# Capital Budgeting

Document Sample

```					Capital Budgeting is the process by which the firm decides which long-term investments
to make. Capital Budgeting projects, i.e., potential long-term investments, are expected to
generate cash flows over several years. The decision to accept or reject a Capital
Budgeting project depends on an analysis of the cash flows generated by the project and
its cost. The following three Capital Budgeting decision rules will be presented:

   Payback Period
   Net Present Value (NPV)
   Internal Rate of Return (IRR)

A Capital Budgeting decision rule should satisfy the following criteria:

   Must consider all of the project's cash flows.
   Must consider the Time Value of Money
   Must always lead to the correct decision when choosing among Mutually
Exclusive Projects.

Concepts
   Payback Period
   Net Present Value (NPV)
   Internal Rate of Return (IRR)
   Equations

Tools and Problems
   Capital Budgeting Calculator
   Capital Budgeting Exercise
   Capital Budgeting Quiz
© 2002 - 2012 by Mark A. Lane, Ph.D.

Net Present Value
(NPV):

Internal Rate of
Return (IRR):

Payback Period:

Ratio Analysis is a form of Financial Statement Analysis that is used to obtain a quick
indication of a firm's financial performance in several key areas. The ratios are
categorized as Short-term Solvency Ratios, Debt Management Ratios, Asset Management
Ratios, Profitability Ratios, and Market Value Ratios.

Ratio Analysis as a tool possesses several important features. The data, which are
provided by financial statements, are readily available. The computation of ratios
facilitates the comparison of firms which differ in size. Ratios can be used to compare a
firm's financial performance with industry averages. In addition, ratios can be used in a
form of trend analysis to identify areas where performance has improved or deteriorated
over time.

Because Ratio Analysis is based upon Accounting information, its effectiveness is limited
by the distortions which arise in financial statements due to such things as Historical Cost
Accounting and inflation. Therefore, Ratio Analysis should only be used as a first step in
financial analysis, to obtain a quick indication of a firm's performance and to identify
areas which need to be investigated further.

The pages below present the most widely used ratios in each of the categories given
above. Please keep in mind that there is not universal agreement as to how many of these
ratios should be calculated. You may find that different books use slightly different
formulas for the computation of many ratios. Therefore, if you are comparing a ratio that
you calculated with a published ratio or an industry average, make sure that you use the
same formula as used in the calculation of the published ratio.
Concepts
   Short-term Solvency Ratios
   Debt Management Ratios
   Asset Management Ratios
   Profitability Ratios
   Market Value Ratios
   Equations

Tools and Problems

   Ratio Analysis Exercise
   Ratio Analysis Quiz

Capital
Budgeting
To begin the calculation of the Payback Period for project A let's add
Concepts           an additional column to the above table which represents the Net
Cash Flow (NCF) for the project in each year.
Payback
Cash      Net Cash
Year
Flow       Flow
Net Present
Value                                    0         -1000        -1000
1          500          -500
Internal Rate of
Return                                   2          400          -100
3          200          100
Capital
Budgeting                                4          200          300
Equations                                5          100          400

Tools &
Problems                                      Payback Period
Payback Period = 2 + (100)/(200) = 2.5 years
Capital
Budgeting
Calculator
Find the Payback Period for the project with the
following cash flows.
Capital
Budgeting
Exercise

Capital
Budgeting Quiz

© 2002 - 2012 by Mark A. Lane, Ph.D.

Capital
Budgeting
Payback Period
Concepts

Payback
The Payback Period represents the amount of time that it takes for a
Capital Budgeting project to recover its initial cost. The use of the
Net Present        Payback Period as a Capital Budgeting decision rule specifies that all
Value              independent projects with a Payback Period less than a specified
number of years should be accepted. When choosing among
Internal Rate of   mutually exclusive projects, the project with the quickest payback is
Return             preferred.
Capital          The calculation of the Payback Period is best illustrated with an
Budgeting        example. Consider Capital Budgeting project A which yields the
Equations        following cash flows over its five year life.

Tools &                                                   Cash
Year
Problems                                                  Flow
0            -1000
Capital
Budgeting                                    1            500
Calculator                                   2            400
Capital                                      3            200
Budgeting                                    4            200
Exercise
5            100
Capital
Budgeting Quiz To begin the calculation of the Payback Period for project A let's add
an additional column to the above table which represents the Net
Cash Flow (NCF) for the project in each year.

Cash       Net Cash
Year
Flow        Flow
0           -1000           -1000
1           500             -500
2           400             -100
3           200             100
4           200             300
5           100             400

Notice that after two years the Net Cash Flow is negative (-1000 +
500 + 400 = -100) while after three years the Net Cash Flow is
positive (-1000 + 500 + 400 + 200 = 100). Thus the Payback Period,
or breakeven point, occurs sometime during the third year. If we
assume that the cash flows occur regularly over the course of the
year, the Payback Period can be computed using the following
equation:

Thus, the Payback Period for project A can be computed as follows:
Payback Period
Payback Period = 2 + (100)/(200) = 2.5 years

Thus, the project will recoup its initial investment in 2.5 years.

As a decision rule, the Payback Period suffers from several flaws.
For instance, it ignores the Time Value of Money, does not consider
all of the project's cash flows, and the accept/reject criterion is
arbitrary.

Example Problems
Find the Payback Period for the project with the
following cash flows.
Year                 Cash Flow

0           \$         -1000

1           \$         500

2           \$         400

3           \$         300

4           \$         200

5           \$         100

Payback:                               years

© 2002 - 2012 by Mark A. Lane, Ph.D.
V
Capital
Budgeting
Net Present Value
Concepts

Payback
Project A Project B
Net Present                                        Cash         Cash
Year
Value                                              Flow         Flow
0        \$-1000       \$-1000
Internal Rate of
Return                                   1          500         100
2          400         200
Capital
Budgeting                                3          200         200
Equations                                4          200         400

Tools &                                  5          100         700
Problems
Thus, if Projects A and B are independent projects then both projects
Capital            should be accepted. On the other hand, if they are mutually exclusive
Budgeting          projects then Project A should be chosen since it has the larger NPV.
Calculator

Capital
Budgeting
Exercise
© 2002 - 2012 by Mark A. Lane, Ph.D.
Capital
Budgeting Quiz
Capital
Budgeting
Net Present Value
Concepts

Payback
The Net Present Value (NPV) of a Capital Budgeting project indicates
the expected impact of the project on the value of the firm. Projects with
Net Present a positive NPV are expected to increase the value of the firm. Thus, the
Value          NPV decision rule specifies that all independent projects with a positive
NPV should be accepted. When choosing among mutually exclusive
Internal Rate of
projects, the project with the largest (positive) NPV should be selected.
Return
The NPV is calculated as the present value of the project's cash inflows
Capital        minus the present value of the project's cash outflows. This relationship
Budgeting      is expressed by the following formula:
Equations

Tools &
Problems

Capital        where
Budgeting
Calculator            CFt = the cash flow at time t and
   r = the cost of capital.
Capital
Budgeting    The example below illustrates the calculation of Net Present Value.
Exercise     Consider Capital Budgeting projects A and B which yield the following
cash flows over their five year lives. The cost of capital for the project is
Capital      10%.
Budgeting Quiz
Project A Project B
Cash         Cash
Year
Flow         Flow
0        \$-1000        \$-1000
1          500          100
2          400          200
3          200          200
4          200          400
5          100          700
Net Present Value
Project A:

Project B:

Example Problems
Find the NPV for the following Capital Budgeting project.
Year                            Cash Flow

0            \$                   -1000

1            \$                   500

2            \$                   400

3            \$                   300

4            \$                   200

5            \$                   100

Cost
of                               10                       %
Capita
l:

NPV:           \$
© 2002 - 2012 by Mark A. Lane, Ph.D.

Capital
Budgeting
Internal Rate of Return
Concepts

Payback
The Internal Rate of Return (IRR) of a Capital Budgeting project is
the discount rate at which the Net Present Value (NPV) of a project
Net Present      equals zero. The IRR decision rule specifies that all independent
Value            projects with an IRR greater than the cost of capital should be
accepted. When choosing among mutually exclusive projects, the
Internal Rate of project with the highest IRR should be selected (as long as the IRR is
Return           greater than the cost of capital).
Capital
Budgeting
Equations

Tools &          where
Problems
    CFt = the cash flow at time t and
Capital
Budgeting        The determination of the IRR for a project, generally, involves trial
Calculator       and error or a numerical technique. Fortunately, financial calculators
greatly simplify this process.
Capital          The example below illustrates the determination of IRR. Consider
Budgeting        Capital Budgeting projects A and B which yield the following cash
Exercise         flows over their five year lives. The cost of capital for both projects is
10%.
Capital
Budgeting Quiz                                   Project A Project B
Cash         Cash
Year
Flow         Flow
0       \$-1000        \$-1000
1         500          100
2         400          200
3         200          200
4         200          400
5         100          700
Internal Rate of Return
Project A:

Project B:

Thus, if Projects A snd B are independent projects then both projects
should be accepted since their IRRs are greater than the cost of
capital. On the other hand, if they are mutually exclusive projects then
Project A should be chosen since it has the higher IRR.

Example Problems
Find the IRR for the following Capital Budgeting project.
Year                       Cash Flow

0             \$              -1000
1            \$              500

2            \$              400

3            \$              300

4            \$              200

5            \$              100

IRR:                                           %

© 2002 - 2012 by Mark A. Lane, Ph.D.

Capital
Budgeting
Internal Rate of Return
Concepts

Payback
The Internal Rate of Return (IRR) of a Capital Budgeting project is
the discount rate at which the Net Present Value (NPV) of a project
Net Present        equals zero. The IRR decision rule specifies that all independent
Value              projects with an IRR greater than the cost of capital should be
accepted. When choosing among mutually exclusive projects, the
Internal Rate of   project with the highest IRR should be selected (as long as the IRR is
Return             greater than the cost of capital).
Capital
Budgeting
Equations

Tools &
Problems          where

Capital                  CFt = the cash flow at time t and
Budgeting
Calculator        The determination of the IRR for a project, generally, involves trial
and error or a numerical technique. Fortunately, financial calculators
Capital           greatly simplify this process.
Budgeting
Exercise       The example below illustrates the determination of IRR. Consider
Capital Budgeting projects A and B which yield the following cash
Capital        flows over their five year lives. The cost of capital for both projects
Budgeting Quiz is 10%.

