Lesson 7
A dollar today is worth more than a dollar tomorrow.
Which would you prefer, a $1,000 today or a $1,000 a
year from now?
$1,000 today!
Lesson 7
Take the money and invest it.
It's a sure thing.
The Time Value of Money
Inflation can negatively affect the value of money
over time.
Money today is worth more than the same amount of
money tomorrow.
That's the time value of money!
1
2
A financial calculator is required for this course.
Hewlett-Packard, HP10bii
Texas Instruments, TI BA II Plus
How much is a $1,000 a year from now
actually worth in today's dollars?
Any calculator that has these keys noted here,
I
N
or
PV
PMT
FV
CFj
and is capable of calculating internal rates of return on
uneven cash flows and will probably get the job done.
How much would you be willing to pay
today for the right to receive $1,000 a
year from now?
In this lesson, all of the examples that we'll be working on
will refer to keystrokes used on the HP (10bii) calculator
and all of the solutions to homework problems will include
both the HP (10bii) and the TI (BA II Plus) keystrokes.
4
3
Set decimal places on the calculator's display:
Clear memory:
1
0
How much would you be willing to
pay today for the right to receive
$1,000 a year from now?
1,000
12
DISP
2
C ALL
N
PMT
FV
: Number of periods involved.
: Annuity payment (a series of two or more
payments made in equal amounts over equal
intervals of time)
: Future value.
I/YR : Interest rate per year
(The present value of a single cash flow or lump
sum of $1,000.)
6
5
7-1
Set decimal places on the calculator's display:
Clear memory:
1
0
Compounding means that the amount of interest
earned during each compounding period is based
on both the amount of the original investment and
any previously earned but unpaid interest to date.
In effect, it means that interest is earned on interest.
1,000
12
1
2
C ALL
: Number of periods involved.
N
PMT
: Annuity payment (a series of two or more
payments made in equal amounts over equal
intervals of time)
: Future value.
FV
I/YR : Interest rate per year
P/YR
: Reset to reflect annual compounding
: Present value.
PV
In this case, we'll calculate the present value of the
$1,000 receivable in one year using 12 percent
interest, compounding annually.
DISP
-892.86
An investment of $892.86 today grows to $1,000 at the end of a year at an
interest rate 12% compounding annually.
$892.86 + (12% x $892.86 x 1 yr)
$892.86 +
$107.14
= $1,000
That $1,000 total could also be characterized as the future value in a year of
an $892.86 single cash outflow today at a rate of 12% compounding annually.
7
Clear memory:
1
0
892.86
+/-
12
1
Determine the future value of the same $892.86 for one year at an interest rate of
12%, but instead of annual compounding, let's assume interest compounds quarterly.
C ALL
: Number of periods involved.
N
PMT
Clear memory:
: Annuity payment (a series of two or more
payments made in equal amounts over equal
intervals of time)
892.86
: Present value.
PV
4
0
+/-
12
I/YR : Interest rate per year
P/YR
8
4
: Reset to reflect annual compounding
PMT
PV
: Number of compounding periods
: Annuity payment
: Present value.
I/YR : Interest rate
P/YR
FV
: Future value.
FV
C ALL
N
: Reset compounding periods per year.
: Future value
1,004.92
1,000.00
Verify compounding mathematically:
$892.86 + (12% x $892.86 x 3/12 yr.) = $ 919.65 1st Quarter
$919.65 + (12% x $919.65 x 3/12 yr.) = $ 947.24 2nd Quarter
$947.24 + (12% x $947.24 x 3/12 yr.) = $ 975.65 3rd Quarter
$975.65 + (12% x $975.65 x 3/12 yr.) = $1,004.92 4th Quarter
Total interest earned on this investment for the entire year:
$1,004.92 - $892.86 = $112.06
$112.06 $892.86 = 12.55% effective rate or APR*
* Annual Percentage Rate
In summary, $892.86 of cash invested today at an interest rate of 12%
compounding annually will grow to $1,000 at the end of a year.
10
9
How much would a person have to invest today in an account that earns 5%
interest compounding monthly, if they wished to accumulate $50,000 at the
start of their daughter's college education in 4½ years.
What would the effective interest rate or APR have been in the previous
example if the12% interest rate had compounded daily rather than quarterly?
Clear memory:
365
0
892.86
+/-
12
365
C ALL
PMT
54
0
50,000
5
12
: Annuity payment
: Present value.
PV
I/YR
P/YR
FV
Clear memory:
: Number of compounding periods (total).
N
: Interest rate.
