2. 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
3. P.V. Viswanath 3
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
4. 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.
5. P.V. Viswanath 5
Present and Future Value
Present Value – earlier money on a time line
Future Value – later money on a time line
0 2 3 5
4 6
1
100 100 100
100 100
100
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.
6. P.V. Viswanath 6
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)
7. P.V. Viswanath 7
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.
8. P.V. Viswanath 8
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.
9. P.V. Viswanath 9
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:
10. 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.
11. P.V. Viswanath 11
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)t
t =1
t = n
12. P.V. Viswanath 12
Cash Flow Types and Discounting
Mechanics
There are five types of cash flows -
simple cash flows,
annuities,
growing annuities
perpetuities and
growing perpetuities
13. P.V. Viswanath 13
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
14. P.V. Viswanath 14
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
16. P.V. Viswanath 16
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 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
17. P.V. Viswanath 17
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%
18. P.V. Viswanath 18
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
19. P.V. Viswanath 19
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]
n
r
r
A
n
r
A
PV
Annuity
an
of
PV
)
1
(
1
1
)
,
,
(
20. P.V. Viswanath 20
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).
PVof $1000 each year for next 5 years = $1000
1 -
1
(1.10)5
.10
$3,791
21. P.V. Viswanath 21
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
22. P.V. Viswanath 22
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
23. P.V. Viswanath 23
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
24. P.V. Viswanath 24
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).
FVof $1, 000 each year for next 5 years = $1000
(1.10)5
- 1
.10
= $6,105
25. P.V. Viswanath 25
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.
26. 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
27. P.V. Viswanath 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%
28. P.V. Viswanath 28
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 –
29. P.V. Viswanath 29
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.
PVof an Annuity = PV(A,r,g,n) = A(1 +g)
1 -
(1+g)
n
(1+r)n
(r - g)
30. P.V. Viswanath 30
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 –
PVof extracted gold = $300* 5000 * (1.03)
1 -
(1.03)20
(1.10)20
.10 - .03
$16,145,980
31. P.V. Viswanath 31
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
32. P.V. Viswanath 32
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
33. P.V. Viswanath 33
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)
34. 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
35. P.V. Viswanath 35
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.
36. P.V. Viswanath 36
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 fromassets,
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)
37. P.V. Viswanath 37
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
38. 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 fromassets,
prior to any debt payments
but after firmhas
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
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.
39. P.V. Viswanath 39
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%.
41. P.V. Viswanath 41
Valuation with Infinite Life
Cash flows
Firm: Pre-debt cash
flow
Equity: After debt
cash flows
Expected Growth
Firm: Growth in
Operating Earnings
Equity: Growth in
Net Income/EPS
CF1 CF2 CF3 CF4 CF5
Forever
Firm is in stable growth:
Grows at constant rate
forever
Terminal Value
CFn
.........
Discount Rate
Firm:Cost of Capital
Equity: Cost of Equity
Value
Firm: Value of Firm
Equity: Value of Equity
DISCOUNTED CASHFLOW VALUATION
Length of Period of High Growth
42. 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.
44. 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
45. 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%.
46. 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
Editor's Notes
This show all cash flows. The objective is to explain changes in the cash balance rather than to measure the health or value of the firm.
Cash flows to debt: interest payments, new debt issues, debt repayments
Cash flows to equity: dividends, stock buybacks, new stock issues
It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts.
All to often, much of what we read about cash flow violates this principle. As I write this note, I am reading that Alex Rodriguez has signed a 10-year contract for $ 252 million. While he will no doubt be a wealthy man, the contract’s value cannot be correctly estimated by adding cash flows over time…
The value of any asset is the present value of the expected cash flows on the assets.
For purely mechanical purposes, this is a way of categorizing cash flows for calculating present value.
T can be a fraction of a year (say 0.25 years) or include fractions of years..
T can even be stated in terms of months or days, as long as the interest rate is specified accordingly (as a monthly or daily rate)
As the holding period lengthens, the end-of-the-period value differences get larger.
The loan sharks of yesteryear were on to something when they charged daily rates… A 1% daily rate translates into an annual rate of 3678%.