Project A Project B
Cash         Cash
Year
Flow         Flow
0         \$-1000      \$-1000
1          500          100
2          400          200
3          200          200
4          200          400
5          100          700
Internal Rate of Return
Project A:

Project B:

Thus, if Projects A snd B are independent projects then both projects
should be accepted since their IRRs are greater than the cost of
capital. On the other hand, if they are mutually exclusive projects
then Project A should be chosen since it has the higher IRR.

Example Problems
Find the IRR for the following Capital Budgeting project.
Year                      Cash Flow

0            \$              -1000

1            \$              500

2            \$              400

3            \$              300

4            \$              200

5            \$              100

IRR:                                           %

© 2002 - 2012 by Mark A. Lane, Ph.D.

Ratio Analysis
Concepts
Short-term Solvency or
Short-term
Solvency         Liquidity Ratios
Ratios

Asset
Management       Short-term Solvency Ratios attempt to measure the ability of a firm
Ratios           to meet its short-term financial obligations. In other words, these
ratios seek to determine the ability of a firm to avoid financial
Debt             distress in the short-run. The two most important Short-term
Management       Solvency Ratios are the Current Ratio and the Quick Ratio. (Note:
Ratios           the Quick Ratio is also known as the Acid-Test Ratio.)

Profitability    Current Ratio
Ratios
The Current Ratio is calculated by dividing Current Assets by
Market Value     Current Liabilities. Current Assets are the assets that the firm expects
Ratios           to convert into cash in the coming year and Current Liabilities
represent the liabilities which have to be paid in cash in the coming
Ratio            year. The appropriate value for this ratio depends on the
Equations        characteristics of the firm's industry and the composition of its
Current Assets. However, at a minimum, the Current Ratio should be
Tools &          greater than one.
Problems

Ratio Analysis
Exercise         Example Problems

Ratio Analysis   Use the information below to calculate the Current Ratio.
Quiz
Current Assets: \$    1500

Current Liabilities: \$   1000

Current Ratio:

Quick Ratio
The Quick Ratio recognizes that, for many firms, Inventories can be
rather illiquid. If these Inventories had to be sold off in a hurry to
meet an obligation the firm might have difficulty in finding a buyer
and the inventory items would likely have to be sold at a substantial
discount from their fair market value.

This ratio attempts to measure the ability of the firm to meet its
obligations relying solely on its more liquid Current Asset accounts
such as Cash and Accounts Receivable. This ratio is calculated by
dividing Current Assets less Inventories by Current Liabilities.

Example Problems
Use the information below to calculate the Quick Ratio.

Current Assets: \$     1500

Inventory: \$    500

Current Liabilities: \$   1000

Quick Ratio:

© 2002 - 2012 by Mark A. Lane, Ph.D.

Asset Management Ratios

Asset Management Ratios attempt to measure the firm's success in managing its assets to
generate sales. For example, these ratios can provide insight into the success of the firm's
credit policy and inventory management. These ratios are also known as Activity or
Turnover Ratios.
Receivables Turnover and Days' Receivables
The Receivables Turnover and Days' Receivables Ratios assess the firm's management of
its Accounts Receivables and, thus, its credit policy. In general, the higher the
Receivables Turnover Ratio the better since this implies that the firm is collecting on its
accounts receivables sooner. However, if the ratio is too high then the firm may be
offering too large of a discount for early payment or may have too restrictive credit terms.
The Receivables Turnover Ratio is calculated by dividing Sales by Accounts
Receivables. (Note: since Accounts Receivables arise from Credit Sales it is more
meaningful to use Credit Sales in the numerator if the data is available.)

The Days' Receivables Ratio is calculated by dividing the number of days in a year, 365,
by the Receivables Turnover Ratio. Therefore, the Days' Receivables indicates how long,
on average, it takes for the firm to collect on its sales to customers on credit. This ratio is
also known as the Days' Sales Outstanding (DSO) or Average Collection Period (ACP).

Example Problems
Use the information below to calculate the Receivables Turnover
and Days' Receivables Ratios.

Sales: \$    1500

Accounts Receivable: \$       150

Receivables Turnover:

Days' Receivables:

Inventory Turnover and Days' Inventory
The Inventory Turnover and Days' Inventory Ratios measure the firm's management of
its Inventory. In general, a higher Inventory Turnover Ratio is indicative of better
performance since this indicates that the firm's inventories are being sold more quickly.
However, if the ratio is too high then the firm may be losing sales to competitors due to
inventory shortages. The Inventory Turnover Ratio is calculated by dividing Cost of
Goods Sold by Inventory. When comparing one firms's Inventory Turnover ratio with
that of another firm it is important to consider the inventory valuation methid used by the
firms. Some firms use a FIFO (first-in-first-out) method, others use a LIFO (last-in-first-
out) method, while still others use a weighted average method.
The Days' Inventory Ratio is calculated by dividing the number of days in a year, 365, by
the Inventory Turnover Ratio. Therefore, the Days' Inventory indicates how long, on
average, an inventory item sits on the shelf until it is sold.

Example Problems
Use the information below to calculate the Inventory Turnover
and Days' Inventory Ratios.

Cost of Goods Sold: \$     1500

Inventory: \$     500

Inventory Turnover:

Days' Inventory:

Seasonal Industries
Many firms, such as department stores, are in seaonal industries in which their assets,
especially current assets, and sales volume vary throughout the year.

This can be of particular concern when comparing the Asset Management Ratios of one
firm with another firm in the same industry. This occurs because, in the calculation of
each of the Asset Management Ratios, a number from the Income Statement is divided
by a number from the Balance Sheet. The Income Statement reports revenues and
expenses over a period of time (usually a year) whereas the Balance Sheet reports the
firms assets and liabilities on a particular date.

Thus, for firms in seasonal industries differences in performance may be detected when
no actual difference exists simply because their Balance Sheets are published on different
dates.

Fixed Assets Turnover
The Fixed Assets Turnover Ratio measures how productively the firm is managing its
Fixed Assets to generate Sales. This ratio is calculated by dividing Sales by Net Fixed
Assets. When comparing Fixed Assets Turnover Ratios of different firms it is important
to keep in mind that the values for Net Fixed Assets reported on the firms' Balance Sheets
are book values which can be very different from market values.
Total Assets Turnover
The Total Assets Turnover Ratio measures how productively the firm is managing all of
its assets to generate Sales. This ratio is calculated by dividing Sales by Total Assets.

Example Problems
Use the information below to calculate the Fixed Assets Turnover
and Total Assets Turnover Ratios.

Sales: \$    1500

Net Fixed Assets: \$     1000

Total Assets: \$     1500

Fixed Assets Turnover:

Total Assets Turnover:

Debt Management Ratios

Debt Management Ratios attempt to measure the firm's use of Financial Leverage and
ability to avoid financial distress in the long run. These ratios are also known as Long-
Term Solvency Ratios.

Debt is called Financial Leverage because the use of debt can improve returns to
stockholders in good years and increase their losses in bad years. Debt generally
represents a fixed cost of financing to a firm. Thus, if the firm can earn more on assets
which are financed with debt than the cost of servicing the debt then these additional
earnings will flow through to the stockholders. Moreover, our tax law favors debt as a
source of financing since interest expense is tax deductible.

With the use of debt also comes the possibility of financial distress and bankruptcy.The
amount of debt that a firm can utilize is dictated to a great extent by the characteristics of
the firm's industry. Firms which are in industries with volatile sales and cash flows
cannot utilize debt to the same extent as firms in industries with stable sales and cash
flows. Thus, the optimal mix of debt for a firm involves a tradeoff between the benefits of
leverage and possibility of financial distress.

Debt Ratio, Debt-Equity Ratio, and Equity Multiplier
The Debt Ratio, Debt-Equity Ratio, and Equity Multiplier are essentially three ways of
looking at the same thing: the firm's use of debt to finance its assets. The Debt Ratio is
calculated by dividing Total Debt by Total Assets. The Debt-Equity Ratio is calculated
by dividing Total Debt by Total Owners' Equity. The Equity Multiplier is calculated by
dividing Total Assets by Total Owners' Equity.

Example Problems
Use the information below to calculate the Debt Ratio, Debt-Equity Ratio,
and Equity Multiplier.

Total Assets: \$     2000

Total Debt: \$     1000

Total Owners' Equity: \$      1000

Debt Ratio:            %

Debt-Equity Ratio:

Equity Multiplier:
© 2002 - 2012 by Mark A. Lane, Ph

Debt Management Ratios

Debt Management Ratios attempt to measure the firm's use of Financial Leverage and
ability to avoid financial distress in the long run. These ratios are also known as Long-
Term Solvency Ratios.

Debt is called Financial Leverage because the use of debt can improve returns to
stockholders in good years and increase their losses in bad years. Debt generally
represents a fixed cost of financing to a firm. Thus, if the firm can earn more on assets
which are financed with debt than the cost of servicing the debt then these additional
earnings will flow through to the stockholders. Moreover, our tax law favors debt as a
source of financing since interest expense is tax deductible.

With the use of debt also comes the possibility of financial distress and bankruptcy. The
amount of debt that a firm can utilize is dictated to a great extent by the characteristics of
the firm's industry. Firms which are in industries with volatile sales and cash flows
cannot utilize debt to the same extent as firms in industries with stable sales and cash
flows. Thus, the optimal mix of debt for a firm involves a tradeoff between the benefits of
leverage and possibility of financial distress.

Debt Ratio, Debt-Equity Ratio, and Equity Multiplier

The Debt Ratio, Debt-Equity Ratio, and Equity Multiplier are essentially three ways of
looking at the same thing: the firm's use of debt to finance its assets. The Debt Ratio is
calculated by dividing Total Debt by Total Assets. The Debt-Equity Ratio is calculated
by dividing Total Debt by Total Owners' Equity. The Equity Multiplier is calculated by
dividing Total Assets by Total Owners' Equity.

Example Problems
Use the information below to calculate the Debt Ratio, Debt-Equity Ratio,
and Equity Multiplier.

Total Assets: \$    2000

Total Debt: \$    1000

Total Owners' Equity: \$     1000

Debt Ratio:           %

Debt-Equity Ratio:

Equity Multiplier:

Profitability Ratios

Profitability Ratios attempt to measure the firm's success in generating income. These
ratios reflect the combined effects of the firm's asset and debt management.

Profit Margin
The Profit Margin indicates the dollars in income that the firm earns on each dollar of
sales. This ratio is calculated by dividing Net Income by Sales.

