Fundamentals of Valuation
P.V. Viswanath
Partly based on Damodaran’s Corporate
Finance
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
P.V. Viswanath
2
The Statement of Cash Flows
Figure 4.3: Statem ent of Cash Flows
Net cash flow from operat ions,
aft er t axes and int erest expenses
Cash Flows From Operations
Includes divest it ure and acquisit ion
of real assets (capit al expenditures)
+ Cash Flows From Investing
and disposal and purchase of
financial asset s. Also includes
acquisitions of ot her firms.
Net cash flow from t he issue and
+ Cash Flows from Financing
repurchase of equity, from t he
issue and repayment of debt and aft er
dividend payment s
= Net Change in Cash Balance
P.V. Viswanath
3
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.
P.V. Viswanath
4
Present and Future Value
 Present Value – earlier money on a time line
 Future Value – later money on a time line
0
100
100
100
100
100
100
1
2
3
4
5
6
 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.
P.V. Viswanath
5
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)
P.V. Viswanath
6
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.
P.V. Viswanath
7
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.
P.V. Viswanath
8
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:
P.V. Viswanath
9
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-futurecashflows.
P.V. Viswanath
10
Discounted Cashflow Valuation
t = n CF
t
Value = 
t
t =1 (1+ r)
where,



n = life of the asset
CFt = cashflow in period t
r = discount rate reflecting the riskiness of the
estimated cashflows
P.V. Viswanath
11
Cash Flow Types and Discounting
Mechanics
 There are five types of cash flows 




simple cash flows,
annuities,
growing annuities
perpetuities and
growing perpetuities
P.V. Viswanath
12
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 isPV 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
P.V. Viswanath
13
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
P.V. Viswanath
14
Holding Period and Value
P.V. Viswanath
15
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 toEffective Interest Rate = 1.052 - 1 = .10125 or 10.25%
The general formula is
Effective Annualized Rate = (1+r/m)m – 1
where m is the frequency of compounding (# times per year), and
r is the stated interest rate (or annualized percentage rate (APR) per
year
P.V. Viswanath
16
The Frequency of Compounding
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
10.52%
Continuous
10%
er-1
10.52%
P.V. Viswanath
17
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,
0
A
|
1
A
|
2
A
|
3
A
|
4
P.V. Viswanath
18
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]
A
1 
PV of an Annuity PV ( A, r, n)  1 
n 
r  (1  r ) 
P.V. Viswanath
19
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 1 

1 5

(1.10) 
PV of $1000 each year for next 5 years
= $1000
 $3,791
 .10



 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).
P.V. Viswanath
20
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).

r

Annuity given Present Value= A(PV,r,n) = PV
1
1 (1+ r)n

P.V. Viswanath




21
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


Mont hly Paym ent on Mortgage
= $200,0 00
P.V. Viswanath
0.0067 

 $1473.11
1
1 
(1.0067)360 

22
Future Value of an Annuity
 The future value of an end-of-the-period annuity
can also be calculated as follows(1 + r)n - 1 
FV of an Annuity = FV(A,r,n) = A 

r


P.V. Viswanath
23
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) 5
(1.10) - 1 
FV of $1,0 00 each year for next 5 years
= $1000
= $6,1 05

 .10

 The notation that will be used for the future value of an
annuity will be FV(A,r,n).
P.V. Viswanath
24
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 
r

Annuity given Future Value= A(FV,r,n) = FV 
n
(1+ r) - 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
25
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
P.V. Viswanath
26
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%
P.V. Viswanath
27
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 –
P.V. Viswanath
28
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:

(1+ g) n 
1 
n

(1+ r) 
PV of an Annuity= P V(A,r ,g,n) = A(1+ g) 

(r
g)






 In that specific case, the present value is equal to the
nominal sums of the annuities over the period, without the
growth effect.
P.V. Viswanath
29
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 –
20

(1.03) 
1 
(1.10)20 
PV of extract ed gold= $300* 5000* (1.03)
 $16,145,980
 .10- .03 




P.V. Viswanath
30
IV. Perpetuity
 A perpetuity is a constant cash flow at regular
intervals forever. The present value of a perpetuity
isPV of Perpetuity=
P.V. Viswanath
A
r
31
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
P.V. Viswanath
32
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 PV of Growing Perpetuity
=
CF1
(r - g)
where



