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FINS5514-fins5514代写

时间：2023-06-30

FINS5514: Capital Budgeting and

Financing Decisions

Lecture 2: Investment Decision-I

Topics Covered Today

_______________________________________________________________________________

• The Time Value of Money

– Future Value and Compounding

– Present Value and Discounting

– Future and Present Values of Multiple Cash Flows

– Valuing Level Cash Flows: Annuities and Perpetuities

– Comparing Rates: The Effect of Compounding

2-2

2-3

Basic Definitions

_________________________________________________________________________________

• Present Value – earlier money on a time line (ie value

today)

• Future Value – later money on a time line (ie value at

some time in the future)

• Interest rate – “exchange rate” between earlier money

and later money

– Discount rate

– Cost of capital

– Opportunity cost of capital

– Required return

• time value of money

• On a time line:

– Time 0 is today,

– Time 1 is one period from today or the end of Period 1

and the start of Period 2, and so on.

• Information on a time line is written as:

– Cash flows are written above the tick marks

– Unknown cash flows are denoted with a question mark

– Interest rates are written above the line and between

the tick marks.

Time lines

_________________________________________________________________________________

• Here there is a cash outflow of $100 at time 0 (note

the minus sign),

• The interest rate is 10% in period 1 (t=0 to t=1) and

5% in both period 2 and period 3.

• There are no cash flows at times 1 and 2. There is a

cash flow at time 3 exists but the value is unknown

0 1 2 3

-100 ?

Time

Cash flows

10% 5%

2-6

Future Values

_________________________________________________________________________________

• Suppose you invest $1000 for one year at 5% per

year. What is the future value in one year?

– Interest = 1000(.05) = 50

– Value in one year = principal + interest = 1000 + 50 = 1050

– Future Value (FV) = 1000(1 + .05) = 1050

• Suppose you leave the money in for another year.

How much will you have two years from now?

– FV = 1050 (1.05) = 1102.50

– ie 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

2-7

Future Values: General Formula

_________________________________________________________________________________

FV = PV(1 + r)t

– FV = future value

– PV = present value

– r = period interest rate, expressed as a decimal

– T = number of periods

• Future value interest factor = (1 + r)t

2-8

Effects of Compounding

_________________________________________________________________________________

• Simple interest

– assumes that interest rate paid is a flat percentage of the

principal (P) each period

• Compound interest

– interest is earned/paid on the principal plus any

accumulated interest determined since the start of the

deposit/loan.

• Consider the previous example

– FV with simple interest = 1000 + 50 + 50 = 1100

– FV with compound interest = 1102.50

– The extra 2.50 comes from the interest of .05(50) = 2.50

earned on the first interest payment

2-9

Future Values – Example 2

_________________________________________________________________________________

• Suppose you invest the $1000 from the previous

example for 5 years. How much would you have?

– FV = 1000(1.05)5 = 1276.28

• The effect of compounding is small for a small number

of periods, but increases as the number of periods

increases. (Simple interest would have a future value of

$1250, for a difference of $26.28.)

2-10

Future Values – Example 3

_________________________________________________________________________________

• Suppose you had a relative deposit $10 at 5.5% interest

200 years ago. How much would the investment be

worth today?

• What is the effect of compounding?

2-11

Present Values

_________________________________________________________________________________

• How much do I have to invest today to have some

amount in the future?

– FV = PV(1 + r)t

– Rearrange to solve for :

PV = FV / (1 + r)t

• Discounting ≈ present value of some future amount

• “value” generally refers present value unless we

specifically indicate that we want the future value.

2-12

Present Value – One Period Example

_________________________________________________________________________________

• Suppose you need $10,000 in one year for the down

payment on a new car. If you can earn 7% annually,

how much do you need to invest today?

PV = 10,000 / (1.07)1 = 9345.79

2-13

Present Values – Example 2

_________________________________________________________________________________

_

• Your parents set up a trust fund for you 10 years ago

that is now worth $19,671.51. If the fund earned 7%

per year, how much did your parents invest?

PV = 19,671.51 / (1.07)10 = 10,000

2-14

Present Value – Important Relationship I

_________________________________________________________________________________

• For a given interest rate – the longer the time

period, the lower the present value

– What is the present value of $500 to be received in 5

years? 10 years? The discount rate is 10%

5 years: PV = 500 / (1.1)5 = 310.46

10 years: PV = 500 / (1.1)10 = 192.77

2-15

Present Value – Important Relationship II

_________________________________________________________________________________

• For a given time period – the higher the interest

rate, the smaller the present value

– What is the present value of $500 received in 5 years

if the interest rate is 10%? 15%?

Rate = 10%: PV = 500 / (1.1)5 = 310.46

Rate = 15%: PV = 500 / (1.15)5 = 248.59

2-16

Discount Rate

_________________________________________________________________________________

_

• Often we will want to know what the implied

interest rate is in an investment

• Rearrange the basic PV equation and solve for r

FV = PV(1 + r)t

r = (FV / PV)1/t – 1

2-17

Discount Rate – Example 1

_________________________________________________________________________________

• You are looking at an investment that will pay $1200

in 5 years if you invest $1000 today. What is the

implied rate of interest?

r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%

2-18

Discount Rate – Example 2

________________________________________________________________________________

• Suppose you have a 1-year old son and you want to

provide $75,000 in 17 years towards his college

education. You currently have $5000 to invest. What

interest rate must you earn to have the $75,000 when

you need it?

