Chapter 5. The Time Value of Money Simple Interest n Interest is earned on principal n $100 invested...

Chapter 5

The Time Value of Money

Simple Interest

Interest is earned on principalInterest is earned on principal $100 invested at 6% per year$100 invested at 6% per year 11stst year year interest is $6.00interest is $6.00 22ndnd year year interest is $6.00interest is $6.00 33rdrd year year interest is $6.00interest is $6.00 Total interest earned:$18Total interest earned:$18

Compound Interest

Interest is earned on previously earned interestInterest is earned on previously earned interest $100 invested at 6% with annual compounding$100 invested at 6% with annual compounding 11stst year year interest is $6.00interest is $6.00 Principal is $106Principal is $106 22ndnd year year interest is $6.36interest is $6.36 Principal is $112.36 Principal is $112.36 33rdrd year year interest is $6.74interest is $6.74 Principal is $119.11Principal is $119.11 Total interest earned: $19.10Total interest earned: $19.10

We know that receiving $1 today is worth We know that receiving $1 today is worth moremore than $1 in the future. This is due than $1 in the future. This is due toto opportunity costsopportunity costs..

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

Today Future

If we can measure this opportunity cost, we can:

Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..

Translate $1 in the future into its equivalent todayTranslate $1 in the future into its equivalent today (discounting)(discounting)..

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Today Future

Today

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Future

Future ValueFuture Value

Future Value - single sums

If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?

0 1

PV =PV = FV = FV =



Calculator Solution:Calculator Solution:

P/Y = 1P/Y = 1 I = 6I = 6

N = 1 N = 1 PV = -100 PV = -100

FV = FV = $106$106

00 1 1

PV = -100PV = -100 FV = FV = 106106



Mathematical Solution:Mathematical Solution:

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

FV = 100 (1.06)FV = 100 (1.06)1 1 = = $106$106

00 1 1

PV = -100PV = -100 FV = FV = 106106


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


P/Y = 1P/Y = 1 I = 6I = 6

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

FV = FV = $133.82$133.82

00 5 5

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


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


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

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

you have in the account after 5 years?

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

in the account after 5 years?

0 ?PV =PV = FV = FV =


P/Y = 4P/Y = 4 I = 6I = 6

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

FV = FV = $134.68$134.68

00 20 20

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




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 sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have




0 ?

PV =PV = FV = FV =


P/Y = 12P/Y = 12 I = 6I = 6

N = 60 N = 60 PV = PV = -100-100

FV = FV = $134.89$134.89

00 60 60

PV = -100PV = -100 FV = FV = 134.134.8989




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

FV = 100 (1.005)FV = 100 (1.005)60 60 = = $134.89$134.89

00 60 60

PV = -100PV = -100 FV = FV = 134.134.8989



Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?

Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?

0 ?

PV =PV = FV = FV =

Present ValuePresent Value

Present Value - single sumsIf you receive $100 one year from now, what is the

PV of that $100 if your opportunity cost is 6%?

0 ?

PV =PV = FV = FV =


P/Y = 1P/Y = 1 I = 6I = 6

N = 1 N = 1 FV = FV = 100100

PV = PV = -94.34-94.34



PV = PV = -94.-94.3434 FV = 100 FV = 100

00 1 1


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

PV = 100 / (1.06)PV = 100 / (1.06)1 1 = = $94.34$94.34



PV = PV = -94.-94.3434 FV = 100 FV = 100

00 1 1

Present Value - single sumsIf you receive $100 five years from now, what is the




0 ?

PV =PV = FV = FV =


P/Y = 1P/Y = 1 I = 6I = 6

N = 5 N = 5 FV = FV = 100100

PV = PV = -74.73-74.73



00 5 5

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


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 sumsWhat is the PV of $1,000 to be received 15 years

from now if your opportunity cost is 7%?



