Valuation of Cash Flows

Chapter 3.1-3.3

Chapter 4

outline

• • • • • • Financial costs and benefits The Time Value of Money Present value and future value Valuation of streams of cash flows Valuation of Annuities and Perpetuities Further Examples

Financial Costs and Benefits

Identifying Costs and Benefits

Can we identify the costs and benefits of the following projects?

• • • Advertising campaign Lowering production cost Management training program What information is required in order to measure or value the associated costs and benefits?

The Important Role of Market Prices

• • Suppose you could trade 10 ounces of platinum and receive 20 ounces of gold, would you trade?

The price of one ounce of gold is $250 The price of one ounce of platinum is $550 What if the jeweler thought that the price of gold was too low?

Market Prices as Cash Value

Suppose the jeweler can produce $10,000 worth of jewelry from 20 ounces of gold but only $6,000 worth of jewelry from 10 ounces of platinum. The Jeweler should still not trade 10 ounces of gold for 20 ounces of platinum – why?

When a good is trades in a “competitive market” (which means that you are free to buy and sell the good at the same price), price determines the cash value of the good

Non-Competitive Markets

When a good is not traded in a competitive market (which means that you buy and sell the good at different prices) its value is determined by your own personal preferences (or benefit) from obtaining the good.

Suppose that just before the holidays your boss surprises you by offering all employees the choice between a brand new gardening tool-kit and $375. A brand-new tool-kit costs $450 in stores and can be sold for $400 through one of the internet websites. What would you choose?

Non-Competitive Markets

If you have a garden at home and you were planning on buying the kit anyway then you would value this gift at $450. Alternatively, if you were not planning on buying the kit in the first place then you would value the gift at $375.

When the market for the good is not competitive, the cash value of the economic opportunity depends on your preferences (and market prices)

The Time Value of Money

Timing of Payments

For most decisions in life (and also financial decisions) the costs and benefits occur at different points in time.

• • •

MBA Marriage Mortgage

• •

Consider the investment opportunity $100,000

today

$105,000

in one year

Are we “comparing apples to oranges”?

How do we compare payments in different times?

Interest Rates

Suppose that the current annual interest rate is 7%. This means that on a current deposit of $100 we will earn a total of $7 of interest over the course of the year leading to a total of $107 in our bank account. We refer to this as the risk-free interest rate and we denote this rate by r

f = 7% or 1+r f =1.07

This means that we can exchange (without any risk) $1

today

(or at time “0”) for $1.07 received

in one year

(or at time “1”)

Comparing Payments Across Time

Back to our investment opportunity. We know that the benefit is $105,000 in one year. Can we calculate the cost of this investment in dollars paid in one year instead of dollars paid today in order to compare the two?

By using the risk-free interest rate of 7%:

Costs

= $100, 000 today = $100, 000 ´ (1 +

r f

) = $107, 000 in one year

The net value of the investment in terms of “ dollars in one year ” $105,000-107,000= - $2,000

Comparing Payments Across Time

Back to our investment opportunity (again…). We will lose $2000 “

dollars in one year

” if we choose this investment opportunity, but how much will we loose in terms of “

dollars today

”?

We know that the cost is $100,000

today

. The benefit of this investment in dollars paid

today

instead of dollars

in one year

is By using the risk-free interest rate of 7%:

Benefits

= $105, 000 in one year = $100, 000 1 +

r f

= $98,130.84 today

The net value of the investment in terms of “ dollars today ” is $98,130.84-100,000= - $1,869.16

Comparing Payments Across Time

We valued the investment opportunity in terms of “

dollars today

” and “

dollars in one year

” and both led us to the conclusion not to invest in the project since it leads to a net loss.

Is it always true that both approaches lead to the same conclusion?

As long as we convert costs and benefits to the same point in time, we can compare them to make a decision

Not surprisingly

The net value of the investment in terms of “

dollars today

” and its value in terms of “

dollars in one year

” satisfy

- $1,869.16 (1.07) = $2,000

The Net Present Value Decision Rule

The term

Present Value today’s dollars

or

PV

refers to the fact that value is calculated in The Net Present Value (NPV) of a project or investment opportunity is the difference between the present value of benefits and the present value of costs

Net Present Value (NPV)

NPV

=

PV

(

Benefits

) -

PV

(

Costs

)

If we use positive cash-flows to represent benefits and negative cash flows to represent costs then NPV

=

PV

(

Benefits

-

Costs

) We accept projects with positive NPV

Present Value & Future Value of Cash Flows

The Time Line

The first (and very important) tool to develop is the ability to represent the stream of cash flows of an economic opportunity on a time line.

The Time Line

Problem: Suppose you must pay tuition of $10,000 per year for the next two years. Your tuition payments must be made in equal installments at the start of each semester. What is the timeline of your tuition payments?

