Aswath Damodaran 1

The Tools of Corporate FinancePresent Value

Financial Statement AnalysisFundamentals of Valuation

Aswath Damodaran 2

Present Value

Aswath Damodaran

Aswath Damodaran 3

Intuition Behind Present Value

There are three reasons why a dollar tomorrow is worth less than a dollar today• Individuals prefer present consumption to future consumption. To induce

people to give up present consumption you have to offer them more in the future.• When there is monetary inflation, the value of currency decreases over

time. The greater the inflation, the greater the difference in value between a dollar today and a dollar tomorrow.

• If there is any uncertainty (risk) associated with the cash flow in the future, the less that cash flow will be valued.

Other things remaining equal, the value of cash flows in future time periods will decrease as• the preference for current consumption increases.• expected inflation increases.• the uncertainty in the cash flow increases.

Aswath Damodaran 4

Discounting and Compounding

• The mechanism for factoring in these elements is the discount rate.

• Discount Rate: The discount rate is a rate at which present and future cash flows are traded off. It incorporates -

(1) Preference for current consumption (Greater ....Higher Discount Rate)

(2) expected inflation (Higher inflation .... Higher Discount Rate)

(3) the uncertainty in the future cash flows (Higher Risk....Higher Discount Rate)

• A higher discount rate will lead to a lower value for cash flows in the future.

• The discount rate is also an opportunity cost, since it captures the returns that an individual would have made on the next best opportunity.

Discounting future cash flows converts them into cash flows in present value dollars. Just a discounting converts future cash flows into present cash flows,

Compounding converts present cash flows into future cash flows.

Aswath Damodaran 5

Present Value Principle 1

Cash flows at different points in time cannot be compared and aggregated. All cash flows have to be brought to the same point in time, before comparisons and aggregations are made.

Aswath Damodaran 6

Time Lines

0 1 2 3

4

$ 100 $ 100 $ 100$ 100

Figure 3.1: A Time Line for Cash Flows: $ 100 in Cash Flows Received

at the End of Each of Next 4 years

Cash Flows

Year

Aswath Damodaran 7

Cash Flow Types and Discounting Mechanics

There are five types of cash flows - simple cash flows, annuities, growing annuities perpetuities and growing perpetuities

Aswath Damodaran 8

I.Simple Cash Flows

A simple cash flow is a single cash flow in a specified future time period.

Cash Flow: CFt

_______________________________________________|

Time Period: t The present value of this cash flow is-

PV of Simple Cash Flow = CFt / (1+r)t

The future value of a cash flow is -

FV of Simple Cash Flow = CF0 (1+ r)t

Aswath Damodaran 9

Application: The power of compounding - Stocks, Bonds and Bills

Between 1926 and 1998, Ibbotson Associates found that stocks on the average made about 11% a year, while government bonds on average made about 5% a year.

If your holding period is one year,the difference in end-of-period values is small:• Value of $ 100 invested in stocks in one year = $ 111

• Value of $ 100 invested in bonds in one year = $ 105

Aswath Damodaran 10

Holding Period and Value

Figure 3.3: Effect of Compounding Periods: Stocks versus T.Bonds

$0.00

$1,000.00

$2,000.00

$3,000.00

$4,000.00

$5,000.00

$6,000.00

$7,000.00

Compounding Periods

Stocks

T. Bonds

$ 100 invested in stocks, earning

11% a year, is worth more than seen

times as much at the end of 40 years

than $ 100 invested in bonds, earning

5% a year.

The compounding effect increases

as the time horizon increases.

Aswath Damodaran 11

Concept Check

Most pension plans allow individuals to decide where their pensions funds will be invested - stocks, bonds or money market accounts.

