Fin 351: lectures 2

Time Value of Money

Present value of future cash flows or payments

Fin 351: lectures 2

This week’s plan or learning goals Review of what we have learned in the last lecture Finance tales Understand the concept of the time value of money Learn how to compare:

• Cash flows or payments you get today• Cash flows or payments you get in the future

• Understand the following terms:• present value (PV)• discount rate (r)• net present value (NPV)• annuity • perpetuity

Fin 351: lectures 2

Today’s plan (2)

Learn how to draw cash flows of projects Learn how to calculate the present value

of annuities and its applications Learn how to calculate the present value

of perpetuities

Fin 351: lectures 2

What have we learned?

Financial markets• What are they?

• What are their function?

The cost of capital• What is it?

Fin 351: lectures 2

Finance Culture

Do you know what flower it is?

Why am I interested in it?

Fin 351: lectures 2

Tulip tales On a crisp August day in 1594, an elderly botanist named Carolus

Clusius planted a handful of tulip bulbs — a flower native to asia — in a small garden at the university of Leiden in the Netherlands, and in the spring of 1594, the first tulips bloomed in Holland. Clusius' planting is considered by the Dutch to be the birth of their famous flower bulb business which celebrates its 403rd birthday in 1997.

Clusius' tulips caused a sensation in 17th century Holland. They became the rage as aristocrats flaunted the exotic flowers as symbols of power and prestige. Soon, Dutch society was swept up in a tulip-trading craze, and hard nosed traders offered sky-high bids for the bulbs.

One early 17th century bill of sale recorded the following transaction for one single tulip bulb: Two loads of wheat, four loads of rye, four fat oxen, eight fat swine, twelve fat sheep, two hogsheads of wine, four barrels of beer, two barrels of butter, 1,000 pounds of cheese, a marriage bed with linens, and a sizeable wagon to haul it all away.

Fin 351: lectures 2

Financial choices

Which would you rather receive today?

• TRL 1,000,000,000 ( one billion Turkish lira )

• USD 652.72 ( U.S. dollars ) Both payments are absolutely

guaranteed. What do we do?

Fin 351: lectures 2

Financial choices

We need to compare “apples to apples” - this means we need to get the TRL:USD exchange rate

From www.bloomberg.com we can see:

• USD 1 = TRL 1,603,500 Therefore TRL 1bn = USD 623

Fin 351: lectures 2

Financial choices with time

Which would you rather receive?• $1000 today

• $1200 in one year

Both payments have no risk, that is, • there is 100% probability that you will be paid

• there is 0% probability that you won’t be paid

Fin 351: lectures 2

Financial choices with time (2)

Why is it hard to compare ?• $1000 today

• $1200 in one year This is not an “apples to apples” comparison.

They have different units $100 today is different from $100 in one year Why?

• A cash flow is time-dated money• It has a money unit such as USD or TRL

• It has a date indicating when to receive money

Fin 351: lectures 2

Present value

In order to have an “apple to apple” comparison, we convert future payments to the present values• this is like converting money in TRL to money in USD

• Certainly, we can also convert the present value to the future value to compare payments we get today with payments we get in the future.

• Although these two ways are theoretically the same, but the present value way is more important and has more applications, as to be shown in stock and bond valuations.

Fin 351: lectures 2

Present value (2)

The formula for converting future cash flows or payments:

= present value at time zero = cash flow in the future (in year i)

= discount rate for the cash flow in year i

tt

ti

it

itt r

CPVor

r

CPV

)1()1( 00

0tPV

itr

itC

itC

Fin 351: lectures 2

Example 1

What is the present value of $100 received in one year (next year) if the discount rate is 7%?• Step 1: draw the cash flow diagram

• Step 2: think !

PV<?> $100

• Step 3: PV=100/(1.07)1 =

Year one

$100

PV=?

Fin 351: lectures 2

Example 2

What is the present value of $100 received in year 5 if the discount rate is 7%?• Step 1: draw the cash flow diagram

• Step 2: think !

PV<?> $

• Step 3: PV=100/(1.07)5 = Year 5

$100

PV=?

Fin 351: lectures 2

Example 3

What is the present value of $100 received in year 20 if the discount rate is 7%?• Step 1: draw the cash flow diagram

• Step 2: think !

PV<?> $

• Step 3: PV=100/(1.07)20 =

Year 20

$100

PV=?

Fin 351: lectures 2

Present value of multiple cash flows For a cash flow received in year one and a

cash flow received in year two, different discount rates must be used.

The present value of these two cash flows is the sum of the present value of each cash flow, since two present value have the same unit: time zero USD.

2

221

11

2010210

)1()1(

)()(),(

rCrC

CPVCPVCCPV ttt

Fin 351: lectures 2

Example 4 John is given the following set of cash flows and

discount rates. What is the PV?

• Step 1: draw the cash flow diagram• Step 2: think ! PV<?> $200• Step 3: PV=100/(1.1)1 + 100/(1.09)2 =

%101 r

Year one

$100

PV=?