Moving Cash Flows Across Time

When comparing cash flows obtained in different points in time we need to first convert all cash flows to one point in time so that we do not “compare apples to oranges”

Rule 1

It is only possible to compare or combine values at the same point in time

Rule 2

To move a cash flow forward in time, we must compound it

Rule 3

To move a cash flow back in time, we must discount it

Compounding Cash Flows

Suppose we have $1,000 today and we want to determine the equivalent amount in one year’s time. If the annual risk free interest rate is 10% then we can calculate: ($1000

today

)(1.10) = $1100

in one year

What if we were interested in the equivalent amount in two year’s time? Given that the annual risk-free rate for two years is also 10% then we can calculate:

Future Value

The value of a cash flow that is moved forward in time is known as its future value When interest rate is 10% then, $1210 is the future value of $1000 two years from today $1100 is the future value of $1000 one year from today

Future Value

If we were interested in the future value of $1000 three years from today we would simply multiply one more period

FV

3 = $1000 ´ (1.10) ´ (1.10) ´ (1.10)

Future Value of a Cash Flow “$C”

“n” years from today, given interest “r”

FV n

= $

C

´ (1 +

r

)

n

The future value is increasing in the interest rate “r” and the number of periods “n”.

Discounting Cash Flows

Suppose we anticipate having $1,000 in one year and we want to compute the value today of this payment. If the annual risk-free interest rate is 10% then we can calculate: ($1000

in one year

)/(1.10) = $909.09

today

Suppose we need the money now…does our preference for money today matter for this decision between $909.09 today or $1000 tomorrow?

Suppose you anticipate receiving $1000 in two years. Given that the annual risk-free rate for two years is also 10% then we can calculate:

PV

= $1000 1.10

2 = 826.45

Present Value

If we were interested in the present value of $1000 received three years from today we would simply multiply one more period

PV

= $1000 1.10

3 = 751.31

Present Value of a Cash Flow “$C”

Received “n” years from today, given interest “r”

PV

= $

C

(1 +

r

)

n

The present value is decreasing in the interest rate “r” and the number of periods “n”

Savings problem

Suppose you plan to save $1000 annually in the next three years starting from today.

Time line of deposits: If the interest rate on your savings account is 10%, then how much money will you have in your account three years from now?

Savings problem

We are asked to find the future value of the stream of cash flows. The future value at time 3 of the payment of $1000 received at time 0: The future value at time 3 of the payment of $1000 received at time 1: The future value at time 3 of the payment of $1000 received at time 2: The future value of the stream of cash flows:

Savings problem

An alternative way to find the future value of the stream of cash flows is to follow the balance on our saving account. At time 0 we have …………. in our bank account.

At time 1 we have ………… in our bank account. At time 2 we have …………. in our bank account.

At time 3 we have accumulated …………….

Savings problem

Now suppose that you wish to invest all the money now instead of $1000 each year. What size investment is required in order to obtain the exact same amount of accumulated funds three years from now?

You are asked to find the present value of the stream of cash flows.

The present value of the stream of cash flows is …………….

PV



n N



n

) 

n N

 (1

C n



n

Stream of Cash Flows Valuation

Present Value of Stream of Cash Flows

The present value of a stream of “n” cash flows of arbitrary amounts is the sum of the present value of all cash flows.

PV



n N



n

) 

n N

 (1

C n



n

Car Loan

Suppose that you need money to purchase a new car. Your uncle will lend you the money so long as you agree to pay him back within four years, and you offer to pay him the rate of interest that he would otherwise get by putting his money in a savings account. Based on your yearnings and living expenses, you think you will be able to pay him $5000 in one year (time t=1), and then $8000 each year for the next three years. If your uncle would otherwise earn 6% per year on his savings, how much can you borrow from him?

Car Loan

You are asked about the present value of the stream of cash flows you plan to pay your uncle.

The present value of the stream of cash flows is the sum of present values of all individual cash flows

PV

= $5000 1.06

+ $8000 1.06

2 = $24,890.65

+ $8000 1.06

3 + $8000 1.06

4

The Future Value Present Value Link

So you are asking you uncle to give up $24,890.65 now and in return receive a stream of cash flows. Your calculation indicated that your uncle should be indifferent between the two alternatives. To verify lets compare the future wealth of your uncle if he had invested instead of leant to you $24,890.65.

FV

4 = $24,890.65

´ 1.06

4 = $31, 423.87

What shall we compare this to?

The Future Value Present Value Link

Future Value of a Cash Flow Stream with Present Value of PV

FV n

=

PV

´

(1

+

r

)

n

We can alternatively calculate the balance on the saving account of your uncle if he was to save your payments for the next four years by calculating the sum of future values of all individual installments:

FV

4 = $5000(1.06) 3 + $8000(1.06) 2 + $8000(1.06) + $8000 = $31,423.88

The Net Present Value

Net Present Value (NPV)

NPV

=

PV

(

Benefits

) -

PV

(

Costs

) =

PV

(

Benefits

-

Costs

) Consider the following investment opportunity: If you invest $1000 today , you will receive $500 at the end of each of the next three years. If you could otherwise earn 10% per year on your money, should you undertake the investment opportunity?