: Reset compounding periods per year.
: Future value.
C ALL
N
PMT
: Annuity payment.
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Reset compounding periods per year.
PV
1,006.68
: Number of compounding periods (total).
: Present value.
-39,944.48
Interest earned on this investment:
$113.82
($1,006.68 - $892.86)
$113.82
= 12.78% APR
$892.86
12
11
7-2
Determine the interest rate that would be required to double a $1,000
investment in 5 years, assuming interest compounds annually.
How long would it take to double our money if the best investment we could
find produced a 10% return, compounding monthly?
Clear memory:
Clear memory:
5
0
2,000
1
1,000
C ALL
: Number of compounding periods (total).
N
N
: Annuity payment.
PMT
I/YR
: Interest rate. 14.87%
P/YR
: Reset compounding periods per year.
12
1,000
: Present value.
PV
: Number of compounding periods (total).
83.52
0
2,000
10
: Future value.
FV
+/-
C ALL
+/-
PMT
12 = 6.97 years
: Annuity payment.
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Reset compounding periods per year.
PV
: Present value.
13
14
Problem 7-1
Problem 7-1 - Answer
Calculations Using Single Cash Flows
Calculations Using Single Cash Flows
Respond to each of the following:
A. Determine the present value of a single future cash flow of $10,000, due
in 20 years at 7% compounding semi-annually.
A. Determine the present value of a single future cash flow of $10,000, due
in 20 years at 7% compounding semi-annually.
Answer: $2,525.72 (The negative sign is ignored in this case.)
B. If an investment account is opened with a deposit of $1,000, how much
will that account be worth in 30 years assuming an expected return on
investment of 12% compounding annually?
HP10bii:
C ALL
40
0
10,000
7
2
C. What rate of return, compounding monthly, would have to be earned
on a $100,000 investment in order to accumulate $1 million in 30 years?
D. How many years would it take to accumulate $1,000,000 on a $100,000
investment, assuming an 8% return compounding monthly?
E. Compute the future value of $100,000 in 30 years at 10% compounding
daily (ignore the effect of leap years).
Press
F. An investor is considering the purchase of a 5-year, $20,000 note
receivable, which bears interest, all due at maturity, at a rate of 8%
compounding annually. If the investor were to buy the note at a time
when there are four years left to maturity, how much would the
investor pay to achieve a 12% rate of return, compounding quarterly?
N
PMT
: Clear memory.
: Number of compounding periods
: Annuity payment.
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Reset compounding periods per year.
PV
: Present value.
16
15
Problem 7-1 - Answer
Problem 7-1 - Answer
A. Determine the present value of a single future cash flow of $10,000, due
in 20 years at 7% compounding semi-annually.
A. Determine the present value of a single future cash flow of $10,000, due
in 20 years at 7% compounding semi-annually.
Answer: $2,525.72 (The negative sign is ignored in this case.)
Answer: $2,525.72 (The negative sign is ignored in this case.)
: Number of compounding periods.
** Note for users of other calculators: Some financial calculators are set to
one compounding period per year and don't allow modification of that
setting. That is easily overcome by always inputting the interest rate as
the interest rate per compounding period, rather the annual interest rate
: Annuity payment.
Other calculators:
TI BAII Plus:
C/CE
2nd
P/Y
2nd
CLR TVM
40
0
10,000
7
2 ENTER
CPT
N
PMT
: Clear all Time-Value-of-Money values.
FV
: Future value.
I/Y
: Interest rate.
C/CE
PV
(See manual) : Clear memory:
40
0
10,000
3.5
: Reset compounding periods per year.
: Present value.
N
PMT
: Number of compounding periods
: Annuity payment.
FV
: Future value.
I
: Interest rate.
PV
: Present value. (See manual for compute function.)
18
17
7-3
Problem 7-1 - Answer
Problem 7-1 - Answer
B. If an investment account is opened with a deposit of $1,000, how much
will that account be worth in 30 years assuming an expected return on
investment of 12% compounding annually?
B. If an investment account is opened with a deposit of $1,000, how much
will that account be worth in 30 years assuming an expected return on
investment of 12% compounding annually?
Answer: $29,959.92
Answer: $29,959.92
HP10bii:
TI BAII Plus:
C ALL
1,000
: Clear memory:
PMT
+/-
PV
12
I/YR
: Interest rate.
P/YR
: Reset compounding periods per year.
1
Press
FV
PMT
+/-
PV
: Present value.
12
I/Y
: Interest rate.