Where would you choose to invest your pension funds? Predominantly or all equity Predominantly or all bonds and money market accounts A Mix of Bonds and Stocks Will your allocation change as you get older? Yes No

Aswath Damodaran 12

The Frequency of Compounding

The frequency of compounding affects the future and present values of cash flows. The stated interest rate can deviate significantly from the true interest rate –

• For instance, a 10% annual interest rate, if there is semiannual compounding, works out to-

Effective Interest Rate = 1.052 - 1 = .10125 or 10.25%

Frequency Rate t Formula Effective Annual Rate

Annual 10% 1 r 10.00%

Semi-Annual 10% 2 (1+r/2)2-1 10.25%

Monthly 10% 12 (1+r/12)12-1 10.47%

Daily 10% 365 (1+r/365)365-1 10.5156%

Continuous 10% expr-1 10.5171%

Aswath Damodaran 13

II. Annuities

An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. Defining A to be the annuity,

A A A A

| | | |

0 1 2 3 4

Aswath Damodaran 14

Present Value of an Annuity

The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years]

PV of an Annuity = PV(A,r, n) = A

1 - 1

(1 + r)n

r

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Aswath Damodaran 15

Example: PV of an Annuity

The present value of an annuity of $1,000 at the end of each year for the next five years, assuming a discount rate of 10% is -

The notation that will be used in the rest of these lecture notes for the present value of an annuity will be PV(A,r,n).

PV of $1000 each year for next 5 years = $1000

1 - 1

(1.10)5

.10

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥= 3,$ 791

Aswath Damodaran 16

Annuity, given Present Value

The reverse of this problem, is when the present value is known and the annuity is to be estimated - A(PV,r,n).

Annuity given Present Value = A(PV, r, n) = PV r

1 - 1

(1 + r)n

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Aswath Damodaran 17

Computing Monthly Payment on a Mortgage

Suppose you borrow $200,000 to buy a house on a 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%. The monthly payments on this loan, with the payments occurring at the end of each month, can be calculated using this equation:• Monthly interest rate on loan = APR/ 12 = 0.08/12 = 0.0067

Monthly Payment on Mortgage = $200,000 0.0067

1 - 1

(1.0067)360

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥=$1473.11

Aswath Damodaran 18

Future Value of an Annuity

The future value of an end-of-the-period annuity can also be calculated as follows-

FV of an Annuity = FV ( A , r , n ) = A

( 1 + r )

n

- 1

r

⎡

⎣

⎢

⎤

⎦

⎥

Aswath Damodaran 19

An Example

Thus, the future value of $1,000 at the end of each year for the next five years, at the end of the fifth year is (assuming a 10% discount rate) -

The notation that will be used for the future value of an annuity will be FV(A,r,n).

FV of $1,000 each year for next 5 years = $1000 (1.10)5 - 1

.10

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = 6,$ 105

Aswath Damodaran 20

Annuity, given Future Value

if you are given the future value and you are looking for an annuity - A(FV,r,n) in terms of notation -

Annuity given Future Value = A(FV, r, n) = FV r

(1+ r)n - 1

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Aswath Damodaran 21

Application : Saving for College Tuition

Assume that you want to send your newborn child to a private college (when he gets to be 18 years old). The tuition costs are $ 16000/year now and that these costs are expected to rise 5% a year for the next 18 years. Assume that you can invest, after taxes, at 8%.• Expected tuition cost/year 18 years from now = 16000*(1.05)18 = $38,506

• PV of four years of tuition costs at $38,506/year = $38,506 * PV(A ,8%,4 years)= $127,537

If you need to set aside a lump sum now, the amount you would need to set aside would be -• Amount one needs to set apart now = $127,357/(1.08)18 = $31,916

If set aside as an annuity each year, starting one year from now -• If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405

Aswath Damodaran 22

Application : How much is an MBA worth?

Assume that you were earning $40,000/year before entering program and that tuition costs are $16000/year. Expected salary is $ 54,000/year after graduation. You can invest money at 8%.For simplicity, assume that the first payment of $16,000 has to be made at

the start of the program and the second payment one year later.

• PV Of Cost Of MBA = $16,000+16,000/1.08 + 40000 * PV(A,8%,2 years) = $102,145

Assume that you will work 30 years after graduation, and that the salary differential ($14000 = $54000-$40000) will continue through this period.• PV of Benefits Before Taxes = $14,000 * PV(A,8%,30 years) = $157,609

