set 2: lesson 7 notes · take the money and invest it. it's a sure thing. inflation can...

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Lesson 7

7-1

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!Take the money and invest it.

It's a sure thing.

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!

How much is a $1,000 a year from now actually worth in today's dollars?

orHow much would you be willing to pay today for the right to receive $1,000 a year from now?

A financial calculator is required for this course.

Hewlett-Packard, HP10bii

Texas Instruments, TI BA II Plus

Any calculator that has these keys noted here,

and is capable of calculating internal rates of return on uneven cash flows and will probably get the job done.

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.

NN II PVPV PMTPMT FVFV CFjCFj

How much would you be willing to pay today for the right to receive $1,000 a year from now?

(The present value of a single cash flow or lump sum of $1,000.)

I/YRI/YR

FVFV

PMTPMT

NN

C ALLC ALL

Set decimal places on the calculator's display:

Clear memory: : Number of periods involved.

: Annuity payment (a series of two or more payments made in equal amounts over equal intervals of time)

: Future value.

: Interest rate per year

2DISPDISP

10

1,00012

Lesson 7The Time Value of Money

7 8

9 10

11 12

7-2

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.

In this case, we'll calculate the present value of the $1,000 receivable in one year using 12 percent interest, compounding annually.

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN

C ALLC ALL

Set decimal places on the calculator's display:

Clear memory: : Number of periods involved.


: Future value.


: Reset to reflect annual compounding

: Present value.

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.

DISPDISP

10

1,00012

1PVPV

-892.86

2

FVFV

PVPV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALLClear memory: : Number of periods involved.


: Present value.


: Reset to reflect annual compounding

: Future value.

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

892.8612

1

1,000.00

+/-+/-

FVFV

PVPV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALL

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.

Clear memory: : Number of compounding periods : Annuity payment : Present value.: Interest rate : Reset compounding periods per year.: Future value

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

40

892.8612

4

1,004.92

+/-+/-

FVFV

PVPV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALL

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: : Number of compounding periods (total).

: Annuity payment : Present value.: Interest rate.

: Reset compounding periods per year.: Future value.

Interest earned on this investment:

3650

892.8612

365

1,006.68

+/-+/-

$113.82($1,006.68 - $892.86)

$113.82 $892.86 = 12.78% APR

PVPV

FVFV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALL

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.

Clear memory: : Number of compounding periods (total).: Annuity payment.

: Future value.: Interest rate. : Reset compounding periods per year.: Present value.

540

50,0005

12

-39,944.48

13 14

15 16

17 18

7-3

PVPV

FVFV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALL

Determine the interest rate that would be required to double a $1,000 investment in 5 years, assuming interest compounds annually.

Clear memory: : Number of compounding periods (total).: Annuity payment. : Future value.: Interest rate. 14.87% : Reset compounding periods per year.: Present value.

50

2,000

11,000 +/-+/-

PVPV

FVFV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALL

How long would it take to double our money if the best investment we could find produced a 10% return, compounding monthly?

Clear memory: : Number of compounding periods (total).

83.52 12 = 6.97 years: Annuity payment. : Future value.: Interest rate.: Reset compounding periods per year.: Present value.

02,000

121,000 +/-+/-

10

Problem 7-1Calculations 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.

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?

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).

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?

Problem 7-1 - AnswerCalculations Using Single Cash Flows

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.)

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN

HP10bii: : Clear memory.

: Number of compounding periods

: Annuity payment.

: Future value.

: Interest rate.

: Reset compounding periods per year.

: Present value.

400

10,0007

2PVPV

C ALLC ALL

Press

Problem 7-1 - Answer

PVPV

ENTERENTER C/CEC/CE

NN

PMTPMT

FVFV

I/YI/Y

CLR TVMCLR TVMC/CEC/CE 2nd2nd

TI BAII Plus:: Clear all Time-Value-of-Money values.

: Number of compounding periods.

: Annuity payment.

: Future value.

: Interest rate.


: Present value.

400

10,0007

22nd2nd P/YP/Y

CPTCPT




PVPV

II



FVFV

PMTPMT

NN

** 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

Other calculators: (See manual) : Clear memory:


: Annuity payment.

: Future value.

: Interest rate.

: Present value. (See manual for compute function.)

400

10,0003.5

19 20

21 22

23 24

7-4


PVPV

FVFV


Answer: $29,959.92

P/YRP/YR

I/YRI/YR

PMTPMT

NN

HP10bii: : Clear memory:


: Annuity payment.

: Present value.

: Interest rate.


: Future value.