Return on Assets (ROA) and Return on Equity (ROE)
The Return on Assets Ratio indicates the dollars in income earned by the firm on its
assets and the Return on Equity Ratio indicates the dollars of income earned by the firm
on its shareholders' equity. It is important to remember that these ratios are based on
Accounting book values and not on market values. Thus, it is not appropriate to compare
these ratios with market rates of return such as the interest rate on Treasury bonds or the
return earned on an investment in a stock.

Example Problems
Use the information below to calculate the Profit Margin, Return on Assets (ROA),
and Return on Equity (ROE).

Sales: \$    2000

Net Income: \$      500

Total Assets: \$     1500

Total Owners' Equity: \$      1000

Profit Margin:            %

Return on Assets:             %

Return on Equity:             %

Profitability Ratios

Profitability Ratios attempt to measure the firm's success in generating income. These
ratios reflect the combined effects of the firm's asset and debt management.

Profit Margin
The Profit Margin indicates the dollars in income that the firm earns on each dollar of
sales. This ratio is calculated by dividing Net Income by Sales.
Return on Assets (ROA) and Return on Equity (ROE)
The Return on Assets Ratio indicates the dollars in income earned by the firm on its
assets and the Return on Equity Ratio indicates the dollars of income earned by the firm
on its shareholders' equity. It is important to remember that these ratios are based on
Accounting book values and not on market values. Thus, it is not appropriate to compare
these ratios with market rates of return such as the interest rate on Treasury bonds or the
return earned on an investment in a stock.

Example Problems
Use the information below to calculate the Profit Margin, Return on Assets (ROA),
and Return on Equity (ROE).

Sales: \$    2000

Net Income: \$     500

Total Assets: \$    1500

Total Owners' Equity: \$     1000

Profit Margin:           %

Return on Assets:            %

Return on Equity:            %

Market Value Ratios
Market Value Ratios relate an observable market value, the stock price, to book values
obtained from the firm's financial statements.

Price-Earnings Ratio (P/E Ratio)
The Price-Earnings Ratio is calculated by dividing the current market price per share of
the stock by earnings per share (EPS). (Earnings per share are calculated by dividing net
income by the number of shares outstanding.)

The P/E Ratio indicates how much investors are willing to pay per dollar of current
earnings. As such, high P/E Ratios are associated with growth stocks. (Investors who are
willing to pay a high price for a dollar of current earnings obviously expect high earnings
in the future.) In this manner, the P/E Ratio also indicates how expensive a particular
stock is. This ratio is not meaningful, however, if the firm has very little or negative
earnings.

where

Market-to-Book Ratio
The Market-to-Book Ratio relates the firm's market value per share to its book value per
share. Since a firm's book value reflects historical cost accounting, this ratio indicates
management's success in creating value for its stockholders. This ratio is used by "value-
based investors" to help to identify undervalued stocks.

where

Example Problems
Use the information below to calculate the Price-Earnings Ratio
and Market-to-Book Ratio.

Net Income: \$     2000
Total Owners' Equity: \$     5000

Stock Price: \$    25

Number of Shares Outstanding:       1000

Price-Earnings Ratio:

Earnings per Share: \$

Market-to-Book Ratio:

Book Value per Share: \$

Market Value Ratios

Market Value Ratios relate an observable market value, the stock price, to book values
obtained from the firm's financial statements.

Price-Earnings Ratio (P/E Ratio)
The Price-Earnings Ratio is calculated by dividing the current market price per share of
the stock by earnings per share (EPS). (Earnings per share are calculated by dividing net
income by the number of shares outstanding.)

The P/E Ratio indicates how much investors are willing to pay per dollar of current
earnings. As such, high P/E Ratios are associated with growth stocks. (Investors who are
willing to pay a high price for a dollar of current earnings obviously expect high earnings
in the future.) In this manner, the P/E Ratio also indicates how expensive a particular
stock is. This ratio is not meaningful, however, if the firm has very little or negative
earnings.

where
Market-to-Book Ratio
The Market-to-Book Ratio relates the firm's market value per share to its book value per
share. Since a firm's book value reflects historical cost accounting, this ratio indicates
management's success in creating value for its stockholders. This ratio is used by "value-
based investors" to help to identify undervalued stocks.

where

Example Problems
Use the information below to calculate the Price-Earnings Ratio
and Market-to-Book Ratio.

Net Income: \$     2000

Total Owners' Equity: \$     5000

Stock Price: \$    25

Number of Shares Outstanding:       1000

Price-Earnings Ratio:

Earnings per Share: \$

Market-to-Book Ratio:

Book Value per Share: \$
Ratio Analysis Exercise
This exercise demonstrates the analysis of financial statements using Ratio Analysis.
Click the "New Problem" button to generate a new problem. Calculate each of the ratios
indicated below. Then click the "Show Answer" button to view the solution. The
worksheet also functions as a calculator. You can enter your own data into the fields and
then click the buttons to view the solutions.

Balance Sheet (\$ in Millions)                    Income Statement (\$ in
Assets              1998 Liabilities and  1998                     Millions)
Owners' Equity                                       1998
Current Assets             Current                       Sales
Liabilities
Cost of Goods Sold
Cash                       Accounts
Accounts                   Notes Payable                 Expenses
Receivable                                               Depreciation
Inventory                  Total Current                 Earnings Before
Liabilities                   Interest and Taxes
Total Current              Long-Term                     Interest Expense
Assets                     Liabilities
Long-Term Debt                Taxable Income

Fixed Assets                 Total Long-Term             Taxes
Liabilities                 Net Income
Property, Plant, and         Owners' Equity              Dividends
Equipment
Less Accumulated             Common Stock                Addition to Retained
Depreciation                 (\$1 Par)                    Earnings
Net Fixed Assests            Capital Surplus                   Other Information
Number of Shares
Retained                    Outstanding (Milions)
Earnings
Price per Share
Total Owners'
Equity
Total Assets                 Total Liab. and
Owners' Equity

Calculate the following ratios:
Times Interest
Current Ratio                                        Return on Equity (ROE)
Earned
Payout and Retention
Quick Ratio                      Debt Ratio
Ratios
Receivables Turnover and Days'   Debt to Equity
Price/Earnings Ratio
Receivables                      Ratio
Inventory Turnover and Days'
Equity Multiplier   Market-to-Book Ratio
Inventory
EPS and Book Value Per
Fixed Assets Turnover            Profit Margin
Share
Return on Assets
Total Assets Turnover
(ROA)

Ratio Equations
Short-term Solvency Ratios
Current
Ratio:

Quick Ratio:

Asset Management Ratios
Receivables
Turnover:
Days'
Receivables:
Inventory
Turnover:
Days'
Inventory:
Fixed Assets
Turnover:
Total Assets
Turnover:
Debt Management Ratios
Times
Interest
Earned
(TIE) Ratio:

Debt Ratio:

Debt-Equity
Ratio:
Equity
Multiplier:
Profitability Ratio
Profit
Margin:
Return on
Assets:
Return on
Equity:
Market Value Ratios
Price/Earnin
gs Ratios:
Market-to-
Book Ratio:
Dividend Ratios
Payout
Ratio:
Retention
Ratio:
Other Equations
Earnings Per
Share:
Book Value
Per Share:
© 2002 - 2012 by Mark A. Lane, Ph.D.

Ratio Equations
Short-term Solvency Ratios
Current
Ratio:

Quick Ratio:

Asset Management Ratios
Receivables
Turnover:
Days'
Receivables:
Inventory
Turnover:
Days'
Inventory:
Fixed Assets
Turnover:
Total Assets
Turnover:
Debt Management Ratios
Times
Interest
Earned
(TIE) Ratio:

Debt Ratio:

Debt-Equity
Ratio:
Equity
Multiplier:
Profitability Ratio
Profit
Margin:
Return on
Assets:
Return on
Equity:
Market Value Ratios
Price/Earnin
gs Ratios:
Market-to-
Book Ratio:
Dividend Ratios
Payout
Ratio:
Retention
Ratio:
Other Equations
Earnings Per
Share:
Book Value
Per Share:

© 2002 - 2012 by Mark A. Lane, Ph.D.

Financial Cash Flow
In Finance decisions are based upon cash flows. The value of any asset, including the
firm's stock, is based upon the cash flows that it is expected to generate. Moreover, a firm
needs cash to acquire inventories, to acquire fixed assets, to pay wages, to pay dividends,
to pay interest, etc.
Accounting income is not cash flow. This results from the conventions of Accrual
Accounting. For example, noncash items such as depreciation are subtracted out in the
calculation of net income. However, Accounting data is readily available and it is an
important source of information for analysts, both inside and outside the firm.

Therefore, it is useful to extract cash flow information from Financial Statements. In this
section, we shall illustrate how to identify the activities of the firm which are generating
cash and which are consuming cash.

In this section, we shall see how to identify the cash flows generated by the firm's assets
and how those cash flows are distributed to the firm's investors.

Concepts

   Cash Flow from Assets
o Operating Cash Flow
o Capital Spending
o Additions to Net Working Capital
   Cash Flow to Investors
o Cash Flow to Debtholders
o Cash Flow to Common Stockholders
o Cash Flow to Preferred Stockholders

Tools and Problems

   Financial Cash Flow Exercise
   Financial Cash Flow Quiz

Financial Cash Flow
In Finance decisions are based upon cash flows. The value of any asset, including the
firm's stock, is based upon the cash flows that it is expected to generate. Moreover, a firm
needs cash to acquire inventories, to acquire fixed assets, to pay wages, to pay dividends,
to pay interest, etc.

Accounting income is not cash flow. This results from the conventions of Accrual
Accounting. For example, non cash items such as depreciation are subtracted out in the
calculation of net income. However, Accounting data is readily available and it is an
important source of information for analysts, both inside and outside the firm.
Therefore, it is useful to extract cash flow information from Financial Statements. In this
section, we shall illustrate how to identify the activities of the firm which are generating
cash and which are consuming cash.

In this section, we shall see how to identify the cash flows generated by the firm's assets
and how those cash flows are distributed to the firm's investors.

Concepts

   Cash Flow from Assets
o Operating Cash Flow
o Capital Spending
o Additions to Net Working Capital
   Cash Flow to Investors
o Cash Flow to Debt holders
o Cash Flow to Common Stockholders
o Cash Flow to Preferred Stockholders

Tools and Problems

   Financial Cash Flow Exercise
   Financial Cash Flow Quiz

Cash Flow to Investors
The Cash Flow to Investors in the firm, i.e., the debtholders and equityholders, indicates
how the cash flow generated by the firm's assets are distributed to the debtholders and
equityholders. The calculations illustrated on this page will refer to the Balance Sheet and
Income Statement which follow.