2-19

Finding the Number of Periods

_________________________________________________________________________________

• Start with basic equation and solve for t (remember

your logs)

FV = PV(1 + r)t

t = ln(FV / PV) / ln(1 + r)

2-20

Number of Periods – Example 1

_________________________________________________________________________________

• You want to purchase a new car and you are willing

to pay $20,000. If you can invest at 10% per year

and you currently have $15,000, how long will it be

before you have enough money to pay cash for the

car?

t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

2-21

Multiple Cash Flows –Future Value Example 1

____________________________________________________________________________________

• You are able to deposit $4000 at the end of the next 3

years in a bank account paying 8% interest. You

currently have $7000 in the account. How much will you

have in 3 years? In 4 years?

• Find the value at year 3 of each cash flow and add them

together.

2-22

cash flow 7000 4000 4000 4000

|______|_____|_____|_____|

period 0 1 2 3 4

Today (year 0): FV = 7000(1.08)3 = 8,817.98

Year 1: FV = 4,000(1.08)2 = 4,665.60

Year 2: FV = 4,000(1.08) = 4,320

Year 3: value = 4,000

Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000

= 21,803.58

• Value at year 4 = 21,803.58(1.08) = 23,547.87

2-23

Multiple Cash Flows – FV Example 2

_________________________________________________________________________________

• Suppose you invest $500 in a mutual fund today and

$600 in one year. If the fund pays 9% annually, how

much will you have in two years?

2-24

Multiple Cash Flows – Example 2 Continued

• How much will you have in 5 years if you make no

further deposits?

• First way:

– FV = 500(1.09)5 + 600(1.09)4 = 1616.26

• Second way – use value at year 2:

– FV = 1248.05(1.09)3 = 1616.26

2-25

Multiple Cash Flows – Present Value Example 3

• You are offered an investment that will pay you

$200 in one year,$400 the next year, $600 the next

year and $800 at the end of the fourth year. You can

earn 12% on very similar investments. What is the

most you should pay for this one?

cash flow 200 400 600 800

|______|_____|_____|_____|

period 0 1 2 3 4

• Find the PV of each cash flows and add them

– Year 1 CF: 200 / (1.12)1 = 178.57

– Year 2 CF: 400 / (1.12)2 = 318.88

– Year 3 CF: 600 / (1.12)3 = 427.07

– Year 4 CF: 800 / (1.12)4 = 508.41

– Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93

2-24

2-24

2-27

Annuities and Perpetuities Defined

________________________________________________________________________________

• Annuity – finite series of equal payments that occur

at regular intervals

– If the first payment occurs at the end of the period, it

is called an ordinary annuity

– If the first payment occurs at the beginning of the

period, it is called an annuity due or annuity in

advance

• Perpetuity – infinite series of equal payments

2-28

Ordinary annuity

_________________________________________________________________________________

• Annuity paid over n periods.

• The first cash flow (R1) occurs at the end of the first

period.

• All cash flows in the annuity need to be discounted to

t=0.

Cash flow R1 R2 R3 R4 ... Rn-1 Rn

|___|___|___|___|___|____________|___|

Period 0 1 2 3 4 n-1 n

2-29

Annuities Basic Formulas

____________________________________________________________________________________

• Annuities:

−+

=

+

−

=

r

rCFV

r

rCPV

t

t

1)1(

)1(

11

2-30

To get the formulas :

• Assume an annuity of $1 for n periods, where the

periodical compound interest rate is r.

Cash flow $1 $1 $1 $1 ... $1 $1

|___|___|___|___|___|____________|___|

Period 0 1 2 3 4 n-1 n

• Then the future value of this stream of cash flows is

FV = 1 + 1(1 + r)1 + 1(1 + r)2 + . . . . . + 1(1 + r)n-1 (A)

• Multiply each side of (A) by (1+r):

FV(1+r) = (1+r)1 + (1+r)2 + (1+r)3 + . . . + (1+r)n (B)

• Subtract (A) from (B)

FV(1 + r) - FV = (1 +r)n - 1

2-31

• and simplifying:

• Using a similar technique we find that the present

value of a series of cash flows of $1 for n periods, at a

rate of r

]1 - ) + 1 ( [1=

n

r

rFV

2-32

Annuity – Sweepstakes Example

_________________________________________________________________________________

• Suppose you win the Publishers Clearinghouse $10

million sweepstakes. The money is paid in equal annual

installments of $333,333.33 over 30 years. If the

appropriate discount rate is 5%, how much is the

sweepstakes actually worth today?

PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29

2-33

Buying a House

_________________________________________________________________________________

• You are ready to buy a house and you have $20,000

for a down payment and closing costs. Closing costs

are estimated to be 4% of the loan value. You have

an annual salary of $36,000 and the bank is willing

to allow your monthly mortgage payment to be

equal to 28% of your monthly income. The interest

rate on the loan is 6% per year with monthly

compounding (.5% per month) for a 30-year fixed

rate loan. How much money will the bank loan you?

How much can you offer for the house?

2-34

Buying a House – Continued

_________________________________________________________________________________

• Bank loan

Monthly income = 36,000 / 12 = 3,000

Maximum payment = .28(3,000) = 840

PV = 840[1 – 1/1.005360] / .005 = 140,105

• Total Price

Closing costs = .04(140,105) = 5,604

Down payment = 20,000 – 5604 = 14,396

Total Price = 140,105 + 14,396 = 154,501

2-35

Finding the Payment

_________________________________________________________________________________

• Suppose you want to borrow $20,000 for a new car. You

can borrow at 8% per year, compounded monthly (8/12

= .66667% per month). If you take a 4 year loan, what is

your monthly payment?

20,000 = C[1 – 1 / 1.006666748] / .0066667

C = 488.26

2-36

Finding the Number of Payments

____________________________________________________________________________________

• Suppose you borrow $2000 at 5% and you are going to

make annual payments of $734.42. How long before you

pay off the loan?