300

1,00012

1

C ALLC ALL

Press

+/-+/-


FVFV

PVPV

ENTERENTER C/CEC/CE

NN

PMTPMT

I/YI/Y


TI BAII Plus:: Clear all Time-Value-of-Money values


: Annuity payment.

: Present value.

: Interest rate.


: Future value.

300

1,00012

12nd2nd P/YP/Y

CPTCPT


Answer: $29,959.92

+/-+/-


FVFV

PVPV

I/YRI/YR


Answer: 7.70%

P/YRP/YR

PMTPMT

NN



: Annuity payment.

: Future value.

: Present value.


: Interest rate.*

3600

1,000,000100,000

12

C ALLC ALL

Press

+/-+/-

* 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.


PVPV

I/YI/Y

FVFV

ENTERENTER C/CEC/CE

NN

PMTPMT




: Annuity payment.

: Future value.

: Present value.


: Interest rate.

3600

1,000,000100,000

122nd2nd P/YP/Y

CPTCPT

+/-+/-


Answer: 7.70%


FVFV

PVPV

PMTPMT

+/-+/-I/YRI/YR

NN


Answer: 28.88 years (346.54 mo. 12)

P/YRP/YR


: Annuity payment.

: Future value.

: Present value

: Interest rate.*



01,000,000

100,0008

12

C ALLC ALL

Press

* For calculators set to one compounding period per year, the interest rate to be input is the monthly interest rate of .6667% (8% 12)


PVPV

FVFV

PMTPMT

+/-+/-I/YI/Y

NN

ENTERENTER C/CEC/CE



: Annuity payment.

: Future value.

: Present value

: Interest rate.



01,000,000

100,0008

122nd2nd P/YP/Y

CPTCPT


Answer: 28.88 years (346.54 mo. 12)

25 26

27 28

29 30

7-5

FVFVPress


PVPV

FVFV


Answer: $2,007,728.58

P/YRP/YR

I/YRI/YR

PMTPMT

NN



: Annuity payment.

: Present value.

: Interest rate.*


: Future value.

10,9500

100,00010

365

C ALLC ALL

Press

+/-+/-

* For calculators set to one compounding period per year, the interest rate to be input is the daily rate of .0274% (10 365)


FVFV

PVPV

ENTERENTER C/CEC/CE

NN

PMTPMT

I/YI/Y




: Annuity payment.

: Present value.

: Interest rate.


: Future value.

10,9500

100,00010

3652nd2nd P/YP/Y

CPTCPT

+/-+/-


Answer: $2,007,728.58


PVPV


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.)

P/YRP/YR

I/YRI/YR

PMTPMT

NN

HP10bii: First: Clear memory:


: Annuity payment.

: Present value.

: Interest rate.


: Future value.

50

20,0008

1

C ALLC ALL

+/-+/-


FVFV




P/YRP/YR

I/YRI/YR

PMTPMT

NN

HP10bii: Then: Clear memory:


: Annuity payment.

: Future value.

: Interest rate.


: Present value.

160

29,386.5612

4

C ALLC ALL

Press


PVPV

ENTERENTER C/CEC/CE

NN

PMTPMT

I/YI/Y


TI BAII Plus: First: Clear all Time-Value-of-Money values


: Annuity payment.

: Present value.

: Interest rate.


: Future value.

50

20,0008

12nd2nd P/YP/Y

+/-+/-





PVPV

FVFV

ENTERENTER C/CEC/CE

NN

PMTPMT

I/YI/Y


TI BAII Plus: Then: Clear all Time-Value-of-Money values


: Annuity payment.

: Future value.

: Interest rate.


: Present value.

160

29,386.5612

42nd2nd P/YP/Y

CPTCPT




PVPV

FVFVCPTCPT

31 32

33 34

35 36

7-6

Annuity(A series of equal cash payments over equal intervals of time.)

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN

C ALLC ALL

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: : Number of payments.

: Annuity payment.

: Present value.

: Interest rate.

: Set compounding periods per year.

: Future value.

360100

07

12

PVPV

121,997.10

+/-+/-

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN

C ALLC ALL

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?

Clear memory: : Number of payments.

: Annuity payment.

: Present value.

: Interest rate.


: Future value.

360300

012

12

PVPV

1,048,489.24

+/-+/-

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN

C ALLC ALL

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 the period by:

: Number of payments.: Annuity payment.: Present value.: Interest rate.: Set compounding periods per year.: Future value.

360300

012

12

PVPV

1,058,974.13

+/-+/-

BEG/ENDBEG/END

This amount is equal to the amount of interest earned on a single investment of $300 for 30 years at 12% compounding monthly.