Balance Sheet (\$ in Millions)                      Income Statement (\$ in
Assets           1998 1997 Liabilities and 1998 1997                    Millions)
Owners'                                                1998
Equity                            Sales                1710
Current Assets               Current                           Cost of Goods Sold 1100
Liabilities
Cash              690   600   Accounts      530     460      Administrative
100
Payable                        Expenses
Accounts          340   300   Notes Payable 240     240      Depreciation           77
Receivable                                                   Earnings Before
423
Inventory         120   100Total Current      770   700      Interest and Taxes
Liabilities                       Interest Expense       50
Total Current    1150 1000 Long-Term                         Taxable Income         373
Assets                     Liabilities                       Taxes                  30
Long-Term          571   710      Net Income             343
Debt
Dividends              201
Fixed Assets               Total Long-        571   710
142
Liabilities                       Earnings
Property, Plant, 1210 1200 Owners'
and Equipment              Equity
Less             357 280 Common               122   120
Accumulated                Stock (\$1 Par)
Depreciation
Net Fixed        853 920 Capital              218   210
Assests                    Surplus
Retained           322   180
Earnings
Total Owners'      662   510
Equity
Total Assets     2003 1920 Total Liab.        2003 1920
and Owners'
Equity

Cash Flow to Debtholders

The Cash Flow to Debtholders is defined as debt service less new long term borrowing.
(Debt service represents interest expense and repayments of principal.) An equivalent
definition which makes use of values which can readily be obtained form the Balance
Sheet and Income Statement is interest expense less net new borrowing. This can be
calculated as follows:

Cash Flow to Debtholders = Interest Expense - Ending Long-term Debt + Beginning
Long-term Debt

Interest Expense is the fundamental cash flow from the firm to its debtholders. Moreover,
if a firm's Long-term Debt increases from one year to the next (i.e., more new debt was
issued than was repaid) then the debtholders supplied the firm with additional cash.
Cash Flow to Debtholders Example
Using information from the above Balance Sheet and Income Statement.

Solution:

Cash Flow to Debtholders = \$50 - \$571 + \$710 = \$189

Cash Flow to Common Stockholders

The principal cash flow from the firm to its Common Stockholders is dividends. Firms
also issue new stock periodically. This represents a cash flow from the Common
Stockholders to the firm. (Some firm also repurchase their own stock, i.e., a cash flow
from the firm to its stockholders. When this occurs the repurchased shares are recorded
on the Balance Sheet as Treasury Stock. (The Balance Sheet used for the examples on this
page indicates that the firm has no Treasury Stock.) The Cash Flow to Common
Stockholders can be calculated as common dividends paid less net new common equity
raised.

Cash Flow to Common
= Dividends Paid
Stockholders
- (Ending Common Stock - Beginning Common
Stock)
- (Ending Capital Surplus - Beginning Capital
Surplus)
+ (Ending Treasury Stock - Beginning Treasure
Stock)
Cash Flow to Common Stockholders Example
Using information from the above Balance Sheet and Income Statement.

Solution:

Cash Flow to Common Stockholders = \$201 - (\$122 - \$120) - (\$218 - \$210) - (\$0 - \$0) =
\$191

Cash Flow to Preferred Stockholders

The Cash Flow to Preferred Stockholders is defined as Preferred Dividends Paid less net
new Preferred Equity raised.
Cash Flow to Preferred
= Preferred Dividends Paid
Stockholders
- (Ending Preferred Stock - Beginning Preferred
Stock)

In the example on this page, there is no Preferred Stock. Thus the Cash Flow to Preferred
Stockholders for this firm is 0.

Cash Flow to Investors (Debtholders and Equityholders)

Once the above items have been determined, the Cash Flow to the Investors in the Firm
can be calculated as follows:

Cash Flow to Investors = Cash Flow to Debtholders
+ Cash Flow to Common Stockholders
+ Cash Flow to Preferred Stockholders
Cash Flow to Investors Example
Using information from the previous examples.

Solution:

Cash Flow to Investors = \$189 + \$191 + \$0 = \$380

Notice that the Cash Flow to Investors equals the Cash Flow from Assets determined on
the previous page. This was not by accident. Thus, just as there is a Balance Sheet
Identity (Total Assets = Total Liabilities and Owners' Equity) there is also a Cash Flow
Identity (Cash Flow from Assets = Cash Flow to Investors).

Cash Flow to Investors
The Cash Flow to Investors in the firm, i.e., the debtholders and equityholders, indicates
how the cash flow generated by the firm's assets are distributed to the debtholders and
equityholders. The calculations illustrated on this page will refer to the Balance Sheet and
Income Statement which follow.
Balance Sheet (\$ in Millions)                  Income Statement (\$ in
Assets           1998 1997 Liabilities and    1998 1997              Millions)
Owners'                                            1998
Equity                        Sales                1710
Current Assets               Current                       Cost of Goods Sold 1100
Cash             690 600 Accounts             530   460                         100
Expenses
Payable                       Depreciation         77
Accounts         340 300 Notes Payable        240   240    Earnings Before
Receivable                                                                      423
Interest and Taxes
Inventory        120 100 Total Current        770   700    Interest Expense     50
Liabilities
Taxable Income       373
Total Current    1150 1000 Long-Term
Assets                       Liabilities                   Taxes                30
Long-Term        571   710    Net Income           343
Debt                          Dividends            201
Fixed Assets                 Total Long-      571   710    Addition to Retained
142
Term                          Earnings
Liabilities
Property, Plant, 1210 1200 Owners'
and Equipment                Equity
Less             357 280 Common               122   120
Accumulated                  Stock (\$1 Par)
Depreciation
Net Fixed        853 920 Capital              218   210
Assests                      Surplus
Retained         322   180
Earnings
Total Owners'    662   510
Equity
Total Assets     2003 1920 Total Liab.        2003 1920
and Owners'
Equity

Cash Flow to Debtholders

The Cash Flow to Debtholders is defined as debt service less new long term borrowing.
(Debt service represents interest expense and repayments of principal.) An equivalent
definition which makes use of values which can readily be obtained form the Balance
Sheet and Income Statement is interest expense less net new borrowing. This can be
calculated as follows:
Cash Flow to Debtholders = Interest Expense - Ending Long-term Debt + Beginning
Long-term Debt

Interest Expense is the fundamental cash flow from the firm to its debtholders. Moreover,
if a firm's Long-term Debt increases from one year to the next (i.e., more new debt was
issued than was repaid) then the debtholders supplied the firm with additional cash.

Cash Flow to Debtholders Example
Using information from the above Balance Sheet and Income Statement.

Solution:

Cash Flow to Debtholders = \$50 - \$571 + \$710 = \$189

Cash Flow to Common Stockholders

The principal cash flow from the firm to its Common Stockholders is dividends. Firms
also issue new stock periodically. This represents a cash flow from the Common
Stockholders to the firm. (Some firm also repurchase their own stock, i.e., a cash flow
from the firm to its stockholders. When this occurs the repurchased shares are recorded
on the Balance Sheet as Treasury Stock. (The Balance Sheet used for the examples on this
page indicates that the firm has no Treasury Stock.) The Cash Flow to Common
Stockholders can be calculated as common dividends paid less net new common equity
raised.

Cash Flow to Common
= Dividends Paid
Stockholders
- (Ending Common Stock - Beginning Common
Stock)
- (Ending Capital Surplus - Beginning Capital
Surplus)
+ (Ending Treasury Stock - Beginning Treasure
Stock)
Cash Flow to Common Stockholders Example
Using information from the above Balance Sheet and Income Statement.

Solution:

Cash Flow to Common Stockholders = \$201 - (\$122 - \$120) - (\$218 - \$210) - (\$0 - \$0) =
\$191
Cash Flow to Preferred Stockholders

The Cash Flow to Preferred Stockholders is defined as Preferred Dividends Paid less net
new Preferred Equity raised.

Cash Flow to Preferred
= Preferred Dividends Paid
Stockholders
- (Ending Preferred Stock - Beginning Preferred
Stock)

In the example on this page, there is no Preferred Stock. Thus the Cash Flow to Preferred
Stockholders for this firm is 0.

Cash Flow to Investors (Debtholders and Equityholders)

Once the above items have been determined, the Cash Flow to the Investors in the Firm
can be calculated as follows:

Cash Flow to Investors = Cash Flow to Debtholders
+ Cash Flow to Common Stockholders
+ Cash Flow to Preferred Stockholders
Cash Flow to Investors Example
Using information from the previous examples.

Solution:

Cash Flow to Investors = \$189 + \$191 + \$0 = \$380

Notice that the Cash Flow to Investors equals the Cash Flow from Assets determined on
the previous page. This was not by accident. Thus, just as there is a Balance Sheet
Identity (Total Assets = Total Liabilities and Owners' Equity) there is also a Cash Flow
Identity (Cash Flow from Assets = Cash Flow to Investors).
Financial Cash Flow Equations
Operating Cash Flow = EBIT + Depreciation - Taxes
Capital Spending = Ending Net Fixed Assets
- Beginning Net Fixed Assets
+ Depreciation
Additions to NWC = Ending NWC - Beginning NWC
Net Working Capital = Current Assets - Current Liabilities
Cash Flow from Assets = Operating Cash Flow
- Capital Spending
Cash Flow to Debtholders = Interest Expense
- Ending Long-term Debt
+ Beginning Long-term Debt
Cash Flow to Common = Dividends Paid
Stockholders - (Ending Common Stock - Beginning Common
Stock)
- (Ending Capital Surplus - Beginning Capital
Surplus)
+ (Ending Treasury Stock - Beginning Treasure
Stock)
Cash Flow to Preferred = Preferred Dividends Paid
Stockholders - (Ending Preferred Stock - Beginning Preferred
Stock)
Cash Flow to Investors = Cash Flow to Debtholders
+ Cash Flow to Common Stockholders
+ Cash Flow to Preferred Stockholders

© 2002 - 2012 by Mark A. Lane, Ph.D.

Financial Forecasting
Financial Forecasting describes the process by which firms think about and prepare for
the future. The forecasting process provides the means for a firm to express its goals and
priorities and to ensure that they are internally consistent. It also assists the firm in
identifying the asset requirements and needs for external financing.

For example, the principal driver of the forecasting process is generally the sales forecast.