2000 = 734.42(1 – 1/1.05t) / .05

0.136161869 = 1 – 1/1.05t

1/1.05t = 0.863838131

1.157624287 = 1.05t

t = ln(1.157624287) / ln(1.05) = 3 years

2-37

Future Values for Annuities

_________________________________________________________________________________

• Suppose you begin saving for your retirement by

depositing $2000 per year in an IRA. If the interest

rate is 7.5%, how much will you have in 40 years?

FV = 2000(1.07540 – 1)/.075 = 454,513.04

2-38

Annuity due

_________________________________________________________________________________

• The first cash flow occurs at the beginning of the

first period. Notice that R1 occurs at t=0.

• It is already discounted or expressed in net present

value terms.

• When calculating the present value of the annuity

only the remaining n-1 payments need to be

discounted then added to the value of the first cash

flow.

cash flow R1 R2 R3 R4 ... Rn

|___|___|___|___|___|____________|___|

period 0 1 2 3 4 n-1 n

2-39

Annuity Due- FV

_________________________________________________________________________________

• You are saving for a new house and you put $10,000

per year in an account paying 8%. The first payment

is made today. How much will you have at the end

of 3 years?

cash flow 10000 10000 10000

|______|_____|_____|

period 0 1 2 3

FV = (10,000[(1.083 – 1) / .08]) *(1.08) = 35,061.12

• If you use the regular annuity formula, the FV will occur

at the same time as the last payment (ie year 2).

• To get the value at the end of the third period, you have

to take it forward one more period.

x (1 + r)

2-38

2-38

• Calculate the present value of an annuity due that

pays $150 each period, lasts for 3 periods and has

an interest rate of 2%

cash flow 150 150 150

|______|_____|_____|

period 0 1 2 3

PV = 150 1 – 1 x (1+0.02) = $441.23

[0.02 * (1.02)3]

0.02

Annuity Due- PV

_________________________________________________________________________________

2-42

Deferred Annuity

_________________________________________________________________________________

• A deferred annuity refers to a stream of fixed payments

which begin sometime in the future, i.e. the first

payment is delayed.

• A typical case would be a pension which does not begin

until age 60 or an inheritance paid in fixed instalments

beginning after a particular date.

2-43

Methods

_________________________________________________________________________________

• Method 1

– Let b be the delay period and assume that the annuity is

an ordinary annuity

– Calculate the present value of the annuity at the beginning

of period b. To calculate the PV of this stream at time zero

(today) discount the above annuity to time 0, by dividing

by (1+r)b-1

cash flow R1 R2 ... Rn

|___|___|___|___|___|____________|

period 0 1 … b-1 b n

• Method 2

– Take the present value of annuity paid for n years and

subtract an annuity paid over b-1 periods

cash flow R1 R2 ... Rn

|___|___|___|___|___|____________|

period 0 1 … b-1 b n

2-41

2-44

2-45

Deferred Annuity example

_________________________________________________________________________________

• Determine the present value of a $250 annuity, payable

annually for five years when the first payment occurs at

the end of the seventh year. Use a discount rate of 11%

p.a.

cash flow 250 250 250 250 250

|___|___|___|___|___|___|___|

period 0 1 … 6 7 8 9 10 11

PV = [250[1 – 1/1.115] / .11]*(1.11)-6 or

PV = [250[1 – 1/1.1111] / .11]-[250[1 – 1/1.116] / .11]

2-46

Perpetuities

_________________________________________________________________________________

• A perpetuity involves the payment of a stream of fixed

payments where cash flows continue forever.

PV = C / r

• The formula is determined by considering that the

number of compounding periods is infinite

(continuously compounded) and finding the limits as n

goes to infinity

2-47

Perpetuity – Example

_________________________________________________________________________________

• Company B wants to sell preference shares at $100

per share. A similar issue of shares already issued

has a price of $40 and offers a dividend of $1 per

quarter. What dividend does the company have to

offer if the shares are going to sell?

PV = $40, CF = $1 per quarter

• Current required return:

40 = 1 / r

r = .025 or 2.5% per quarter

• Dividend for new preferred:

100 = C / .025

C = 2.50 per quarter

• Often annuities and perpetuities grow over

time.

• For example, income from an investment

could start at a set amount $C and then grow

by a fixed percentage (g) every year.

• In this situation, the growth rate must be

taken into account

Constant Growth Rates

_________________________________________________________________________________

62

• A growing stream of cash flows that lasts forever

0 1 2 3

……

C C×(1+g) C ×(1+g)2

+ + +L

C ×(1+ g)2

(1+ r) (1+ r)2 (1+ r)3

C C ×(1+ g)

PV =

• The formula for the present value of a growing

perpetuity is (if gPVGrowingPerpetuity =

C

r -g

Growing Perpetuity

_________________________________________________________________________________

63

Growing perpetuity - example

0

…

1 2 3

$1.30 $1.30×(1.05) $1.30 ×(1.05)2

= $26.00

0.10 − 0.05

$1.30PV =

• The expected dividend of Company XYZ next year is

$1.30 and dividends are expected to grow at 5%

forever. If the discount rate is 10%, what is the present

value of this dividend stream?

64

0 1

C

C×(1+ g)T −1C C×(1+ g)

(1+ r) (1+ r)2 (1+ r)TPV = + +L+

r − g

PV =

C

( 1+ r)

(1+ g)T

1−

L

2 3

C×(1+g) C ×(1+g)2 C×(1+g)T-1

T

• A growing stream of cash flows with a fixed maturity

• The formula for the present value of a growing annuity is:

Growing Annuity

_________________________________________________________________________________

65

Growing annuity - example

$20,000

= $265,121.57$20,000

(1.10 )

(1.03) 401−

0.10−0.03

PV =

$20,000×(1.03)

• A retirement plan offers to pay $20,000 the first year, and to

increase the annual payment by 3% each year until the person

dies. Assume the person will die in 40 years. What is the present

value at retirement if the discount rate is 10%?