For a cashflow to be an annuity, it has to be
The same amount in each period
The periods have to remain fixed (monthly, annual)
Good place to link up to the financial calculator. The present value button does wonders…
May help to separate the second term in the equation and state it as a present value factor.
In the example above, the present value factor is 3.791. For those using present value tables, this makes the link to the present value equation.
What if the cash flows had been at the beginning of each year?
Present value = $ 1000 + $ 1000 (PV(A, 10%, 4 years) = $ 3,791 (1.1)
Reverses the process. Here, we look for an annuity given a present value (Examples: Figuring out the monthly payment on a mortgage)
Follow-up questions:
If you pay at the beginning of each month, rather than the end, how much could you shorten the mortgage period?
$1473 + 1473(PV(A,.67%,n)) = 200000; Trial and error yields n=350: Save 10 months of payments
This is how much the value will be on the last day (when the last annuity comes due)
Again, worth asking how much the future value would be if the cash flows were at the beginning of each year.
Reverses the problem
The assumptions of constant tuition each year for 4 years, and that the payments are made at the end of each year (no university will accept that payment plan) are purely for simplicity.
A straight bond is a combination of an annuity and a simple cash flow. The interest rate used should reflect the risk of the bond. If it is a government bond, it will be a default-free interest rate. If it is a corporate bond, it should be a rate that reflects the default risk.
We have used one interest rate to discount all of the cash flows. A more precise valuation may be obtained by discounting each cash flow at a rate that reflects when that cash flow comes in.. A 1-year rate for the 1-year cash flow, a 2-year rate for the 2-year cash flow…
This is usually a cash flow that cannot be found on a financial calculator but is very common with equity investments…
This formula will work for all cases, except when g=r. When g>r, both the numerator and denominator become negative allowing you to still compute the present value. When g=r, the present value= nA (the growth and discount rate effects cancel out)
You could always compute the present value by taking each cash flow and discounting back to the present. You should get the same answer.
The best way to think about a perpetuity is not as forever but as a very long time.. The present value of an annuity that last 50 or 60 years converges on this value…
There are British and Canadian console bonds still in existence…
This is a cash flow growing at a constant rate forever. Here, g<r. Since the growth rate is forever, it is constrained to be less than or equal to the growth rate of the economy. If we allow for that constraint, g will always be less than r.
Note that we use dividends from next year in the numerator.
The discount rate can be measured from the perspective of just equity investors or all suppliers of capital in the firm.
The value of equity is the present value of cash flows to the equity investors discounted back at the rate of return that those equity investors need to make to break even (the cost of equity).
In the strictest sense of the word, the only cash flow stockholders in a publicly traded firm get from their investment is dividends, and the dividend discount model is the simplest and most direct version of an equity valuation model.
Reinvestment can be computed as change in non-cash working capital plus capital expenditures less depreciation
Note that these cash flows are after debt payments and the discount rate used is the cost of equity.
A firm includes not just the equity, but all claim holders. The cash flow to the firm is the collective cash flow that all claim holders make from the firm, and it is discounted at the weighted average of their different costs.
This asset has a finite life. Since the cash flows are prior to debt payments, the cash flows are firm cash flows and the cost of capital is the appropriate discount rate.
The value of this store is $ 10.066 million.
With an asset with an infinite life, you need to stop estimating cash flows at some point and estimate a terminal value. In a discounted cash flow framework, this can be done when the growth rate in cash flows becomes perpetual (less than or equal to the growth rate of the economy)
Sets up the basic inputs:
1. Discount rates
2. Cash flows
3. Expected Growth
4. Length of the period that they can sustain a growth rate higher than the growth rate of the economy.
The growth rate and reinvestment rate assumptions are linked.
The free cash flow to equity is the cash flow after net debt payment or issues and reinvestment. Note that the debt creates a cash inflow for equity investors because it reduces how much they need to reinvest back into the firm.
The terminal value of equity is discounted back to today at the cost of equity. Note that the cost of equity of 9.78% is used to estimate the terminal value of equity.
These are the basic inputs for the valuation.
The cost of capital for Boeing of 9.17% is used as the discount rate for the cash flows. This is the value of Boeing as a firm and I would need to subtract out the outstanding debt to get to the value of equity.