NPV

= $1000 + $500 1.1

+ $500 1.1

2 + $500 1.1

3

Special Stream of Cash Flows Valuation

Annuities and Perpetuities

• • • • • Several of real-life situations involve streams of cash flows with “standard” structure.

Mortgages Car loans Retirement savings Corporate and treasury bonds stocks • • We will study the valuation of Annuity: periodic stream of cash flows with finite life ( Growing annuity has growing cash flows) Perpetuity: annuity with infinite life ( Growing perpetuity has growing cash flows)

Perpetuity

The PV of a Perpetuity equals the amount of funds required today in order to generate this perpetuity. How much is required today in order to generate an annual payment of $1 starting one year from now and continuing forever if the interest rate is 8%?

The present value of a perpetuity is: =

C r

Perpetuities

Perpetuities

Annuities

The PV of an Annuity equals the amount of funds required today in order to generate this annuity. How much is required today in order to generate an annual payment of $1 starting one year from now and continuing for 10 years if the interest rate is 8%?

The present value of an “n” year annuity is:

PV

(

C

in annuity) =

C r

æ ç è 1 1 ( )

n

ö ÷ ø

Annuities

Annuities

Annuities

Annuities

Growing Perpetuity

A growing perpetuity with growth rate “g” : The Present Value of a Growing Perpetuity

PV

(growing perpetuity)

=

r C

-

g

Growing Perpetuities

Growing Perpetuities

Growing Annuities

The present value of an “n” year growing annuity is:

PV

(growing annuity) =

r C

-

g

ç è 1 è 1 + 1 +

g

ö

n r

ø

Growing Annuities

Growing Annuities

Further examples

Loan Payment

Loan Payment

Loan Term

Loan Term

Mortgage Example

Q21(2 nd edition): When you purchased your house, you took our a 30-year annual payment mortgage with an interest rate of 6% per year. The annual payment of the mortgage is $12,000. You have just made a payment and have now decided to pay the mortgage off by repaying the outstanding balance. What is the payoff amount if: a.

b.

c.

You have lived in the house for 12 years (there are 18 years left on the mortgage) You have lived in the house for 20 years (there are 10 years left on the mortgage) You have lived in the house for 12 years (there are 18 years left on the mortgage) and you decide to pay off the mortgage immediately before the twelfth payment is due?

Amortization Schedule

a. The PV(18 remaining payments)=

PV

= $12, 000 0.06

ç 1 1 1.06

18 ö ø $129, 931 b. The PV(10 remaining payments)=

PV

= $12, 000 0.06

ç 1 1 1.06

10 ö ø $88, 321 c. The PV(remaining payments just before 12 th payment)=

PV

= $12, 000 + $129, 931 = $141, 931

Cost of schooling

Q27(2 nd edition): [modified] Your oldest daughter is about to start kindergarten at a private school. Tuition is $10,000 per year, payable at the beginning of the school year. You expect to keep your daughter in private school through high school. You expect tuition to increase at a rate of 5% per year over the 13 years of her schooling. What is the present value of the tuition payments if the interest rate is 6% per year? How much would you need to have in the bank now to fund all 13 years of tuition?

What would your answers to the above questions be if interest rate is 5% per year?

Cost of Schooling

When the interest rate is 6% you would need to have in the bank:

PV

= $10, 000 + $10, 500 6% 5% æ 1 æ è 1.05

1.06

ö ø 12 ö ø $122,890 If interest rate is 5% we cannot use the standard formula

PV

=

13

´

$10, 000

Growing annuity when growth rate is larger than interest rate The standard formula is applicable for all cases in which the interest rate is either larger or smaller than the growth rate

PV

=

r CF

1 -

g

æ 1 æ è 1 + 1 +

r g

ö ø

n

ö ø

Investment in Art

Q33(2 nd edition): You are thinking about buying a piece of art that costs $50,000. The art dealer is proposing the following deal: He will lend you the money, and you will repay the loan by making the same payment every two years for the next 20 years (i.e., a total of 10 payments).

If the interest rate is 4%, how much will you have to pay every two years?

The interest rate over a period of 2 years is: 1.04

2 = 1.0816

Calculating biennial payment: $50, 000 =

X

0.0816

æ è 1 1 1.0816

10 ø

X

= $7, 505.3

Retirement

Q35(2 nd edition): You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30 including your 65 th th birthday, and you decide, starting today and continuing on every birthday up to and birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65 th birthday?

The present value of $2 million received at age 65

FV

= $2, 000, 000 Þ

PV

= 2000000 1.05

30 Calculating annual installment: $462, 754.9

=

X

0.05

æ è 1 1 1.05

30 ø

X

= = $462, 754.9

$30,102.9