Chapter 5 Time Value of Money Handout

8/15/2019 Chapter 5 Time Value of Money Handout

http://slidepdf.com/reader/full/chapter-5-time-value-of-money-handout 1/12Page

Chapter 5 - The Time

Value of Money

©©©© 2005, Pearson Prentice Hall

The Time Value of Money

Compounding and

Discounting Single Sums

We know that receiving P 1.00 today is worth

more than P 1.00 in the future. This is due

to opportunity costs.

The opportunity cost of receiving P 1.00 in the

future is the interest we could have earned if

we had received the P 1.00 sooner.

Today Future

If we can measure this opportunity

cost, we can:

Translate P 1.00 today into its equivalent in the

future (compounding).

Translate P 1.00 in the future into its equivalent

today (discounting).

?

Today Future

Today

?

Future

Compound Interest

and Future Value

Future Value - single sums

If you deposit P 100.00 in an account earning 6.0%, how

much would you have in the account after 1 year?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 1 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1 = P 106.00

0 1

PV = -100 FV = 106



Future Value - single sums

If you deposit P 100.00 in an account earning 6.0%, how

much would you have in the account after 5 years?



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

FV = PV (1 + i)n

FV = 100 (1.06)5 = P 133.82

0 5

PV = -100 FV = 133.82



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

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

FV = 100 (1.015)20 = P 134.68

0 20

PV = -100 FV = 134.68

Future Value - single sumsIf you deposit P 100.00 in an account earning 6.0% with

quarterly compounding, how much would you have in

the account after 5 years?



FV = 100 (FVIF .005, 60 ) (can’t use FVIF table)

FV = PV (1 + i/m)

m x n

FV = 100 (1.005)60 = P 134.89

0 60

PV = -100 FV = 134.89

Future Value - single sumsIf you deposit P 100.00 in an account earning 6.0% with

monthly compounding, how much would you have in the

account after 5 years?

0 100

PV = -1000 FV = 2.98M

Future Value - continuous compoundingWhat is the FV of P 1,000.00 earning 8.0% with

continuous compounding, after 100 years?


FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = P 2,980,957.99

Present Value Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .06, 1 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)1 = P 94.34

PV = -94.34 FV = 100

0 1

Present Value - single sumsIf you receive P 100.00 one year from now, what is the

PV of that P 100.00 if your opportunity cost is 6.0%?






PV = FV / (1 + i)n

PV = 100 / (1.06)5 = P 74.73

Present Value - single sumsIf you receive P 100.00 five years from now, what is the

PV of that P 100.00 if your opportunity cost is 6.0%?

0 5

PV = -74.73 FV = 100




PV = FV / (1 + i)n

PV = 100 / (1.07)15 = P 362.45

Present Value - single sumsWhat is the PV of P 1,000.00 to be received 15

years from now if your opportunity cost is 7.0%?

0 15

PV = -362.45 FV = 1000

Calculator Solution:

P/Y = 1 N = 5

PV = -5,000 FV = 11,933

I = 19%

0 5

PV = -5,000 FV = 11,933

Present Value - single sumsIf you sold land for P 11,933.00 that you bought 5

years ago for P 5,000.00, what is your annual rate

of return?



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

PV = FV / (1 + i)n

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

.419 = ((1/ (1+i)5)

2.3866 = (1+i)5

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

Present Value - single sumsIf you sold land for P 11,933.00 that you bought 5 years

ago for P 5,000.00, what is your annual rate of return?

Present Value - single sumsSuppose you placed P 100.00 in an account that pays

9.6% interest, compounded monthly. How long will it

take for your account to grow to P 500.00?


PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)

1.60944 = .007968 N N = 202 months

Hint for single sum problems:

In every single sum present value andfuture value problem, there are four

variables:FV, PV, i and n.

When doing problems, you will be giventhree variables and you will solve for thefourth variable.

Keeping this in mind makes solving timevalue problems much easier!




Compounding and Discounting

Cash Flow Streams

0 1 2 3 4

Annuity: a sequence of equal cash

flows, occurring at the end of each

period.

0 1 2 3 4

Annuities

If you buy a bond, you will

receive equal semi-annual coupon

interest payments over the life of

the bond.

If you borrow money to buy a

house or a car, you will pay a

stream of equal payments.

Examples of Annuities:


P/Y = 1 I = 8 N = 3

PMT = -1,000

FV = P 3,246.40

Future Value - annuityIf you invest P 1,000.00 each year at 8.0%, how

much would you have after 3 years?

0 1 2 3

1000 1000 1000


FV = PMT (FVIFA i, n )FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or)

FV = PMT (1 + i)n - 1

i

FV = 1,000 (1.08)3 - 1 = P 3,246.40

0.08

Future Value - annuityIf you invest P 1,000.00 each year at 8.0%, how

much would you have after 3 years?