Since most Balance Sheet and Income Statement accounts are related to sales, the
forecasting process can help the firm assess the increase in Current and Fixed Assets
which will be needed to support the forecasted sales level. Similarly, the external
financing which will be needed to pay for the forecasted increase in assets can be
determined.

Firms also have goals related to Capital Structure (the mix of debt and equity used to
finance the firms assets), Dividend Policy, and Working Capital Management. Therefore,
the forecasting process allows the firm to determine if its forecasted sales growth rate is
consistent with its desired Capital Structure and Dividend Policy.

The forecasting approach presented in this section is the Percentage of Sales method. It
forecasts the Balance Sheet and Income Statement by assuming that most accounts
maintain a fixed proportion of Sales. This approach, although fairly simple, illustrates
many of the issues related to forecasting and can readily be extended to allow for a more
flexible technique, such as forecasting items on an individual basis.

Concepts

   Percentage of Sales Method
   External Financing Needed (EFN)
   Financial Forecasting Equations

Tools and Problems

   Financial Forecasting Exercise
   Financial Forecasting Quiz

© 2002 - 2012 by Mark A. Lane, Ph.D.
Percentage of Sales Method
The Percentage of Sales Method is a Financial Forecasting approach which is based on
the premise that most Balance Sheet and Income Statement Accounts vary with sales.
Therefore, the key driver of this method is the Sales Forecast and based upon this, Pro-
Forma Financial Statements (i.e., forecasted) can be constructed and the firms needs for
external financing can be identified. The calculations illustrated on this page will refer to
the Balance Sheet and Income Statement which follow. The forecasted Sales growth rate
in this example is 25%

Balance Sheet (\$ in Millions)                      Income Statement (\$ in
Assets              1999 Liabilities and          1999                 Millions)
Owners' Equity                                          1999
Current Assets             Current Liabilities              Sales                  1200
Cash                200 Accounts Payable          400       Cost of Goods Sold     900
Accounts            400 Notes Payable             400       Taxable Income         300
Receivable                                                  Taxes                  90
Inventory           600 Total Current             800       Net Income             210
Liabilities                      Dividends              70
Total Current       1200 Long-Term                          Addition to Retained
Assets                     Liabilities                                             140
Earnings
Long-Term Debt         500
Fixed Assets               Total Long-Term        500
Liabilities
Net Fixed Assests 800 Owners' Equity
Common Stock (\$1       300
Par)
Retained Earnings      400
Total Owners'          700
Equity
Total Assets        2000 Total Liab. And          2000
Owners' Equity

Percentages of Sales

The first step is to express the Balance Sheet and Income Statement accounts which vary
directly with Sales as percentages of Sales. This is done by dividing the balance for these
accounts for the current year (1999) by sales revenue for the current year.

The Balance Sheet accounts which generally vary closely with Sales are Cash, Accounts
Receivable, Inventory, and Accounts Payable. Fixed Assets are also often tied closely to
Sales, unless there is excess capacity. (The issue of excess capacity will be addressed in
External Financing Needed section.) For this example, we will assume that Fixed Assets
are currently at full capacity and, thus, will vary directly will sales.

Retained Earnings on the Balance Sheet represent the cumulative total of the firm's
earnings which have been reinvested in the firm. Thus, the change in this account is
linked to Sales; however, the link comes from relationship between Sales growth and
Earnings

The Notes Payable, Long-Term Debt, and Common Stock accounts do not vary
automatically with Sales. The changes in these accounts depend upon how the firm
chooses to raise the funds needed to support the forecasted growth in Sales.

On the Income Statement, Costs are expressed as a percentage of Sales. Since we are
assuming that all costs remain at a fixed percentage of Sales, Net Income can be
expressed as a percentage of Sales. This indicates the Profit Margin.

Taxes are expressed as a percentage of Taxable Income (to determine the tax rate).
Dividends and Addition to Retained Earnings are expressed as a percentage of Net
Income to determine the Payout and Retention Ratios respectively.

Percentage of Sales Calculations
The examples in this box illustrate the calculations which were used to determine the
percentages provided in the following Balance Sheet and Income Statement.

Cash Cash/Sales = \$200/\$1200 = .1667 = 16.67%
Inventory Inventory/Sales = \$600/\$1200 = .5 = 50%
Accounts Payable (Accounts Payable)/Sales = \$400/\$1200 = .3333 = 33.33%
Costs Costs/Sales = \$900/\$1200 = .75 = 75%
Taxes Taxes/(Taxable Income) = \$90/\$300 = .3 = 30%
Net Income (Net Income)/Sales = \$210/\$1200 = .175 = 17.5%
Dividends Dividends/(Net Income) = \$70/\$210 = .3333 = 33.33%

Balance Sheet (\$ in Millions)                   Income Statement (\$ in
Assets         1999 %        Liabilities 1999 %                     Millions)
and Owners'                                 1999 %
Equity                        Sales         1200
Current                      Current                       Cost of Goods
Assets                       Liabilities                                 900 75%
Sold
Cash          200 16.67% Accounts           400 33.33%      Taxable
300 25%
Payable                            Income
Accounts      400 33.33% Notes              400 N/A         Taxes          90 30%*
Receivable               Payable                            Net Income     210 17.5%
Inventory     600 50.00% Total              800             Dividends      70 33.33%*
Liabilities                        Retained       140 66.67%*
Total Current 1200       Long-Term                          Earnings
Assets                   Liabilities
Long-Term          500 N/A
Debt
Fixed Assets             Total Long-        500
Term
Liabilities
Net Fixed     800 66.67% Owners'
Assests                  Equity
Common             300 N/A
Stock (\$1
Par)
Retained           400 N/A*
Earnings
Total              700
Owners'
Equity
Total Assets 2000        Total Liab.        2000
and Owners'
Equity

Partial Pro-Forma

The next step is to construct the Partial Pro-forma Financial Statements. First, determine
the forcasted Sales level. This is done my multiplying Sales for the current year (1999) by
one plus the forecasted growth rate in Sales.

S1= S0(1 + g) = \$1200(1 + .25) = \$1500

where

    S1 = the forecasted Sales level,
    S0 = the current Sales level, and
    g = the forecasted growth rate in Sales.
Once the forecastes Sales level has been determined, the Balance Sheet and Income
Statement accounts which vary directly with Sales can be determined by multiplying the
percentages by the Sales forecast. The accounts which do not vary directly with Sales are
simply transferred to the Partial Pro-Forma Financial Statements at their current levels.

Retained Earnings on the Balance Sheet are the one item whose amount is determined
using a slightly different procedure. The Partial Pro-Forma balance for Reatined Earnings
equals Retained Earnings in the current year plus the forecasted Addition to Retained
Earnings from the Partial Pro-Forma Income Statement. The balances for summary
accounts, such as Total Current Assets and Total Current Liabilities, are determined by
summing their constituent accounts.

Partial Pro-Forma Calculations
The examples in this box illustrate the calculations which were used to derive the
following Partial Pro-Forma Balance Sheet and Income Statement.

Cash (Cash%)(Sales Forecast) = (16.67%)(\$1500) = \$250
Inventory (Inventory%)(Sales Forecast) = 50%(\$1500) = \$750
Costs (Costs%)(Sales Forecast) = 75%(1500) = \$1200
Retained 66.67%(\$262.5) = \$175
Earnings
Retained Retained Earnings + Addition to Retained Earnings Forecast = \$400 +
Earnings \$175
(Balance Sheet)

Balance Sheet (\$ in Millions)                     Income Statement (\$ in
Assets         1999 2000 Liabilities and     1999 2000                Millions)
Owners'                                          1999 2000
Equity                          Sales            1200 1500
Current Assets              Current                         Cost of Goods
Liabilities                                      900 1125
Sold
Cash           200 250 Accounts              400    500     Taxable Income 300 375
Payable                         Taxes            90 112.5
Accounts       400 500 Notes Payable         400    400     Net Income       210 262.5
Receivable
Dividends        70 87.5
Inventory      600 750 Total Current         800    900
Total Current     1200 1500 Long-Term                       Retained
Assets                      Liabilities                     Earnings
Long-Term           500   500
Debt
Fixed Assets                Total Long-         500   500
Term
Liabilities
Net Fixed         800 1000 Owners'
Assets                      Equity
Common              300   300
Stock (\$1 Par)
Retained            400   575
Earnings
Total Owners'       700   875
Equity
Total Assets      2000 2500 Total Liab.         2000 2275
and Owners'
Equity

External Financing Needed (EFN)

The External Financing Needed (EFN) can be determined from the Partial Pro-Forma
Balance Sheet. It is simply equal to the difference between Partial Pro-Forma Total
Assets and Partial Pro-Forma Total Liabilities and Owners' Equity.

EFN = \$2500 - \$2275 = \$225

Please note that the External Financing Needed section explores the calculation of EFN
when there is excess capacity.

Pro-Forma Financial Statements

The final step is to determine how the EFN is to be raised. Firms can choose to raise the
EFN by borrowing on short-term basis (Notes Payable), borrowing on a long-term basis
(Long-Term Debt), issuing equity (Common Stock), or some combination of the above.
The chosen method is called the Plug.

In this example we shall assume that the EFN is to be raised through long-term
borrowing. Thus the plug is Long-Term Debt. To determine the Pro-Forma Financial
Statements simply increase Long-Term Debt by the EFN of \$225 determined in the
previous step.

Balance Sheet (\$ in Millions)                  Income Statement (\$ in
Assets         1999 2000 Liabilities and 1999 2000                      Millions)
Owners'                                               1999 2000
Equity
Sales            1200 1500
Current Assets           Current
Cost of Goods
Liabilities                                           900 1125
Sold
Cash           200 250 Accounts          400 500
Taxable Income 300 375
Payable
Taxes            90 112.5
Accounts       400 500 Notes Payable 400 400
Receivable                                                    Net Income       210 262.5
Inventory      600 750 Total Current 800 900                  Dividends        70 87.5
Total Current 1200 1500 Long-Term                             Retained         140 175
Assets                   Liabilities                          Earnings
Long-Term       500 500
Debt
Fixed Assets             Total Long- 725 725
Term
Liabilities
Net Fixed      800 1000 Owners'
Assets                   Equity
Common          300 300
Stock (\$1 Par)
Retained        400 575
Earnings
Total Owners' 700 875
Equity
Total Assets   2000 2500 Total Liab.     2000 2500
and Owners'
Equity

© 2002 - 2012 by Mark A. Lane, Ph.D.

