the time value of money compounding and discounting single sums and annuities 1999, prentice hall,...

The Time Value of Money The Time Value of Money

Compounding and Discounting Compounding and Discounting Single Sums and AnnuitiesSingle Sums and Annuities

1999, Prentice Hall, Inc.

Notes:Notes: Although it is easiest to use your Although it is easiest to use your

financial calculator to solve time value financial calculator to solve time value problems, you problems, you MUSTMUST understand understand what you are doing. This will require a what you are doing. This will require a lot of practice to eliminate mistakes.lot of practice to eliminate mistakes.

Understanding the concept of Time Understanding the concept of Time Value of Money Value of Money NOWNOW is is extremely extremely important because all the remaining important because all the remaining chapters will require TVM concept chapters will require TVM concept application.application.

Notes:Notes: In your Test, you will be In your Test, you will be REQUIREDREQUIRED to to

show both your show both your financial calculator financial calculator solutionsolution and and Mathematical solutionMathematical solution (either using the (either using the formulaformula or the or the Financial Financial TablesTables))

Only in multiple choice questions will you Only in multiple choice questions will you not be required to show your calculation. not be required to show your calculation. Therefore, you can use the financial Therefore, you can use the financial calculator alone. You must however, calculator alone. You must however, make sure that you know how to use your make sure that you know how to use your calculator properly; otherwise you will calculator properly; otherwise you will easily make mistakes with the use of a easily make mistakes with the use of a financial calculator.financial calculator.

Using your Financial Calculators (TI BAII Using your Financial Calculators (TI BAII Plus and Sharp EL-733A)Plus and Sharp EL-733A)

For the For the first timefirst time that you are using your calculator, that you are using your calculator, perform the following:perform the following:– TI BAII PlusTI BAII Plus users, Set Payment Frequency and users, Set Payment Frequency and

Compounding Frequency to 1 (Compounding Frequency to 1 (press 2press 2ndnd, press , press P/Y, press 1, press ENTER, press down arrow, P/Y, press 1, press ENTER, press down arrow, press 1, press ENTER, press 2press 1, press ENTER, press 2ndnd, press QUIT, press QUIT))

– Sharp EL-733ASharp EL-733A users, Set to FIN Mode if not yet users, Set to FIN Mode if not yet in FIN Mode ( in FIN Mode ( presspress 22ndnd F, press Mode F, press Mode). You ). You should see FIN on the Display.should see FIN on the Display.

Set Decimal to 4 placesSet Decimal to 4 places– TI BAII PlusTI BAII Plus users, users, press 2press 2ndnd, press Format, , press Format,

press 4,press 4, press ENTER , press 2press ENTER , press 2ndnd, press QUIT, press QUIT.. – Sharp EL-733ASharp EL-733A users, users, press 2press 2ndndF, press TAB, F, press TAB,

press 4.press 4.

Using your Financial Calculators (TI BAII Plus Using your Financial Calculators (TI BAII Plus and Sharp EL-733A)and Sharp EL-733A)

To start To start eacheach calculation calculation,,– TI BAII PlusTI BAII Plus users, users, press CE/Cpress CE/C,, press 2press 2ndnd, press , press

CLR TVM, press 2CLR TVM, press 2ndnd,, press CLR Work. press CLR Work. BAII BAII Plus has a continuous memory. Turning-off the Plus has a continuous memory. Turning-off the calculator does not erase what was previously calculator does not erase what was previously stored in its memory, although turning it on again stored in its memory, although turning it on again resets the display to zero. Therefore, it is extremely resets the display to zero. Therefore, it is extremely important to clear memory before each calculation. important to clear memory before each calculation.

– Sharp EL-733ASharp EL-733A users, users, press 2press 2ndndF, press CA.F, press CA. To erase the previously entered number, To erase the previously entered number,

– TI BAII PlusTI BAII Plus users, simply users, simply press CE/Cpress CE/C..– Sharp EL-733ASharp EL-733A users, simply users, simply press C CE.press C CE.