CF1 is the expected cash flow next year,
g is the constant growth rate and
r is the discount rate.
P.V. Viswanath
33
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
P.V. Viswanath
34
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.
P.V. Viswanath
35
Equity Valuation
Figure 5.5: Equity Valuation
Assets
Cash flows considered are
cashflows from assets,
after debt payments and
after making reinvestments
needed for future growth
Liabilities
Assets in P lace
Growth Assets
Debt
Equity
Discount rate reflects only the
cost of raising equity financing
P resent 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
P.V. Viswanath
36
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%.
 (1.05)12 

850,000 (1.05)
1 - (1.0978)12 



1,000,000
Value of Equit y in St ore=
+
= $8,053,999
(.0978-.05)
(1.0978)12
P.V. Viswanath
37
Firm Valuation
Figure 5.6: Firm Valuation
Assets
Cash flows considered are
cashflows from assets,
prior to any debt payments
but after firm has
reinvested to create growth
assets
Liabilities
Assets in P lace
Growth Assets
Debt
Equity
Discount rate reflects the cost
of raising both debt and equity
financing, in proportion to their
use
P resent 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
Reinvestment
Net 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.
P.V. Viswanath
38
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%.
P.V. Viswanath
39
Expected Cash Flows and present
value
Year
Expecte d Cash Flows
Value at End
PV at 9 .5 1%
1
$
1,050 ,0 00
$
958 ,8 17
2
$
1,102 ,5 0 0
$
919 ,3 29
3
$
1,157 ,6 25
$
881 ,4 68
4
$
1,215 ,5 06
$
845 ,1 66
5
$
1,276 ,2 82
$
810 ,3 59
6
$
1,340 ,0 96
$
776 ,9 86
7
$
1,407 ,1 00
$
744 ,9 87
8
$
1,477 ,4 55
$
714 ,3 06
9
$
1,551 ,3 28
$
684 ,8 88
10
$
1,628 ,8 95
$
656 ,6 82
11
$
1,710 ,3 39
$
629 ,6 38
12
$
1,795 ,8 56
$
1 ,4 44 ,1 24
$
2,500 ,0 00
Value of St ore =
P.V. Viswanath
$ 10 ,0 66 ,7 49
40
Valuation with Infinite Life
DISCOUNTED CASHFLOW VALUATION
Expecte d Gr ow th
Firm: Grow th in
Operating Earnings
Equity: Grow th in
Net Income/EPS
Cas h flow s
Firm: Pre-debt cash
f low
Equity: After debt
cash flow s
Firm is in stable grow th:
Grow s at constant rate
f orever
Terminal Value
Value
Firm: V alue of Firm
CF 1
CF 2
CF 3
CF 4
CF 5
CF n
.........
Forever
Equity: Value of Equity
Le ngth of Pe r iod of High Gr ow th
Dis count Rate
Firm:Cost of Capital
Equity: Cost of Equity
P.V. Viswanath
41
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
42
Estimating cash flows to equity: The
Home Depot
Year
1
2
3
4
5
6
7
8
9
10
Net Income
$
$
$
$
$
$
$
$
$
$
1,856
2,135
2,455
2,823
3,246
3,733
4,293
4,937
5,678
6,530
Reinvestment Needs Net Debt Paid
$
1,392
$
$
1,601
$
$
1,841
$
$
2,117
$
$
2,435
$
$
2,800
$
$
3,220
$
$
3,703
$
$
4,258
$
$
4,897
$
Sum of PV of FCFE =
P.V. Viswanath
(139)
(160)
(184)
(212)
(243)
(280)
(322)
(370)
(426)
(490)
FCFE
$
$
$
$
$
$
$
$
$
$
603
694
798
917
1,055
1,213
1,395
1,605
1,845
2,122
PV of FCFE
$
$
$
$
$
$
$
$
$
$
549
576
603
632
662
693
726
761
797
835
$6,833
43
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
44
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
45
Expected Cash Flows and Firm Value
 Terminal Value = $ 1710 (1.05)/(.0917-.05) = $ 43,049
million
Year
Cash Flow
Terminal
Value
1
$978
$895
2
3
4
5
$1,124
$1,293
$1,487
$1,710
$943
$994
$1,047
$28,864
$43,049
Value of Boeing as a firm =
P.V. Viswanath
Present
Value
$32,743
46