0 1 2 40

L

$20,000×(1.03)39

2-53

Effective Annual Rate (EAR)

_________________________________________________________________________________

• This is the actual rate paid (or received) after

accounting for compounding that occurs during the

year

• If you want to compare two alternative investments

with different compounding periods you need to

compute the EAR and use that for comparison.

2-54

Annual Percentage Rate

_________________________________________________________________________________

• This is the annual rate that is quoted by law-

nominal or quoted rate

• APR = period rate times the number of periods per

year

• Consequently, to get the period rate we rearrange

the APR equation:

Period rate = APR / number of periods per year

• You should NEVER divide the effective rate by the

number of periods per year – it will NOT give you

the period rate

2-55

Computing APRs

_________________________________________________________________________________

• What is the APR if the monthly rate is 0.5%?

0.5(12) = 6%

• What is the APR if the semiannual rate is 0.5%?

0.5(2) = 1%

• What is the monthly rate if the APR is 12% with

monthly compounding?

12 / 12 = 1%

– Can you divide the above APR by 2 to get the

semiannual rate? NO!!! You need an APR based on

semiannual compounding to find the semiannual rate.

2-56

Things to Remember

_________________________________________________________________________________

• You ALWAYS need to make sure that the interest rate

and the time period match.

– If you are looking at annual periods, you need an

annual rate.

– If you are looking at monthly periods, you need a

monthly rate.

• If you have an APR based on monthly compounding,

you have to use monthly periods for lump sums, or

adjust the interest rate appropriately if you have

payments other than monthly

2-57

Computing EARs – Example

_________________________________________________________________________________

• Suppose you can earn 1% per month on $1 invested

today.

– What is the APR? 1(12) = 12%

– How much are you effectively earning?

• FV = 1(1.01)12 = 1.1268

• Rate = (1.1268 – 1) / 1 = .1268 = 12.68%

• Suppose if you put it in another account, you earn

3% per quarter.

– What is the APR?

– How much are you effectively earning?

2-58

EAR – Formula

_________________________________________________________________________________

1

m

APR

1 EAR

m

−

+=

- APR is the quoted rate

- m is the number of compounding periods per year

2-59

Decisions, Decisions II

_________________________________________________________________________________

• You are looking at two savings accounts. One pays

5.25%, with daily compounding. The other pays 5.3%

with semiannual compounding. Which account should

you use?

– First account:

EAR = (1 + .0525/365)365 – 1 = 5.39%

– Second account:

EAR = (1 + .053/2)2 – 1 = 5.37%

• Which account should you choose and why?

2-60

Decisions, Decisions II Continued

_________________________________________________________________________________

• Let’s verify the choice. Suppose you invest $100 in each

account. How much will you have in each account in one

year?

– First Account:

Daily rate = .0525 / 365 = .00014383562

FV = 100(1.00014383562)365 = 105.39

– Second Account:

Semiannual rate = .0539 / 2 = .0265

FV = 100(1.0265)2 = 105.37

• You have more money in the first account.

2-61

Computing APRs from EARs

_________________________________________________________________________________

• If you have an effective rate, how can you compute the

APR?

• Rearrange the EAR equation and you get:

+= 1 - EAR) (1 m APR m1

2-62

APR – Example

_________________________________________________________________________________

• Suppose you want to earn an effective rate of 12% and

you are looking at an account that compounds on a

monthly basis. What APR must they pay?

= 12 (1 + .12)/ − 1

= .1138655152 or 11.39%

2-63

Computing Payments with APRs

_________________________________________________________________________________

• Suppose you want to buy a new computer system

and the store is willing to sell it to allow you to make

monthly payments. The entire computer system

costs $3500. The loan period is for 2 years and the

interest rate is 16.9% with monthly compounding.

What is your monthly payment?

Monthly rate = .169 / 12 = .01408333333

Number of months = 2(12) = 24

3500 = C[1 – 1 / 1.01408333333)24] / .01408333333

C = 172.88

2-64

Future Values with Monthly Compounding

____________________________________________________________________________________

• Suppose you deposit $50 a month into an account that

has an APR of 9%, based on monthly compounding. How

much will you have in the account in 35 years?

Monthly rate = .09 / 12 = .0075

Number of months = 35(12) = 420

FV = 50[1.0075420 – 1] / .0075 = 147,089.22

2-65

Continuous Compounding

_________________________________________________________________________________

• Sometimes investments or loans are figured based on

continuous compounding

EAR = eq – 1

– The e is a special function on the calculator normally

denoted by ex

• Example: What is the effective annual rate of 7%

compounded continuously?

– EAR = e.07 – 1 = .0725 or 7.25%

• When loans are made, the borrower is obliged to

repay both the principal amount and to pay

interest.

• There are many different ways this can be done.

• Here we will consider three popular approaches:

– Pure discount loans

– Interest only loans

– Amortized loans

Loans and Amortization

_________________________________________________________________________________

2-66

• A pure discount loan is one in which the

principal is repaid in one lump sum.

• Valuing a pure discount loan requires

calculating its present value.

• Calculate the present value of a $50,000 pure

discount loan with a ten year life span and a 5%

interest rate:

Pure Discount Loans

_________________________________________________________________________________

2-67

• An interest only loan is one in which interest is paid in

each period and then the principal is repaid in one lump

sum at the end of the loan.

• Consider a $20,000 loan with interest of 10% and a 3

year life span.

– In each year, the interest is $2000

– At the end of year 3, the $20,000 is repaid.

Interest Only Loans

_________________________________________________________________________________

2-68

• In an amortized loan, interest if paid every

period and the principal is also paid off at

regular intervals.

– Most amortized loans have a single fixed payment

each month – a form of ordinary annuity

• Valuing these loans uses the same formula for

the present value of an ordinary annuity as

before.