P/Y = 1 I = 8 N = 3

PMT = -1,000

PV = P 2,577.10

0 1 2 3

1000 1000 1000

Present Value - annuityWhat is the PV of P 1,000.00 at the end of each of

the next 3 years, if the opportunity cost is 8.0%?




PV = PMT (PVIFA i, n )

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

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = P 2,577.10

0.08

Present Value - annuityWhat is the PV of P 1,000.00 at the end of each of

the next 3 years, if the opportunity cost is 8.0%?

Other Cash Flow Patterns

0 1 2 3


Perpetuities

Suppose you will receive a fixed

payment every period (month, year,

etc.) forever. This is an example of

a perpetuity.

You can think of a perpetuity as an

annuity that goes on forever.

Present Value of a

Perpetuity

When we find the PV of an annuity,

we think of the following

relationship:


Mathematically,

(PVIFA i, n ) =

We said that a perpetuity is an

annuity where n = infinity. What

happens to this formula when n

gets very, very large?

1 -1

(1 + i)n

i

When n gets very large,

this becomes zero.

So we’re left with PVIFA =

1

i

1 -1

(1 + i)n

i



PMT

iPV =

So, the PV of a perpetuity is very

simple to find:

Present Value of a Perpetuity What should you be willing to pay in

order to receive P 10,000.00

annually forever, if you require

8.0% per year on the investment?

PMT P 10,000.00

i 0.08PV = =

= P 125,000.00

Ordinary Annuity

vs.

Annuity Due

P 1,000.00 P 1,000.00 P 1,000.00

4 5 6 7 8

Begin Mode vs. End Mode

P 1,000.00 P 1,000.00 P 1,000.00

4 5 6 7 8year year year

5 6 7

PVin

END

Mode

FVin

END

Mode

Begin Mode vs. End Mode

P 1,000.00 P 1,000.00 P 1,000.00

4 5 6 7 8

year year year

6 7 8

PVin

BEGIN

Mode

FVin

BEGIN

Mode

Earlier, we examined this

“ordinary” annuity:

Using an interest rate of 8.0%, we

find that:

The Future Value (at 3) is

P 3,246.40.

The Present Value (at 0) is

P 2,577.10.

0 1 2 3

1,000 1,000 1,000



What about this annuity?

Same 3-year time line,

Same 3 P 1,000.00 cash flows, but

The cash flows occur at the

beginning of each year, rather

than at the end of each year.

This is an “annuity due.”

0 1 2 3

1000 1000 1000


Mode = BEGIN P/Y = 1 I = 8

N = 3 PMT = -1,000

FV = P 3,506.11

0 1 2 3

-1000 -1000 -1000

Future Value - annuity dueIf you invest P 1,000.00 at the beginning of each of

the next 3 years at 8.0%, how much would you

have at the end of year 3?

0 1 2 3

-1000 -1000 -1000






N = 3 PMT = -1,000

FV = P 3,506.11




Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

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

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

FV = PMT (1 + i)n - 1

i

FV = 1,000 (1.08)3 - 1 = P 3,506.11

.08

(1 + i)

(1.08)



N = 3 PMT = 1,000

PV = P 2,783.26

0 1 2 3

1,000 1,000 1,000

Present Value - annuity dueWhat is the PV of P 1,000.00 at the beginning of each of

the next 3 years, if your opportunity cost is 8.0%?

Present Value - annuity due

Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

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

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

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = P 2,783.26

.08

(1 + i)

(1.08)



Is this an annuity?

How do we find the PV of a cash flow

stream when all of the cash flows are

different? (Use a 10% discount rate.)

Uneven Cash Flows

0 1 2 3 4

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

Sorry! There’s no quickie for this one.

We have to discount each cash flow

back separately.

0 1 2 3 4

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

Uneven Cash Flows

period CF PV (CF)

0 -P 10,000.00 -P 10,000.00

1 2,000.00 1,818.18

2 4,000.00 3,305.79

3 6,000.00 4,507.89

4 7,000.00 4,781.09PV of Cash Flow Stream: P 4,412.95

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000Annual Percentage Yield (APY)

Which is the better loan:

8% compounded annually, or

7.85% compounded quarterly?

We can’t compare these nominal (quoted)

interest rates, because they don’t include the

same number of compounding periods per

year!

We need to calculate the APY.

Annual Percentage Yield (APY)

Find the APY for the quarterly loan:

The quarterly loan is more expensive than

the 8.0% loan with annual compounding!