• This has to be discounted back two years - $157,609/1.082 = $135,124

• The present value of getting an MBA is = $135,124 - $102,145 = $32,979

Aswath Damodaran 23

Some Follow-up Questions

1. How much would your salary increment have to be for you to break even on your MBA?

2. Keeping the increment constant, how many years would you have to work to break even?

Aswath Damodaran 24

Application: Savings from Refinancing Your Mortgage

Assume that you have a thirty-year mortgage for $200,000 that carries an interest rate of 9.00%. The mortgage was taken three years ago. Since then, assume that interest rates have come down to 7.50%, and that you are thinking of refinancing. The cost of refinancing is expected to be 2.50% of the loan. (This cost includes the points on the loan.) Assume also that you can invest your funds at 6%.Monthly payment based upon 9% mortgage rate (0.75% monthly rate)

= $200,000 * A(PV,0.75%,360 months)

= $1,609

Monthly payment based upon 7.50% mortgage rate (0.625% monthly rate)

= $200,000 * A(PV,0.625%,360 months)

= $1,398 Monthly Savings from refinancing = $1,609 - $1,398 = $211

Aswath Damodaran 25

Refinancing: The Trade Off

If you plan to remain in this house indefinitely,

Present Value of Savings (at 6% annually; 0.5% a month)

= $211 * PV(A,0.5%,324 months)

= $33,815• The savings will last for 27 years - the remaining life of the existing mortgage.

• You will need to make payments for three additional years as a consequence of the refinancing -

Present Value of Additional Mortgage payments - years 28,29 and 30

= $1,398 * PV(A,0.5%,36 months)/1.0627

= $9,532 Refinancing Cost = 2.5% of $200,000 = $5,000 Total Refinancing Cost = $9,532 + $5,000 = $14,532 Net Effect = $ 33,815 - $ 14,532 = $ 19,283: Refinance

Aswath Damodaran 26

Follow-up Questions

1. How many years would you have to live in this house for you break even on this refinancing?

2. We've ignored taxes in this analysis. How would it impact your decision?

Aswath Damodaran 27

Valuing a Straight Bond

You are trying to value a straight bond with a fifteen year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%.PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) + 1000/1.08515 = $

1186.85 If interest rates rise to 10%,

PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.1015 = $1,057.05

Percentage change in price = -10.94% If interest rate fall to 7%,

PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.0715 = $1,341.55

Percentage change in price = +13.03% This asymmetric response to interest rate changes is called convexity.

Aswath Damodaran 28

Contrasting Short Term and Long Term Bonds

Price Changes as a function of Bond Maturities

Bond Maturity

% Change in Price

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

1 5 15 30

% Change if rate drops

to 7%

% Change if rate

increases to 10%

Aswath Damodaran 29

Bond Pricing Proposition 1

The longer the maturity of a bond, the more sensitive it is to changes in interest rates.

Aswath Damodaran 30

Contrasting Low-coupon and High-coupon Bonds

Bond Price Changes as a function of Coupon Rates

Coupon Rate

% Price Change

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

0% 5% 10.75% 12%

% Change if rate

drops to 7%

% Change if rate

increases to 10%

Aswath Damodaran 31

Bond Pricing Proposition 2

The lower the coupon rate on the bond, the more sensitive it is to changes in interest rates.

Aswath Damodaran 32

III. Growing Annuity

A growing annuity is a cash flow growing at a constant rate for a specified period of time. If A is the current cash flow, and g is the expected growth rate, the time line for a growing annuity looks as follows –

0 1 2 3

A(1+g)2

Figure 3.8: A Growing Annuity

A(1+g)3

A(1+g)nA(1+g)

n...........

Aswath Damodaran 33

Present Value of a Growing Annuity

The present value of a growing annuity can be estimated in all cases, but one - where the growth rate is equal to the discount rate, using the following model:

In that specific case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect.

€

PV of an Annuity = PV(A,r,g,n) = A(1+g) 1 -

(1+g)n

(1+r)n

(r -g)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

Aswath Damodaran 34

The Value of a Gold Mine

Consider the example of a gold mine, where you have the rights to the mine for the next 20 years, over which period you plan to extract 5,000 ounces of gold every year. The price per ounce is $300 currently, but it is expected to increase 3% a year. The appropriate discount rate is 10%. The present value of the gold that will be extracted from this mine can be estimated as follows –

PV of extracted gold = $300* 5000 * (1.03)