Amortized Loans

_________________________________________________________________________________

2-69

Financing Decisions

Lecture 2: Investment Decision-I

Topics Covered Today

_______________________________________________________________________________

• The Time Value of Money

– Future Value and Compounding

– Present Value and Discounting

– Future and Present Values of Multiple Cash Flows

– Valuing Level Cash Flows: Annuities and Perpetuities

– Comparing Rates: The Effect of Compounding

2-2

2-3

Basic Definitions

_________________________________________________________________________________

• Present Value – earlier money on a time line (ie value

today)

• Future Value – later money on a time line (ie value at

some time in the future)

• Interest rate – “exchange rate” between earlier money

and later money

– Discount rate

– Cost of capital

– Opportunity cost of capital

– Required return

• time value of money

• On a time line:

– Time 0 is today,

– Time 1 is one period from today or the end of Period 1

and the start of Period 2, and so on.

• Information on a time line is written as:

– Cash flows are written above the tick marks

– Unknown cash flows are denoted with a question mark

– Interest rates are written above the line and between

the tick marks.

Time lines

_________________________________________________________________________________

• Here there is a cash outflow of $100 at time 0 (note

the minus sign),

• The interest rate is 10% in period 1 (t=0 to t=1) and

5% in both period 2 and period 3.

• There are no cash flows at times 1 and 2. There is a

cash flow at time 3 exists but the value is unknown

0 1 2 3

-100 ?

Time

Cash flows

10% 5%

2-6

Future Values

_________________________________________________________________________________

• Suppose you invest $1000 for one year at 5% per

year. What is the future value in one year?

– Interest = 1000(.05) = 50

– Value in one year = principal + interest = 1000 + 50 = 1050

– Future Value (FV) = 1000(1 + .05) = 1050

• Suppose you leave the money in for another year.

How much will you have two years from now?

– FV = 1050 (1.05) = 1102.50

– ie 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

2-7

Future Values: General Formula

_________________________________________________________________________________

FV = PV(1 + r)t

– FV = future value

– PV = present value

– r = period interest rate, expressed as a decimal

– T = number of periods

• Future value interest factor = (1 + r)t

2-8

Effects of Compounding

_________________________________________________________________________________

• Simple interest

– assumes that interest rate paid is a flat percentage of the

principal (P) each period

• Compound interest

– interest is earned/paid on the principal plus any

accumulated interest determined since the start of the

deposit/loan.

• Consider the previous example

– FV with simple interest = 1000 + 50 + 50 = 1100

– FV with compound interest = 1102.50

– The extra 2.50 comes from the interest of .05(50) = 2.50

earned on the first interest payment

2-9

Future Values – Example 2

_________________________________________________________________________________

• Suppose you invest the $1000 from the previous

example for 5 years. How much would you have?

– FV = 1000(1.05)5 = 1276.28

• The effect of compounding is small for a small number

of periods, but increases as the number of periods

increases. (Simple interest would have a future value of

$1250, for a difference of $26.28.)

2-10

Future Values – Example 3

_________________________________________________________________________________

• Suppose you had a relative deposit $10 at 5.5% interest

200 years ago. How much would the investment be

worth today?

• What is the effect of compounding?

2-11

Present Values

_________________________________________________________________________________

• How much do I have to invest today to have some

amount in the future?

– FV = PV(1 + r)t

– Rearrange to solve for :

PV = FV / (1 + r)t

• Discounting ≈ present value of some future amount

• “value” generally refers present value unless we

specifically indicate that we want the future value.

2-12

Present Value – One Period Example

_________________________________________________________________________________

• Suppose you need $10,000 in one year for the down

payment on a new car. If you can earn 7% annually,

how much do you need to invest today?

PV = 10,000 / (1.07)1 = 9345.79

2-13

Present Values – Example 2

_________________________________________________________________________________

_

• Your parents set up a trust fund for you 10 years ago

that is now worth $19,671.51. If the fund earned 7%

per year, how much did your parents invest?

PV = 19,671.51 / (1.07)10 = 10,000

2-14

Present Value – Important Relationship I

_________________________________________________________________________________

• For a given interest rate – the longer the time

period, the lower the present value

– What is the present value of $500 to be received in 5

years? 10 years? The discount rate is 10%

5 years: PV = 500 / (1.1)5 = 310.46

10 years: PV = 500 / (1.1)10 = 192.77

2-15

Present Value – Important Relationship II

_________________________________________________________________________________

• For a given time period – the higher the interest

rate, the smaller the present value

– What is the present value of $500 received in 5 years

if the interest rate is 10%? 15%?

Rate = 10%: PV = 500 / (1.1)5 = 310.46

Rate = 15%: PV = 500 / (1.15)5 = 248.59

2-16

Discount Rate

_________________________________________________________________________________

_

• Often we will want to know what the implied

interest rate is in an investment

• Rearrange the basic PV equation and solve for r

FV = PV(1 + r)t

r = (FV / PV)1/t – 1

2-17

Discount Rate – Example 1

_________________________________________________________________________________

• You are looking at an investment that will pay $1200

in 5 years if you invest $1000 today. What is the

implied rate of interest?

r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%

2-18

Discount Rate – Example 2

________________________________________________________________________________

• Suppose you have a 1-year old son and you want to

provide $75,000 in 17 years towards his college

education. You currently have $5000 to invest. What

interest rate must you earn to have the $75,000 when

you need it?

2-19

Finding the Number of Periods

_________________________________________________________________________________

• Start with basic equation and solve for t (remember

your logs)

FV = PV(1 + r)t

t = ln(FV / PV) / ln(1 + r)

2-20

Number of Periods – Example 1

_________________________________________________________________________________

• You want to purchase a new car and you are willing

to pay $20,000. If you can invest at 10% per year

and you currently have $15,000, how long will it be

before you have enough money to pay cash for the

car?

t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

2-21

Multiple Cash Flows –Future Value Example 1

____________________________________________________________________________________

• You are able to deposit $4000 at the end of the next 3

years in a bank account paying 8% interest. You

currently have $7000 in the account. How much will you

have in 3 years? In 4 years?