ENTER
C/CE
CPT
FV
: Annuity payment.
1,000
2nd
P/Y
1
: Future value.
: Clear all Time-Value-of-Money values
30
0
: Number of compounding periods
: Present value.
CLR TVM
2nd
C/CE
30
0
N
N
: Number of compounding periods
: Annuity payment.
: Reset compounding periods per year.
: Future value.
19
20
Problem 7-1 - Answer
Problem 7-1 - Answer
C. What rate of return, compounding monthly, would have to be earned
on a $100,000 investment in order to accumulate $1 million in 30 years?
C. What rate of return, compounding monthly, would have to be earned
on a $100,000 investment in order to accumulate $1 million in 30 years?
Answer: 7.70%
Answer: 7.70%
HP10bii:
TI BAII Plus:
C ALL
360
0
1,000,000
100,000 +/12
N
PMT
: Clear memory:
: Annuity payment.
FV
: Future value.
PV
: Present value.
P/YR
C/CE
: Number of compounding periods.
: Reset compounding periods per year.
2nd
2nd
CLR TVM
360
0
1,000,000
100,000 +/P/Y 12 ENTER
Press I/YR : Interest rate.*
CPT
N
PMT
: Clear all Time-Value-of-Money values
: Number of compounding periods.
: Annuity payment.
FV
: Future value.
PV
: Present value.
C/CE
I/Y
: Reset compounding periods per year.
: Interest rate.
* For calculators set to one compounding period per year, then the solution
will appear as .64%, which is a monthly interest rate and must be
multiplied by 12 to get the 7.70% annual rate.
22
21
Problem 7-1 - Answer
Problem 7-1 - Answer
D. How many years would it take to accumulate $1,000,000 on a $100,000
investment, assuming an 8% return compounding monthly?
Answer: 28.88 years (346.54 mo.
D. How many years would it take to accumulate $1,000,000 on a $100,000
investment, assuming an 8% return compounding monthly?
12)
Answer: 28.88 years (346.54 mo.
HP10bii:
12)
TI BAII Plus:
C ALL
0
1,000,000
100,000 +/8
12
Press
PMT
: Clear memory:
FV
: Future value.
PV
: Present value
I/YR
: Interest rate.*
P/YR
: Reset compounding periods per year.
N
C/CE
: Annuity payment.
2nd
: Number of compounding periods.
2nd
CLR TVM
0
1,000,000
100,000 +/8
P/Y 12 ENTER
CPT
PMT
: Clear all Time-Value-of-Money values
: Annuity payment.
FV
: Future value.
PV
: Present value
I/Y
: Interest rate.
C/CE
N
: Reset compounding periods per year.
: Number of compounding periods.
* For calculators set to one compounding period per year, the interest rate to
be input is the monthly interest rate of .6667% (8% 12)
24
23
7-4
Problem 7-1 - Answer
Problem 7-1 - Answer
E. Compute the future value of $100,000 in 30 years at 10% compounding
daily (ignore the effect of leap years).
E. Compute the future value of $100,000 in 30 years at 10% compounding
daily (ignore the effect of leap years).
Answer: $2,007,728.58
Answer: $2,007,728.58
HP10bii:
TI BAII Plus:
C ALL
10,950
0
100,000 +/10
365
PV
: Present value.
I/YR
: Interest rate.*
P/YR
: Reset compounding periods per year.
10,950
0
100,000 +/10
P/Y
365 ENTER
2nd
: Future value.
FV
CLR TVM
2nd
C/CE
: Annuity payment.
PMT
Press
: Clear memory:
: Number of compounding periods.
N
CPT
N
PMT
: Clear all Time-Value-of-Money values
: Number of compounding periods
: Annuity payment.
PV
: Present value.
I/Y
: Interest rate.
C/CE
FV
: Reset compounding periods per year.
: Future value.
* For calculators set to one compounding period per year, the interest rate to
be input is the daily rate of .0274% (10 365)
25
26
Problem 7-1 - Answer
Problem 7-1 - Answer
F. An investor is considering the purchase of a 5-year, $20,000 note
receivable, which bears interest, all due at maturity, at a rate of 8%
compounding annually. If the investor were to buy the note at a time
when there are four years left to maturity, how much would the
investor pay to achieve a 12% rate of return, compounding quarterly?
F. An investor is considering the purchase of a 5-year, $20,000 note
receivable, which bears interest, all due at maturity, at a rate of 8%
compounding annually. If the investor were to buy the note at a time
when there are four years left to maturity, how much would the
investor pay to achieve a 12% rate of return, compounding quarterly?