Enter Outflow Value as negative. To enter it as Enter Outflow Value as negative. To enter it as negative negative enter the value/s, press +/-. DO NOT use enter the value/s, press +/-. DO NOT use the minus sign key.the minus sign key.

Using your Financial Calculators (TI BAII Plus Using your Financial Calculators (TI BAII Plus and Sharp EL-733A)and Sharp EL-733A)

The order in which data (PV, n, I, etc) are entered does not The order in which data (PV, n, I, etc) are entered does not matter.matter.

To compute for the result,To compute for the result, press CPT press CPT forfor TI BAII PlusTI BAII Plus users, users, press COMPpress COMP for for Sharp EL-733ASharp EL-733A users. Then users. Then press press whatever variable you are computing for (PV. FV, etc)whatever variable you are computing for (PV. FV, etc)

To perform calculations involving annuity dues, payment To perform calculations involving annuity dues, payment must be set to the begin mode. must be set to the begin mode. – TI BAII PlusTI BAII Plus users, users, press 2press 2ndnd, press BGN, press , press BGN, press

22ndnd,press SET, press 2,press SET, press 2ndnd, press QUIT., press QUIT.– Sharp EL-733ASharp EL-733A users, users, press BGNpress BGN..

It is important to reset the mode back to ENDIt is important to reset the mode back to END. Most . Most payment problems are made at the end of each year payment problems are made at the end of each year (ordinary annuities).(ordinary annuities).

We know that receiving $1 today is worth We know that receiving $1 today is worth more than $1 in the future. This is duemore than $1 in the future. This is due to theto the time value of moneytime value of money..

The cost of receiving $1 in the future is The cost of receiving $1 in the future is thethe interestinterest we could have earned if we we could have earned if we had received the $1 sooner.had received the $1 sooner.

Today Future

If we can MEASURE this If we can MEASURE this interest cost, we can:interest cost, we can:

?

Translate $1 today into its equivalent in Translate $1 today into its equivalent in the futurethe future (COMPOUNDING)(COMPOUNDING)..

Today Future

If we can MEASURE this If we can MEASURE this interest cost, we can:interest cost, we can:

Translate $1 today into its equivalent in Translate $1 today into its equivalent in the futurethe future (COMPOUNDING)(COMPOUNDING)..

Translate $1 in the future into its Translate $1 in the future into its equivalent todayequivalent today (DISCOUNTING)(DISCOUNTING)..

?

?

Today Future

Today Future

Future Value – Future Value – single sumsingle sum

Future Value - single sumsFuture Value - single sums

If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?

Calculator Solution:Calculator Solution:

I/Y = i = 6 I/Y = i = 6 N = n = 5 N = n = 5

PV = -100PV = -100

FV = FV = $133.82$133.82

00 5 5

PV = -100PV = -100 FV = FV =

Future Value - single sumsFuture Value - single sums

If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?

Calculator Solution:Calculator Solution:

I/Y = i = 6I/Y = i = 6

N = n = 5 N = n = 5 PV = -100 PV = -100

FV = FV = $133.82$133.82

00 5 5

PV = -100PV = -100 FV = FV = 133.133.8282

Future Value - single sumsFuture Value - single sums

If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?

Mathematical Solution:Mathematical Solution:

FV = PV (FVIF FV = PV (FVIF i, ni, n ))

FV = 100 (FVIF FV = 100 (FVIF .06, 5.06, 5 ) (use FVIF table, or)) (use FVIF table, or)

FV = PV (1 + i)FV = PV (1 + i)nn

FV = 100 (1.06)FV = 100 (1.06)5 5 = = $$133.82133.82

00 5 5

PV = -100PV = -100 FV = FV = 133.133.8282

Calculator Solution:Calculator Solution:

I/Y = i = 1.5I/Y = i = 1.5

N = n = 20 N = n = 20 PV = PV = -100-100

FV = FV = $134.68$134.68

00 20 20

PV = -100PV = -100 FV = FV =

Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have

in the account after 5 years?in the account after 5 years?

Calculator Solution:Calculator Solution:

I/Y = i = 1.5I/Y = i = 1.5

N = n = 20 N = n = 20 PV = PV = -100-100

FV = FV = $134.68$134.68

00 20 20

PV = -100PV = -100 FV = FV = 134.134.6868

Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have

in the account after 5 years?in the account after 5 years?

