MBA 620 Cost of Capital, Risk/Return, and Capital Budgeting

Scenario

Your team’s work with Largo Global Inc. (LGI) is nearly complete. In your consulting role, you have recommended steps for improving the company’s financial health. You have offered advice on a revenue target, recommended steps for optimizing operations, and suggested investments that will improve LGI’s competitive position. In your final project, you will continue working on capital budgeting with a focus on the best way for LGI to finance the investments you recommend.

Your Project 5 business report will focus on ensuring LGI’s capital structure is sound and that the company is on a financially sustainable path. You will recommend a plan for financing investments that does not expose LGI to unnecessary risk. By the end of this project, the company’s financial statements should demonstrate that it has returned to a competitive position.

Complete the Analysis Calculation

Your team has provided you with an Excel workbook containing LGI’s financials. You will use the

Project 5 Excel workbook

to perform advanced capital budgeting techniques to assess the viability of the investment you made in the previous project.

Complete the analysis calculation for the project:

Download the

Project 5 Excel Workbook

,      click the Instructions tab, and read the instructions.

Calculate cost of debt and equity as well as      weighted average cost of capital (WACC).

Apply the capital asset pricing model (CAPM).

Develop a capital budget.

Prepare the Analysis Report

Project 5 Report
Instructions
Answer the five questions below. They focus entirely on the financing, risk/return, Cost of
Capital, and Budget Forecasting of Largo Global Inc. (LGI) based on the investing activities that
took place in Project 4. Base your analysis on the data provided and calculated in the Excel
workbook. Support your reasoning from the readings in Project 5, Step 1, and the discussion in
Project 5, Step 3. Be sure to cite your sources.
Provide a detailed response below each question. Use 12-point font and double spacing.
Maintain the existing margins in this document. Your final Word document, including the
questions, should be at most five pages. Include a title page in addition to the five pages. Any
tables and graphs you include are also excluded from the five-page limit. Name your document
as follows: P5_Final_lastname_Report_date.
You must address all five questions and fully use the information on all tabs of Project 5 and
data in other Excel workbooks (e.g., from Project 1: ratio, common size, and cash flow analysis).
You are strongly encouraged to exceed the requirements by refining your analysis. Consider
other tools and techniques that were discussed in the required and recommended reading for
Project 5. This means adding an in-depth explanation of what happened in that year for which
data was provided to make precise recommendations to LGI.
Title Page
Name
Course and section number
Faculty name
Submission date
Questions
1. How would you assess the evolution of the capital structure of LGI? Reflecting on your work
in Project 1, would you consider the risk exposure under control? If not, what are your
recommendations?
[insert your answer here]
2. What kind of information do you find valuable in CAPM to guide you in assessing the risk of
LGI compared to other firms and the market in general?
[insert your answer here]
3. Identify and differentiate the stakeholders of LGI and explain how each one should perceive
and weigh the firm’s risk and return.
[insert your answer here]
4. Would you consider the investment made in Project 4 optimally financed, considering the
proportion of debt that LGI bears? How did the current WACC in Project 5 depart from the state
of the firm in Project 1?
[insert your answer here]
5. If you had to advise a potential investor interested in having a minority stake in LGI, what
information would you provide to help the investor decide? Would you be bullish or have
reservations? Support your answer with facts and data from all MBA 620 projects and your
budget projections.
[insert your answer here]
INSTRUCTIONS
Complete the Cost of Capital tab
o Find the cost of Equity using the Capital Asset Pricing Model (CAPM)
o Find the Weighted Average Cost of Capital (WACC)
Complete the Payback tab
o Complete the After-tax Cash Flow re-evaluation table
o Complete the DCF Payback timeline
o Complete the questions on the tab
Complete the Budget Projections tab
o Revenue increases 4% annually
o Expense increases 2¾% annually
Model (CAPM)
Instructions:
1 Find the cost of Equity using the Capital Asset Pricing Model (CAPM)
2 Find the Weighted Average Cost of Equity (WACC)
1
CAPM Information from Largo Global Cost of Equity
RF
Risk-free rate of return = 2.20 percent1
𝛽
Beta = 1.12.
RM
Expected Return of the Market = 7.05 percent2
RP
Market premium = RM – RF
____
1 current U.S. 10-yr Treasury Yield. Source: U.S.Treasury.gov. Mar, 2022
2 The S&P 500 long-term average when holding the S&P 500 index.
Source: https://ycharts.com/indicators/sp_500_1_year_return Jan, 2022
CAPM = Risk Free Rate + Beta x Market Premium
𝐶𝐴𝑃𝑀 =
RF
ꞵ
————————————–
RM
𝑅𝐹
+
( 𝛽 × (𝑅ത𝑀 − 𝑅𝐹 ))
= CAPM
2 WACC Information from Largo Global
a. As of today, Largo Global market capitalization (E) is $6,373,341,000.1
b. Largo Global’s Market value of debt is $761,000,000.
c. Cost of Equity = CAPM from question 1
d. Cost of Debt = Last Fiscal Year End Interest Expense2 / Market Value of Debt (D).
e. Use the tax rates given in Project 4 Tab 3.
_________
1 Market value of equity (E), also known as market cap, is calculated using the following equation:
Market Cap = Share Price x Shares Outstanding from Project 1
2 From Project 1. Note that the Cost of Debt formula expressed at here is different from the cost of debt formula
introduced in most textbooks. Most textbooks only consider the long-term debt (i.e., bond) as the debt and use the
bond valuation formula when calculating the cost of debt and WACC.
E
D
Total Capital (V)
Last Fiscal Year End
Interest Expense
Tax Rate (TC)
$
–
𝑬
𝑽
1. Find the weight of equity = E / (E + D).
𝑫
𝑽
2. Find the weight of debt = D / (E + D).
Re
3. Find the cost of equity using CAPM.
Rd
4. Find the cost of debt.
WACC
5. Find the weighted average cost of capital.
After-Tax Cash Flow Re-evaluation and Payback Timelines Instructions
Technologically advanced distribution equipment proposal re-evaluation
The CFO has asked you to re-evaluate the cash flow projections associated with the equipment purchase proposal due to the proposed loan agreement, and
recommend whether the purchase should go forward. Table 1 shows the data and Table 2 shows projections of the cash inflows and outflows that would occur
during the first eight years using the new equipment.
Keep the following in mind: Row 34 has a suggested Excel function to use. Complete all the blank cells within the tables.
I. In the Data Table:
A. Use the WACC calculated on the Cost of Capital tab
B. Calculate the loan amount with a 10% down payment
II. In the After-tax Cash Flow:
C. Complete the Depreciation Expense from Project 4 (straight line, $0 Salvage)
D. Complete the interest expense using the loan interest rate.
E. Complete the After-tax Cash Flow Table including the interest expense
F. Compute the PV, NPV1, IRR, and adjusted NPV2
III. In the Payback Timeline View:
G. Complete the discounted cash flow Payback Timeline View of Discounted Cash Flows
i) complete the timeline amounts based on the DCF (DCF is the same as PV)
ii) complete the timeline amounts for the Cumulative DCF
iii) calculate the payback period in years and months
IV. Answer the following questions:
1. What is the total depreciation for tax purposes?
2. What is the total PV of the Cash Flows using the WACC rate?
3. What is the NPV using the WACC rate?
4. What is the NPV using the alternative rate?
5. What is the IRR?
6. What is the payback period using the DCF?
7. Should the project be accepted? Why?
Payback Table View
Table 1 – Data
191.10 million
26.0%
8.0%
Cost of new equipment (at year 0)
Corporate income tax rate – Federal
Corporate income tax rate – State of Maryland
Discount rate for the project using WACC
Loan Amount
Loan Interest rate (Prime + 2)
million
5.25%
Table 2 – After-tax Cash Flow Table
(all figures in $ millions)
Year
Projected Cash Projected Cash
Depreciation
Inflows from Outflows from
Expense
Operations
Operations
Excel function to use :
0
1
2
3
4
5
6
7
8
$850.0
$900.0
$990.0
$1,005.0
$1,200.0
$1,300.0
$1,350.0
$1,320.0
$840.0
$810.0
$870.0
$900.0
$1,100.0
$1,150.0
$1,300.0
$1,300.0
SLN
Interest
Expense
Projected
Taxable
Income
Projected Projected Projected
Federal
State
After-tax
Income
Income
Cash
Taxes
Taxes
Flows
IPMT
$23.89
$0.00
PV
PV
($13.89)
($3.61)
($1.11)
$14.72
PV
NPV
NPV1 – calculated NPV including interest expense
NPV2 – calculated NPV at the lower discount rate of 5.02%
IRR
Payback Timeline View Example of Actual Cash Flows
0
1
2
3
4
5
6
7
|
|
|
|
|
|
|
|
Cash Flow
($191.10)
$8.76
$62.18
$82.63
$73.42
$70.84
$104.60
$39.40
Cumulative Cash
Flow
($191.10)
($182.34)
($120.16)
($37.53)
$35.89
$106.73
$211.33
$250.73
Payback Period
3 years
6
0
1
2
3
4
5
6
7
|
|
|
|
|
|
|
|
Discounted Cash
Flow (DCF)
Cumulative DCF
Payback Period
ANSWER THESE QUESTIONS:
1. What is the total depreciation for tax purposes?
2. What is the total PV of the Cash Flows using the WACC rate?
3. What is the NPV using the WACC rate?
4. What is the NPV using the alternative rate?
5. What is the IRR?
6. What is the payback period using the DCF?
years
7. Should the project be accepted? Why?
n agreement, and
ws that would occur
NPV1
IRR
NPV2
NPV
IRR
NPV
IRR
8
|
$20.44
$271.17
$271.17
months
8
PV
|
$0.00
$0.00
$0.00
months
INSTRUCTIONS:
1). Complete the budget projections for years 2024-2027 using the following information
Revenue increases 4% annually
Expense increases 2¾% annually
For Depreciation and Interest expenses assume the Actual 2023 figure as the base for the budget and
forecast then add the amount calculated in the Payback tab for both budget and forecast.
2). Answer the question below the forecast.
1).
Largo Global Income Statement of December 31, 2023 (millions)
Sales (net sales)
Cost of goods sold
Gross profit
Selling, general, and administrative
expenses
Earnings before Interest, taxes,
depreciation, and amortization
(EBITDA)
Depreciation and amortization
Earning before interest and taxes
(EBIT) Operating income (loss)
Interest expense
Earnings before taxes (EBT)
Taxes (34%)
Net earnings (loss)/Net Income
ACTUAL BUDGET
2023
2024
$2,013
1400
613
0
FORECAST
2026
2025
0
0
0
0
0
0
0
0
0
0
0
0
0
0
125
488
174
314
$
141
173
59
114
2). Based on the changes suggested throughout the 5 projects, is Largo Global in a better financial position?
base for the budget and
d forecast.
illions)
FORECAST
2027
0
0
0
0
0
etter financial position?
Project 5: Review and
Practice Guide
UMGC
MBA 620: Financial
Decision Making
Project 5: Review and
Practice Guide
Cost of Capital, Risk/Return & Capital Budgeting
Contents
Topic 1: Capital Budgeting ……………………………………………………………………………………………………………. 3
The Finance Balance Sheet ……………………………………………………………………………………………………….. 3
Introduction to Capital Budgeting ……………………………………………………………………………………………… 3
What is CAPEX? ……………………………………………………………………………………………………………………….. 3
Key Reasons for Making Capital Expenditures……………………………………………………………………………… 3