1 - (1.03)20

(1.10)20

.10 - .03

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

=$16,145,980

Aswath Damodaran 35

PV of Extracted Gold as a Function of Expected Growth Rate

Present Value of Extracted Gold as a function of Growth Rate

$-

$5,000,000

$10,000,000

$15,000,000

$20,000,000

$25,000,000

$30,000,000

$35,000,000

$40,000,000

$45,000,000

$50,000,000

0% 1% 2% 3% 4% 5% 6% 7% 8% 9%

10% 11% 12% 13% 14% 15%

Growth Rate in Gold Prices

Present Value of Extracted Gold

Aswath Damodaran 36

Concept Check

If both the growth rate and the discount rate go up by 1%, will the present value of the gold to be extracted from this mine increase or decrease?

Aswath Damodaran 37

IV. Perpetuity

A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is-

PV of Perpetuity = A

r

Aswath Damodaran 38

Valuing a Console Bond

A console bond is a bond that has no maturity and pays a fixed coupon. Assume that you have a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows -

Value of Console Bond = $60 / .09 = $667

Aswath Damodaran 39

V. Growing Perpetuities

A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is -

where• CF1 is the expected cash flow next year,

• g is the constant growth rate and

• r is the discount rate.

PV of Growing Perpetuity = CF1

(r - g)

Aswath Damodaran 40

Valuing a Stock with Growing Dividends

Southwestern Bell paid dividends per share of $2.73 in 1992. Its earnings and dividends have grown at 6% a year between 1988 and 1992, and are expected to grow at the same rate in the long term. The rate of return required by investors on stocks of equivalent risk is 12.23%.

Current Dividends per share = $2.73

Expected Growth Rate in Earnings and Dividends = 6%

Discount Rate = 12.23%

Value of Stock = $2.73 *1.06 / (.1223 -.06) = $46.45

Aswath Damodaran 41

Financial Statement Analysis

Aswath Damodaran 42

Questions we would like answered…

Assets Liabilities

Assets in Place Debt

Equity

What is the value of the debt?

How risky is the debt?

What is the value of the equity?

How risky is the equity?

Growth Assets

What are the assets in place?

How valuable are these assets?

How risky are these assets?

What are the growth assets?

How valuable are these assets?

Aswath Damodaran 43

Basic Financial Statements

The balance sheet, which summarizes what a firm owns and owes at a point in time.

The income statement, which reports on how much a firm earned in the period of analysis

The statement of cash flows, which reports on cash inflows and outflows to the firm during the period of analysis

Aswath Damodaran 44

The Balance Sheet

Assets Liabilities

Fixed Assets

Debt

Equity

Short-term liabilities of the firm

Intangible Assets

Long Lived Real Assets

Assets which are not physical,

like patents & trademarks

Current Assets

Financial InvestmentsInvestments in securities &

assets of other firms

Short-lived Assets

Equity investment in firm

Debt obligations of firm

Current

Liabilties

Other

LiabilitiesOther long-term obligations

Figure 4.1: The Balance Sheet

Aswath Damodaran 45

A Financial Balance Sheet

Assets Liabilities

Assets in Place Debt

Equity

Fixed Claim on cash flows

Little or No role in management

Fixed Maturity

Tax Deductible

Residual Claim on cash flows

Significant Role in management

Perpetual Lives

Growth Assets

Existing Investments

Generate cashflows today

Includes long lived (fixed) and

short-lived(working

capital) assets

Expected Value that will be

created by future investments

Aswath Damodaran 46

The Income Statement

Figure 4.2: Income Statement

Revenues

Gross revenues from sale

of products or services

- Operating Expenses

Expenses associates with

generating revenues

= Operating IncomeOperating income for the

period

- Financial ExpensesExpenses associated with

borrowing and other financing

- TaxesTaxes due on taxable income

= Net Income before extraordinary items

Earnings to Common &

Preferred Equity for

Current Period

- (+) Extraordinary Losses (Profits)Profits and Losses not

associated with operations

- Income Changes Associated with Accounting ChangesProfits or losses associated

with changes in accounting

rules

- Preferred DividendsDividends paid to preferred

stockholders

= Net Income to Common Stockholders

Aswath Damodaran 47

Modifications to Income Statement

There are a few expenses that consistently are miscategorized in financial statements.In particular,• Operating leases are considered as operating expenses by accountants but

they are really financial expenses

• R &D expenses are considered as operating expenses by accountants but they are really capital expenses.