Mathematical Solution:Mathematical Solution:

FV = PV (FVIF FV = PV (FVIF i, ni, n ))

FV = 100 (FVIF FV = 100 (FVIF .015, 20.015, 20 ) ) (can’t use FVIF table)(can’t use FVIF table)

FV = PV (1 + i/m) FV = PV (1 + i/m) m x nm x n

FV = 100 (1.015)FV = 100 (1.015)20 20 = = $134.68$134.68

00 20 20

PV = -100PV = -100 FV = FV = 134.134.6868

Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have

in the account after 5 years?in the account after 5 years?

Present Value – Present Value – single sumsingle sum

Calculator Solution:Calculator Solution:

I/Y = i =6I/Y = i =6

N = n = 5 N = n = 5 FV = FV = 100100

PV = PV = -74.73-74.73

00 5 5

PV = PV = FV = 100 FV = 100

Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is

the PV of that $100 if the interest rate is 6%?the PV of that $100 if the interest rate is 6%?

Mathematical Solution:Mathematical Solution:

PV = FV (PVIF PV = FV (PVIF i, ni, n ))

PV = 100 (PVIF PV = 100 (PVIF .06, 5.06, 5 ) (use PVIF table, or)) (use PVIF table, or)

PV = FV / (1 + i)PV = FV / (1 + i)nn

PV = 100 / (1.06)PV = 100 / (1.06)5 5 = = $74.73$74.73

00 5 5

PV = PV = -74.-74.7373 FV = 100 FV = 100

Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is

the PV of that $100 if the interest rate is 6%?the PV of that $100 if the interest rate is 6%?

Calculator Solution:Calculator Solution:

N = n = 5N = n = 5

PV = -5,000 PV = -5,000 FV = 11,933FV = 11,933

I/Y = i =I/Y = i =19%19%

00 5 5

PV = -5,000PV = -5,000 FV = 11,933 FV = 11,933

Present Value - single sumsPresent Value - single sumsIf you sold land for $11,933 that you bought 5 years If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?ago for $5,000, what is your annual rate of return?

Mathematical Solution:Mathematical Solution:

PV = FV (PVIF PV = FV (PVIF i, ni, n ) )

5,000 = 11,933 (PVIF 5,000 = 11,933 (PVIF ?, 5?, 5 ) )

PV = FV / (1 + i)PV = FV / (1 + i)nn

5,000 = 11,933 / (1+ i)5,000 = 11,933 / (1+ i)5 5

.419 = ((1/ (1+i).419 = ((1/ (1+i)55))

2.3866 = (1+i)2.3866 = (1+i)55

(2.3866)(2.3866)1/51/5 = (1+i) = (1+i) i = .19i = .19

Present Value - single sumsPresent Value - single sumsIf you sold land for $11,933 that you bought 5 years If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?ago for $5,000, what is your annual rate of return?

Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, 9.6% interest, compounded monthlycompounded monthly. How long . How long

will it take for your account to grow to $500?will it take for your account to grow to $500?

00

PV = PV = FV = FV =

Calculator Solution:Calculator Solution: FV = 500FV = 500 I/Y = i = 0.8I/Y = i = 0.8 PV = -100PV = -100 N = n = N = n = 202 months202 months

Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, 9.6% interest, compounded monthlycompounded monthly. How long . How long

will it take for your account to grow to $500?will it take for your account to grow to $500?

00 ? ?

PV = -100PV = -100 FV = 500 FV = 500

Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, 9.6% interest, compounded monthlycompounded monthly. How long . How long

will it take for your account to grow to $500?will it take for your account to grow to $500?