1001 C

%92 r1002 C

$100

Year two

Fin 351: lectures 2

Example 5 John is given the following set of cash flows and

discount rates. What is the PV?

• Step 1: draw the cash flow diagram• Step 2: think ! PV<?> $350• Step 3: PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 =

503 C

1.01 r

Yr 1

$100

PV=?

1001 C

09.02 r2002 C

$50

Yr 3

07.03 r

Yr 2

$200

Fin 351: lectures 2

Projects

A “project” is a term that is used to describe the following activity:• spend some money today

• receive cash flows in the future A stylized way to draw project cash flows is

as follows:

Initial investment(negative cash flows)

Expected cash flows in year one (probably positive)

Expected cash flows in year two (probably positive)

Fin 351: lectures 2

Examples of projects An entrepreneur starts a company:

• initial investment is negative cash outflow.

• future net revenue is cash inflow . An investor buys a share of IBM stock

• cost is cash outflow; dividends are future cash inflows. A lottery ticket:

• investment cost: cash outflow of $1

• jackpot: cash inflow of $20,000,000 (with some very small probability…)

Thus projects can range from real investments, to financial investments, to gambles (the lottery ticket).

Fin 351: lectures 2

Firms or companies

A firm or company can be regarded as a set of projects.• capital budgeting is about choosing the best

projects in real asset investments.

How do we know one project is worth taking?

Fin 351: lectures 2

Net present value

A net present value (NPV) is the sum of the initial investment (usually made at time zero) and the PV of expected future cash flows.

T

tt

t

t

T

r

CC

CCPVCNPV

10

10

)1(

)(

Fin 351: lectures 2

NPV rule

If NPV > 0, the manager should go ahead to take the project; otherwise, the manager should not.

Fin 351: lectures 2

Example 6

Given the data for project A, what is the NPV?

• Step1: draw the cash flow graph

• Step 2: think! NPV<?>10

• Step 3: NPV=-50+50/(1.075)+10/(1.08)2 =

%0.810

%5.750

50

22

11

0

rC

rC

C

Yr 0

Yr 1 Yr 2

$10$50-$50

Fin 351: lectures 2

Example 7 John got his MBA from SFSU. When he was interviewed by a

big firm, the interviewer asked him the following question: A project costs 10 m and produces future cash flows, as shown

in the next slide, where cash flows depend on the state of the economy.

In a “boom economy” payoffs will be high• over the next three years, there is a 20% chance of a boom• In a “normal economy” payoffs will be medium• over the next three years, there is a 50% chance of normal

In a “recession” payoffs will be low• over the next 3 years, there is a 30% chance of a recession

In all three states, the discount rate is 8% over all time horizons.

Tell me whether to take the project or not

Fin 351: lectures 2

Cash flows diagram in each state

Boom economy

Normal economy

Recession

-$10 m$8 m $3 m $3 m

-$10 m

-$10 m

$2 m$7 m

$0.9 m$1 m$6 m

$1.5 m

Fin 351: lectures 2

Example 7 (continues)

The interviewer then asked John:• Before you tell me the final decision, how do

you calculate the NPV?• Should you calculate the NPV at each economy or

take the average first and then calculate NPV

• Can your conclusion be generalized to any situations?

Fin 351: lectures 2

Calculate the NPV at each economy

In the boom economy, the NPV is• -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36

In the average economy, the NPV is• -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613

In the bust economy, the NPV is • -10+ 6/1.08 + 1/1.082 + 0.9/1.083 =-$2.87The expected NPV is 0.2*2.36+0.5*(-.613)+0.3*(-2.87)=-$0.696

Fin 351: lectures 2

Calculate the expected cash flows at each time

At period 1, the expected cash flow is• C1=0.2*8+0.5*7+0.3*6=$6.9

At period 2, the expected cash flow is• C2=0.2*3+0.5*2+0.3*1=$1.9

At period 3, the expected cash flows is• C3=0.2*3+0.5*1.5+0.3*0.9=$1.62

The NPV is• NPV=-10+6.9/1.08+1.9/1.082+1.62/1.083

• =-$0.696

Fin 351: lectures 2

Perpetuities We are going to look at the PV of a perpetuity starting one year from

now. Definition: if a project makes a level, periodic payment into perpetuity,

it is called a perpetuity. Let’s suppose your friend promises to pay you $1 every year, starting

in one year. His future family will continue to pay you and your future family forever. The discount rate is assumed to be constant at 8.5%. How much is this promise worth?

PV???