The degree of discretion granted to firms on revenue recognition and extraordinary items is used to manage earnings and provide misleading pictures of profitability.

Aswath Damodaran 48

Dealing with Operating Lease Expenses

Debt Value of Operating Leases = PV of Operating Lease Expenses at the pre-tax cost of debt

This now creates an asset - the value of which is equal to the debt value of operating leases. This asset now has to be depreciated over time.

Finally, the operating earnings has to be adjusted to reflect these changes:• Adjusted Operating Earnings = Operating Earnings + Operating Lease

Expense - Depreciation on the leased asset

• If we assume that depreciation = principal payment on the debt value of operating leases, we can use a short cut:

Adjusted Operating Earnings = Operating Earnings + Debt value of Operating leases * Cost of debt

Aswath Damodaran 49

Operating Leases at Boeing and The Home Depot in 1998

Boeing Home Depot

Year Operating Lease Expense Present Value at

5.5%

Operating

Lease Expense

Present Value

at 5.8%

1 $ 205 $ 194.31 $ 294 $ 277.88

2 $ 167 $ 150.04 $ 291 $ 259.97

3 $ 120 $ 102.19 $ 264 $ 222.92

4 $ 86 $ 69.42 $ 245 $ 195.53

5 $ 61 $ 46.67 $ 236 $ 178.03

Yr 6 -15 $ - $ - $ 270 $ 1,513.37

PV of Operating Lease Expenses $ 562.64 $ 2,647.70

Aswath Damodaran 50

Imputed Interest Expenses on Operating Leases

Boeing The Home DepotPV of Operating Leases $ 562.64 $ 2647.70Interest rate on Debt 5.50% 5.80%Imputed interest expense on PV of operating leases $ 30.95 $ 153.57

Aswath Damodaran 51

The Effects of Capitalizing Operating Leases

Debt : will increase, leading to an increase in debt ratios used in the cost of capital and levered beta calculation

Operating income: will increase, since operating leases will now be before the imputed interest on the operating lease expense

Net income: will be unaffected since it is after both operating and financial expenses anyway

Return on Capital will generally decrease since the increase in operating income will be proportionately lower than the increase in book capital invested

Aswath Damodaran 52

R&D Expenses: Operating or Capital Expenses

Accounting standards require us to consider R&D as an operating expense even though it is designed to generate future growth. It is more logical to treat it as capital expenditures.

To capitalize R&D,• Specify an amortizable life for R&D (2 - 10 years)

• Collect past R&D expenses for as long as the amortizable life

• Sum up the unamortized R&D over the period. (Thus, if the amortizable life is 5 years, the research asset can be obtained by adding up 1/5th of the R&D expense from five years ago, 2/5th of the R&D expense from four years ago...:

Aswath Damodaran 53

Capitalizing R&D Expenses: Boeing

Year R&D UnamortizedPortion

Value

1989 $754 0.10 $75

1990 $827 0.20 $165

1991 $1,417 0.30 $425

1992 $1,846 0.40 $738

1993 $1,661 0.50 $831

1994 $1,704 0.60 $1,022

1995 $1,300 0.70 $910

1996 $1,633 0.80 $1,306

1997 $1,924 0.90 $1,732

1998 $1,895 1.00 $1,895

Capitalized Value of R& D Expenses = $9,100

Aswath Damodaran 54

Boeing’s Corrected Operating Income

BoeingOperating Income $1,720 + Research and Development Expenses $1,895 - Amortization of Research Asset $1,382 + Imputed Interest Expense on OperatingLeases

$ 31

= Adjusted Operating Income $2,264

Aswath Damodaran 55

The Effect of Capitalizing R&D

Operating Income will generally increase, though it depends upon whether R&D is growing or not. If it is flat, there will be no effect since the amortization will offset the R&D added back. The faster R&D is growing the more operating income will be affected.

Net income will increase proportionately, depending again upon how fast R&D is growing

Book value of equity (and capital) will increase by the capitalized Research asset

Capital expenditures will increase by the amount of R&D; Depreciation will increase by the amortization of the research asset; For all firms, the net cap ex will increase by the same amount as the after-tax operating income.