Answer: In this problem you first calculate the future value of the note
receivable at maturity ($29,386.56) and then determine the
present value of that future amount.
Answer: In this problem you first calculate the future value of the note
receivable at maturity ($29,386.56) and then determine the
present value of that future amount.
$18,312.73 (Ignore the negative sign in this case.)
$18,312.73 (Ignore the negative sign in this case.)
HP10bii: First
HP10bii: Then
C ALL
5
0
20,000
1
Press
16
0
29,386.56
12
4
: Present value.
PV
8
C ALL
: Annuity payment.
PMT
+/-
: Clear memory:
: Number of compounding periods.
N
I/YR
: Interest rate.
P/YR
: Reset compounding periods per year.
: Annuity payment.
PMT
Press
: Future value.
FV
: Clear memory:
: Number of compounding periods.
N
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Reset compounding periods per year.
: Present value.
PV
28
27
Problem 7-1 - Answer
Problem 7-1 - Answer
F. An investor is considering the purchase of a 5-year, $20,000 note
receivable, which bears interest, all due at maturity, at a rate of 8%
compounding annually. If the investor were to buy the note at a time
when there are four years left to maturity, how much would the
investor pay to achieve a 12% rate of return, compounding quarterly?
F. An investor is considering the purchase of a 5-year, $20,000 note
receivable, which bears interest, all due at maturity, at a rate of 8%
compounding annually. If the investor were to buy the note at a time
when there are four years left to maturity, how much would the
investor pay to achieve a 12% rate of return, compounding quarterly?
Answer: In this problem you first calculate the future value of the note
receivable at maturity ($29,386.56) and then determine the
present value of that future amount.
Answer: In this problem you first calculate the future value of the note
receivable at maturity ($29,386.56) and then determine the
present value of that future amount.
$18,312.73 (Ignore the negative sign in this case.)
$18,312.73 (Ignore the negative sign in this case.)
TI BAII Plus: First
C/CE
2nd
20,000
2nd
P/Y
1
TI BAII Plus: Then
CLR TVM
: Clear all Time-Value-of-Money values
C/CE
: Number of compounding periods.
16
0
29,386.56
12
P/Y
4 ENTER
5
0
PMT
+/-
PV
: Present value.
8
I/Y
: Interest rate.
N
ENTER
C/CE
CPT
FV
: Annuity payment.
: Reset compounding periods per year.
2nd
: Future value.
2nd
CLR TVM
CPT
N
PMT
: Clear all Time-Value-of-Money values
: Number of compounding periods.
: Annuity payment.
FV
: Future value.
I/Y
: Interest rate.
C/CE
PV
: Reset compounding periods per year.
: Present value.
30
29
7-5
How much would you have for retirement in 30 years, if you invested
$100 at the end of each month at an interest rate of 7% compounding
monthly?
In other words, what's the future value of this $100 annuity?
HP 10bii calculator:
Clear memory:
C ALL
360
Annuity
100
+/-
PMT
0
7
(A series of equal cash payments over equal intervals of time.)
: Number of payments.
N
: Annuity payment.
: Present value.
PV
I/YR : Interest rate.
12
P/YR
: Set compounding periods per year.
: Future value.
FV
121,997.10
31
32
How much would accumulate if you could afford $300 at the end of
each month and could somehow find an investment that generated a
12% return, compounding monthly?
Calculate the future value of the $300 annuity assuming payments at
the beginning rather than the end of each month for 30 years.
Clear memory:
Reset the end of the period payment schedule to the beginning of
BEG/END
the period by:
360
300
N
+/-
PMT
0
12
PV
12
Clear memory:
C ALL
: Number of payments.
360
: Annuity payment.
300
: Present value.
I/YR : Interest rate.
P/YR
FV
: Set compounding periods per year.
N
+/-
PMT
0
12
PV
: Number of payments.
: Annuity payment.
: Present value.
I/YR : Interest rate.
12
: Future value.
C ALL
P/YR
FV
1,048,489.24
: Set compounding periods per year.
: Future value.
1,058,974.13
$1,058,974.13 vs. $1,048,489.24
($10,484.89 additional interest)
This amount is equal to the amount of interest earned on a single
investment of $300 for 30 years at 12% compounding monthly.
34
33
Assume you wish to set up an investment account from which you'll be
able to withdraw $10,000 at the end of each year for the next 20 years.
Assuming the account will earn interest at a rate of 8% compounding
annually how much will have to be invested today to accommodate
those future withdrawals?