Mathematical Solution:Mathematical Solution:

PV = FV / (1 + i)PV = FV / (1 + i)nn

100 = 500 / (1+ .008)100 = 500 / (1+ .008)NN

5 = (1.008)5 = (1.008)NN

ln 5 = ln (1.008)ln 5 = ln (1.008)NN

ln 5 = N ln (1.008)ln 5 = N ln (1.008)

1.60944 = .007968 N1.60944 = .007968 N N = 202 monthsN = 202 months

Hint for single sum problems:Hint for single sum problems:

In every single sum future value In every single sum future value and present value problem, there and present value problem, there are 4 variables: are 4 variables:

FVFV, , PVPV, , ii, and , and nn When doing problems, you will be When doing problems, you will be

given 3 of these variables and given 3 of these variables and asked to solve for the 4th variable.asked to solve for the 4th variable.

Keeping this in mind makes “time Keeping this in mind makes “time value” problems much easier!value” problems much easier!

The Time Value of MoneyThe Time Value of Money

Compounding and DiscountingCompounding and Discounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4

AnnuitiesAnnuities

Annuity: a sequence of equal cash Annuity: a sequence of equal cash flows, occurring at the end of each flows, occurring at the end of each period.period.

0 1 2 3 4

Examples of Annuities:Examples of Annuities:

If you buy a bond, you will If you buy a bond, you will receive equal coupon interest receive equal coupon interest payments over the life of the payments over the life of the bond.bond.

If you borrow money to buy a If you borrow money to buy a house or a car, you will pay a house or a car, you will pay a stream of equal payments.stream of equal payments.

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

0 1 2 3

Calculator Solution:Calculator Solution:

I/Y= i = 8I/Y= i = 8 N = n = 3N = n = 3

PMT = -1,000 PMT = -1,000

FV = FV = $3,246.40$3,246.40

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

0 1 2 3

10001000 10001000 1000 1000

Calculator Solution:Calculator Solution:

I/Y= i = 8I/Y= i = 8 N = n = 3N = n = 3

PMT = -1,000 PMT = -1,000

FV = FV = $3,246.40$3,246.40

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

0 1 2 3

10001000 10001000 1000 1000

Mathematical Solution:Mathematical Solution:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

Mathematical Solution:Mathematical Solution:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

Mathematical Solution:Mathematical Solution:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii

FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 = - 1 = $3246.40$3246.40

.08 .08

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the

next 3 years, if the discount rate is 8%?next 3 years, if the discount rate is 8%?

0 1 2 3

Calculator Solution:Calculator Solution:

I/Y = i = 8I/Y = i = 8 N = n = 3N = n = 3

PMT = -1,000 PMT = -1,000

PV = PV = $2,577.10$2,577.10

0 1 2 3

10001000 10001000 1000 1000

Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the

next 3 years, if the discount rate is 8%?next 3 years, if the discount rate is 8%?

Calculator Solution:Calculator Solution:

I/Y = i = 8I/Y = i = 8 N = n = 3N = n = 3

PMT = -1,000 PMT = -1,000

PV = PV = $2,577.10$2,577.10

0 1 2 3

10001000 10001000 1000 1000

Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the

next 3 years, if the discount rate is 8%?next 3 years, if the discount rate is 8%?

Mathematical Solution:Mathematical Solution:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)

Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the

next 3 years, if the discount rate is 8%?next 3 years, if the discount rate is 8%?

Mathematical Solution:Mathematical Solution:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)

11

PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii

Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the

next 3 years, if the discount rate is 8%?next 3 years, if the discount rate is 8%?

Mathematical Solution:Mathematical Solution:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)

11

PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii

11

PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,577.10$2,577.10

.08.08

Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the

next 3 years, if the discount rate is 8%?next 3 years, if the discount rate is 8%?

Ordinary AnnuityOrdinary Annuity vs. vs.

Annuity Due Annuity Due

Earlier, we examined this Earlier, we examined this “ordinary” annuity:“ordinary” annuity:

Using an interest rate of 8%, we find Using an interest rate of 8%, we find that:that:

The The Future ValueFuture Value (at 3) is (at 3) is $3,246.40.$3,246.40.

The The Present ValuePresent Value (at 0) is (at 0) is $2,577.10.$2,577.10.

0 1 2 3

10001000 10001000 1000 1000

What about this annuity?What about this annuity?