Methods to Inform Capital Expenditure Decisions ……………………………………………………………………….. 3
Net Present Value Method: Calculating the NPV of a Project ………………………………………………………… 4
Net Present Value (NPV): Calculation Example ……………………………………………………………………………. 4
NPV: The Best Capital-Budgeting Technique ……………………………………………………………………………….. 5
NPV: The Five-step Approach ……………………………………………………………………………………………………. 5
Summary of NPV Method …………………………………………………………………………………………………………. 5
Key Advantages ……………………………………………………………………………………………………………………. 5
Key Disadvantage …………………………………………………………………………………………………………………. 5
Internal Rate of Return (IRR) Method…………………………………………………………………………………………. 6
Excel Function: IRR …………………………………………………………………………………………………………………… 6
IRR Example ……………………………………………………………………………………………………………………………. 6
NPV or IRR? …………………………………………………………………………………………………………………………….. 7
Unconventional Cash Flows ………………………………………………………………………………………………………. 7
Payback Period Method ……………………………………………………………………………………………………………. 7
Computing Payback Period ……………………………………………………………………………………………………….. 7
Payback Period Calculation: Example and Formula ………………………………………………………………………. 8
Discounted Payback Period……………………………………………………………………………………………………….. 8
Discounted Payback Period Cash Flows and Calculations ……………………………………………………………… 8
Evaluating the Payback Rule ……………………………………………………………………………………………………… 8
Topic 2: Cost of Capital and Financing Decisions ……………………………………………………………………………… 9
What is Capital? ………………………………………………………………………………………………………………………. 9
Sources of Capital for a Start-up ………………………………………………………………………………………………… 9
Estimating the Cost of Capital ……………………………………………………………………………………………………. 9
Cost of Capital and Risks …………………………………………………………………………………………………………. 10
Risk Is Uncertainty………………………………………………………………………………………………………………….. 10
Risk and Return Trade-off ……………………………………………………………………………………………………….. 10
How to Measure Return …………………………………………………………………………………………………………. 10
Example: Holding Period Return ………………………………………………………………………………………………. 11
Example: Expected Return ………………………………………………………………………………………………………. 11
Four Measures of Risk…………………………………………………………………………………………………………….. 11
Measuring Risk: Calculate Variance ………………………………………………………………………………………….. 11
Risk and Diversification …………………………………………………………………………………………………………… 12
Diversification: Individuals vs Companies ………………………………………………………………………………….. 12
Equity Securities…………………………………………………………………………………………………………………….. 12
Estimating the Cost of Equity …………………………………………………………………………………………………… 13
Method 1: Capital Asset Price Model (CAPM) ……………………………………………………………………………. 13
Estimating Beta ……………………………………………………………………………………………………………………… 13
Market-Risk Premium (Rm − Rf) ………………………………………………………………………………………………… 13
Assumptions of CAPM…………………………………………………………………………………………………………….. 14
Method 2: Constant-Growth Dividend Model ……………………………………………………………………………. 14
Bank Loans and Corporate Bonds …………………………………………………………………………………………….. 14
Yield to Maturity (YTM)…………………………………………………………………………………………………………… 15
Taxes and the Cost-of-Debt Equation ……………………………………………………………………………………….. 15
Weighted Average Cost of Capital (WACC) ………………………………………………………………………………… 15
Limitations of WACC ………………………………………………………………………………………………………………. 15
Alternatives to WACC……………………………………………………………………………………………………………… 15
Problems/Exercises ……………………………………………………………………………………………………………………. 16
What to Do ……………………………………………………………………………………………………………………………. 16
Self Study Problems ……………………………………………………………………………………………………………….. 16
Advanced Problems and Questions 10.36 …………………………………………………………………………………. 16
Solution to Problem 10.36 ………………………………………………………………………………………………………. 16
Advanced Problems and Questions 10.38 …………………………………………………………………………………. 18
Solution to Problem 10.38 ………………………………………………………………………………………………………. 19
Advanced Problems and Questions 10.39 …………………………………………………………………………………. 20
Solution to Problem 10.39 ………………………………………………………………………………………………………. 21
References ……………………………………………………………………………………………………………………………. 23
Project 5 Review and Practice Guide
Back to Table of Contents
Topic 1: Capital Budgeting
The Finance Balance Sheet
Based on Parrino et al. (2012)
(2012(2012)
Introduction to Capital Budgeting
•
•
•
Capital Budgeting determines the long-term productive assets that will create wealth.
Capital investments are large cash outlays and long-term commitments not easily reversed that
affect performance in the long run.
Capital-budgeting techniques help management systematically analyze potential opportunities
to decide which are worth undertaking (Parrino et al., 2012).
What is CAPEX?
CAPEX—capital expenditures, or a company’s key long-term expenses
Key Reasons for Making Capital Expenditures
•
•
•
•
•
Renewal—equipment repair, overhaul, rebuilding, or retrofitting. Usually does not require an
elaborate analysis and are made on a routine basis.
Replacement—to address equipment malfunction or obsolescence. Typically involves decisions at
the plant level.
Expansion—involves strategic decisions requiring complex, detailed analysis.
Regulatory
Other
Methods to Inform Capital Expenditure Decisions
•
•
•
Net present value (NPV)—Use the cost of capital to discount the cash flow of a project. Choose
the project with positive net present value.
Internal Rate of Return (IRR)—Calculate the rate of return of a project. Choose the highest one
or the one that has a higher rate of return than the cost of capital.
Payback Period—Calculate how long does it take for a project to pay back itself. Choose the one
with shorted payback period.
3
Project 5 Review and Practice Guide
Back to Table of Contents
Net Present Value Method: Calculating the NPV of a Project
NPV = NCF0 +
NCF1 NCF2
NCFn
+
+
…
+
1 + k (1 + k) 2
(1 + k) n
(10.1)
n
NCFt
t
t = 0 (1 + k)
=
Source: Parrino et al. (2012)
Net Present Value (NPV): Calculation Example
You can use the cash flow timeline below and Excel to calculate NPV for this project in which the cost of
capital is 15%:
P = −$300 +
B
$80
$80
$80
$80
$80 + $30
+
+
+
+
(1.15 ) (1.15 ) (1.15 ) (1.15 )
(1.15 )
1
2
3
4
= −$300 + $69.57 + $60.49 + $52.60 + $45.74 + 54.69
= −$16.91
Based on Parrino et al. (2012)
4
5
Project 5 Review and Practice Guide
Back to Table of Contents
NPV: The Best Capital-Budgeting Technique
•
•
•
•
Net Present Value (NPV) represents the current value of the project after accounting for
expected cash flow and the cost of capital.
The NPV of a project is the difference between the present values of its expected cash inflows
and expected cash outflows.
NPV is the best capital-budgeting technique because it is consistent with goal of maximizing
shareholder wealth.
Positive NPV projects increase shareholder wealth and negative NPV projects decrease
shareholder wealth. (Parrino et al., 2012)
NPV: The Five-step Approach
1. Estimate project cost
o Identify and add the present value of expenses related to the project.
o There are projects whose entire cost occurs at the start of the project, but many
projects have costs occurring beyond the first year.
o The cash flow in year 0 (NCF0) on the timeline is negative, indicating an outflow
2. Estimate project net cash flows
o Both cash inflows (CIF) and cash outflows (COF) are likely in each year. Estimate the net
cash flow (NCFn) = CIFn − COFn for each year.
o Include salvage value of the project in its terminal year.
3. Determine project risk and estimate cost of capital
o The cost of capital is the discount rate (k) used to determine the present value of
expected net cash flows.
o The riskier a project, the higher its cost of capital (Parrino et al., 2012)
4. Compute the project’s NPV
o Determine the difference between the present values of the expected net cash flows
from the project and the expected cost of the project.
5. Make a decision
o Accept a project if it has a positive NPV, reject it if the NPV is negative.
Summary of NPV Method
•
Decision rule
o NPV > 0 Accept the project
o NPV < 0 Reject the project Key Advantages o o o Uses the discounted cash flow valuation technique to adjust for the time value of money Provides a direct (dollar) measure of how much a capital project will increase the value of a company Is Consistent with the goal of maximizing stockholder value Key Disadvantage o Can be difficult to understand without an accounting and finance background (Parrino et al., 2012). 5 Project 5 Review and Practice Guide Back to Table of Contents Internal Rate of Return (IRR) Method • • • IRR is the discount rate at which a project has an NPV equal to zero. A project is acceptable if its IRR is greater than the firm's cost of capital. The IRR is an important and legitimate alternative to the NPV method (Parrino et al., 2012). NPV = NCF0 + NCF1 NCF2 NCFn + + ... + 2 1 + k (1 + k) (1 + k) n (10.1) n NCFt t t = 0 (1 + k) = Excel Function: IRR IRR(values, [guess]) The IRR function syntax has the following arguments: Values Required. An array or a reference to cells that contain numbers for which you want to calculate the internal rate of return. • Values must contain at least one positive value and one negative value to calculate the internal rate of return. • IRR uses the order of values to interpret the order of cash flows. Be sure to enter your payment and income values in the sequence you want. • If an array or reference argument contains text, logical values, or empty cells, those values are ignored. Guess Optional. A number that you guess is close to the result of IRR. (Microsoft, n.d.) IRR Example Expected Cash Flow from a CAPEX Based on Parrino et al. (2012) 6 Project 5 Review and Practice Guide Back to Table of Contents NPV or IRR? Is NPV or IRR a better measure for capital budgeting? Consider these points: • IRR is the discount rate when NPV is zero, when the project is breakeven, so many times the two methods yield consistent results. • IRR's biggest strength is also its limitation: A single discount rate does not consider the change in interest rate level. • IRR is ineffective when the projects have strings of positive and negative cash flows. • IRR has the advantage of summarizing the rate of return of the project in one number, so it is a popular method. Unconventional Cash Flows The IRR technique may provide more than one rate of return, making the calculation unreliable. Therefore, it should not be used to determine whether to accept or reject a project. Following are examples of unconventional cash flows: • Positive cash flow followed by negative net cash flows. • Simultaneous positive and negative net cash flows. • Conventional followed by a negative net cash flow at the end of a project's life. Payback Period Method Payback period—the time it takes for the sum of the net cash flows from a project to equal the project's initial investment o One of the most popular tools for evaluating capital projects o Can serve as a risk indicator—the quicker a project's cost recovery, the less risky the project o Decision criteria: payback period shorter than a specific amount of time (Parrino et al., 2012) Computing Payback Period The following timeline shows the net and cumulative net cash flow (NCF) for a proposed capital project with an initial cost of $80,000. This data is used to compute the payback period, 2.5 years. 