• Find the value at year 3 of each cash flow and add them

together.

2-22

cash flow 7000 4000 4000 4000

|______|_____|_____|_____|

period 0 1 2 3 4

Today (year 0): FV = 7000(1.08)3 = 8,817.98

Year 1: FV = 4,000(1.08)2 = 4,665.60

Year 2: FV = 4,000(1.08) = 4,320

Year 3: value = 4,000

Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000

= 21,803.58

• Value at year 4 = 21,803.58(1.08) = 23,547.87

2-23

Multiple Cash Flows – FV Example 2

_________________________________________________________________________________

• Suppose you invest $500 in a mutual fund today and

$600 in one year. If the fund pays 9% annually, how

much will you have in two years?

2-24

Multiple Cash Flows – Example 2 Continued

• How much will you have in 5 years if you make no

further deposits?

• First way:

– FV = 500(1.09)5 + 600(1.09)4 = 1616.26

• Second way – use value at year 2:

– FV = 1248.05(1.09)3 = 1616.26

2-25

Multiple Cash Flows – Present Value Example 3

• You are offered an investment that will pay you

$200 in one year,$400 the next year, $600 the next

year and $800 at the end of the fourth year. You can

earn 12% on very similar investments. What is the

most you should pay for this one?

cash flow 200 400 600 800

|______|_____|_____|_____|

period 0 1 2 3 4

• Find the PV of each cash flows and add them

– Year 1 CF: 200 / (1.12)1 = 178.57

– Year 2 CF: 400 / (1.12)2 = 318.88

– Year 3 CF: 600 / (1.12)3 = 427.07

– Year 4 CF: 800 / (1.12)4 = 508.41

– Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93

2-24

2-24

2-27

Annuities and Perpetuities Defined

________________________________________________________________________________

• Annuity – finite series of equal payments that occur

at regular intervals

– If the first payment occurs at the end of the period, it

is called an ordinary annuity

– If the first payment occurs at the beginning of the

period, it is called an annuity due or annuity in

advance

• Perpetuity – infinite series of equal payments

2-28

Ordinary annuity

_________________________________________________________________________________

• Annuity paid over n periods.

• The first cash flow (R1) occurs at the end of the first

period.

• All cash flows in the annuity need to be discounted to

t=0.

Cash flow R1 R2 R3 R4 ... Rn-1 Rn

|___|___|___|___|___|____________|___|

Period 0 1 2 3 4 n-1 n

2-29

Annuities Basic Formulas

____________________________________________________________________________________

• Annuities:

−+

=

+

−

=

r

rCFV

r

rCPV

t

t

1)1(

)1(

11

2-30

To get the formulas :

• Assume an annuity of $1 for n periods, where the

periodical compound interest rate is r.

Cash flow $1 $1 $1 $1 ... $1 $1

|___|___|___|___|___|____________|___|

Period 0 1 2 3 4 n-1 n

• Then the future value of this stream of cash flows is

FV = 1 + 1(1 + r)1 + 1(1 + r)2 + . . . . . + 1(1 + r)n-1 (A)

• Multiply each side of (A) by (1+r):

FV(1+r) = (1+r)1 + (1+r)2 + (1+r)3 + . . . + (1+r)n (B)

• Subtract (A) from (B)

FV(1 + r) - FV = (1 +r)n - 1

2-31

• and simplifying:

• Using a similar technique we find that the present

value of a series of cash flows of $1 for n periods, at a

rate of r

]1 - ) + 1 ( [1=

n

r

rFV

2-32

Annuity – Sweepstakes Example

_________________________________________________________________________________

• Suppose you win the Publishers Clearinghouse $10

million sweepstakes. The money is paid in equal annual

installments of $333,333.33 over 30 years. If the

appropriate discount rate is 5%, how much is the

sweepstakes actually worth today?

PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29

2-33

Buying a House

_________________________________________________________________________________

• You are ready to buy a house and you have $20,000

for a down payment and closing costs. Closing costs

are estimated to be 4% of the loan value. You have

an annual salary of $36,000 and the bank is willing

to allow your monthly mortgage payment to be

equal to 28% of your monthly income. The interest

rate on the loan is 6% per year with monthly

compounding (.5% per month) for a 30-year fixed

rate loan. How much money will the bank loan you?

How much can you offer for the house?

2-34

Buying a House – Continued

_________________________________________________________________________________

• Bank loan

Monthly income = 36,000 / 12 = 3,000

Maximum payment = .28(3,000) = 840

PV = 840[1 – 1/1.005360] / .005 = 140,105

• Total Price

Closing costs = .04(140,105) = 5,604

Down payment = 20,000 – 5604 = 14,396

Total Price = 140,105 + 14,396 = 154,501

2-35

Finding the Payment

_________________________________________________________________________________

• Suppose you want to borrow $20,000 for a new car. You

can borrow at 8% per year, compounded monthly (8/12

= .66667% per month). If you take a 4 year loan, what is

your monthly payment?

20,000 = C[1 – 1 / 1.006666748] / .0066667

C = 488.26

2-36

Finding the Number of Payments

____________________________________________________________________________________

• Suppose you borrow $2000 at 5% and you are going to

make annual payments of $734.42. How long before you

pay off the loan?

2000 = 734.42(1 – 1/1.05t) / .05

0.136161869 = 1 – 1/1.05t

1/1.05t = 0.863838131

1.157624287 = 1.05t

t = ln(1.157624287) / ln(1.05) = 3 years

2-37

Future Values for Annuities

_________________________________________________________________________________

• Suppose you begin saving for your retirement by

depositing $2000 per year in an IRA. If the interest

rate is 7.5%, how much will you have in 40 years?