$1,058,974.13 vs. $1,048,489.24($10,484.89 additional interest)

PVPV

FVFV

P/YRP/YR

I/YRI/YR

PMTPMT

NN

C ALLC ALL

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? Clear memory:

Reset to end of the period payments:: Number of payments.

: Annuity payments.

: Future value.

: Interest rate.


: Present value.

2010,000

08

1

-98,181.47

BEG/ENDBEG/END

FVFV

PMTPMT

PVPV

P/YRP/YR

I/YRI/YR

NN

C ALLC ALL

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:

Reset to end of the period payments:: Number of payments.

: Future value.

: Present value.

: Interest rate.


: Annuity payment.

21650,000

08

12

-103.46

BEG/ENDBEG/END

37 38

39 40

41 42

7-7

+/-+/-

FVFV

PMTPMT

PVPV

P/YRP/YR

I/YRI/YR

NN

C ALLC ALL

How much longer will it take to accumulate the $50,000 given a monthly payment of $75 a month?Clear memory:

Check display to verify beginning of the period payments are set.: Number of payments.

254.22 12 = 21.19 years: Future value.

: Present value.

: Interest rate.


: Annuity payment.

50,00008

1275 +/-+/-

FVFV

PMTPMT

PVPV

P/YRP/YR

I/YRI/YR

NN

C ALLC ALL

Calculate the rate that would have to be achieved to meet the original 18-year timetable given payments of $75 a month.Clear memory:

Check display to verify beginning of the period payments are set.: Number of payments.

: Future value.

: Present value.

: Interest rate.

10.83: Set compounding periods per year.

: Annuity payment.

50,0000

1275

216

FVFV

PMTPMT

PVPV

P/YRP/YR

I/YRI/YR

NN

C ALLC ALL

Assume your considering the purchase of a $170,000 home with a $20,000 cash down payment and a $150,000 mortgage loan.

What's the monthly mortgage payment?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:

Reset to end of the period payments:: Number of payments.: Future value.: Present value.: Interest rate.: Set compounding periods per year.: Annuity payment.

3600

150,0007

12

-997.95

BEG/ENDBEG/END

Problem 7-2Calculations with Annuities

Respond to each of the following: A. Determine the present value of an annuity of $1,000 at the end of each quarter

for 5 years at 9% compounding quarterly.

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?

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.

E. Determine the fixed interest rate that will produce a monthly mortgage payment of $750 on a $120,000, 30 year, fully amortizing mortgage.

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?

Problem 7-2 - AnswerCalculations 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.

Answer: $15,963.71 (ignore the negative sign)

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN

HP10bii: Check display to make sure end of the period payments are set. ("Begin" does not appear.)

If this needs to be changed then enter:

: Clear memory

: Number of payments.

: Annuity payment.

: Future value.

: Interest rate.


: Present value.

201,000

09

4PVPV

C ALLC ALL

Press

BEG/ENDBEG/END

Problem 7-2 - AnswerCalculations 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.


TI BAII Plus: Check display to make sure end of the period payments are set. ("BGN" does not appear.)


: Clear all Time-Value-of-Money values.


: Annuity payment.

: Future value.

: Interest rate.


: Present value.PVPVCPTCPT

ENTERENTER C/CEC/CE42nd2nd P/YP/Y

I/YI/Y9FVFV0

PMTPMT1,000NN20


2nd2nd BGNBGN 2nd2nd SETSET C/CEC/CE

43 44

45 46

47 48

7-8


FVFV

PVPV



P/YRP/YR

I/YRI/YR

PMTPMT

NN

HP10bii: Check display to make sure beginning of the period payments are set. ("Begin" appears.)


: Clear memory


: Annuity payment.

: Present value.

: Interest rate.


: Future value.

12010007

12

C ALLC ALL

Press

BEG/ENDBEG/END


FVFV

PVPV

TI BAII Plus: Check display to make sure beginning of the period payments are set. ("BGN" appears.)




: Annuity payment.

: Present value.

: Interest rate.


: Future value.CPTCPT


I/YI/Y70

PMTPMT100NN120






PMTPMT

FVFV

PVPV





: Present value.

: Future value.

: Interest rate.


: Annuity payment.CPTCPT

ENTERENTER C/CEC/CE1I/YI/Y10

200,0000

NN10CLR TVMCLR TVMC/CEC/CE 2nd2nd



Answer: -$11,408.25

2nd2nd P/YP/Y



Answer: -$665.30



: Clear memory


: Present value.

: Future value.

: Interest rate.


: Annuity payment.

BEG/ENDBEG/END

PMTPMT

FVFV

PVPV

P/YRP/YR

I/YRI/YR

NN360100,000

07

12

C ALLC ALL

Press


PMTPMT

FVFV

PVPV


Answer: -$11,408.25

P/YRP/YR

I/YRI/YR

NN



: Clear memory


: Present value.