0 1 2 3 4 NCF -$80,000 $35,000 $35,000 $20,000 $25,000 Cumulative NCF -$80,000 -$45,000 -$10,000 $10,000 $35,000 Based on Parrino et al. (2012) 7 Year Project 5 Review and Practice Guide Back to Table of Contents Payback Period Calculation: Example and Formula = 2 years + $10,000 $20,000 = 2 years + 0.5 = 2.5 years To compute the payback period, estimate a project's cost and its future net cash flows: PB = Years before cost recovery + Remaining cost to recover Cash flow during the year Discounted Payback Period • • Future cash flows are discounted by the firm's cost of capital. The major advantage of the discounted payback is that it tells management how long it takes a project to reach a positive NPV (Parrino et al., 2012). Discounted Payback Period Cash Flows and Calculations 1 0 2 Net cash flow (NCF) -$80,000 $40,000 $40,000 Cumulative NCF -$80,000 -$40,000 $0,0000 Discounted NCF (at 10%) -$80,000 $36,364 $33,058 Cumulative discount NCF -$80,000 -$43,636 -$10,578 Payback Period =2 years + $0/$40,000 = 2 years Discounted payback period = 2 years + $10,578/$30,053 = 2.35 years NPV = $99,475 - $80,000 = $19,475 Cost of capital = 10% 3 Year $40,000 $40,000 $30,053 $19,478 Based on Parrino et al. (2012) Evaluating the Payback Rule The ordinary payback period is easy to calculate and provides a simple measure of an investment's liquidity risk. However, it • ignores the time value of money, • has no economic rationale that makes the payback method consistent with shareholder wealth maximization, and • is biased against long-term projects, such as R&D or new product development. Its biggest weakness is the failure to consider cash flows after the payback period (Parrino et al., 2012). 8 Project 5 Review and Practice Guide Back to Table of Contents Topic 2: Cost of Capital and Financing Decisions What is Capital? Capital—money available to pay for day-to-day operations and future growth funding Sources of Capital for a Start-up • • • • • • • • Self, friends, family Loans or lines of credit: small and short-term loans Small business grants from foundations and government Incubators that provide resources in exchange for equity Angel investors (typically 25k to 250k) Venture capital, typically above 1 million (investors exert control of the start-up) Crowdfunding through an online platform that enables small contributions from many investors (Clendenen, 2020) Commitment by a major customer for which the customer receives priority to buy the product of their investment before other customers (Zwilling, 2010) Estimating the Cost of Capital The cost of capital can be estimated using the weighted average cost of each security issued by the company. For a project, the cost of capital includes the following (Parrino et al., 2012): • Discount rate used to calculate NPV (see the table on the following page) • Required rate of return • The opportunity cost to the holders of a company's securities To determine the weighted average cost of capital (WACC), divide the costs of capital into debt and equity and use the following equations: • 𝑘𝐹𝑖𝑟𝑚 = 𝑥𝐸𝑞𝑢𝑖𝑡𝑦 𝑘𝐸𝑞𝑢𝑖𝑡𝑦 + 𝑥𝐷𝑒𝑏𝑡 𝑘𝐷𝑒𝑏𝑡 where 𝑥𝐷𝑒𝑏𝑡 is the percentage of debt and 𝑥𝐸𝑞𝑢𝑖𝑡𝑦 is the percentage of equity Kdebt is the cost of debt Kequity is the cost of equity • WACC after tax is o 𝑘𝐹𝑖𝑟𝑚,𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 = 𝑥𝐸𝑞𝑢𝑖𝑡𝑦 𝑘𝐸𝑞𝑢𝑖𝑡𝑦 + 𝑥𝐷𝑒𝑏𝑡 𝑘𝐷𝑒𝑏𝑡,𝑝𝑟𝑒−𝑡𝑎𝑥 (1 - tax rate) o = 𝑥𝑝𝑠 𝑘𝑝𝑠 + 𝑥𝑐𝑠 𝑘𝑐𝑠 + 𝑥𝐷𝑒𝑏𝑡 𝑘𝐷𝑒𝑏𝑡 𝑝𝑟𝑒𝑡𝑎𝑥 (1 − 𝑡) 9 Project 5 Review and Practice Guide Back to Table of Contents Cost of Capital and Risks When the Cost of Capital is used as discount rate to evaluate a new project (CAPEX), the risk of a project should be considered (Brealy & Myers, 2003). Category Discount Rate Speculative ventures 30% New Products 20% Expansion of existing business 15% (company cost of capital) Cost improvement, known technology 10% Risk Is Uncertainty • • Of future cash inflows due to o Market risk—market conditions that affect revenue o Credit risk—customer's availability to pay o Operational risk—production unpredictability o Interest rate risk—changes in interest rate level Of future cash outflows due to o Liquidity risk—the company's ability to pay o Interest rate risk—changes in interest rate level Risk and Return Trade-off People do not want to lose money! Therefore, a higher-risk investment must offer a potential return high enough to make it as attractive as the lower-risk alternative. At the same time, an investor must accept a higher level of risk to achieve higher gains (Chen, 2020). • The potential return required depends on the amount of risk—the probability of being dissatisfied with an outcome. • The higher the risk, the higher the required rate-of-return (Parrino et al., 2012) How to Measure Return • Expected vs realized return o Expected return ▪ estimated or predicted before the outcome is known ▪ Expected Return (ER) = (Probability 1 * Return 1) + (Probability 2 * Return 2) + … o Realized return ▪ calculated after the outcome is known ▪ Realized Return = (Selling price − Purchase price) / Purchase Price 10 Project 5 Review and Practice Guide Back to Table of Contents Example: Holding Period Return Ella buys a stock for $26.00. After one year, the stock price is $29.00 and she receives a dividend of $0.80. What is her return for the period? 𝛥𝑃 + 𝐶𝐹1 𝑃0 ($29.00 − $26.00) + $0.80 = $26.00 $3.80 = = 0.14615 𝑜𝑟 14.62% $26.00 𝑅𝑡 = 𝑅𝑐𝑎 + 𝑅𝑖 = Example: Expected Return o There is 30% chance the total return on Dell stock will be -3.45%, a 30% chance it will be +5.17% , a 30% chance it will be +12.07% and a 10% chance that it will be +24.14%. Calculate the expected return. 𝐸(𝑅𝐷𝑒𝑙𝑙 ) = [. 30 × (−0.0345)] + (. 30 × 0.0517) + (. 30 × 0.1207) + (. 10 × 0.2414) = −0.010305 + 0.01551 + 0.03621 + 0.02414 = 0.0655 𝑜𝑟 6.55% Four Measures of Risk • Four most commonly used terms in measuring risks o Volatility o Variance o Standard Deviation o Beta Measuring Risk: Calculate Variance 1. Square the difference between each possible outcome and the mean 2. Multiply each squared difference by its probability of occurring 3. Add 𝑛 𝑉𝑎𝑟(𝑅 ) = 𝜎𝑅2 = ∑{𝑝𝑖 × [𝑅𝑖 − 𝐸(𝑅)]2 } 𝑖=1 o If all possible outcomes are equally likely, the formula becomes 𝜎𝑅2 = o ∑𝑛𝑖=1[𝑅𝑖 − 𝐸(𝑅)]2 𝑛 Standard deviation is the square root of the variance √𝜎𝑅2 = 𝜎 Source: Parrino et al. (2012) 11 Project 5 Review and Practice Guide Back to Table of Contents Risk and Diversification Diversification eliminates unique risk but not market risk. • • • Investing in two or more assets with returns that do not always move in the same direction at the same time can reduce the risk in an investment portfolio. Diversification can nearly eliminate unique risk to individual assets, but the risk common to all assets in the market remains. Risk that cannot be diversified away is non-diversifiable, or systematic risk. This is the risk inherent in the market or economy (Brealey & Myers, 2003). Diversification: Individuals vs Companies • • • Individual investors can diversify easily and cheaply but it is much more expensive for a company to diversify product lines. Individual investors do not pay extra to companies that are diversified or less to companies that are not diversified. A company's value neither increases nor decreases based on its degree of diversification. The value of a company is the sum of its parts, no more no less (Brealey & Myers, 2003). Equity Securities Common stock and preferred stock—two types of ownership interest in a corporation; the most prevalent types of equity securities o Dividend payments do not affect a company's taxes o Have limited liability for stockholders; claims made against the corporation cannot include a stockholder's personal assets o Generally viewed as perpetuities; do not have maturity dates o Dividends are promised rather than guaranteed to preferred stockholders (Parrino et al., 2012) 12 Project 5 Review and Practice Guide Back to Table of Contents Estimating the Cost of Equity Market information is used to estimate the cost of equity. There are two other methods for estimating the cost of common stock: o Method 1: Capital Asset Pricing Model (CAPM) o Method 2: Constant-Growth Dividend Model The most appropriate method depends on the availability and reliability of information (Parrino et al., 2012) Method 1: Capital Asset Price Model (CAPM) Expected Return of an asset can be broken down into risk- free rate and compensation to investors for taking more risk: 𝐸(𝑅𝑖 ) = 𝑅𝑟𝑓 + 𝛽𝑖 [𝐸(𝑅𝑚 ) − 𝑅𝑟𝑓 ] • • • Expected Return = Risk Free Rate + Beta × Market Risk Beta measures the risks of a stock relative to its market o Beta > 1: The stock is more volatile than the market
o Beta < 1: The stock is less volatile than the market Betas by sector and the companies included in each industry are available online (Damodaran, 2021) Estimating Beta To estimate beta for a non-publicly traded firm o Identify a "comparable" company with publicly traded stock that is in the same business and that has a similar amount of debt o use an average of the betas for the public firms in the same industry Levered vs unlevered beta o Levered beta is the market beta, considering the capital structure of the firm as is. o Unlevered beta removes the influence of debt, enabling comparison between companies. Market-Risk Premium (Rm − Rf) • • Market-risk premium cannot be observed: the rate of return investors expect is unknown Market-risk premium usually estimates the average risk premium investors have earned in the past as an indication of the risk premium they might require today o From 1926 through 2015, the US stock market exceeded actual returns on long-term US government bonds by an average of 5.92% per year o If a financial analyst believes that the market-risk premium in the past is a reasonable estimate of the risk premium today, then he or she might use a similar percentage as the market risk premium for the future (Parrino et al., 2012) 13 Project 5 Review and Practice Guide Back to Table of Contents Assumptions of CAPM The following assumptions underlie the capital asset pricing model (CAPM): • Many investors who are all price takers, i.e., financial markets, are competitive. • All investors plan to invest over the same time horizon. • There are no distortionary taxes or transaction costs. • All investors can borrow and lend at same risk-free rate. • Investors care only about their expected return (like) and variance (dislike). • All investors have same information and beliefs about the distribution of returns. • The market portfolio that determines beta consists of all publicly traded assets (Parrino et al., 2012). Method 2: Constant-Growth Dividend Model The constant-growth dividend model is useful for a company that pays dividends that will grow at a constant rate—such as an electric utility—rather than a fast-growing high-tech firm. 𝐶𝐹 • Present Value of Perpetuity: 𝑃𝑉𝑃0 = 𝑖 𝐷 • Value of stocks with fixed dividend level: 𝑃0 = 𝑅 • Value of stocks with dividend growing at the rate of g: 𝑃0 = • The equation above can be rearranged to solve for the required rate of return (R), which is also the cost of common stock (KCS) 𝐷1 𝑘𝑐𝑠 = +𝑔 𝑃0 𝐷1 𝑅−𝑔 In practice, most people use the CAPM to estimate the cost of equity if the result is going to be used in the discount rate for evaluating a project (Parrino et al., 2012). Bank Loans and Corporate Bonds • • • • • • Investors' required rate of return is often not directly observable. The market value (price) of securities is often used to estimate the required rate of return. To estimate the cost of debt, long-term debt (i.e., maturity longer than one year) is of particular interest; long-term debt can be considered permanent, since companies often issue new debt to pay off the old (Parrino et al., 2012). The interest rate (or historical interest rate determined when the debt was issued) the firm is paying on its outstanding debt does not necessarily reflect its current cost of debt. The current cost of long-term debt is the appropriate cost of debt for weighted average cost of capital (WACC) calculations; WACC is the opportunity cost of capital for the firm's investors as of today. Use yield to maturity (YTM) to determine the cost of debt and adjust for the tax deductibility of interest on debt (Parrino et al., 2012). 