Same 3-year time line,Same 3-year time line, Same 3 $1000 cash flows, butSame 3 $1000 cash flows, but The cash flows occur at the The cash flows occur at the

beginningbeginning of each year, rather of each year, rather than at the than at the endend of each year. of each year.

This is an This is an “annuity due.”“annuity due.”

0 1 2 3

10001000 1000 1000 1000 1000

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

0 1 2 3

Calculator Solution:Calculator Solution:

Mode = BEGIN I/Y = i = 8Mode = BEGIN I/Y = i = 8

N = n = 3N = n = 3 PMT = -1,000 PMT = -1,000

FV = FV = $3,506.11$3,506.11

0 1 2 3

-1000-1000 -1000 -1000 -1000 -1000

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

0 1 2 3

-1000-1000 -1000 -1000 -1000 -1000

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

Calculator Solution:Calculator Solution:

Mode = BEGIN I/Y = i = 8Mode = BEGIN I/Y = i = 8

N = n = 3N = n = 3 PMT = -1,000 PMT = -1,000

FV = FV = $3,506.11$3,506.11

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) (1 + i)) (1 + i)

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(use FVIFA table, or)

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) (1 + i)) (1 + i)

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(use FVIFA table, or)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii(1 + i)(1 + i)

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) (1 + i)) (1 + i)

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(use FVIFA table, or)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii

FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 = - 1 = $3,506.11$3,506.11

.08 .08

(1 + i)(1 + i)

(1.08)(1.08)

Present Value - annuity duePresent Value - annuity due What is the PV of $1,000 at the beginning of each What is the PV of $1,000 at the beginning of each

of the next 3 years, if the discount rate is 8%? of the next 3 years, if the discount rate is 8%?

0 1 2 3

Calculator Solution:Calculator Solution:

Mode = BEGIN I/Y = i = 8Mode = BEGIN I/Y = i = 8

N = n= 3N = n= 3 PMT = 1,000 PMT = 1,000

PV = PV = $2,783.26$2,783.26

0 1 2 3

10001000 1000 1000 1000 1000

Present Value - annuity duePresent Value - annuity due What is the PV of $1,000 at the beginning of each What is the PV of $1,000 at the beginning of each

of the next 3 years, if the discount rate is 8%? of the next 3 years, if the discount rate is 8%?

Calculator Solution:Calculator Solution:

Mode = BEGIN I/Y = i = 8Mode = BEGIN I/Y = i = 8

N = n= 3N = n= 3 PMT = 1,000 PMT = 1,000

PV = PV = $2,783.26$2,783.26

0 1 2 3

10001000 1000 1000 1000 1000

Present Value - annuity duePresent Value - annuity due What is the PV of $1,000 at the beginning of each What is the PV of $1,000 at the beginning of each

of the next 3 years, if the discount rate is 8%? of the next 3 years, if the discount rate is 8%?

Present Value - annuity duePresent Value - annuity due

Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)

Present Value - annuity duePresent Value - annuity due

Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)

11

PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii(1 + i)(1 + i)

Present Value - annuity duePresent Value - annuity due

Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)

11

PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii

11

PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,783.26$2,783.26

.08.08

(1 + i)(1 + i)

(1.08)(1.08)

Other Cash Flow PatternsOther Cash Flow Patterns

0 1 2 3

Is this an annuity?Is this an annuity? How do we find the PV of a cash flow How do we find the PV of a cash flow

stream when all of the cash flows are stream when all of the cash flows are different? (Use a 10% discount rate).different? (Use a 10% discount rate).

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

Uneven Cash FlowsUneven Cash Flows

Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.

Uneven Cash FlowsUneven Cash Flows

Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

Uneven Cash FlowsUneven Cash Flows

Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

periodperiod CF CF PV (CF)PV (CF)

00 -10,000 -10,000 -10,000.00-10,000.00

11 2,000 2,000 1,818.181,818.18

22 4,000 4,000 3,305.793,305.79

33 6,000 6,000 4,507.894,507.89

44 7,000 7,000 4,781.094,781.09

PV of Cash Flow Stream: $ 4,412.95PV of Cash Flow Stream: $ 4,412.95

00 11 22 33 44

-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000

ExampleExample Cash flows from an investment are Cash flows from an investment are

expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?