Percentage of Sales Method
The Percentage of Sales Method is a Financial Forecasting approach which is based on
the premise that most Balance Sheet and Income Statement Accounts vary with sales.
Therefore, the key driver of this method is the Sales Forecast and based upon this, Pro-
Forma Financial Statements (i.e., forecasted) can be constructed and the firms needs for
external financing can be identified. The calculations illustrated on this page will refer to
the Balance Sheet and Income Statement which follow. The forecasted Sales growth rate
in this example is 25%

Balance Sheet (\$ in Millions)                     Income Statement (\$ in
Assets              1999 Liabilities and         1999                 Millions)
Owners' Equity                                         1999
Current Assets             Current Liabilities             Sales                  1200
Cash                200 Accounts Payable         400       Cost of Goods Sold     900
Accounts            400 Notes Payable            400       Taxable Income         300
Receivable                                                 Taxes                  90
Inventory           600 Total Current            800       Net Income             210
Liabilities                     Dividends              70
Total Current       1200 Long-Term                         Addition to Retained
Assets                     Liabilities                                            140
Earnings
Long-Term Debt        500
Fixed Assets               Total Long-Term       500
Liabilities
Net Fixed Assests 800 Owners' Equity
Common Stock (\$1      300
Par)
Retained Earnings     400
Total Owners'         700
Equity
Total Assets        2000 Total Liab. And         2000
Owners' Equity

Percentages of Sales

The first step is to express the Balance Sheet and Income Statement accounts which vary
directly with Sales as percentages of Sales. This is done by dividing the balance for these
accounts for the current year (1999) by sales revenue for the current year.

The Balance Sheet accounts which generally vary closely with Sales are Cash, Accounts
Receivable, Inventory, and Accounts Payable. Fixed Assets are also often tied closely to
Sales, unless there is excess capacity. (The issue of excess capacity will be addressed in
External Financing Needed section.) For this example, we will assume that Fixed Assets
are currently at full capacity and, thus, will vary directly will sales.

Retained Earnings on the Balance Sheet represent the cumulative total of the firm's
earnings which have been reinvested in the firm. Thus, the change in this account is
linked to Sales; however, the link comes from relationship betwen Sales growth and
Earnings
The Notes Payable, Long-Term Debt, and Common Stock accounts do not vary
automatically with Sales. The changes in these accounts depend upon how the firm
chooses to raise the funds needed to support the forecasted growth in Sales.

On the Income Statement, Costs are expressed as a percentage of Sales. Since we are
assuming that all costs remain at a fixed percentage of Sales, Net Income can be
expressed as a percentage of Sales. This indicates the Profit Margin.

Taxes are expressed as a percentage of Taxable Income (to determine the tax rate).
Dividends and Addition to Retained Earnings are expressed as a percentage of Net
Income to determine the Payout and Retention Ratios respectively.

Percentage of Sales Calculations
The examples in this box illustrate the calculations which were used to determine the
percentages provided in the following Balance Sheet and Income Statement.

Cash Cash/Sales = \$200/\$1200 = .1667 = 16.67%
Inventory Inventory/Sales = \$600/\$1200 = .5 = 50%
Accounts Payable (Accounts Payable)/Sales = \$400/\$1200 = .3333 = 33.33%
Costs Costs/Sales = \$900/\$1200 = .75 = 75%
Taxes Taxes/(Taxable Income) = \$90/\$300 = .3 = 30%
Net Income (Net Income)/Sales = \$210/\$1200 = .175 = 17.5%
Dividends Dividends/(Net Income) = \$70/\$210 = .3333 = 33.33%

Balance Sheet (\$ in Millions)                      Income Statement (\$ in
Assets      1999 %        Liabilities 1999 %                         Millions)
and Owners'                                     1999 %
Equity                           Sales          1200
Current                   Current                          Cost of Goods
Assets                    Liabilities                                     900 75%
Sold
Cash        200 16.67% Accounts         400 33.33%         Taxable
Payable                                         300 25%
Income
Accounts    400 33.33% Notes            400 N/A            Taxes          90 30%*
Receivable                Payable                          Net Income 210 17.5%
Inventory   600 50.00% Total            800                Dividends      70 33.33%*
Current
140 66.67%*
Retained
Total Current 1200      Long-Term                           Earnings
Assets                  Liabilities
Long-Term           500 N/A
Debt
Fixed Assets            Total Long-         500
Term
Liabilities
Net Fixed    800 66.67% Owners'
Assests                 Equity
Common              300 N/A
Stock (\$1
Par)
Retained            400 N/A*
Earnings
Total               700
Owners'
Equity
Total Assets 2000       Total Liab.         2000
and Owners'
Equity

Partial Pro-Forma

The next step is to construct the Partial Pro-forma Financial Statements. First, determine
the forcasted Sales level. This is done my multiplying Sales for the current year (1999) by
one plus the forecasted growth rate in Sales.

S1= S0(1 + g) = \$1200(1 + .25) = \$1500

where

    S1 = the forecasted Sales level,
    S0 = the current Sales level, and
    g = the forecasted growth rate in Sales.

Once the forecastes Sales level has been determined, the Balance Sheet and Income
Statement accounts which vary directly with Sales can be determined by multiplying the
percentages by the Sales forecast. The accounts which do not vary directly with Sales are
simply transferred to the Partial Pro-Forma Financial Statements at their current levels.

Retained Earnings on the Balance Sheet are the one item whose amount is determined
using a slightly different procedure. The Partial Pro-Forma balance for Reatined Earnings
equals Retained Earnings in the current year plus the forecasted Addition to Retained
Earnings from the Partial Pro-Forma Income Statement. The balances for summary
accounts, such as Total Current Assets and Total Current Liabilities, are determined by
summing their constituent accounts.

Partial Pro-Forma Calculations
The examples in this box illustrate the calculations which were used to derive the
following Partial Pro-Forma Balance Sheet and Income Statement.

Cash (Cash%)(Sales Forecast) = (16.67%)(\$1500) = \$250
Inventory (Inventory%)(Sales Forecast) = 50%(\$1500) = \$750
Costs (Costs%)(Sales Forecast) = 75%(1500) = \$1200
Retained 66.67%(\$262.5) = \$175
Earnings
Retained Retained Earnings + Addition to Retained Earnings Forecast = \$400 +
Earnings \$175
(Balance Sheet)

Balance Sheet (\$ in Millions)                     Income Statement (\$ in
Assets         1999 2000 Liabilities and     1999 2000                Millions)
Owners'                                          1999 2000
Equity                          Sales            1200 1500
Current Assets              Current                         Cost of Goods
Liabilities                                      900 1125
Sold
Cash           200 250 Accounts              400    500     Taxable Income 300 375
Payable                         Taxes            90 112.5
Accounts       400 500 Notes Payable         400    400     Net Income       210 262.5
Receivable
Dividends        70 87.5
Inventory      600 750 Total Current         800    900
Retained         140 175
Total Current 1200 1500 Long-Term                           Earnings
Assets                      Liabilities
Long-Term        500    500
Debt
Fixed Assets                Total Long-      500    500
Term
Liabilities
Net Fixed      800 1000 Owners'
Assests                   Equity
Common             300    300
Stock (\$1 Par)
Retained           400    575
Earnings
Total Owners'      700    875
Equity
Total Assets    2000 2500 Total Liab.        2000 2275
and Owners'
Equity

External Financing Needed (EFN)

The External Financing Needed (EFN) can be determined from the Partial Pro-Forma
Balance Sheet. It is simply equal to the difference between Partial Pro-Forma Total
Assets and Partial Pro-Forma Total Liabilities and Owners' Equity.

EFN = \$2500 - \$2275 = \$225

Please note that the External Financing Needed section explores the calculation of EFN
when there is excess capacity.

Pro-Forma Financial Statements

The final step is to determine how the EFN is to be raised. Firms can choose to raise the
EFN by borrowing on short-term basis (Notes Payable), borrowing on a long-term basis
(Long-Term Debt), issuing equity (Common Stock), or some combination of the above.
The chosen method is called the Plug.

In this example we shall assume that the EFN is to be raised through long-term
borrowing. Thus the plug is Long-Term Debt. To determine the Pro-Forma Financial
Statements simply increase Long-Term Debt by the EFN of \$225 determined in the
previous step.

Balance Sheet (\$ in Millions)                     Income Statement (\$ in
Assets         1999 2000 Liabilities and 1999 2000                   Millions)
Owners'                                         1999 2000
Equity                          Sales           1200 1500
Current Assets              Current                         Cost of Goods
Liabilities                                     900 1125
Sold
Cash           200 250 Accounts            400 500          Taxable Income 300 375
Payable                         Taxes           90 112.5
Accounts       400 500 Notes Payable 400 400
Receivable                                               Net Income       210 262.5
Inventory       600   750 Total Current    800   900     Dividends        70 87.5
Total Current   1200 1500 Long-Term                      Retained         140 175
Assets                    Liabilities                    Earnings
Long-Term        500   500
Debt
Fixed Assets              Total Long-      725   725
Term
Liabilities
Net Fixed       800 1000 Owners'
Assests                   Equity
Common           300   300
Stock (\$1 Par)
Retained         400   575
Earnings
Total Owners'    700   875
Equity
Total Assets    2000 2500 Total Liab.      2000 2500
and Owners'
Equity

© 2002 - 2012 by Mark A. Lane, Ph.D.
Risk and Return

Investors purchase financial assets such as shares of stock because they desire to increase
their wealth, i.e., earn a positive rate of return on their investments. The future, however,
is uncertain; investors do not know what rate of return their investments will realize.

In finance, we assume that individuals base their decisions on what they expect to happen
and their assessment of how likely it is that what actually occurs will be close to what
they expected to happen. When evaluating potential investments in financial assets, these
two dimensions of the decision making process are called expected return and risk.

The concepts presented in this section include the development of measures of expected
return and risk on an indivdual financial asset and on a portfolio of financial assets, the
principle of diversification, and the Captial Asset Pricing Model (CAPM).

Concepts
   Expected Return
   Measures of Risk - Variance and Standard Deviation
   Portfolio Risk and Return
   Diversification
   Capital Asset Pricing Model - (CAPM)
   Equations

Tools and Problems
   Expected Return Calculator
   Expected Return Exercise
   Expected Return Quiz
   Two Asset Portfolio Calculator
   Two Asset Portfolio Exercise
   Two Asset Portfolio Quiz
   CAPM Calculator
   CAPM Exercise
   CAPM Quiz

© 2002 - 2012 by Mark A. Lane, Ph.D.