14 Project 5 Review and Practice Guide Back to Table of Contents Yield to Maturity (YTM) Yield to maturity is the internal rate of return (IRR) of a bond investment if the investor holds the bond to maturity and all payments are made as scheduled. • YTM accounts for the time value of money (TVM) and the bond purchase price. • The current YTM of a bond reflects the required rate of return for the bondholder. • YTM is used to estimate the cost of debt to a company (Parrino et al., 2012). Taxes and the Cost-of-Debt Equation The after-tax cost of interest payments equals the pre-tax cost times 1, minus the tax rate (Parrino et al., 2012): 𝑘𝐷𝑒𝑏𝑡 𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 = 𝑘𝐷𝑒𝑏𝑡 𝑝𝑟𝑒−𝑡𝑎𝑥 × (1 − 𝑡) Weighted Average Cost of Capital (WACC) • After-tax weighted-average cost of capital equation: 𝑊𝐴𝐶𝐶 = 𝑥𝐷𝑒𝑏𝑡 𝑘𝐷𝑒𝑏𝑡 𝑝𝑟𝑒𝑡𝑎𝑥 (1 − 𝑡) + 𝑥𝑝𝑠 𝑘𝑝𝑠 + 𝑥𝑐𝑠 𝑘𝑐𝑠 • • Use market values, not book values, to calculate WACC. A list of estimated WACC per industry in the US is at https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/wacc.htm Limitations of WACC WACC can be used as a discount rate for evaluating projects under the following conditions (Parrino et al., 2012): 1. The level of systematic risk for the project is the same as that of the portfolio of projects that currently comprise the firm. 2. The project uses the same financing mix—proportions of debt, preferred shares, and common shares—as the firm as a whole. Alternatives to WACC • • Using a public company in a similar, or pure-play comparable business (often difficult to find) Classifying projects into categories based on their systematic risks and specifying a discount rate for each Category Discount Rate Speculative ventures 30% New products 20% Expansion of existing business 15% Cost Improvement, known technology 10% Source: Parrino et al. (2012) 15 Project 5 Review and Practice Guide Back to Table of Contents Problems/Exercises What to Do Complete all the practice exercises from the book and the custom exercise that follows to gain the knowledge and skills to complete the final Project 5 deliverable. The answers are provided, so you can check your work. Self Study Problems • • • Chapter 7 Self Study problems (all) Chapter 10 Self Study problems (all) Chapter 13 Self Study problems (all) Advanced Problems and Questions 10.36 10.36 Quasar Tech Co. is investing $6 million in new machinery to produce next-generation routers. Sales will amount to $1.75 million for the next three years and increase to $2.4 million for the three years after that. The project is expected to last six years. Operating costs, excluding depreciation, will be $898,620 annually. The machinery will be depreciated to a salvage value of $0 over 6 years using the straight-line method. The company's tax rate is 30 percent, and the cost of capital is 16 percent. A. B. C. D. What is the payback period? What is the average accounting return (ARR)? Calculate the project NPV. What is the IRR for the project? Solution to Problem 10.36 A. Project 1 Cash Year Net Income Depreciation 0 Cumulative CF Flows $(6,000,000) $(6,000,000) 1 $(104,034) $1,000,000 895,966 (5,104,034) 2 $(104,034) $1,000,000 895,966 (4,208,068) 3 $(104,034) $1,000,000 895,966 (3,312,102) 4 350,966 $1,000,000 1,350,966 (1,961,136) 5 350,966 $1,000,000 1,350,966 (610,170) 6 350,966 $1,000,000 1,350,966 740,796 PB = Years before cost recovery + (Remaining cost to recover/ Cash flow during the year) = 5 + ($610,170 / $1,350,966) = 5.45 years 16 Project 5 Review and Practice Guide Back to Table of Contents B. Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 $ 1,750,000 $ 1,750,000 $ 1,750,000 $ 2,400,000 $ 2,400,000 $ 2,400,000 898,620 898,620 898,620 898,620 898,620 898,620 Depreciation 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 EBIT $(1,48,620) $(1,48,620) $(1,48,620) $ 501,380 $ 501,380 $ 501,380 Taxes (30%) 44,586 44,586 44,586 (150,414) (150,414) (150,414) Net income $ (104,034) $ (104,034) $ (104,034) $ 350,966 $ 350,966 $ 350,966 Beginning BV 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 Less: Depreciation 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 $ 5,000,000 $ 4,000,000 $ 3,000,000 $ 2,000,000 $ 1,000,000 Sales Expenses Ending BV Average after-tax income = $123,466 Average book value of equipment = $3,000,000 Accounting rate of return = Average after-tax income Average book value = $ $123,466 = 4.1% $3,000,000 C. Cost of this project = $6,000,000 Required rate of return = k =16% Length of project = n = 6 years 1 1 1− 1− 𝑁𝐶𝐹𝑡 1 (1.16)3 (1.16)3 𝑁𝑃𝑉 = ∑ = −$6,000,000 + $895,966 × [ ] + $1,350,966 × [ ]× 𝑡 (1 + 𝑘) 0.16 0.16 (1.16)3 𝑛 𝑡=0 = −$6,000,000 + $2,012,241 + $1,943,833 = −$2,043,927 17 0 Project 5 Review and Practice Guide Back to Table of Contents D. To compute the IRR, try rates lower than 16 percent. Try IRR = 3%. 𝑛 𝑁𝑃𝑉 = 0 = ∑ 𝑡=0 𝑁𝐶𝐹𝑡 0 (1 + 𝐼𝑅𝑅)𝑡 1 1 1− 1 (1.03)3 (1.03)3 ] + $1,350,966 × [ ]× 0.03 0.03 (1.03)3 1− = −$6,000,000 + $895,966 × [ = −$6,000,000 + $2,534,340 + $3,497,084 = $31,424 Try IRR = 3.1%. 𝑛 𝑁𝑃𝑉 = 0 − ∑ 𝑡=0 𝑁𝐶𝐹𝑡 0 (1 + 𝐼𝑅𝑅)𝑡 1 1 1− 3 1 (1.031) (1.031)3 ] + $1,350,966 × [ ]× 0.031 0.031 (1.031)3 1− = −$6,000,000 + $895,966 × [ = −$6,000,000 + $2,529,475 + $3,480,225 = $9,700 The IRR of the project is approximately 3.1%. Advanced Problems and Questions 10.38 10.38 Trident Corp. is evaluating two independent projects. The following table lists the costs and expected cash flows. The cost of capital is 10 percent. A. B. C. D. Year A B 0 -$312,500 -$395,000 1 121,450 153,552 2 121,450 158,711 3 121,450 166,220 4 121,450 132,000 5 121,450 122,00 Calculate the projects' NPV. Calculate the projects' IRR. Which project should be chosen based on NPV? Based on IRR? Is there a conflict? If you are the decision maker for the firm, which project or projects will be accepted? Explain your reasoning. 18 Project 5 Review and Practice Guide Back to Table of Contents Solution to Problem 10.38 A. Project A: Cost of this project = $312,500 Annual cash flows = $121,450 Required rate of return = k = 10% Length of project = n = 5 years 1 1− 𝑁𝐶𝐹𝑡 (1.10)5 𝑁𝑃𝑉 = ∑ = −312,500 + $121,450 × [ ] (1 + 𝑘)𝑡 0.10 𝑛 𝑡=0 = −$312,500 + 460,391 = $147,891 Project B: Cost of this project = $395,000 Required rate of return = k = 10% Length of project = n = 5 years 𝑛 𝑁𝑃𝑉 = ∑ 𝑡=0 𝑁𝐶𝐹𝑡 $153,552 $158,711 $166,220 $132,000 $122,000 = −$395,000 + + + + + 𝑡 (1 + 𝑘) (1.10)1 (1.10)2 (1.10)3 (1.10)4 (1.10)5 = −395,000 + $139,593 + $131,166 + $124,884 + 90,158 + 75,752 = $166,553 B. Project A: Since NPV > 0, to compute the IRR, try rates higher than 10%.
Try IRR = 27%.
1
1−
𝑁𝐶𝐹𝑡
(1.27)5
𝑁𝑃𝑉 = 0 = ∑
0 = −312,500 + $121,450 × [
]
(1 + 𝐼𝑅𝑅)𝑡
0.27
𝑛
𝑡=0
= −$312,500 + 313,666 ≠ $1,166
Try IRR = 27.2%
1
1−
𝑁𝐶𝐹𝑡
(1.272)5
𝑁𝑃𝑉 = 0 = ∑
0 = −312,500 + $121,450 × [
]
(1 + 𝐼𝑅𝑅)𝑡
0.272
𝑛
𝑡=0
= −$312,500 + 312,418 = −$82 ≅ 0
The IRR of Project A is approximately 27.2 percent. Using a financial calculator, we find that the IRR is
27.187 percent.
19
Project 5 Review and Practice Guide
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Project B:
Since NPV > 0, to compute the IRR, try rates higher than 10 percent. Try IRR = 26%
The IRR of Project A is approximately 27.2 percent. Using a financial calculator, the solution reached is
27.187 percent.
Try IRR = 26.1%
𝑛
𝑁𝑃𝑉 = 0 = ∑
𝑡=0
𝑁𝐶𝐹𝑡
$153,552 $158,711 $166,220 $132,000 $122,000
0 = −$395,000 +
+
+
+
+
𝑡
(1 + 𝐼𝑅𝑅)
(1.261)1
(1.261)2
(1.261)3
(1.261)4
(1.261)5
= −$395,000 + $121,770 + $99,811 + $82,897 + $52,205 + $38,263 = −$54 ≅ 0
The IRR of Project B is approximately 26.1 percent.
C.
There is no conflict between the NPV and IRR decisions. Using NPV decision criteria, both projects have
positive NPVs; they are independent projects; both should be accepted. Using IRR decision criteria, both
projects have IRRs greater than the cost of capital; both will be accepted.
D.
Based on NPV, both projects will be accepted.
Advanced Problems and Questions 10.39
10.39 Tyler, Inc., is considering switching to a new production technology. The cost of the required
equipment will be $4 million. The discount rate is 12%. The cash flows the firm expects the new
technology to generate are as follows.
Years
CF
1-2
0
3-5
$ 845,000
6-9
$ 1,845,000
A. Compute the payback and discounted payback periods for the project.
B. What is the NPV for the project? Should the firm proceed with the project?
C. What is the IRR? Based on IRR, what would the decision be?
20
Project 5 Review and Practice Guide
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Solution to Problem 10.39
A.
Cumulative CF
Year
0
Cash Flows
Cumulative PVCF
PVCF
$(4,000,000)
$(4,000,000)
$(4,000,000)
$(4,000,000)
1
—
—
(4,000,000)
(4,000,000)
2
—
—
(4,000,000)
(4,000,000)
3
845,000
601,454
(3,155,000)
(3,398,546)
4
845,000
537,013
(2,310,000)
(2,861,533)
5
845,000
479,476
(1,465,000)
(2,382,057)
6
1,450,000
734,615
(15,000)
(1,647,442)
7
1,450,000
655,906
1,435,000
(991,536)
8
1,450,000
585,631
2,885,000
(405,905)
9
1,450,000
522,885
4,335,000
116,979
𝑃𝑎𝑦𝑏𝑎𝑐𝑘 𝑝𝑒𝑟𝑖𝑜𝑑 = 𝑌𝑒𝑎𝑟𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑐𝑜𝑠𝑡 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 +
=6+
𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑡𝑜 𝑟𝑒𝑐𝑜𝑣𝑒𝑟
𝐶𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑑𝑢𝑟𝑖𝑛𝑔 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
$15,000
= 6.01 𝑦𝑒𝑎𝑟𝑠
$1,450,000
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑 𝑃𝑎𝑦𝑏𝑎𝑐𝑘 𝑝𝑒𝑟𝑖𝑜𝑑 = 𝑌𝑒𝑎𝑟𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑐𝑜𝑠𝑡 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 +
= 8+
21
$405,905
= 8.8 𝑦𝑒𝑎𝑟𝑠
$522,885
𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑡𝑜 𝑟𝑒𝑐𝑜𝑣𝑒𝑟
𝐶𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑑𝑢𝑟𝑖𝑛𝑔 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
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B.
Cost of this project = $4,000,000
Required rate of return = k = 12%
Length of project = n = 9 years
𝑛
𝑁𝑃𝑉 = ∑
𝑡=0
𝑁𝐶𝐹𝑡
(1 + 𝑘)𝑡
1
1
(1.12)3
]×
0.12
(1.12)2
1−
= −$4,000,000 + 0 + 0 + $845,000 × [
1
1−
1
(1.12)4
+ $1,450,000 × [
]×
= −$4,000,000 + 0 + 0 + $1,617,943 + $2,499,037
0.12
(1.12)5
= $116,980
Since NPV > 0, the project should be accepted.
C.
Given a positive NPV, to compute the IRR, one should try rates higher than 12%.
Try IRR = 12.5%.
𝑛
𝑁𝑃𝑉 = ∑
𝑡=0
𝑁𝐶𝐹𝑡
(1 + 𝑘)𝑡
1
1
(1.125)3
]×
0.125
(1.125)2
1−
= −$4,000,000 + 0 + 0 + $845,000 × [
1
1−
1
(1.125)4
+ $1,450,000 × [
]×
0.125
(1.125)5
= −$4,000,000 + 0 + 0 + $1,589,915 + $2,418,479 = $8,394
Using a financial calculator and 12.5% returns an IRR of 12.539%. The IRR exceeds the cost of capital
(12%); the project should be accepted.
22
Project 5 Review and Practice Guide
Back to Table of Contents
References
Brealey, R. A. & Myers, S. C. (2003). Principles of Corporate Finance 7th ed. McGraw-Hill.