FV = 2000(1.07540 – 1)/.075 = 454,513.04

2-38

Annuity due

_________________________________________________________________________________

• The first cash flow occurs at the beginning of the

first period. Notice that R1 occurs at t=0.

• It is already discounted or expressed in net present

value terms.

• When calculating the present value of the annuity

only the remaining n-1 payments need to be

discounted then added to the value of the first cash

flow.

cash flow R1 R2 R3 R4 ... Rn

|___|___|___|___|___|____________|___|

period 0 1 2 3 4 n-1 n

2-39

Annuity Due- FV

_________________________________________________________________________________

• You are saving for a new house and you put $10,000

per year in an account paying 8%. The first payment

is made today. How much will you have at the end

of 3 years?

cash flow 10000 10000 10000

|______|_____|_____|

period 0 1 2 3

FV = (10,000[(1.083 – 1) / .08]) *(1.08) = 35,061.12

• If you use the regular annuity formula, the FV will occur

at the same time as the last payment (ie year 2).

• To get the value at the end of the third period, you have

to take it forward one more period.

x (1 + r)

2-38

2-38

• Calculate the present value of an annuity due that

pays $150 each period, lasts for 3 periods and has

an interest rate of 2%

cash flow 150 150 150

|______|_____|_____|

period 0 1 2 3

PV = 150 1 – 1 x (1+0.02) = $441.23

[0.02 * (1.02)3]

0.02

Annuity Due- PV

_________________________________________________________________________________

2-42

Deferred Annuity

_________________________________________________________________________________

• A deferred annuity refers to a stream of fixed payments

which begin sometime in the future, i.e. the first

payment is delayed.

• A typical case would be a pension which does not begin

until age 60 or an inheritance paid in fixed instalments

beginning after a particular date.

2-43

Methods

_________________________________________________________________________________

• Method 1

– Let b be the delay period and assume that the annuity is

an ordinary annuity

– Calculate the present value of the annuity at the beginning

of period b. To calculate the PV of this stream at time zero

(today) discount the above annuity to time 0, by dividing

by (1+r)b-1

cash flow R1 R2 ... Rn

|___|___|___|___|___|____________|

period 0 1 … b-1 b n

• Method 2

– Take the present value of annuity paid for n years and

subtract an annuity paid over b-1 periods

cash flow R1 R2 ... Rn

|___|___|___|___|___|____________|

period 0 1 … b-1 b n

2-41

2-44

2-45

Deferred Annuity example

_________________________________________________________________________________

• Determine the present value of a $250 annuity, payable

annually for five years when the first payment occurs at

the end of the seventh year. Use a discount rate of 11%

p.a.

cash flow 250 250 250 250 250

|___|___|___|___|___|___|___|

period 0 1 … 6 7 8 9 10 11

PV = [250[1 – 1/1.115] / .11]*(1.11)-6 or

PV = [250[1 – 1/1.1111] / .11]-[250[1 – 1/1.116] / .11]

2-46

Perpetuities

_________________________________________________________________________________

• A perpetuity involves the payment of a stream of fixed

payments where cash flows continue forever.

PV = C / r

• The formula is determined by considering that the

number of compounding periods is infinite

(continuously compounded) and finding the limits as n

goes to infinity

2-47

Perpetuity – Example

_________________________________________________________________________________

• Company B wants to sell preference shares at $100

per share. A similar issue of shares already issued

has a price of $40 and offers a dividend of $1 per

quarter. What dividend does the company have to

offer if the shares are going to sell?

PV = $40, CF = $1 per quarter

• Current required return:

40 = 1 / r

r = .025 or 2.5% per quarter

• Dividend for new preferred:

100 = C / .025

C = 2.50 per quarter

• Often annuities and perpetuities grow over

time.

• For example, income from an investment

could start at a set amount $C and then grow

by a fixed percentage (g) every year.

• In this situation, the growth rate must be

taken into account

Constant Growth Rates

_________________________________________________________________________________

62

• A growing stream of cash flows that lasts forever

0 1 2 3

……

C C×(1+g) C ×(1+g)2

+ + +L

C ×(1+ g)2

(1+ r) (1+ r)2 (1+ r)3

C C ×(1+ g)

PV =

• The formula for the present value of a growing

perpetuity is (if g

C

r -g

Growing Perpetuity

_________________________________________________________________________________

63

Growing perpetuity - example

0

…

1 2 3

$1.30 $1.30×(1.05) $1.30 ×(1.05)2

= $26.00

0.10 − 0.05

$1.30PV =

• The expected dividend of Company XYZ next year is

$1.30 and dividends are expected to grow at 5%

forever. If the discount rate is 10%, what is the present

value of this dividend stream?

64

0 1

C

C×(1+ g)T −1C C×(1+ g)

(1+ r) (1+ r)2 (1+ r)TPV = + +L+

r − g

PV =

C

( 1+ r)

(1+ g)T

1−

L

2 3

C×(1+g) C ×(1+g)2 C×(1+g)T-1

T

• A growing stream of cash flows with a fixed maturity

• The formula for the present value of a growing annuity is:

Growing Annuity

_________________________________________________________________________________

65

Growing annuity - example

$20,000

= $265,121.57$20,000

(1.10 )

(1.03) 401−

0.10−0.03

PV =

$20,000×(1.03)

• A retirement plan offers to pay $20,000 the first year, and to

increase the annual payment by 3% each year until the person

dies. Assume the person will die in 40 years. What is the present

value at retirement if the discount rate is 10%?