Assume your expecting your first child and want to invest an equal
amount at the beginning of each month for 18 years to help cover the
anticipated costs of college. Assuming an 8% return on investment,
compounding monthly, how much must the monthly investment be to
have $50,000 at the end of that 18-year period?
Clear memory:
Clear memory:
C ALL
Reset to end of the period payments:
20
10,000
0
8
1
N
PMT
FV
Reset to end of the period payments:
BEG/END
216
50,000
0
8
12
: Number of payments.
: Annuity payments.
: Future value.
I/YR : Interest rate.
P/YR
PV
C ALL
: Set compounding periods per year.
: Present value.
-98,181.47
N
: Number of payments.
FV
: Future value.
PV
: Present value.
BEG/END
I/YR : Interest rate.
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
-103.46
36
35
7-6
How much longer will it take to accumulate the $50,000 given a
monthly payment of $75 a month?
Calculate the rate that would have to be achieved to meet the original
18-year timetable given payments of $75 a month.
Clear memory:
Clear memory:
C ALL
Check display to verify beginning of the period payments are set.
Check display to verify beginning of the period payments are set.
N
50,000
0
8
12
75 +/-
216
50,000
0
: Number of payments.
254.22
12 = 21.19 years
FV
: Future value.
PV
: Present value.
C ALL
N
: Number of payments.
FV
: Future value.
PV
: Present value.
I/YR : Interest rate.
10.83
I/YR : Interest rate.
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
12
75
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
+/-
37
Problem 7-2
Assume your considering the purchase of a $170,000 home with a
$20,000 cash down payment and a $150,000 mortgage loan.
Calculations with Annuities
Respond to each of the following:
What's the monthly mortgage payment?
A. Determine the present value of an annuity of $1,000 at the end of each quarter
for 5 years at 9% compounding quarterly.
Assuming a traditional 30-year, fixed rate, fully amortizing mortgage, an
equal monthly payment is established in an amount that pays off the entire
principal and interest due on the loan over the 30-year period. This fixed
monthly payment is an annuity that can be easily determined using a
financial calculator.
Clear memory:
360
0
150,000
7
12
B. Determine the future value of an annuity of $100 at the beginning of each
month for 10 years, at 7% compounding monthly.
C. If $200,000 is needed for retirement in 10 years, how much must be invested at
the beginning of each year, at an interest rate of 10% compounding annually, to
reach that goal?
C ALL
Reset to end of the period payments:
D. Determine the amount of the equal monthly mortgage payment on a $100,000,
30-year, fully amortizing mortgage, bearing interest at a fixed 7% rate,
compounding monthly. (Mortgage payments are made at the end of each month.)
Then make the journal entries to record the first two monthly payments.
BEG/END
N
: Number of payments.
FV
: Future value.
PV
: Present value.
38
E. Determine the fixed interest rate that will produce a monthly mortgage payment
of $750 on a $120,000, 30 year, fully amortizing mortgage.
I/YR : Interest rate.
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
F. The parents of a newborn daughter anticipate they'll need $10,000 at the
beginning of each year for four years to pay her annual college tuition beginning
on her 18th birthday. How much must be invested at the beginning of each year
for 18 years (beginning on her date of birth) to accumulate the funds necessary
to make those annual payments assuming a 7% return on investment,
compounding annually?
-997.95
40
39
Problem 7-2 - Answer
Problem 7-2 - Answer
Calculations with Annuities
Calculations with Annuities
A. Determine the present value of an annuity of $1,000 at the end of each quarter
for 5 years at 9% compounding quarterly.
A. Determine the present value of an annuity of $1,000 at the end of each quarter
for 5 years at 9% compounding quarterly.
Answer: $15,963.71 (ignore the negative sign)
Answer: $15,963.71 (ignore the negative sign)
HP10bii:
TI BAII Plus:
Check display to make sure end of the period payments are set.
("Begin" does not appear.)
Check display to make sure end of the period payments are set.
("BGN" does not appear.)
If this needs to be changed then enter:
If this needs to be changed then enter:
C ALL
20
1,000
0
9
4
Press
N
PMT
BEG/END
: Clear memory
: Number of payments.
: Annuity payment.
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Set compounding periods per year.
PV
C/CE
2nd
: Present value.
P/Y
2nd
CLR TVM
20
1,000
0
9
4 ENTER
CPT
N
PMT
2nd
SET
C/CE
: Annuity payment.
: Future value.
I/Y
: Interest rate.
PV
2nd
: Number of payments.