Chen, J. (2020, February 3). Risk-return tradeoff. Investopedia. Retrieved August 9, 2021, from
https://www.investopedia.com/terms/r/riskreturntradeoff.asp
Clendenen, D. (2020, March 19). 23 of the best fundraising websites for small business. LendGenius.
https://www.lendgenius.com/blog/fundraising-websites/
Damodaran, A. (2021, January). Total betas by sector (for computing private company cost of equity)—
US. Damodaran Online. Retrieved August 9, 2021, from
https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/totalbeta.html
Microsoft. (n.d.). Excel functions (by category). Retrieved July 22, 2021, from
https://support.microsoft.com/en-us/office/excel-functions-by-category-5f91f4e946d2-9bd1-63f26a86c0eb
7b42-
Parrino, R., Kidwell, D. S., & Bates, T. W. (2012). Fundamentals of corporate finance. Wiley.
Zwilling, M. (2010, February 12). Top 10 sources of funding for start-ups. Forbes. Retrieved August 9,
2021, from https://www.forbes.com/2010/02/12/funding-for-startups-entrepreneurs-financezwilling.html?sh=46bb828b160f
Now that you have read this Review and Practice Guide and completed the exercises, you are
ready to participate in the assignment in Step 3.
23
ALL CHAPTERS BEL0W
Required Reading
Parrino, R., Kidwell, D. S., & Bates, T. W. (2012). Fundamentals of
Corporate Finance (2nd ed.) Wiley.
Chapter 7: Risk and Return
•
Section 7.1 to 7.7
Chapter 10: The Fundamentals of Capital Budgeting
•
Sections 10.1 to 10.6
Chapter 13: The Cost of Capital
•
Sections 13.1 to 13.4
Recommended Reading
Clayman, M. R., Fridson, M. S., & Troughton, G. H. (2012). Corporate
Finance: A Practical Approach (2nd. ed.). Wiley.
Chapter 7: Risk and Return
•
Section 7.1 to 7.7
7
Risk and Return
© David Young-Wolff/PhotoEdit
Learning Objectives
Explain the relation between risk and return.
Describe the two components of a total holding period return, and calculate this return for an
asset.
Explain what an expected return is and calculate the expected return for an asset.
Explain what the standard deviation of returns is and why it is very useful in finance, and
calculate it for an asset.
Explain the concept of diversification.
Discuss which type of risk matters to investors and why.
Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to evaluate
whether the expected return of an asset is sufficient to compensate an investor for the risks
associated with that asset.
When Blockbuster Inc. filed for bankruptcy protection on Thursday, September 23, 2010,
its days as the dominant video rental firm were long gone. Netflix had become the most
successful competitor in the video rental market through its strategy of renting videos
exclusively online and avoiding the high costs associated with operating video rental
stores.
The bankruptcy filing passed control of Blockbuster to a group of bondholders, including
the famous billionaire investor Carl Icahn, and the shares owned by the old stockholders
became virtually worthless. The bondholders planned to reorganize the company and
restructure its financing so that it had a chance of competing more effectively with Netflix
in the future.
Over the previous five years, Blockbuster stockholders had watched the value of their
shares steadily decline as, year after year, the company failed to respond effectively to the
threat posed by Netflix. From September 23, 2005 to September 23, 2010, the price of
Blockbuster shares fell from $4.50 to $0.04. In contrast, the price of Netflix shares rose
from $24.17 to $160.47 over the same period. While the Blockbuster stockholders were
losing almost 100 percent of their investments, Netflix stockholders were earning an
average return of 46 percent per year!
This chapter discusses risk, return, and the relation between them. The difference in the
returns earned by Blockbuster and Netflix stockholders from 2005 to 2010 illustrates a
challenge faced by all investors. The shares of both of these companies were viewed as
risky investments in 2005, and yet an investor who put all of his or her money in
Blockbuster lost virtually everything, while an investor who put all of his or her money in
Netflix earned a very high return. How should have investors viewed the risks of investing
in these companys’ shares in 2005? How is risk related to the returns that investors might
expect to earn? How does diversification reduce the overall risk of an investor’s portfolio?
These are among the topics that we discuss in this chapter.
CHAPTER PREVIEW
Up to this point, we have often mentioned the rate of return that we use to discount cash flows, but
we have not explained how that rate is determined. We have now reached the point where it is time
to examine key concepts underlying the discount rate. This chapter introduces a quantitative
framework for measuring risk and return. This framework will help you develop an intuitive
understanding of how risk and return are related and what risks matter to investors. The relation
between risk and return has implications for the rate we use to discount cash flows because the
time value of money that we discussed in Chapters 5 and 6 is directly related to the returns that
investors require. We must understand these concepts in order to determine the correct present
value for a series of cash flows and to be able to make investment decisions that create value for
stockholders.
We begin this chapter with a discussion of the general relation between risk and return to
introduce the idea that investors require a higher rate of return from riskier assets. This is one of
the most fundamental relations in finance. We next develop the statistical concepts required to
quantify holding period returns, expected returns, and risk. We then apply these concepts to
portfolios with a single asset and with more than one asset to illustrate the benefit of
diversification. From this discussion, you will see how investing in more than one asset enables an
investor to reduce the total risk associated with his or her investment portfolio, and you will learn
how to quantify this benefit.
Once we have discussed the concept of diversification, we examine what it means for the relation
between risk and return. We find that the total risk associated with an investment consists of two
components: (1) unsystematic risk and (2) systematic risk. Diversification enables investors to
eliminate the unsystematic risk associated with an individual asset. Investors do not require higher
returns for the unsystematic risk that they can eliminate through diversification. Only systematic
risk—risk that cannot be diversified away—affects expected returns on an investment. The
distinction between unsystematic and systematic risk and the recognition that unsystematic risk
can be diversified away are extremely important in finance. After reading this chapter, you will
understand precisely what the term risk means in finance and how it is related to the rates of
return that investors require.
7.1 RISK AND RETURN
•
The rate of return that investors require for an investment depends on the risk associated
with that investment. The greater the risk, the larger the return investors require as
compensation for bearing that risk. This is one of the most fundamental relations in
finance. The rate of return is what you earn on an investment, stated in percentage terms.
We will be more specific later, but for now you might think of risk as a measure of how
certain you are that you will receive a particular return. Higher risk means you are less
certain.
To get a better understanding of how risk and return are related, consider an example. You
are trying to select the best investment from among the following three stocks:
BUILDING INTUITION MORE RISK MEANS A HIGHER EXPECTED RETURN
The greater the risk associated with an investment, the greater the return investors expect from it.
A corollary to this idea is that investors want the highest return for a given level of risk or the
lowest risk for a given level of return. When choosing between two investments that have the same
level of risk, investors prefer the investment with the higher return. Alternatively, if two
investments have the same expected return, investors prefer the less risky alternative.
Which would you choose? If you were comparing only Stocks A and B, you should choose
Stock A. Both stocks have the same expected return, but Stock A has less risk. It does not
make sense to invest in the riskier stock if the expected return is the same. Similarly, you
can see that Stock C is clearly superior to Stock B. Stocks B and C have the same level of
risk, but Stock C has a higher expected return. It would not make sense to accept a lower
return for taking on the same level of risk.
But what about the choice between Stocks A and C? This choice is less obvious. Making it
requires understanding the concepts that we discuss in the rest of this chapter.
7.2 QUANTITATIVE MEASURES OF RETURN
•
Before we begin a detailed discussion of the relation between risk and return, we should
define more precisely what these terms mean. We begin with measures of return.
Holding Period Returns
total holding period return
the total return on an asset over a specific period of time or holding period
When people refer to the return from an investment, they are generally referring to the
total return over some investment period, or holding period. The total holding period
return consists of two components: (1) capital appreciation and (2) income. The capital
appreciation component of a return, RCA, arises from a change in the price of the asset over
the investment or holding period and is calculated as follows:
where P0 is the price paid for the asset at time zero and P1 is the price at a later point in
time.
The income component of a return arises from income that an investor receives from the
asset while he or she owns it. For example, when a firm pays a cash dividend on its stock,
the income component of the return on that stock, RI, is calculated as follows:
where CF1 is the cash flow from the dividend.
The total holding period return, RT, is simply the sum of the capital appreciation and
income components of return:
You can download actual realized investment returns for a large number of stock market
indexes at the Callan Associates Web site, http://www.callan.com/research/periodic/.
Let’s consider an example of calculating the total holding period return on an investment.
One year ago today, you purchased a share of Dell Inc. stock for $12.50. Today it is worth
$13.90. Dell paid no dividend on its stock. What total return did you earn on this stock over
the past year?
If Dell paid no dividend and you received no other income from holding the stock, the total
return for the year equals the return from the capital appreciation. The total return is
calculated as follows:
What return would you have earned if Dell had paid a $1 dividend and today’s price was
$12.90? With the $1 dividend and a correspondingly lower price, the total return is the
same:
You can see from this example that a dollar of capital appreciation is worth the same as a
dollar of income.
APPLICATION 7.1 LEARNING BY DOING
Calculating the Return on an Investment
PROBLEM: You purchased a beat-up 1974 Datsun 240Z sports car a year ago for $1,500. Datsun is
what Nissan, the Japanese car company, was called in the 1970s. The 240Z was the first in a series