: Future value.

: Interest rate.


: Annuity payment.

100

200,00010

1

C ALLC ALL

Press

BEG/ENDBEG/END






: Present value.

: Future value.

: Interest rate.


: Annuity payment.



Answer: -$665.30

PMTPMT

FVFV

PVPV

CPTCPT


0100,000


2nd2nd P/YP/Y

49 50

51 52

53 54

7-9


I/YRI/YR

PVPV

FVFV

PMTPMT


Answer: 6.39%



: Clear memory


: Annuity payment.

: Future value.

: Present value.


: Interest rate.

BEG/ENDBEG/END

P/YRP/YR

NN360750

0120,00012

C ALLC ALL

Press

+/-+/-


I/YI/Y





: Annuity payment.

: Future value.

: Present value


: Interest rate.


PMTPMT

FVFV

PVPV

CPTCPT

ENTERENTER C/CEC/CE12120,000

0750


2nd2nd P/YP/Y


Answer: 6.39%

+/-+/-


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?

Answer: -$996.27 (ignore the negative sign)

(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)

See pages that follow for calculation:




First: Clear memory


: Annuity payment.

: Future value.

: Interest rate.


: Present value.

BEG/ENDBEG/END

P/YRP/YR

I/YRI/YR

FVFV

PMTPMT

NN410,000

07

1PVPV

C ALLC ALL

Press

-36,243.16


PVPV

FVFV

HP10bii:

Then: Clear memory


: Future value.

: Present value.

: Interest rate.


: Annuity payment.PMTPMT

P/YRP/YR

I/YRI/YR

NN1836,243.16

07

1

C ALLC ALL

Press

-996.27




First: Clear all Time-Value-of-Money values.


: Annuity payment.

: Future value.

: Interest rate.


: Present value.


PVPVCPTCPT


I/YI/Y7FVFV0

PMTPMT10,000NN4


-36,243.16

55 56

57 58

59 60

7-10


PVPV

FVFV

TI BAII Plus:

Then: Clear all Time-Value-of-Money values.


: Future value.

: Present value.

: Interest rate.


: Annuity payment. PMTPMTCPTCPT


036,243.16


2nd2nd P/YP/Y

-996.27

The present or future value of multiple cash flows is simply the sum of the present or future values of all cash flows involved.

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.

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.

-$3,000

FV = $1,331-$1,000-$2,000 FV = $2,420

FV = $3,300$7,051

FV = ?

-$1,000-$1,000-$1,000

FV = $3,641-$1,000 -$1,000-$1,000 FV = $2,310

FV = $1,100$7,051

FV = ?

-$2,000-$1,000

FV = $1,331-$1,000-$2,000 FV = $4,620

FV = $1,100$7,051

FV = ?

$3,000$2,000$1,000

$3,000$2,000$1,000

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.

10,0001,000 1,000

PV = ? 1 2 3 72 Months71

1,0001,000 1,000

706160 624 5

PV = 11,572.42PV = -7,767.53PV = -6,197.70

-13,965.23

Problem 7-3PV and FV Calculations with Uneven Cash Flows

Respond to each of the following:

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.)

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.)

Problem 7-3PV and FV Calculations with Uneven Cash Flows

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

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.

10,0001,000 1,000

PV = ? 1 2 3 72 Months71

1,0001,000 1,000

706160 624 5

PV = -671.21PV = -666.76PV = -662.35

.

.

.-13,965.23

61 62

63

7-11

Problem 7-3 - AnswerPV and FV Calculations with Uneven Cash Flows

Respond to each of the following:

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.)

Answer: -$30,277.98 (ignore the negative sign)

$12,000$15,000

$10,000

PV = ?

PV = $ 9,090.90PV = $ 9,917.36PV = $11,269.72

$30,277.98

$10,000$ 2,000

$12,000

$10,000

$ 5,000$15,000

$10,000

$10,000

PV = ?

PV = $24,868.52PV = $ 1,652.89PV = $ 3,756.57 $30,277.98


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.)

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.

25,00017,000 18,000

0 1 2 16 26 years24

16,000? 15,000

232019 2217 18

PV = -15,000.00PV = -14,953.27PV = -14,848.46PV = -14,693.36PV = -14,550.23

-74,045.32

21 25

? ? ? ?


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 = $882,478.44FV of -$250 annuity at the beginning of 300 months = $474,408.77FV of -$500 annuity at the beginning of 180 months = $252,288.00

$1,609,175.21

set 2: lesson 7 notes · take the money and invest it. it's a sure thing. inflation can...

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