Aswath Damodaran 56

The Statement of Cash Flows

Cash Flows From Operations

+ Cash Flows From Investing

+ Cash Flows from Financing

Net cash flow from operations,

after taxes and interest expenses

Includes divestiture and acquisition

of real assets (capital expenditures)

and disposal and purchase of

financial assets. Also includes

acquisitions of other firms.

Net cash flow from the issue and

repurchase of equity, from the

issue and repayment of debt and after

dividend payments

= Net Change in Cash Balance

Figure 4.3: Statement of Cash Flows

Aswath Damodaran 57

The Financial perspective on cash flows

In financial analysis, we are much more concerned about • Cash flows to the firm or operating cash flows, which are before cash

flows to debt and equity)

• Cash flows to equity, which are after cash flows to debt but prior to cash flows to equity

Aswath Damodaran 58

Fundamentals of Valuation

Aswath Damodaran 59

Discounted Cashflow Valuation: Basis for Approach

• where,

• n = Life of the asset

• CFt = Cashflow in period t

• r = Discount rate reflecting the riskiness of the estimated cashflows

Value = CFt

(1+ r)tt =1

t = n∑

Aswath Damodaran 60

Two Measures of Cash Flows

Cash flows to Equity: Thesea are the cash flows generated by the asset after all expenses and taxes, and also after payments due on the debt. This cash flow, which is after debt payments, operating expenses and taxes, is called the cash flow to equity investors.

Cash flow to Firm: There is also a broader definition of cash flow that we can use, where we look at not just the equity investor in the asset, but at the total cash flows generated by the asset for both the equity investor and the lender. This cash flow, which is before debt payments but after operating expenses and taxes, is called the cash flow to the firm

Aswath Damodaran 61

Two Measures of Discount Rates

Cost of Equity: This is the rate of return required by equity investors on an investment. It will incorporate a premium for equity risk -the greater the risk, the greater the premium.

Cost of capital: This is a composite cost of all of the capital invested in an asset or business. It will be a weighted average of the cost of equity and the after-tax cost of borrowing.

Aswath Damodaran 62

Equity Valuation

Assets Liabilities

Assets in Place Debt

Equity

Discount rate reflects only the

cost of raising equity financing

Growth Assets

Figure 5.5: Equity Valuation

Cash flows considered are

cashflows from assets,

after debt payments and

after making reinvestments

needed for future growth

Present value is value of just the equity claims on the firm

Aswath Damodaran 63

Valuing Equity in a Finite Life Asset

Assume that you are trying to value the Home Depot’s equity investment in a new store.

Assume that the cash flows from the store after debt payments and reinvestment needs are expected will be $ 850,000 a year, growing at 5% a year for the next 12 years.

In addition, assume that the salvage value of the store, after repaying remaining debt will be $ 1 million.

Finally, assume that the cost of equity is 9.78%.

€

Value of Equity in Store =

850,000 (1.05) 1-(1.05)12

(1.0978)12

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

(.0978-.05) +

1,000,000(1.0978)12 = $8,053,999

Aswath Damodaran 64

Firm Valuation

Assets Liabilities

Assets in Place Debt

Equity

Discount rate reflects the cost

of raising both debt and equity

financing, in proportion to their

use

Growth Assets

Figure 5.6: Firm Valuation

Cash flows considered are

cashflows from assets,

prior to any debt payments

but after firm has

reinvested to create growth

assets

Present value is value of the entire firm, and reflects the value of

all claims on the firm.

Aswath Damodaran 65

Valuing a Finite-Life Asset

Consider the Home Depot's investment in a proposed store. The store is assumed to have a finite life of 12 years and is expected to have cash flows before debt payments and after reinvestment needs of $ 1 million, growing at 5% a year for the next 12 years.

The store is also expected to have a value of $ 2.5 million at the end of the 12th year (called the salvage value).

The Home Depot's cost of capital is 9.51%.

