Fundamentals of Valuation

P.V. Viswanath

Partly based on Damodaran’s Corporate Finance

P.V. Viswanath 2

Cash Flows:The Accountant’s Approach

The objective of the Statement of Cash Flows, prepared by accountants, is to explain changes in the cash balance rather than to measure the health or value of the firm

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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 acquisitionof real assets (capital expenditures)and disposal and purchase of financial assets. Also includes acquisitions of other firms.

Net cash flow from the issue andrepurchase of equity, from theissue and repayment of debt and afterdividend payments

= Net Change in Cash Balance

Figure 4.3: Statement of Cash Flows

P.V. Viswanath 4

Cash Flows:The Financial Analyst’s Approach

In financial analysis, we are much more concerned about Cash flows to Equity: These are the cash flows generated by the

asset after all expenses and taxes, and also after payments due on the debt. Cash flows to equity, which are after cash flows to debt but prior to cash flows to equity

Cash flow to Firm: This cash flow is before debt payments but after operating expenses and taxes. This looks 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.

These cash flow measures can be used to value assets, the firm’s equity and the entire firm itself.

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Present and Future Value

Present Value – earlier money on a time line Future Value – later money on a time line

0 2 3 54 61

100 100 100100 100100

If a project yields $100 a year for 6 years, we may want to know the value of those flows as of year 1; then the year 1 value would be a present value.

If we want to know the value of those flows as of year 6, that year 6 value would be a future value.

If we wanted to know the value of the year 4 payment of $100 as of year 2, then we are thinking of the year 4 money as future value, and the year 2 dollars as present value.

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Rates and Prices

A rate is a “price” used to convert earlier money into later money, and vice-versa.

If $1 of today’s money is equal in value to $1.05 of next period’s money, then the conversion rate is 0.05 or 5%.

Equivalently, the price of today’s dollar in terms of next period money is 1.05. The excess of next period’s monetary value over this period’s value (1.05 – 1.00 or 0.05) is often referred to, as interest.

The price of next period’s money in terms of today’s money would be 1/1.05 or 95.24 cents.

This price reflects two elements:(1) Preference for current consumption (Greater =>Higher Discount Rate)(2) the uncertainty in the future cash flows (Higher Risk =>Higher Discount Rate)

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Rate Terminology

Interest rate – “exchange rate” between earlier money and later money (normally the later money is certain).

Discount Rate – rate used to convert future value to present value. Compounding rate – rate used to convert present value to future

value. Cost of capital – the rate at which the firm obtains funds for

investment. Opportunity cost of capital – the rate that the firm has to pay

investors in order to obtain an additional $ of funds. Required rate of return – the rate of return that investors demand

for providing the firm with funds for investment.

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Relation between rates

If capital markets are in equilibrium, the rate that the firm has to pay to obtain additional funds will be equal to the rate that investors will demand for providing those funds. This will be “the” market rate.

Hence this is the rate that should be used to convert future values to present values and vice-versa.

Hence this should be the discount rate used to convert future project (or security) cashflows into present values.

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Two essential concepts

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.

2. The concept of a Time Line:

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

P.V. Viswanath 10

Discount Rates and Risk

In reality there is no single discount rate that can be used to evaluate all future cashflows.

The reason is that future cashflows differ not only in terms of when they occur, but also in terms of riskiness.

Hence, one needs to either convert future risky cashflows into certainty-equivalent cashflows, or, as is more commonly done, add a risk premium to the “certain-future-cashflows” discount rate to get the discount rate appropriate for risky-future-cashflows.

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Discounted Cashflow Valuation

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

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Cash Flow Types and Discounting Mechanics

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

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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

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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

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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

StocksT. 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 increasesas the time horizon increases.

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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%

The general formula isEffective Annualized Rate = (1+r/m)m – 1where m is the frequency of compounding (# times per year), andr is the stated interest rate (or annualized percentage rate (APR) per year

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The Frequency of Compounding

Frequency Rate t FormulaEffective 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 10.52%

Continuous 10%   er-1 10.52%

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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

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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]

nrr

AnrAPVAnnuityanofPV

)1(

11),,(

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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

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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

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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

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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

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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

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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

Note, however, that the two formulas, Annuity, given Future Value and Present Value, given annuity can be derived from each other, quite easily. You may want to simply work with a single formula.

P.V. Viswanath 26

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

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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%

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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...........

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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)

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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

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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

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Valuing a Consol Bond

A consol 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 Consol Bond = $60 / .09 = $667

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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)

P.V. Viswanath 34

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

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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. This is used to value equity.

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. This is used to value the entire firm.

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Equity Valuation

Assets Liabilities

Assets in Place Debt

Equity

Discount rate reflects only the cost of raising equity financingGrowth 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

Free Cash Flow to Equity = Net Income – Net Reinvestment – Net Debt Paid (or + Net Debt Issued), where

Net Reinvestment = Incr in Working Capital + Cap Exp – Depreciation

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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

P.V. Viswanath 38

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 paymentsbut 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.

Free Cash Flow to the Firm = Earnings before Interest and Taxes (1-tax rate) – Net ReinvestmentNet Reinvestment is defined as actual expenditures on short-term and long-term assets less depreciation.The tax benefits of debt are not included in FCFF because they are taken into account in the firm’s cost of capital.

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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%.

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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

P.V. Viswanath 41

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

CF1 CF2 CF3 CF4 CF5

Forever

Firm is in stable growth:Grows at constant rateforever

Terminal Value

CFn.........

Discount RateFirm:Cost of Capital

Equity: Cost of Equity

ValueFirm: Value of Firm

Equity: Value of Equity

DISCOUNTED CASHFLOW VALUATION

Length of Period of High Growth

P.V. Viswanath 42

Valuing the Home Depot’s Equity

Assume that we expect the free cash flows to equity at 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.

P.V. Viswanath 43

Estimating cash flows to equity: The Home Depot

Year Net I ncome Reinvestment Needs Net Debt Paid 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

P.V. Viswanath 44

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

P.V. Viswanath 45

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%.

P.V. Viswanath 46

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