of cars that led to the Nissan 370Z that is being sold today. Recognizing that a mint-condition 240Z
is a much sought-after car, you invested $7,000 and a lot of your time fixing up the car. Last week,
you sold it to a collector for $18,000. Not counting the value of the time you spent restoring the car,
what is the total return you earned on this investment over the one-year holding period?
APPROACH: Use Equation 7.1 to calculate the total holding period return. To calculate RT using
Equation 7.1, you must know P0, P1, and CF1. In this problem, you can assume that the $7,000 was
spent at the time you bought the car to purchase parts and materials. Therefore, your initial
investment, P0, was $1,500 $7,000 $8,500. Since there were no other cash inflows or outflows
between the time that you bought the car and the time that you sold it, CF1 equals $0.
SOLUTION: The total holding period return is:
Expected Returns
•
Suppose that you are a senior who plays college baseball and that your team is in the
College World Series. Furthermore, suppose that you have been drafted by the Washington
Nationals and are coming up for what you expect to be your last at-bat as a college player.
The fact that you expect this to be your last at-bat is important because you just signed a
very unusual contract with the Nationals. Your signing bonus will be determined solely by
whether you get a hit in your final collegiate at-bat. If you get a hit, then your signing bonus
will be $800,000. Otherwise, it will be $400,000. This past season, you got a hit 32.5
percent of the times you were at bat (you did not get a hit 67.5 percent of the time), and
you believe this percentage reflects the likelihood that you will get a hit in your last
collegiate at-bat.1
What is the expected value of your bonus? If you have taken a statistics course, you might
recall that an expected value represents the sum of the products of the possible outcomes
and the probabilities that those outcomes will be realized. In our example the expected
value of the bonus can be calculated using the following formula:
where E(Bonus) is your expected bonus, pH is the probability of a hit, pNH is the probability
of no hit, BH is the bonus you receive if you get a hit, and BNH is the bonus you receive if you
get no hit. Since pH equals 0.325, pNH equals 0.675, BH equals $800,000, and BNH equals
$400,000, the expected value of your bonus is:
Notice that the expected bonus of $530,000 is not equal to either of the two possible
payoffs. Neither is it equal to the simple average of the two possible payoffs. This is because
the expected bonus takes into account the probability of each event occurring. If the
probability of each event had been 50 percent, then the expected bonus would have
equaled the simple average of the two payoffs:
However, since it is more likely that you will not get a hit (a 67.5 percent chance) than that
you will get a hit (a 32.5 percent chance), and the payoff is lower if you do not get a hit, the
expected bonus is less than the simple average.
What would your expected payoff be if you got a hit 99 percent of the time? We intuitively
know that the expected bonus should be much closer to $800,000 in this case. In fact, it is:
The key point here is that the expected value reflects the relative likelihoods of the possible
outcomes.
We calculate an expected return in finance in the same way that we calculate any
expected value. The expected return is a weighted average of the possible returns from an
investment, where each of these returns is weighted by the probability that it will occur. In
general terms, the expected return on an asset, E (RAsset), is calculated as follows:
expected return
an average of the possible returns from an investment, where each return is weighted by
the probability that it will occur
where Ri is possible return i and pi is the probability that you will actually earn Ri. The
summation symbol in this equation
is mathematical shorthand indicating that n values are added together. In Equation 7.2,
each of the n possible returns is multiplied by the probability that it will be realized, and
these products are then added together to calculate the expected return.
It is important to make sure that the sum of the n individual probabilities, the pi’s, always
equals 1, or 100 percent, when you calculate an expected value. The sum of the
probabilities cannot be less than 100 percent because you must account for all possible
outcomes in the calculation. On the other hand, as you may recall from statistics, the sum of
the probabilities of all possible outcomes cannot exceed 100 percent. For example, notice
that the sum of the pi’s equals 1 in each of the expected bonus calculations that we
discussed earlier (0.325 0.625 in the first calculation, 0.5 0.5 in the second, and 0.99 0.01 in
the third).
The expected return on an asset reflects the return that you can expect to receive from
investing in that asset over the period that you plan to own it. It is your best estimate of this
return, given the possible outcomes and their associated probabilities.
Note that if each of the possible outcomes is equally likely (that is, p1 = p2 = p3 = … = pn = p =
1/n), this formula reduces to the formula for a simple (equally weighted) average of the
possible returns:
To see how we calculate the expected return on an asset, suppose you are considering
purchasing Dell, Inc. stock for $13.90 per share. You plan to sell the stock in one year. You
estimate that there is a 30 percent chance that Dell stock will sell for $13.40 at the end of
one year, a 30 percent chance that it will sell for $14.90, a 30 percent that it will sell for
$15.40, and a 10 percent chance that it will sell for $16.00. If Dell pays no dividends on its
shares, what is the return that you expect from this stock in the next year?
Since Dell pays no dividends, the total return on its stock equals the return from capital
appreciation:
Therefore, we can calculate the return from owning Dell stock under each of the four
possible outcomes using the approach we used for the similar Dell problem we solved
earlier in the chapter. These returns are calculated as follows:
Applying Equation 7.2, the expected return on Dell stock over the next year is therefore
5.83 percent, calculated as follows:
Notice that the negative return is entered into the formula just like any other. Also notice
that the sum of the pi’s equals 1.
APPLICATION 7.2 LEARNING BY DOING
Calculating Expected Returns
PROBLEM: You have just purchased 100 railroad cars that you plan to lease to a large railroad
company. Demand for shipping goods by rail has recently increased dramatically due to the rising
price of oil. You expect oil prices, which are currently at $98.81 per barrel, to reach $115.00 per
barrel in the next year. If this happens, railroad shipping prices will increase, thereby driving up the
value of your railroad cars as increases in demand outpace the rate at which new cars are being
produced.
Given your oil price prediction, you estimate that there is a 30 percent chance that the value of your
railroad cars will increase by 15 percent, a 40 percent chance that their value will increase by 25
percent, and a 30 percent chance that their value will increase by 30 percent in the next year. In
addition to appreciation in the value of your cars, you expect to earn 10 percent on your investment
over the next year (after expenses) from leasing the railroad cars. What total return do you expect
to earn on your railroad car investment over the next year?
APPROACH: Use Equation 7.1 first to calculate the total return that you would earn under each of
the three possible outcomes. Next use these total return values, along with the associated
probabilities, in Equation 7.2 to calculate the expected total return.
SOLUTION: To calculate the total returns using Equation 7.1,
you must recognize that ΔP/P0 is the capital appreciation under each outcome and that
CF1/P0 equals the 10 percent that you expect to receive from leasing the rail cars. The expected
returns for the three outcomes are:
You can then use Equation 7.2 to calculate the expected return for your rail car investment:
Alternatively, since there is a 100 percent probability that the return from leasing the railroad cars
is 10 percent, you could have simply calculated the expected increase in value of the railroad cars:
and added the 10 percent to arrive at the answer of 33.5 percent. Of course, this simpler approach
only works if the return from leasing is known with certainty.
EXAMPLE 7.1 DECISION MAKING
Using Expected Values in Decision Making
SITUATION: You are deciding whether you should advertise your pizza business on the radio or on
billboards placed on local taxicabs. For $1,000 per month, you can either buy 20 one-minute ads on
the radio or place your ad on 40 taxicabs.
There is some uncertainty regarding how many new customers will visit your restaurant after
hearing one of your radio ads. You estimate that there is a 30 percent chance that 35 people will
visit, a 45 percent chance that 50 people will visit, and a 25 percent chance that 60 people will visit.
Therefore, you expect the following number of new customers to visit your restaurant in response
to each radio ad:
This means that you expect 20 one-minute ads to bring in 20 × 48 = 960 new customers.
Similarly, you estimate that there is a 20 percent chance you will get 20 new customers in response
to an ad placed on a taxi, a 30 percent chance you will get 30 new customers, a 30 percent chance
that you will get 40 new customers, and a 20 percent chance that you will get 50 new customers.
Therefore, you expect the following number of new customers in response to each ad that you place
on a taxi:
Placing ads on 40 taxicabs is therefore expected to bring in 40 35 1,400 new customers.
Which of these two advertising options is more attractive? Is it cost effective?
DECISION: You should advertise on taxicabs. For a monthly cost of $1,000, you expect to attract
1,400 new customers with taxicab advertisements but only 960 new customers if you advertise on
the radio.
The answer to the question of whether advertising on taxicabs is cost effective depends on how
much the gross profits (profits after variable costs) of your business are increased by those 1,400
customers. Monthly gross profits will have to increase by $1,000, or average 72 cents per new
customer ($1,000/1,400 $0.72) to cover the cost of the advertising campaign.
> BEFORE YOU GO ON
1. What are the two components of a total holding period return?
2. How is the expected return on an investment calculated?
7.3 THE VARIANCE AND STANDARD DEVIATION AS
MEASURES OF RISK
•
We turn next to a discussion of the two most basic measures of risk used in finance—the
variance and the standard deviation. These are the same variance and standard deviation
measures that you studied if you took a course in statistics.
Calculating the Variance and Standard Deviation
Let’s begin by returning to our College World Series example. Recall that you will receive a
bonus of $800,000 if you get a hit in your final collegiate at-bat and a bonus of $400,000 if
you do not. The expected value of your bonus is $530,000. Suppose you want to measure
the risk, or uncertainty, associated with the bonus. How can you do this? One approach
would be to compute a measure of how much, on average, the bonus payoffs deviate from