APY = ( 1 + ) m - 1quoted rate

m

APY = ( 1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4

First Group Project



Instructions

Get the best commercial bank interest rate forthe following initial deposits: P 1,000.00,

P 5,000.00, and P 10,000.00 for one (1) year. The best rate shall come from a survey/canvass

of at least three banks.

List down the following information: name ofbank, branch, the name of the authorizedrepresentative of the bank, and the contactnumber.

Instructions

On the next meeting, each group will reveal

the rates for the three initial deposits.

The group with the best rates will get thehighest score (10 points).

Each group will submit a one page report for

the interest rates obtained to be submitted on

the next meeting.

The next meeting will be next Wednesday.

Practice Problems Example

0 1 2 3 4 5 6 7 8

P 0 0 0 0 40 40 40 40 40

Cash flows from an investment are

expected to be P 40,000.00 per year at

the end of years 4, 5, 6, 7, and 8. If you

require a 20.0% rate of return, what is

the PV of these cash flows?

This type of cash flow sequence is

often called a “deferred annuity.”

0 1 2 3 4 5 6 7 8

P 0 0 0 0 40 40 40 40 40

2) Find the PV of the annuity:

PV3: End mode; P/YR = 1; I = 20;

PMT = 40,000; N = 5

PV3= P 119,624.00

0 1 2 3 4 5 6 7 8

P 0 0 0 0 40 40 40 40 40



Then discount this single sum back to

time 0.

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

N = 3; FV = 119,624;

Solve: PV = P 69,226.00

P 119,624.00

0 1 2 3 4 5 6 7 8

P 0 0 0 0 40 40 40 40 40

The PV of the cash flow

stream is P 69,226.00.

P 69,226.00

0 1 2 3 4 5 6 7 8

P 0 0 0 0 40 40 40 40 40

P 119,624.00

Retirement Example

After graduation, you plan to invest

P 400.00 per month in the stock

market. If you earn 12.0% per year

on your stocks, how much will you

have accumulated when you retire in

30 years?

0 1 2 3 . . . 360

400 400 400 400

Using your calculator,

P/YR = 12

N = 360

PMT = -400

I%YR = 12

FV = P 1,397,985.65

0 1 2 3 . . . 360

400 400 400 400

Retirement Example

If you invest P 400.00 at the end of each month for




FV = PMT (FVIFA i, n )

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

FV = PMT (1 + i)n - 1

i

Retirement Example

If you invest P 400.00 at the end of each month for




FV = PMT (FVIFA i, n )

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

FV = PMT (1 + i)n - 1

i

FV = 400 (1.01)360 - 1 = P 1,397,985.65

.01



House Payment Example

If you borrow P 100,000.00 at 7.0%

fixed interest for 30 years in order

to buy a house, what will be your

monthly house payment?

0 1 2 3 . . . 360

? ? ? ?


P/YR = 12

N = 360

I%YR = 7

PV = P 100,000.00PMT = -P 665.30

0 1 2 3 . . . 360

? ? ? ? House Payment Example



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

1

PV = PMT 1 - (1 + i)n

i

1

100,000 = PMT 1 - (1.005833 )360

PMT=P 665.30.005833

Team Assignment

Upon retirement, your goal is to spend 5

years traveling around the world. To

travel in style will require P 250,000.00per year at the beginning of each year.

If you plan to retire in 30 years, what are

the equal monthly payments necessary

to achieve this goal? The funds in your

retirement account will compound at

10.0% annually.

How much do we need to have by

the end of year 30 to finance thetrip?

PV30 = PMT (PVIFA .10, 5) (1.10) =

= 250,000 (3.7908) (1.10) =

= P 1,042,470.00

27 28 29 30 31 32 33 34 35

250 250 250 250 250


http://slidepdf.com/reader/full/chapter-5-time-value-of-money-handout 12/12P


Mode = BEGIN

PMT = -P 250,000.00

N = 5

I%YR = 10

P/YR = 1

PV = P 1,042,466.00

27 28 29 30 31 32 33 34 35

250 250 250 250 250

Now, assuming 10.0% annual

compounding, what monthly

payments will be required for you

to have P 1,042,466.00 at the end of

year 30?

27 28 29 30 31 32 33 34 35

250 250 250 250 250

P 1,042,466

• Using your calculator,

Mode = END

N = 360

I%YR = 10

P/YR = 12

FV = P 1,042,466.00PMT = -P 461.17

27 28 29 30 31 32 33 34 35

250 250 250 250 250

P 1,042,466.00

So, you would have to place P 461.17 in

your retirement account, which earns

10.0% annually, at the end of each of the

next 360 months to finance the 5-yearworld tour.

Chapter 5 Time Value of Money Handout

Documents

Transcript of Chapter 5 Time Value of Money Handout