FV
C/CE
BGN
: Clear all Time-Value-of-Money values.
: Set compounding periods per year.
: Present value.
42
41
7-7
Problem 7-2 - Answer
Problem 7-2 - Answer
B. Determine the future value of an annuity of $100 at the beginning of each
month for 10 years, at 7% compounding monthly.
B. Determine the future value of an annuity of $100 at the beginning of each
month for 10 years, at 7% compounding monthly.
Answer: $17,409.45 (ignore the negative sign)
Answer: $17,409.45 (ignore the negative sign)
HP10bii:
TI BAII Plus:
Check display to make sure beginning of the period payments are set.
("Begin" appears.)
Check display to make sure beginning of the period payments are set.
("BGN" appears.)
If this needs to be changed then enter:
If this needs to be changed then enter:
C ALL
120
100
0
7
12
Press
N
PMT
PV
BEG/END
: Clear memory
C/CE
CLR TVM
120
100
0
7
: Number of payments.
: Annuity payment.
: Present value.
I/YR
: Interest rate.
P/YR
: Set compounding periods per year.
FV
2nd
2nd
P/Y
12
: Future value.
N
PMT
2nd
BGN
: Present value.
: Interest rate.
CPT
FV
C/CE
: Annuity payment.
I/Y
C/CE
SET
: Number of payments.
PV
ENTER
2nd
: Clear all Time-Value-of-Money values.
: Set compounding periods per year.
: Future value.
43
44
Problem 7-2 - Answer
Problem 7-2 - Answer
C. If $200,000 is needed for retirement in 10 years, how much must be invested at
the beginning of each year, at an interest rate of 10% compounding annually, to
reach that goal?
C. If $200,000 is needed for retirement in 10 years, how much must be invested at
the beginning of each year, at an interest rate of 10% compounding annually, to
reach that goal?
Answer: -$11,408.25
Answer: -$11,408.25
HP10bii:
TI BAII Plus:
Check display to make sure beginning of the period payments are set.
("Begin" appears.)
Check display to make sure beginning of the period payments are set.
("BGN" appears.)
If this needs to be changed then enter:
If this needs to be changed then enter:
C ALL
10
0
200,000
10
1
Press
BEG/END
: Clear memory
C/CE
N
: Number of payments.
PV
: Present value.
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
2nd
P/Y
2nd
CLR TVM
10
0
200,000
10
1 ENTER
CPT
2nd
BGN
2nd
: Number of payments.
PV
: Present value.
FV
: Future value.
I/Y
: Interest rate.
PMT
C/CE
: Clear all Time-Value-of-Money values.
N
C/CE
SET
: Set compounding periods per year.
: Annuity payment.
46
45
Problem 7-2 - Answer
Problem 7-2 - Answer
D. Determine the amount of the equal monthly mortgage payment on a $100,000,
30-year, fully amortizing mortgage, bearing interest at a fixed 7% rate,
compounding monthly. (Mortgage payments are made at the end of each month.)
Then make the journal entries to record the first two monthly payments.
D. Determine the amount of the equal monthly mortgage payment on a $100,000,
30-year, fully amortizing mortgage, bearing interest at a fixed 7% rate,
compounding monthly. (Mortgage payments are made at the end of each month.)
Then make the journal entries to record the first two monthly payments.
Answer: -$665.30
Answer: -$665.30
HP10bii:
TI BAII Plus:
Check display to make sure end of the period payments are set.
("Begin" does not appear.)
Check display to make sure end of the period payments are set.
("BGN" does not appear.)
If this needs to be changed then enter:
If this needs to be changed then enter:
C ALL
360
100,000
0
7
12
Press
BEG/END
: Clear memory
N
: Number of payments.
PV
: Present value.
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
C/CE
2nd
P/Y
2nd
CLR TVM
360
100,000
0
7
12 ENTER
CPT
2nd
BGN
2nd
N
: Number of payments.
PV
: Present value.
FV
: Future value.
I/Y
: Interest rate.
C/CE
PMT
SET
C/CE
: Clear all Time-Value-of-Money values.
: Set compounding periods per year.
: Annuity payment.
48
47
7-8
Problem 7-2 - Answer
Problem 7-2 - Answer
E. Determine the fixed interest rate that will produce a monthly mortgage payment
of $750 on a $120,000, 30 year, fully amortizing mortgage.
E. Determine the fixed interest rate that will produce a monthly mortgage payment
of $750 on a $120,000, 30 year, fully amortizing mortgage.