the expected value. The underlying intuition here is that the greater the difference between
the actual bonus and the expected value, the greater the risk. For example, you might
calculate the difference between each possible bonus payment and the expected value, and
sum these differences. If you do this, you will get the following result:
Unfortunately, using this calculation to obtain a measure of risk presents two problems.
First, since one difference is positive and the other difference is negative, one difference
partially cancels the other. As a result, you are not getting an accurate measure of total risk.
Second, this calculation does not take into account the number of potential outcomes or the
probability of each outcome.
variance (σ2)
a measure of the uncertainty associated with an outcome
A better approach would be to square the differences (squaring the differences makes all
the numbers positive) and multiply each squared difference by its associated probability
before summing them up. This calculation yields the variance (σ2) of the possible
outcomes. The variance does not suffer from the two problems mentioned earlier and
provides a measure of risk that has a consistent interpretation across different situations
or assets. For the original bonus arrangement, the variance is:
Note that the square of the Greek symbol sigma, σ2, is generally used to represent the
variance.
Because it is somewhat awkward to work with units of squared dollars, in a calculation
such as this we would typically take the square root of the variance. The square root gives
us the standard deviation (σ) of the possible outcomes. For our example, the standard
deviation is:
standard deviation (σ)
the square root of the variance
As you will see when we discuss the normal distribution, the standard deviation has a
natural interpretation that is very useful for assessing investment risks.
The general formula for calculating the variance of returns can be written as follows:
Equation 7.3 simply extends the calculation illustrated above to the situation where there
are n possible outcomes. Like the expected return calculation (Equation 7.2), Equation 7.3
can be simplified if all of the possible outcomes are equally likely. In this case it becomes:
In both the general case and the case where all possible outcomes are equally likely, the
standard deviation is simply the square root of the variance
.
Interpreting the Variance and Standard Deviation
The variance and standard deviation are especially useful measures of risk for variables
that are normally distributed—those that can be represented by a normal distribution.
The normal distribution is a symmetric frequency distribution that is completely
described by its mean (average) and standard deviation. Exhibit 7.1 illustrates what this
distribution looks like. Even if you have never taken a statistics course, you have already
encountered the normal distribution. It is the “bell curve” on which instructors often base
their grade distributions. SAT scores and IQ scores are also based on normal distributions.
normal distribution
a symmetric frequency distribution that is completely described by its mean and standard
deviation; also known as a bell curve due to its shape
This distribution is very useful in finance because the returns for many assets are
approximately normally distributed. This makes the variance and standard deviation
practical measures of the uncertainty associated with investment returns. Since the
standard deviation is more easily interpreted than the variance, we will focus on the
standard deviation as we discuss the normal distribution and its application in finance.
In Exhibit 7.1, you can see that the normal distribution is symmetric: the left and right sides
are mirror images of each other. The mean falls directly in the center of the distribution,
and the probability that an outcome is less than or greater than a particular distance from
the mean is the same whether the outcome is on the left or the right side of the distribution.
For example, if the mean is 0, the probability that a particular outcome is 3 or less is the
same as the probability that it is + 3 or more (both are 3 or more units from the mean). This
enables us to use a single measure of risk for the normal distribution. That measure is the
standard deviation.
EXHIBIT 7.1 Normal Distribution
The normal distribution is a symmetric distribution that is completely described by its
mean and standard deviation. The mean is the value that defines the center of the
distribution, and the standard deviation, s, describes the dispersion of the values centered
around the mean.
The standard deviation tells us everything we need to know about the width of the normal
distribution or, in other words, the variation in the individual values. This variation is what
we mean when we talk about risk in finance. In general terms, risk is a measure of the
range of potential outcomes. The standard deviation is an especially useful measure of risk
because it tells us the probability that an outcome will fall a particular distance from the
mean, or within a particular range. You can see this in the following table, which shows the
fraction of all observations in a normal distribution that are within the indicated number of
standard deviations from the mean.
Since the returns on many assets are approximately normally distributed, the standard
deviation provides a convenient way of computing the probability that the return on an
asset will fall within a particular range. In these applications, the expected return on an
asset equals the mean of the distribution, and the standard deviation is a measure of the
uncertainty associated with the return.
For example, if the expected return for a real estate investment in Miami, Florida, is 10
percent with a standard deviation of 2 percent, there is a 90 percent chance that the actual
return will be within 3.29 percent of 10 percent. How do we know this? As shown in the
table, 90 percent of all outcomes in a normal distribution have a value that is within 1.645
standard deviations of the mean value, and 1.645 × 2 percent = 3.29 percent. This tells us
that there is a 90 percent chance that the realized return on the investment in Miami will
be between 6.71 percent (10 percent − 3.29 percent = 6.71 percent) and 13.29 percent (10
percent + 3.29 percent = 13.29 percent), a range of 6.58 percent (13.29 percent − 6.71
percent = 6.58 percent).
You may be wondering what is standard about the standard deviation. The answer is that
this statistic is standard in the sense that it can be used to directly compare the
uncertainties (risks) associated with the returns on different investments. For instance,
suppose you are comparing the real estate investment in Miami with a real estate
investment in Fresno, California. Assume that the expected return on the Fresno
investment is also 10 percent. If the standard deviation for the returns on the Fresno
investment is 3 percent, there is a 90 percent chance that the actual return is within 4.935
percent (1.645 × 3 percent = 4.935 percent) of 10 percent. In other words, 90 percent of the
time, the return will be between 5.065 percent (10 percent − 4.935 percent = 5.065
percent) and 14.935 percent (10 percent + 4.935 percent = 14.935 percent), a range of 9.87
percent (14.935 percent − 5.065 percent = 9.87 percent).
This range is exactly 9.87 percent/6.58 percent = 1.5 times as large as the range for the
Miami investment opportunity. Notice that the ratio of the two standard deviations also
equals 1.5 (3 percent/2 percent = 1.5). This is not a coincidence. We could have used the
standard deviations to directly compute the relative uncertainty associated with the Fresno
and Miami investment returns. The relation between the standard deviation of returns and
the width of a normal distribution (the uncertainty) is illustrated in Exhibit 7.2.
Let’s consider another example of how the standard deviation is interpreted. Suppose
customers at your pizza restaurant have complained that there is no consistency in the
number of slices of pepperoni that your cooks are putting on large pepperoni pizzas. One
night you decide to work in the area where the pizzas are made so that you can count the
number of pepperoni slices on the large pizzas to get a better idea of just how much
variation there is. After counting the slices of pepperoni on 50 pizzas, you estimate that, on
average, your pies have 18 slices of pepperoni and that the standard deviation is 3 slices.
With this information, you estimate that 95 percent of the large pepperoni pizzas sold in
your restaurant have between 12.12 and 23.88 slices. You are able to estimate this range
because you know that 95 percent of the observations in a normal distribution fall within
1.96 standard deviations of the mean. With a standard deviation of three slices, this implies
that the number of pepperoni slices on 95 percent of your pizzas is within 5.88 slices of the
mean (3 slices × 1.96 = 5.88 slices). This, in turn, indicates a range of 12.12 (18 − 5.88 =
12.12) to 23.88 (18 + 5.88 = 23.88) slices.
Since you put only whole slices of pepperoni on your pizzas, 95 percent of the time the
number of slices is somewhere between 12 and 24. No wonder your customers are up in
arms! In response to this information, you decide to implement a standard policy regarding
the number of pepperoni slices that go on each type of pizza.
EXHIBIT 7.2 Standard Deviation and Width of the Normal Distribution
The larger standard deviation for the return on the Fresno investment means that the
Fresno investment is riskier than the Miami investment. The actual return for the Fresno
investment is more likely to be further from its expected return.
APPLICATION 7.3 LEARNING BY DOING
Understanding the Standard Deviation
PROBLEM: You are considering investing in a share of Google Inc., stock and want to evaluate how
risky this potential investment is. You know that stock returns tend to be normally distributed, and
you have calculated the expected return on Google stock to be 4.67 percent and the standard
deviation of the annual return to be 23 percent. Based on these statistics, within what range would
you expect the return on this stock to fall during the next year? Calculate this range for a 90 percent
level of confidence (that is, 90 percent of the time, the returns will fall within the specified range).
APPROACH: Use the values in the previous table or Exhibit 7.1 to compute the range within which
Google’s stock return will fall 90 percent of the time. First, find the number of standard deviations
associated with a 90 percent level of confidence in the table or Exhibit 7.1 and then multiply this
number by the standard deviation of the annual return for Google’s stock. Then subtract the
resulting value from the expected return (mean) to obtain the lower end of the range and add it to
the expected return to obtain the upper end.
SOLUTION: From the table, you can see that we would expect the return over the next year to be
within 1.645 standard deviations of the mean 90 percent of the time. Multiplying this value by the
standard deviation of Google’s stock (23 percent) yields 23 percent × 1.645 = 37.835 percent. This
means that there is a 90 percent chance that the return will be between −33.165 percent (4.67