Risk and Return

Investors purchase financial assets such as shares of stock because they desire to increase
their wealth, i.e., earn a positive rate of return on their investments. The future, however,
is uncertain; investors do not know what rate of return their investments will realize.

In finance, we assume that individuals base their decisions on what they expect to happen
and their assessment of how likely it is that what actually occurs will be close to what
they expected to happen. When evaluating potential investments in financial assets, these
two dimensions of the decision making process are called expected return and risk.

The concepts presented in this section include the development of measures of expected
return and risk on an indivdual financial asset and on a portfolio of financial assets, the
principle of diversification, and the Captial Asset Pricing Model (CAPM).

Concepts
   Expected Return
   Measures of Risk - Variance and Standard Deviation
   Portfolio Risk and Return
   Diversification
   Capital Asset Pricing Model - (CAPM)
   Equations

Tools and Problems
   Expected Return Calculator
   Expected Return Exercise
   Expected Return Quiz
   Two Asset Portfolio Calculator
   Two Asset Portfolio Exercise
   Two Asset Portfolio Quiz
   CAPM Calculator
   CAPM Exercise
   CAPM Quiz

© 2002 - 2012 by Mark A. Lane, Ph.D.

Risk and Return

Investors purchase financial assets such as shares of stock because they desire to increase
their wealth, i.e., earn a positive rate of return on their investments. The future, however,
is uncertain; investors do not know what rate of return their investments will realize.

In finance, we assume that individuals base their decisions on what they expect to happen
and their assessment of how likely it is that what actually occurs will be close to what
they expected to happen. When evaluating potential investments in financial assets, these
two dimensions of the decision making process are called expected return and risk.

The concepts presented in this section include the development of measures of expected
return and risk on an indivdual financial asset and on a portfolio of financial assets, the
principle of diversification, and the Captial Asset Pricing Model (CAPM).

Concepts
   Expected Return
   Measures of Risk - Variance and Standard Deviation
   Portfolio Risk and Return
   Diversification
   Capital Asset Pricing Model - (CAPM)
   Equations

Tools and Problems
   Expected Return Calculator
   Expected Return Exercise
   Expected Return Quiz
   Two Asset Portfolio Calculator
   Two Asset Portfolio Exercise
   Two Asset Portfolio Quiz
   CAPM Calculator
   CAPM Exercise
   CAPM Quiz

Expected Return:             %

© 2002 - 2012 by Mark A. Lane, Ph.D.
Expected Return:             %

© 2002 - 2012 by Mark A. Lane, Ph.D.

Measures of Risk - Variance and
Standard Deviation

Risk reflects the chance that the actual return on an investment may be very different than
the expected return. One way to measure risk is to calculate the variance and standard
deviation of the distribution of returns.

Consider the probability distribution for the returns on stocks A and B provided below.

Return on Return on
State Probability      Stock A   Stock B
1        20%            5%         50%
2        30%            10%        30%
3        30%            15%        10%
3        20%            20%       -10%

The expected returns on stocks A and B were calculated on the Expected Return page.
The expected return on Stock A was found to be 12.5% and the expected return on Stock
B was found to be 20%.

Given an asset's expected return, its variance can be calculated using the following
equation:

where

    N = the number of states,
    pi = the probability of state i,
    Ri = the return on the stock in state i, and
    E[R] = the expected return on the stock.

The standard deviation is calculated as the positive square root of the variance.

Variance and Standard Deviation on Stocks A and B
Note: E[RA] = 12.5% and E[RB] = 20%

Stock A

Stock B
Although Stock B offers a higher expected return than Stock A, it also is riskier since its
variance and standard deviation are greater than Stock A's. This, however, is only part of
the picture because most investors choose to hold securities as part of a diversified
portfolio.

Example Problems
Find the Expected Return, Variance, and Standard Deviation on
a stock given the following probability distribution of returns
for the stock.
State      Probability                Return

1          20                         5
%                           %

2          30                         10
%                           %

3          30                         15
%                           %

4          20                         20
%                           %

Expected Return:             %

Variance:

Standard Deviation:              %

© 2002 - 2012 by Mark A. Lane, Ph.D.

Measures of Risk - Variance and
Standard Deviation

Risk reflects the chance that the actual return on an investment may be very different than
the expected return. One way to measure risk is to calculate the variance and standard
deviation of the distribution of returns.

Consider the probability distribution for the returns on stocks A and B provided below.
Return on Return on
State Probability      Stock A   Stock B
1        20%            5%         50%
2        30%            10%        30%
3        30%            15%        10%
3        20%            20%       -10%

The expected returns on stocks A and B were calculated on the Expected Return page.
The expected return on Stock A was found to be 12.5% and the expected return on Stock
B was found to be 20%.

Given an asset's expected return, its variance can be calculated using the following
equation:

where

    N = the number of states,
    pi = the probability of state i,
    Ri = the return on the stock in state i, and
    E[R] = the expected return on the stock.

The standard deviation is calculated as the positive square root of the variance.

Variance and Standard Deviation on Stocks A and B
Note: E[RA] = 12.5% and E[RB] = 20%

Stock A

Stock B
Although Stock B offers a higher expected return than Stock A, it also is riskier since its
variance and standard deviation are greater than Stock A's. This, however, is only part of
the picture because most investors choose to hold securities as part of a diversified
portfolio.

Example Problems
Find the Expected Return, Variance, and Standard Deviation on
a stock given the following probability distribution of returns
for the stock.
State      Probability                Return

1          20                         5
%                           %

2          30                         10
%                           %

3          30                         15
%                           %

4          20                         20
%                           %

Expected Return:             %

Variance:

Standard Deviation:              %

The example on the Portfolio Risk and Return page illustrated that a portfolio formed
from risky securities can have a lower standard deviation than either of the individual
securities. On this page we shall explore this concept further to demonstrate that the
benefits of diversification, i.e., the reduction in risk, depends upon the correlation
coefficient (or covariance) between the returns on the securities comprising the portfolio.

Consider stocks C and D. Stock C has an expected return of 8% and a standard deviation
of 10%. Stock D has an expected return of 16% and a standard deviation of 20%. The
concept of diversification will be illustrated by forming portfolios of stocks C and D
under three different assumptions regarding the correlation coefficient between the
returns on stocks C and D.
Case 1: Correlation Coefficient = 1

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals 1.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%         8%          10%
90%        8.8%         11%
80%        9.6%         12%
70%       10.4%         13%
60%       11.2%         14%
50%        12%          15%
40%       12.8%         16%
30%       13.6%         17%
20%       14.4%         18%
10%       15.2%         19%
0%        16%          20%

Opportunity Set and Efficient Set
Opportunity Set - The opportunity set depicts the set of risk return choices that can be
achieved by forming a portfolio of stocks C and D. It is represented by the entire curves

Efficient Set - The efficent set (or efficient frontier) is the positively sloped portion of
the opportunity set. It is the set of risk return choices which offer the highest expected
return for a given level of risk.

When the correlation coefficient between the returns on two securities is equal to +1 the
returns are said to be perfectly positively correlated. As can be seen from the table and
the plot of the opportunity set, when the returns on two securities are perfectly positively
correlated, none of the risk of the individual stocks can be eliminated by diversification.
In this case, forming a portfolio of stocks C and D simply provides additional risk/return
choices for investors.
Case 2: Correlation Coefficient = -1

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals -1.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%        8%          10%
90%        8.8%         7%
80%        9.6%         4%
70%       10.4%         1%
66.67
10.67%         0%
%
60%       11.2%         2%
50%        12%          5%
40%       12.8%         8%
30%       13.6%        11%
20%       14.4%        14%
10%       15.2%        17%
0%        16%         20%

When the correlation coefficient between the returns on two securities is equal to -1 the
returns are said to be perfectly negatively correlated or perfectly inversely correlated.
When this is the case, all risk can be eliminated by investing a positive amount in the two
stocks. This is shown in the table above when the weight of Stock C is 66.67%.

Case 3: Correlation Coefficient = 0

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals 0.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock       d         Deviatio
C       Return         n
100%        8%          10%
90%        8.8%       9.22%
80%        9.6%       8.94%
70%       10.4%       9.22%
60%       11.2%        10%
50%        12%       11.18%
40%       12.8%      12.65%
30%       13.6%      14.32%
20%       14.4%      16.12%
10%       15.2%      18.03%
0%        16%         20%

When the correlation coefficient between the returns on two securities is equal to 0 the
returns are said to be uncorrelated. In this case, some risk can be eliminated via
diversification. Notice that when the weight of Stock C is between 100% and 60% the
portfolios have a higher expected return than Stock C and a lower standard deviation than
either Stocks C or D. This is depicted in the graph by the inward curve in the opportunity
set.

The Real World

In practice, the correlation coefficient between most stocks ranges between 0.5 to 0.7.
When this is the case, the opportunity set will have a similar shape to that shown in the
case in which the returns were uncorrelated. Thus, risk can be reduced via diversification.
You can utilize the Two Asset Portfolio Calculator to explore this relationship. Moreover,
the benefits of diversification increase as more stocks are added to the portfolio.

© 2002 - 2012 by Mark A. Lane, Ph.D.

The example on the Portfolio Risk and Return page illustrated that a portfolio formed
from risky securities can have a lower standard deviation than either of the individual
securities. On this page we shall explore this concept further to demonstrate that the
benefits of diversification, i.e., the reduction in risk, depends upon the correlation
coefficient (or covariance) between the returns on the securities comprising the portfolio.

Consider stocks C and D. Stock C has an expected return of 8% and a standard deviation
of 10%. Stock D has an expected return of 16% and a standard deviation of 20%. The
concept of diversification will be illustrated by forming portfolios of stocks C and D
under three different assumptions regarding the correlation coefficient between the
returns on stocks C and D.

Case 1: Correlation Coefficient = 1

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals 1.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%         8%          10%
90%        8.8%         11%
80%        9.6%         12%
70%       10.4%         13%
60%       11.2%         14%
50%        12%          15%
40%       12.8%         16%
30%       13.6%         17%
20%       14.4%         18%
10%       15.2%         19%
0%        16%          20%

Opportunity Set and Efficient Set
Opportunity Set - The opportunity set depicts the set of risk return choices that can be
achieved by forming a portfolio of stocks C and D. It is represented by the entire curves

Efficient Set - The efficent set (or efficient frontier) is the positively sloped portion of
the opportunity set. It is the set of risk return choices which offer the highest expected
return for a given level of risk.