ExampleExample

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

Cash flows from an investment are Cash flows from an investment are expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?

This type of cash flow sequence is This type of cash flow sequence is often called a often called a “deferred annuity.”“deferred annuity.”

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

How to solve:How to solve:

1) 1) Discount each cash flow back to time Discount each cash flow back to time 0 separately.0 separately.

Or,Or,

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

2) 2) Find the PV of the annuity:Find the PV of the annuity:

PVPV3:3: End mode; I/YR = i = 20; End mode; I/YR = i = 20;

PMT = 40,000; N = n = 5 PMT = 40,000; N = n = 5

PVPV33= = $119,624$119,624

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

119,624119,624

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

Then discount this single sum back to Then discount this single sum back to time 0.time 0.

PV: End mode; I/YR = i = 20; PV: End mode; I/YR = i = 20;

N = n = 3; FV = 119,624; N = n = 3; FV = 119,624;

Solve: PV = $69,226Solve: PV = $69,226

119,624119,624

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

119,624119,62469,22669,226

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

119,624119,62469,22669,226

The PV of the cash flow The PV of the cash flow stream is $69,226.stream is $69,226.

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

ExampleExample

After graduation, you plan to invest After graduation, you plan to invest $400$400 per month per month in the stock market. in the stock market. If you earn If you earn 12%12% per year per year on your on your stocks, how much will you have stocks, how much will you have accumulated when you retire in accumulated when you retire in 30 30 yearsyears??

Retirement ExampleRetirement Example

After graduation, you plan to invest After graduation, you plan to invest $400$400 per month in the stock market. per month in the stock market. If you earn If you earn 12%12% per year on your per year on your stocks, how much will you have stocks, how much will you have accumulated when you retire in 30 accumulated when you retire in 30 years?years?

00 11 22 33 . . . 360. . . 360

400 400 400 400400 400 400 400

00 11 22 33 . . . 360. . . 360

400 400 400 400400 400 400 400

Using your calculator,Using your calculator,

N = n = 360 N = n = 360

PMT = -400PMT = -400

I/Y = i = 1I/Y = i = 1

FV = $1,397,985.65FV = $1,397,985.65

00 11 22 33 . . . 360. . . 360

400 400 400 400400 400 400 400

Retirement ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at

the end of year 30? the end of year 30?

Mathematical Solution:Mathematical Solution:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )

FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(can’t use FVIFA table)

Retirement ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at

the end of year 30? the end of year 30?

Mathematical Solution:Mathematical Solution:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )

FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(can’t use FVIFA table)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii

Retirement ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at

the end of year 30? the end of year 30?

Mathematical Solution:Mathematical Solution:

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )

FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(can’t use FVIFA table)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii

FV = 400 (1.01)FV = 400 (1.01)360360 - 1 = - 1 = $1,397,985.65$1,397,985.65

.01 .01

If you borrow If you borrow $100,000 at 7%$100,000 at 7% fixed fixed interest for interest for 30 years30 years in order to in order to buy a house, what will be your buy a house, what will be your

monthly house paymentmonthly house payment??

House Payment ExampleHouse Payment Example

House Payment ExampleHouse Payment Example

If you borrow $100,000 at 7% fixed If you borrow $100,000 at 7% fixed interest for 30 years in order to interest for 30 years in order to buy a house, what will be your buy a house, what will be your

monthly house payment?monthly house payment?

0 1 2 3 . . . 360

? ? ? ?

Using your calculator,Using your calculator,

N = n = 360N = n = 360

I/Y = i = 0.5833I/Y = i = 0.5833

PV = $100,000PV = $100,000

PMT = -$665.30PMT = -$665.30

00 11 22 33 . . . 360. . . 360

? ? ? ?? ? ? ?

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )

100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )

100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)

11

PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )

100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)

11

PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii

11

100,000 = PMT 1 - (1.005833 )100,000 = PMT 1 - (1.005833 )360360 PMT=$665.30PMT=$665.30

.005833.005833