Answer: 6.39%
Answer: 6.39%
HP10bii:
TI BAII Plus:
Check display to make sure end of the period payments are set.
("Begin" does not appear.)
Check display to make sure end of the period payments are set.
("BGN" does not appear.)
If this needs to be changed then enter:
If this needs to be changed then enter:
C ALL
360
750
+/-
0
120,000
12
N
PMT
PV
: Present value.
Press I/YR
360
750
: Annuity payment.
: Future value.
CLR TVM
2nd
C/CE
: Number of payments.
FV
P/YR
BEG/END
: Clear memory
: Set compounding periods per year.
2nd
P/Y
+/-
0
120,000
12 ENTER
: Interest rate.
CPT
N
PMT
2nd
BGN
C/CE
: Annuity payment.
: Future value.
PV
: Present value
I/Y
SET
: Number of payments.
FV
C/CE
2nd
: Clear all Time-Value-of-Money values.
: Set compounding periods per year.
: Interest rate.
49
50
Problem 7-2 - Answer
Problem 7-2 - Answer
HP10bii:
F. The parents of a newborn daughter anticipate they'll need $10,000 at the
beginning of each year for four years to pay her annual college tuition beginning
on her 18th birthday. How much must be invested at the beginning of each year
for 18 years (beginning on her date of birth) to accumulate the funds necessary
to make those annual payments assuming a 7% return on investment,
compounding annually?
Check display to make sure beginning of the period payments are set.
("Begin" appears.)
If this needs to be changed then enter:
BEG/END
First
Answer: -$996.27 (ignore the negative sign)
C ALL
(This solution requires a two-part process. First, the present value of a
$10,000 annual annuity with payments made at the beginning of each year
must be determined at a 7% rate compounding annually. Then that present
value will be used as the future value amount in determining the annual
investment required at the beginning of each year for 18 years at the same
7% rate)
4
10,000
0
7
1
See pages that follow for calculation:
Press
: Clear memory
: Number of payments.
N
: Annuity payment.
PMT
FV
: Future value.
I/YR
: Interest rate.
P/YR
: Set compounding periods per year.
: Present value.
PV
-36,243.16
52
51
Problem 7-2 - Answer
Problem 7-2 - Answer
HP10bii:
TI BAII Plus:
Then
Check display to make sure beginning of the period payments are set.
("BGN" appears.)
C ALL
18
36,243.16
0
7
1
Press
: Clear memory
N
: Number of payments.
FV
: Future value.
PV
: Present value.
I/YR
: Interest rate.
P/YR
: Set compounding periods per year.
PMT
: Annuity payment.
If this needs to be changed then enter:
2nd
BGN
2nd
SET
C/CE
First
C/CE
-996.27
2nd
P/Y
2nd
CLR TVM
4
10,000
0
7
1 ENTER
CPT
N
PMT
: Clear all Time-Value-of-Money values.
: Number of payments.
: Annuity payment.
FV
: Future value.
I/Y
: Interest rate.
C/CE
PV
: Set compounding periods per year.
: Present value.
-36,243.16
54
53
7-9
Problem 7-2 - Answer
The present or future value of multiple cash flows is simply the
sum of the present or future values of all cash flows involved.
TI BAII Plus:
Then
C/CE
2nd
CLR TVM
18
36,243.16
0
7
P/Y
1 ENTER
2nd
CPT
N
: Number of payments.
FV
: Future value.
PV
: Present value.
I/Y
: Interest rate.
C/CE
PMT
Determine the future value of an investment at the end of three years that includes contributions
of $1,000 today, $2,000 a year from now, and $3,000 a year after that, assuming the investment
earns a 10% return compounding annually.
: Clear all Time-Value-of-Money values.
FV = ?
-$1,000
FV = $1,331
FV = $2,420
FV = $3,300
$7,051
-$2,000
-$3,000
FV = ?
: Set compounding periods per year.
-$1,000
-$1,000
-$1,000
$1,000
$2,000
: Annuity payment.
-996.27
-$1,000
-$1,000
-$1,000
$3,000
FV = $3,641
FV = $2,310
FV = $1,100
$7,051
-$2,000
-$1,000
$3,000
FV = $1,331
FV = $4,620
FV = $1,100
$7,051
FV = ?
-$1,000
-$2,000
$1,000
$2,000
The future value of uneven cash flows is simply the sum of the future values of each single cash
flow or any combination of single or annuity cash flows involved. That's also true when applied
to present values.