When the correlation coefficient between the returns on two securities is equal to +1 the
returns are said to be perfectly positively correlated. As can be seen from the table and
the plot of the opportunity set, when the returns on two securities are perfectly positively
correlated, none of the risk of the individual stocks can be eliminated by diversification.
In this case, forming a portfolio of stocks C and D simply provides additional risk/return
choices for investors.

Case 2: Correlation Coefficient = -1

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals -1.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%         8%          10%
90%        8.8%          7%
80%        9.6%          4%
70%        10.4%         1%
66.67
10.67%         0%
%
60%        11.2%         2%
50%         12%          5%
40%        12.8%         8%
30%        13.6%        11%
20%        14.4%        14%
10%        15.2%        17%
0%         16%         20%

When the correlation coefficient between the returns on two securities is equal to -1 the
returns are said to be perfectly negatively correlated or perfectly inversely correlated.
When this is the case, all risk can be eliminated by investing a positive amount in the two
stocks. This is shown in the table above when the weight of Stock C is 66.67%.
Case 3: Correlation Coefficient = 0

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals 0.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%        8%          10%
90%       8.8%        9.22%
80%       9.6%        8.94%
70%       10.4%       9.22%
60%       11.2%        10%
50%        12%       11.18%
40%       12.8%      12.65%
30%       13.6%      14.32%
20%       14.4%      16.12%
10%       15.2%      18.03%
0%        16%         20%

When the correlation coefficient between the returns on two securities is equal to 0 the
returns are said to be uncorrelated. In this case, some risk can be eliminated via
diversification. Notice that when the weight of Stock C is between 100% and 60% the
portfolios have a higher expected return than Stock C and a lower standard deviation than
either Stocks C or D. This is depicted in the graph by the inward curve in the opportunity
set.

The Real World

In practice, the correlation coefficient between most stocks ranges between 0.5 to 0.7.
When this is the case, the opportunity set will have a similar shape to that shown in the
case in which the returns were uncorrelated. Thus, risk can be reduced via diversification.
You can utilize the Two Asset Portfolio Calculator to explore this relationship. Moreover,
the benefits of diversification increase as more stocks are added to the portfolio.
© 2002 - 2012 by Mark A. Lane, Ph.D.

The example on the Portfolio Risk and Return page illustrated that a portfolio formed
from risky securities can have a lower standard deviation than either of the individual
securities. On this page we shall explore this concept further to demonstrate that the
benefits of diversification, i.e., the reduction in risk, depends upon the correlation
coefficient (or covariance) between the returns on the securities comprising the portfolio.

Consider stocks C and D. Stock C has an expected return of 8% and a standard deviation
of 10%. Stock D has an expected return of 16% and a standard deviation of 20%. The
concept of diversification will be illustrated by forming portfolios of stocks C and D
under three different assumptions regarding the correlation coefficient between the
returns on stocks C and D.

Case 1: Correlation Coefficient = 1

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals 1.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%         8%         10%
90%        8.8%        11%
80%        9.6%        12%
70%       10.4%        13%
60%       11.2%        14%
50%        12%         15%
40%       12.8%        16%
30%       13.6%        17%
20%       14.4%        18%
10%       15.2%        19%
0%        16%         20%

Opportunity Set and Efficient Set
Opportunity Set - The opportunity set depicts the set of risk return choices that can be
achieved by forming a portfolio of stocks C and D. It is represented by the entire curves

Efficient Set - The efficent set (or efficient frontier) is the positively sloped portion of
the opportunity set. It is the set of risk return choices which offer the highest expected
return for a given level of risk.

When the correlation coefficient between the returns on two securities is equal to +1 the
returns are said to be perfectly positively correlated. As can be seen from the table and
the plot of the opportunity set, when the returns on two securities are perfectly positively
correlated, none of the risk of the individual stocks can be eliminated by diversification.
In this case, forming a portfolio of stocks C and D simply provides additional risk/return
choices for investors.

Case 2: Correlation Coefficient = -1

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals -1.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%         8%          10%
90%        8.8%          7%
80%        9.6%          4%
70%       10.4%          1%
66.67
10.67%         0%
%
60%       11.2%          2%
50%        12%           5%
40%       12.8%          8%
30%       13.6%         11%
20%       14.4%         14%
10%       15.2%         17%
0%        16%         20%

When the correlation coefficient between the returns on two securities is equal to -1 the
returns are said to be perfectly negatively correlated or perfectly inversely correlated.
When this is the case, all risk can be eliminated by investing a positive amount in the two
stocks. This is shown in the table above when the weight of Stock C is 66.67%.

Case 3: Correlation Coefficient = 0

The table below provides the expected return and standard deviation for portfolios
formed from stocks C and D under the assumption that the correlation coefficient
between their returns equals 0.

Weigh Portfolio Portfolio
t of Expecte Standard
Stock    d      Deviatio
C    Return      n
100%        8%          10%
90%        8.8%       9.22%
80%        9.6%       8.94%
70%       10.4%       9.22%
60%       11.2%        10%
50%        12%       11.18%
40%       12.8%      12.65%
30%       13.6%      14.32%
20%       14.4%      16.12%
10%       15.2%      18.03%
0%        16%         20%

When the correlation coefficient between the returns on two securities is equal to 0 the
returns are said to be uncorrelated. In this case, some risk can be eliminated via
diversification. Notice that when the weight of Stock C is between 100% and 60% the
portfolios have a higher expected return than Stock C and a lower standard deviation than
either Stocks C or D. This is depicted in the graph by the inward curve in the opportunity
set.
The Real World

In practice, the correlation coefficient between most stocks ranges between 0.5 to 0.7.
When this is the case, the opportunity set will have a similar shape to that shown in the
case in which the returns were uncorrelated. Thus, risk can be reduced via diversification.
You can utilize the Two Asset Portfolio Calculator to explore this relationship. Moreover,
the benefits of diversification increase as more stocks are added to the portfolio.

© 2002 - 2012 by Mark A. Lane, Ph.D.

Capital Asset Pricing Model (CAPM)

Because investors are risk averse, they will choose to hold a portfolio of securities to take
advantage of the benefits of Diversification. Therefore, when they are deciding whether
or not to invest in a particular stock, they want to know how the stock will contribute to
the risk and expected return of their portfolios.

The standard deviation of an individual stock does not indicate how that stock will
contribute to the risk and return of a diversified portfolio. Thus, another measure of risk
is needed; a measure of a security's sytematic risk. This measure is provided by the
Capital Asset Pricing Model (CAPM).

Systematic and Unsystematic Risk
An asset's total risk consists of both systematic and unsystematic risk.

Systematic risk, which is also called market risk or undiversifiable risk, is the portion of
an asset's risk that cannot be eliminated via diversification. The systematic risk indicates
how including a particular asset in a diversified portfolio wil contribute to the riskiness of
the portfolio

Unsystematic risk, which is also called firm-specific or diversifiable risk, is the portion of
an asset's total risk that can be eliminated by including the security as part of a
diversifiable portfolio.

The Capital Asset Pricing Model (CAPM) provides an expression which relates the
expected return on an asset to its systematic risk. The relationship is known as the
Security Market Line (SML) equation and the measure of systematic risk in the CAPM
is called Beta.

The Security Market Line (SML)

The SML equation is expressed as follows:

where

    E[Ri] = the expected return on asset i,
    Rf = the risk-free rate,
    E[Rm] = the expected return on the market portfolio,
    i = the Beta on asset i, and
    E[Rm] - Rf = the market risk premium.

The graph below depicts the SML. Note that the slope of the SML is equal to (E[Rm] - Rf)
which is the market risk premium and that the SML intercepts the y-axis at the risk-free
rate.
In capital market equilibrium, the required return on an asset must equal its expected
return. Thus, the SML equation can also be used to determine an asset's required return
given its Beta.

The Beta (i)

The beta for a stock is defined as follows:

where

    im = the Covariance between the returns on asset i and the market portfolio and
    2m = the Variance of the market portfolio.

Note that, by definition, the beta of the market portfolio equals 1 and the beta of the risk-
free asset equals 0.

An asset's systematic risk, therefore, depends upon its covariance with the market
portfolio. The market portfolio is the most diversified portfolio possible as it consists of
every asset in the economy held according to its market portfolio weight.

Example Problems
1. Find the expected return on a stock given that the risk-free rate is 6%, the expected
return on the market portfolio is 12%, and the beta of the stock is 2.

2. Find the beta on a stock given that its expected return is 16%, the risk-free rate is 4%,
and the expected return on the market portfolio is 12%.

The CAPM Exercise provides interactive example problems based upon the SML
equation.

© 2002 - 2012 by Mark A. Lane, Ph.D.
The Capital Asset Pricing Model (CAPM) provides an expression which relates the
expected return on an asset to its systematic risk. The relationship is known as the
Security Market Line (SML) equation and the measure of systematic risk in the CAPM
is called Beta.

The Security Market Line (SML)

The SML equation is expressed as follows:

where

    E[Ri] = the expected return on asset i,
    Rf = the risk-free rate,
    E[Rm] = the expected return on the market portfolio,
    i = the Beta on asset i, and
    E[Rm] - Rf = the market risk premium.

The graph below depicts the SML. Note that the slope of the SML is equal to (E[Rm] - Rf)
which is the market risk premium and that the SML intercepts the y-axis at the risk-free
rate.

In capital market equilibrium, the required return on an asset must equal its expected
return. Thus, the SML equation can also be used to determine an asset's required return
given its Beta.

The Beta (i)

The beta for a stock is defined as follows:
where

    im = the Covariance between the returns on asset i and the market portfolio and
    2m = the Variance of the market portfolio.

Note that, by definition, the beta of the market portfolio equals 1 and the beta of the risk-
free asset equals 0.

An asset's systematic risk, therefore, depends upon its covariance with the market
portfolio. The market portfolio is the most diversified portfolio possible as it consists of
every asset in the economy held according to its market portfolio weight.

Example Problems
1. Find the expected return on a stock given that the risk-free rate is 6%, the expected
return on the market portfolio is 12%, and the beta of the stock is 2.

2. Find the beta on a stock given that its expected return is 16%, the risk-free rate is 4%,
and the expected return on the market portfolio is 12%.

The CAPM Exercise provides interactive example problems based upon the SML
equation.

Risk and Return Equations

Expected Return:

Variance:
Standard Deviation:

Covariance:

Correlation Coefficient:

Portfolio Expected Return:

Two-Asset Portfolio Variance:

Security Market Line (SML):

Beta:

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