$1 $1$1 $1 $1 $1

Yr1 Yr2 Yr3 Yr4 Yr5 Time=infinity

Fin 351: lectures 2

Perpetuities (continue)

Calculating the PV of the perpetuity with a level cash flow C in each period and the first piece of cash flow starting in period one could be hard:

1

21

)1(

1

)1()1()1(

iir

C

r

C

r

C

r

CPV

Fin 351: lectures 2

Perpetuities (continue)

To calculate the PV of perpetuities, we can have some math exercise as follows:

rrr

S

SS

S

S

r

1)1/(11

)1/(11

1)1(

1

32

2

1

Fin 351: lectures 2

Perpetuities (continue)

Calculating the PV of the perpetuity could also be easy if you ask George

rC

SCCr

C

r

C

r

C

r

CPV

i

i

ii

..)1(

1

)1()1()1(

11

21

Fin 351: lectures 2

Calculate the PV of the perpetuity

Consider the perpetuity of one dollar every period your friend promises to pay you. The interest rate or discount rate is 8.5%.

Then PV =1/0.085=$11.765, not a big gift.

Fin 351: lectures 2

Perpetuity (continue)

What is the PV of a perpetuity of paying $C every year, starting from year t +1, with a constant discount rate of r ?

C CC C C C

t+1 t+2 t+3 t+4 T+5 Time=t+infYr0

)1()1()1( 21 r

C

r

C

r

CPV

tt

Fin 351: lectures 2

Perpetuity (continue)

What is the PV of a perpetuity of paying $C every year, starting from year t +1, with a constant discount rate of r ?

rr

Crr

C

rr

C

rrrr

C

r

C

r

C

r

CPV

tti

it

t

tt

)1(

1.

)1()1(

1

)1(

)1(

1

)1(

1

)1(

1

)1(

)1()1()1(

1

21

21

Fin 351: lectures 2

Perpetuity (alternative method)

What is the PV of a perpetuity that pays $C every year, starting in year t+1, at constant discount rate “r”?• Alternative method: we can think of PV of a perpetuity

starting year t+1. The normal formula gives us the value AS OF year “t”. We then need to discount this value to account for periods “1 to t”

That is

rr

C

r

VPV

V

ttt

rC

t

)1()1(

Fin 351: lectures 2

Annuities

Well, a project might not pay you forever. Instead, consider a project that promises to pay you $C every year, for the next “T” years. This is called an annuity.

Can you think of examples of annuities in the real world?

PV???

C CC C C C

Yr1 Yr2 Yr3 Yr4 Yr5 Time=T

Fin 351: lectures 2

Value the annuity

Think of it as the difference between two perpetuities• add the value of a perpetuity starting in yr 1

• subtract the value of perpetuity starting in yr T+1

rrrC

rr

C

r

CPV

TT )1(

11

)1(

Fin 351: lectures 2

Example for annuities

you win the million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won (in PV-terms) ?

Fin 351: lectures 2

My solution

Using the formula for the annuity

71.700,529$07.0*07.1

1

07.0

1*000,50

20

PV

Fin 351: lectures 2

Example

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

Fin 351: lectures 2

Solution

10.774,12$

005.1005.

1

005.

1300Cost Lease 48

Cost

Fin 351: lectures 2

Lottery example

Paper reports: Today’s JACKPOT = $20mm !!• paid in 20 annual equal installments.

• payment are tax-free.

• odds of winning the lottery is 13mm:1

Should you invest $1 for a ticket?• assume the risk-adjusted discount rate is 8%

Fin 351: lectures 2

My solution

Should you invest ? Step1: calculate the PV

Step 2: get the expectation of the PV

Pass up this this wonderful opportunity

mm

mmmmmmPV

818.9$

)08.1(

0.1

)08.1(

0.1)08.1(

0.1202

1$76.0$

0*)13

11(818.9*

13

1][

mm

mmmm

PVE

Fin 351: lectures 2

Mortgage-style loans

Suppose you take a $20,000 3-yr car loan with “mortgage style payments”• annual payments

• interest rate is 7.5% “Mortgage style” loans have two main

features:• They require the borrower to make the same payment

every period (in this case, every year)

• The are fully amortizing (the loan is completely paid off by the end of the last period)

Fin 351: lectures 2

Mortgage-style loans

The best way to deal with mortgage-style loans is to make a “loan amortization schedule”

The schedule tells both the borrower and lender exactly:• what the loan balance is each period (in this case -

year)

• how much interest is due each year ? ( 7.5% )

• what the total payment is each period (year) Can you use what you have learned to figure

out this schedule?

Fin 351: lectures 2

My solution

year Beginningbalance

Interest payment

Principlepayment

Total payment

Ending balance

0

1

2

3

$20,000

13,809

7,154

$1,500 $6,191 $7,691 $13,809

1,036 6,655

537 7,154 7,691 0

7,691 7,154

Fin 351: lectures 2

Future value

The formula for converting the present value to future value:

= present value at time zero = future value in year i

= discount rate during the i years

iii

iittit

rPVFV

rPVFV

)1(

)1(

0

0

0tPV

itr

itFV

itC

Fin 351: lectures 2

Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%.

Fin 351: lectures 2

Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal?

trillion

FV

979.75$

)08.1(24$ 374374

To answer, determine $24 is worth in the year 2003, compounded at 8%.

FYI - The value of Manhattan Island land is FYI - The value of Manhattan Island land is well below this figure.well below this figure.