55
56
Determine how much would have to be invested in an account today, if, at the
beginning of the 5th year following investment, you wished to withdraw $1,000 a
month for 12 months plus the lump-sum amount of $10,000 at the end of that 12th
month. Assume an 8% return on investment, compounding monthly.
Determine how much would have to be invested in an account today, if, at the
beginning of the 5th year following investment, you wished to withdraw $1,000 a
month for 12 months plus the lump-sum amount of $10,000 at the end of that 12th
month. Assume an 8% return on investment, compounding monthly.
PV = ?
PV = ?
1
2
3
4
5
60
PV = -671.21
PV = -666.76
PV = -662.35
.
.
.
-13,965.23
61
62
1,000 1,000 1,000
70
71
72 Months
1,000 1,000 10,000
1
2
PV = -7,767.53
PV = -6,197.70
-13,965.23
3
4
5
60
61
62
1,000 1,000 1,000
70
71
72 Months
1,000 1,000 10,000
PV = 11,572.42
58
57
Problem 7-3
Problem 7-3
PV and FV Calculations with Uneven Cash Flows
PV and FV Calculations with Uneven Cash Flows
Respond to each of the following:
C. If you open an investment account and expect to earn 12% compounding
monthly, how much will you have for retirement in 30 years if you invest
the following amounts at the beginning of each month?
A. If withdrawals of $10,000, $12,000 and $15,000 are needed from an
investment account at the end of each year for the next three years,
respectively, how much must be invested today assuming a 10% return on
investment, compounding annually. (Make this calculation two ways. Use
an annuity in at least in one of your computations.)
$ 250/mo. for the first 5 years
$ 500/mo. for the next 10 years
$1,000/mo. for the final 15 years
B. The parents of a newborn daughter anticipate they'll need the following
amounts to fund their daughters' future college education and wedding:
$15,000 at 18th birthday
$16,000 at 19th birthday
$17,000 at 20th birthday
$18,000 at 21st birthday
$25,000 at 26th birthday
How much will have to be invested at the beginning of each year for 18
years (starting at the date of birth) to accumulate the funds necessary to
meet these anticipated future obligations? (Assume a 7% return on
investment, compounding annually.)
59
7-10
60
Problem 7-3 - Answer
Problem 7-3 - Answer
PV and FV Calculations with Uneven Cash Flows
B. The parents of a newborn daughter anticipate they'll need the following amounts
to fund their daughters' future college education and wedding:
Respond to each of the following:
$15,000 at 18th birthday
$16,000 at 19th birthday
$17,000 at 20th birthday
$18,000 at 21st birthday
$25,000 at 26th birthday
A. If withdrawals of $10,000, $12,000 and $15,000 are needed from an
investment account at the end of each year for the next three years,
respectively, how much must be invested today assuming a 10% return on
investment, compounding annually. (Make this calculation two ways. Use
an annuity in at least in one of your computations.)
How much will have to be invested at the beginning of each year for 18 years
(starting at the date of birth) to accumulate the funds necessary to meet these
anticipated future obligations? (Assume a 7% return on investment,
compounding annually.)
Answer: -$30,277.98 (ignore the negative sign)
PV = ?
PV = $ 9,090.90
PV = $ 9,917.36
PV = $11,269.72
$30,277.98
Answer: $2,035.39 at the beginning of each year for 18 years, with interest
at 7% compounding annually produces a FV of $74,095.32 at the
end of the 18th year.
$10,000
$12,000
$15,000
PV = ?
PV = $24,868.52
PV = $ 1,652.89
PV = $ 3,756.57
$30,277.98
$10,000
$10,000
$ 2,000
$10,000
$12,000
0
1
2
16
17
?
?
?
?
?
18
19
20
21
15,000 16,000 17,000 18,000
22
23
24
25
26 years
25,000
PV = -15,000.00
PV = -14,953.27
PV = -14,848.46
PV = -14,693.36
PV = -14,550.23
-74,045.32
$10,000
$ 5,000
$15,000
61
Problem 7-3 - Answer
C. If you open an investment account and expect to earn 12% compounding
monthly, how much will you have for retirement in 30 years if you invest
the following amounts at the beginning of each month?
$ 250/mo. for the first 5 years
$ 500/mo. for the next 10 years
$1,000/mo. for the final 15 years
Answer: $1,609,175.21
FV of -$250 annuity at the beginning of 360 months = $882,478.44
FV of -$250 annuity at the beginning of 300 months = $474,408.77
FV of -$500 annuity at the beginning of 180 months = $252,288.00
$1,609,175.21
63
7-11
62