Aswath Damodaran 66

Expected Cash Flows and present value

Year Expected Cash Flows Value at End PV at 9.51%

1 $ 1,050,000 $ 958,817

2 $ 1,102,500 $ 919,329

3 $ 1,157,625 $ 881,468

4 $ 1,215,506 $ 845,166

5 $ 1,276,282 $ 810,359

6 $ 1,340,096 $ 776,986

7 $ 1,407,100 $ 744,987

8 $ 1,477,455 $ 714,306

9 $ 1,551,328 $ 684,888

10 $ 1,628,895 $ 656,682

11 $ 1,710,339 $ 629,638

12 $ 1,795,856 $ 2,500,000 $ 1,444,124

Value of Store = $ 10,066,749

Aswath Damodaran 67

Valuation with Infinite Life

Cash flowsFirm: Pre-debt cash flowEquity: After debt cash flows

Expected GrowthFirm: Growth in Operating EarningsEquity: Growth in Net Income/EPS

CF1CF2CF3CF4CF5ForeverFirm is in stable growth:Grows at constant rateforever

Terminal ValueCFn.........Discount RateFirm:Cost of Capital

Equity: Cost of Equity

ValueFirm: Value of Firm

Equity: Value of Equity

DISCOUNTED CASHFLOW VALUATIONLength of Period of High Growth

Aswath Damodaran 68

Valuing the Home Depot’s Equity

Assume that we expect the free cash flows to equity at th Home Depot to grow for the next 10 years at rates much higher than the growth rate for the economy. To estimate the free cash flows to equity for the next 10 years, we make the following assumptions:• The net income of $1,614 million will grow 15% a year each year for the

next 10 years.• The firm will reinvest 75% of the net income back into new investments

each year, and its net debt issued each year will be 10% of the reinvestment.

• To estimate the terminal price, we assume that net income will grow 6% a year forever after year 10. Since lower growth will require less reinvestment, we will assume that the reinvestment rate after year 10 will be 40% of net income; net debt issued will remain 10% of reinvestment.

Aswath Damodaran 69

Estimating cash flows to equity: The Home Depot

Year Net I ncome Reinvestment Needs Net DebtIssued

FCFE PV of FCFE

1 $ 1,856 $ 1,392 $ (139) $ 603 $ 549

2 $ 2,135 $ 1,601 $ (160) $ 694 $ 576

3 $ 2,455 $ 1,841 $ (184) $ 798 $ 603

4 $ 2,823 $ 2,117 $ (212) $ 917 $ 632

5 $ 3,246 $ 2,435 $ (243) $ 1,055 $ 662

6 $ 3,733 $ 2,800 $ (280) $ 1,213 $ 693

7 $ 4,293 $ 3,220 $ (322) $ 1,395 $ 726

8 $ 4,937 $ 3,703 $ (370) $ 1,605 $ 761

9 $ 5,678 $ 4,258 $ (426) $ 1,845 $ 797

10 $ 6,530 $ 4,897 $ (490) $ 2,122 $ 835

Sum of PV of FCFE = $6,833

Aswath Damodaran 70

Terminal Value and Value of Equity today

FCFE11 = Net Income11 – Reinvestment11 – Net Debt Paid (Issued)11

= $6,530 (1.06) – $6,530 (1.06) (0.40) – (-277) = $ 4,430 million

Terminal Price10 = FCFE11/(ke – g)

= $ 4,430 / (.0978 - .06) = $117,186 million The value per share today can be computed as the sum of the present

values of the free cash flows to equity during the next 10 years and the present value of the terminal value at the end of the 10th year.

Value of the Stock today = $ 6,833 million + $ 117,186/(1.0978)10

= $52,927 million

Aswath Damodaran 71

Valuing Boeing as a firm

Assume that you are valuing Boeing as a firm, and that Boeing has cash flows before debt payments but after reinvestment needs and taxes of $ 850 million in the current year.

Assume that these cash flows will grow at 15% a year for the next 5 years and at 5% thereafter.

Boeing has a cost of capital of 9.17%.

Aswath Damodaran 72

Expected Cash Flows and Firm Value

Terminal Value = $ 1710 (1.05)/(.0917-.05) = $ 43,049 million

Year Cash Flow Terminal Value Present Value

1 $978 $895

2 $1,124 $943

3 $1,293 $994

4 $1,487 $1,047

5 $1,710 $43,049 $28,864

Value of Boeing as a firm = $32,743