0 1 2 40

L

$20,000×(1.03)39

2-53

Effective Annual Rate (EAR)

_________________________________________________________________________________

• This is the actual rate paid (or received) after

accounting for compounding that occurs during the

year

• If you want to compare two alternative investments

with different compounding periods you need to

compute the EAR and use that for comparison.

2-54

Annual Percentage Rate

_________________________________________________________________________________

• This is the annual rate that is quoted by law-

nominal or quoted rate

• APR = period rate times the number of periods per

year

• Consequently, to get the period rate we rearrange

the APR equation:

Period rate = APR / number of periods per year

• You should NEVER divide the effective rate by the

number of periods per year – it will NOT give you

the period rate

2-55

Computing APRs

_________________________________________________________________________________

• What is the APR if the monthly rate is 0.5%?

0.5(12) = 6%

• What is the APR if the semiannual rate is 0.5%?

0.5(2) = 1%

• What is the monthly rate if the APR is 12% with

monthly compounding?

12 / 12 = 1%

– Can you divide the above APR by 2 to get the

semiannual rate? NO!!! You need an APR based on

semiannual compounding to find the semiannual rate.

2-56

Things to Remember

_________________________________________________________________________________

• You ALWAYS need to make sure that the interest rate

and the time period match.

– If you are looking at annual periods, you need an

annual rate.

– If you are looking at monthly periods, you need a

monthly rate.

• If you have an APR based on monthly compounding,

you have to use monthly periods for lump sums, or

adjust the interest rate appropriately if you have

payments other than monthly

2-57

Computing EARs – Example

_________________________________________________________________________________

• Suppose you can earn 1% per month on $1 invested

today.

– What is the APR? 1(12) = 12%

– How much are you effectively earning?

• FV = 1(1.01)12 = 1.1268

• Rate = (1.1268 – 1) / 1 = .1268 = 12.68%

• Suppose if you put it in another account, you earn

3% per quarter.

– What is the APR?

– How much are you effectively earning?

2-58

EAR – Formula

_________________________________________________________________________________

1

m

APR

1 EAR

m

−

+=

- APR is the quoted rate

- m is the number of compounding periods per year

2-59

Decisions, Decisions II

_________________________________________________________________________________

• You are looking at two savings accounts. One pays

5.25%, with daily compounding. The other pays 5.3%

with semiannual compounding. Which account should

you use?

– First account:

EAR = (1 + .0525/365)365 – 1 = 5.39%

– Second account:

EAR = (1 + .053/2)2 – 1 = 5.37%

• Which account should you choose and why?

2-60

Decisions, Decisions II Continued

_________________________________________________________________________________

• Let’s verify the choice. Suppose you invest $100 in each

account. How much will you have in each account in one

year?

– First Account:

Daily rate = .0525 / 365 = .00014383562

FV = 100(1.00014383562)365 = 105.39

– Second Account:

Semiannual rate = .0539 / 2 = .0265

FV = 100(1.0265)2 = 105.37

• You have more money in the first account.

2-61

Computing APRs from EARs

_________________________________________________________________________________

• If you have an effective rate, how can you compute the

APR?

• Rearrange the EAR equation and you get:

+= 1 - EAR) (1 m APR m1

2-62

APR – Example

_________________________________________________________________________________

• Suppose you want to earn an effective rate of 12% and

you are looking at an account that compounds on a

monthly basis. What APR must they pay?

= 12 (1 + .12)/ − 1

= .1138655152 or 11.39%

2-63

Computing Payments with APRs

_________________________________________________________________________________

• Suppose you want to buy a new computer system

and the store is willing to sell it to allow you to make

monthly payments. The entire computer system

costs $3500. The loan period is for 2 years and the

interest rate is 16.9% with monthly compounding.

What is your monthly payment?

Monthly rate = .169 / 12 = .01408333333

Number of months = 2(12) = 24

3500 = C[1 – 1 / 1.01408333333)24] / .01408333333

C = 172.88

2-64

Future Values with Monthly Compounding

____________________________________________________________________________________

• Suppose you deposit $50 a month into an account that

has an APR of 9%, based on monthly compounding. How

much will you have in the account in 35 years?

Monthly rate = .09 / 12 = .0075

Number of months = 35(12) = 420

FV = 50[1.0075420 – 1] / .0075 = 147,089.22

2-65

Continuous Compounding

_________________________________________________________________________________

• Sometimes investments or loans are figured based on

continuous compounding

EAR = eq – 1

– The e is a special function on the calculator normally

denoted by ex

• Example: What is the effective annual rate of 7%

compounded continuously?

– EAR = e.07 – 1 = .0725 or 7.25%

• When loans are made, the borrower is obliged to

repay both the principal amount and to pay

interest.

• There are many different ways this can be done.

• Here we will consider three popular approaches:

– Pure discount loans

– Interest only loans

– Amortized loans

Loans and Amortization

_________________________________________________________________________________

2-66

• A pure discount loan is one in which the

principal is repaid in one lump sum.

• Valuing a pure discount loan requires

calculating its present value.

• Calculate the present value of a $50,000 pure

discount loan with a ten year life span and a 5%

interest rate:

Pure Discount Loans

_________________________________________________________________________________

2-67

• An interest only loan is one in which interest is paid in

each period and then the principal is repaid in one lump

sum at the end of the loan.

• Consider a $20,000 loan with interest of 10% and a 3

year life span.

– In each year, the interest is $2000

– At the end of year 3, the $20,000 is repaid.

Interest Only Loans

_________________________________________________________________________________

2-68

• In an amortized loan, interest if paid every

period and the principal is also paid off at

regular intervals.

– Most amortized loans have a single fixed payment

each month – a form of ordinary annuity

• Valuing these loans uses the same formula for

the present value of an ordinary annuity as

before.

Amortized Loans

_________________________________________________________________________________

2-69