchase
  a used car. He agrees to make payments at the end
  of each month for the next 5 years. If the
  interest rate on this loan is 6, what is the
  amount of the payments?

74
Present Value of an Annuity
Example 1a

• Bob borrows 5,000 from his children to purchase
  a used car. He agrees to make payments at the end
  of each month for the next 5 years. If the
  interest rate on this loan is 6, what is the
  amount of the payments?

0 1 5
5,000
75
Present Value of an Annuity
Example 1a

• Bob borrows 5,000 from his children to purchase
  a used car. He agrees to make payments at the end
  of each month for the next 5 years. If the
  interest rate on this loan is 6, what is the
  amount of the payments?

0 1 5
5,000
96.66
I/YR
PV
PMT
FV
N
60 0.5 5,000 ?
76
Present Value of an Annuity
Example 1a

• Bob borrows 5,000 from his children to purchase
  a used car. He agrees to make payments at the end
  of each month for the next 5 years. If the
  interest rate on this loan is 6, what is the
  amount of the payments?

0 1 5
5,000
96.66
I/YR
PV
PMT
FV
N
60 0.5 5,000 ?
77
Annuity Due
Two Types of Annuities

• Ordinary Annuity - Payments (or deposits) occur
  at the end of the period

FV 205
0
100
100

• Annuity Due - Payments (or deposits) occur at the
  beginning of the period

100
100
FV ?
Each payment (or deposit) for an annuity due
earns one additional period interest.
78
Annuity Due
Solving Annuity Due

• Annuity Due - Payments (or deposits) occur at the
  beginning of the period

100
100
FV ?
FV AD FV (ordinary) (1i)
PV AD PV (ordinary) (1i)
79
215.25
BEGIN
BEGIN
N
I/YR
PV
PMT
FV
2 5 100 ?
1
2
80
Additional Problems
Problem 1

• Compute the monthly payments on a 30 year
  mortgage for a 120,000 loan at 8 annual
  interest, compounded monthly.

81
Additional Problems
Problem 2

• You have determined that your budget will only
  allow you to make a 700 monthly mortgage
  payment. If interest rates are currently 6 and
  mortgage terms are typically 30 years, what price
  range home should you be searching for if your
  downpayment is 15,000?