00 15 15

PV = PV = FV = FV =


P/Y = 1P/Y = 1 I = 7I = 7

N = 15 N = 15 FV = FV = 1,0001,000

PV = PV = -362.45-362.45



00 15 15

PV = PV = -362.-362.4545 FV = 1000 FV = 1000


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

PV = 1000/ (1.07)PV = 1000/ (1.07)15 15 = = $362.45$362.45



00 15 15

PV = PV = -362.-362.4545 FV = 1000 FV = 1000

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


00 5 5

PV = PV = FV = FV =


P/Y = 1P/Y = 1 N = 5N = 5

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

I = I = 19%19%

00 5 5

PV = -5000PV = -5000 FV = 11,933 FV = 11,933



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 = i = .19.19


Hint for single sum problems:

In every single sum future value and In every single sum future value and present value problem, there are 4 present value problem, there are 4 variables: 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 asked to given 3 of these variables and asked to solve for the 4th variable.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!


Compounding and DiscountingCompounding and Discounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4

Annuities

Annuity:Annuity: a sequence of a sequence of equalequal cash cash flowsflows, occurring at the , occurring at the endend of each of each period.period.

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

0 1 2 3 4

Annuities

Examples of Annuities:

If you buy a bond, you will If you buy a bond, you will receive equal semi-annual coupon receive equal semi-annual coupon interest payments over the life of interest payments over the life of the bond.the 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 - annuityIf you invest $1,000 each year at 8%, how much

would you have after 3 years?



0 1 2 3


P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3

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

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



0 1 2 3

10001000 10001000 1000 1000


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



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

next 3 years, if the opportunity cost is 8%?

0 1 2 3


P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3

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

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

0 1 2 3

10001000 10001000 1000 1000




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



Other Cash Flow Patterns

0 1 2 3


Perpetuities

Suppose you will receive a fixed Suppose you will receive a fixed payment every period (month, year, payment every period (month, year, etc.) forever. This is an example of etc.) forever. This is an example of a perpetuity.a perpetuity.

You can think of a perpetuity as an You can think of a perpetuity as an annuityannuity that goes on that goes on foreverforever..

PMT i

PV =

The PV of a perpetuity is very The PV of a perpetuity is very simple to find:simple to find:

Present Value of a PerpetuityPresent Value of a Perpetuity

What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?


PMT $10,000PMT $10,000 i .08 i .08

PV = =PV = =


PMT $10,000PMT $10,000 i .08 i .08

= $125,000= $125,000

PV = =PV = =

Ordinary Annuity vs.

Annuity Due

$1000 $1000 $1000

4 5 6 7 8

Begin Mode vs. End Mode

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8


1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 5 6 7


1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 5 6 7

PVPVinin

ENDENDModeMode

FVFVinin

ENDENDModeMode


1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 6 7 8


1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 6 7 8

PVPVinin

BEGINBEGINModeMode

FVFVinin

BEGINBEGINModeMode

Earlier, we examined this “ordinary” annuity:

0 1 2 3

10001000 10001000 1000 1000

Earlier, we examined this “ordinary” annuity:

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

Is this an Is this an annuityannuity?? 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 Flows

00 1 1 2 2 3 3 4 4

-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.

00 1 1 2 2 3 3 4 4

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

Uneven Cash Flows

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 1 1 2 2 3 3 4 4

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

Example

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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 annuitydeferred annuity.”.”

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

How to solve:How to solve:

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

Or,Or,

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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

PVPV3:3: End mode; P/YR = 1; I = 20; End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PMT = 40,000; N = 5

PVPV33= = $119,624$119,624

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

119,624119,624

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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

PV: End mode; P/YR = 1; I = 20; PV: End mode; P/YR = 1; I = 20;

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

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

119,624119,624

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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

69,22669,226

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

119,624119,624

Retirement Example

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 3030 yearsyears??

Retirement 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,

P/YR = 12P/YR = 12

N = 360 N = 360

PMT = -400PMT = -400

I%YR = 12I%YR = 12

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

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

400 400 400 400400 400 400 400

If you borrow If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years 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 Example

House Payment Example

If you borrow If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years in order to years in order to buy a house, what will be your buy a house, what will be your

monthly house payment?monthly house payment?

Using your calculator,Using your calculator,

P/YR = 12P/YR = 12

N = 360N = 360

I%YR = 7I%YR = 7

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

PMT = PMT = -$665.30-$665.30

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

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Chapter 5. The Time Value of Money Simple Interest n Interest is earned on principal n $100 invested...

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Transcript of Chapter 5. The Time Value of Money Simple Interest n Interest is earned on principal n $100 invested...