percent − 37.835 percent = −33.165 percent) and 42.505 percent (4.67 percent + 37.835 percent =
42.505 percent).
While the expected return of 4.67 percent is relatively low, the returns on Google stock vary
considerably, and there is a reasonable chance that the stock return in the next year could be quite
high or quite low (even negative). As you will see shortly, this wide range of possible returns is
similar to the range we observe for typical shares in the U.S. stock market.
Historical Market Performance
Now that we have discussed how returns and risks can be measured, we are ready to
examine the characteristics of the historical returns earned by securities such as stocks and
bonds. Exhibit 7.3 illustrates the distributions of historical returns for some securities in
the United States and shows the average and standard deviations of these annual returns
for the period from 1926 to 2009.
Note that the statistics reported in Exhibit 7.3 are for indexes that represent
total average returns for the indicated types of securities, not total returns on individual
securities. We generally use indexes to represent the performance of the stock or bond
markets. For instance, when news services report on the performance of the stock market,
they often report that the Dow Jones Industrial Average (an index based on 30 large
stocks), the S&P 500 Index (an index based on 500 large stocks), or the NASDAQ Composite
Index (an index based on all stocks that are traded on NASDAQ) went up or down on a
particular day. These and other indexes are discussed in Chapter 9.
The plots in Exhibit 7.3 are arranged in order of decreasing risk, which is indicated by the
decreasing standard deviation of the annual returns. The top plot shows returns for a
small-stock index that represents the 10 percent of U.S. firms that have the lowest total
equity value (number of shares multiplied by price per share). The second plot shows
returns for the S&P 500 Index, representing large U.S. stocks. The remaining plots show
three different types of government debt: Long-term government bonds that mature in 20
years, intermediate-term government bonds that mature in five years, and U.S. Treasury
bills, which are short-term debts of the U.S. government, that mature in 30 days.
EXHIBIT 7.3 Distributions of Annual Total Returns for U.S. Stocks and Bonds from
1926 to 2009
Higher standard deviations of returns have historically been associated with higher
returns. For example, between 1926 and 2009, the standard deviation of the annual
returns for small stocks was higher than the standard deviations of the returns earned by
other types of securities, and the average return that investors earned from small stocks
was also higher. At the other end of the spectrum, the returns on Treasury bills had the
smallest standard deviation, and Treasury bills earned the smallest average return.
Source: Data from Morningstar, 2010 SBBI Yearbook
The key point to note in Exhibit 7.3 is that, on average, annual returns have been higher for
riskier securities. Small stocks, which have the largest standard deviation of total returns,
at 32.79 percent, also have the largest average annual return, 16.57 percent. On the other
end of the spectrum, Treasury bills have the smallest standard deviation, 3.08 percent, and
the smallest average annual return, 3.71 percent. Returns for small stocks in any particular
year may have been higher or lower than returns for the other types of securities, but on
average, they were higher. This is evidence that investors require higher returns for
investments with greater risks.
The statistics in Exhibit 7.3 describe actual investment returns, as opposed to expected
returns. In other words, they represent what has happened in the past. Financial analysts
often use historical numbers such as these to estimate the returns that might be expected
in the future. That is exactly what we did in the baseball example earlier in this chapter. We
used the percentage of at-bats in which you got a hit this past season to estimate the
likelihood that you would get a hit in your last collegiate at-bat. We assumed that your past
performance was a reasonable indicator of your future performance.
To see how historical numbers are used in finance, let’s suppose that you are considering
investing in a fund that mimics the S&P 500 Index (this is what we call an index fund) and
that you want to estimate what the returns on the S&P 500 Index are likely to be in the
future. If you believe that the 1926 to 2009 period provides a reasonable indication of what
we can expect in the future, then the average historical return on the S&P 500 Index of
11.84 percent provides a perfectly reasonable estimate of the return you can expect from
your investment in the S&P 500 Index fund. In Chapter 13 we will explore in detail how
historical data can be used in this way to estimate the discount rate used to evaluate
projects in the capital budgeting process.
Comparing the historical returns for an individual stock with the historical returns for an
index can also be instructive. Exhibit 7.4 shows such a comparison for Apple Inc. and the
S&P 500 Index using monthly returns for the period from September 2005 to September
2010. Notice in the exhibit that the returns on Apple stock are much more volatile than the
average returns on the firms represented in the S&P 500 Index. In other words, the
standard deviation of returns for Apple stock is higher than that for the S&P 500 Index.
This is not a coincidence; we will discuss shortly why returns on individual stocks tend to
be riskier than returns on indexes.
One last point is worth noting while we are examining historical returns: the value of a
$1.00 investment in 1926 would have varied greatly by 2009, depending on where that
dollar was invested. Exhibit 7.5 shows that $1.00 invested in U.S. Treasury bills in 1926
would have been worth $20.53 by 2009. In contrast, that same $1.00 invested in small
stocks would have been worth $12,231.13 by 2009!2 Over a long period of time, earning
higher rates of return can have a dramatic impact on the value of an investment. This huge
difference reflects the impact of compounding of returns (returns earned on returns), much
like the compounding of interest we discussed in Chapter 5.
EXHIBIT 7.4 Monthly Returns for Apple Inc. stock and the S&P 500 Index from
September 2005 through September 2010
The returns on shares of individual stocks tend to be much more volatile than the returns
on portfolios of stocks, such as the S&P 500.
EXHIBIT 7.5 Cumulative Value of $1 Invested in 1926
The value of a $1 investment in stocks, small or large, grew much more rapidly than the
value of a $1 investment in bonds or Treasury bills over the 1926 to 2009 period. This
graph illustrates how earning a higher rate of return over a long period of time can affect
the value of an investment portfolio. Although annual stock returns were less certain
between 1926 and 2009, the returns on stock investments were much greater.
Source: Data from Morningstar, 2010 SBBI Yearbook
> BEFORE YOU GO ON
1. What is the relation between the variance and the standard deviation?
2. What relation do we generally observe between risk and return when we examine
historical returns?
3. How would we expect the standard deviation of the return on an individual stock to
compare with the standard deviation of the return on a stock index?
7.4 RISK AND DIVERSIFICATION
•
It does not generally make sense to invest all of your money in a single asset. The reason is
directly related to the fact that returns on individual stocks tend to be riskier than returns
on indexes. By investing in two or more assets whose values do not always move in the
same direction at the same time, an investor can reduce the risk of his or her collection of
investments, or portfolio. This is the idea behind the concept of diversification.
portfolio
the collection of assets an investor owns
diversification
Reducing risk by investing in two or more assets whose values do not always move in the
same direction at the same time
This section develops the tools necessary to evaluate the benefits of diversification. We
begin with a discussion of how to quantify risk and return for a single-asset portfolio, and
then we discuss more realistic and complicated portfolios that have two or more assets.
Although our discussion focuses on stock portfolios, it is important to recognize that the
concepts discussed apply equally well to portfolios that include a range of assets, such as
stocks, bonds, gold, art, and real estate, among others.
Single-Asset Portfolios
Returns for individual stocks from one day to the next have been found to be largely
independent of each other and approximately normally distributed. In other words, the
return for a stock on one day is largely independent of the return on that same stock the
next day, two days later, three days later, and so on. Each daily return can be viewed as
having been randomly drawn from a normal distribution where the probability associated
with the return depends on how far it is from the expected value. If we know what the
expected value and standard deviation are for the distribution of returns for a stock, it is
possible to quantify the risks and expected returns that an investment in the stock might
yield in the future.
To see how we might do this, assume that you are considering investing in one of two
stocks for the next year: Advanced Micro Devices (AMD) or Intel. Also, to keep things
simple, assume that there are only three possible economic conditions (outcomes) a year
from now and that the returns on AMD and Intel under each of these outcomes are as
follows:
With this information, we can calculate the expected returns for AMD and Intel by using
Equation 7.2:
and
Similarly, we can calculate the standard deviations of the returns for AMD and Intel in the
same way that we calculated the standard deviation for our baseball bonus example in
Section 7.2:
and
Having calculated the expected returns and standard deviations for the expected returns
on AMD and Intel stock, the natural question to ask is which provides the highest riskadjusted return. Before we answer this question, let’s return to the example at the
beginning of Section 7.1. Recall that, in this example, we proposed choosing among three
stocks: A, B, and C. We stated that investors would prefer the investment that provides the
highest expected return for a given level of risk or the lowest risk for a given expected
return. This made it fairly easy to choose between Stocks A and B, which had the same
return but different risk levels, and between Stocks B and C, which had the same risk but
different returns. We were stuck when trying to choose between Stocks A and C, however,
because they differed in both risk and return. Now, armed with tools for quantifying
expected returns and risk, we can at least take a first pass at comparing stocks such as
these.
The coefficient of variation (CV) is a measure that can help us in making comparisons
such as that between Stocks A and C. The coefficient of variation for stock i is calculated as:
coefficient of variation (CV)
a measure of the risk associated with an investment for each one percent of expected
return
In this equation, CV is a measure of the risk associated with an investment for each 1
percent of expected return.
Recall that Stock A has an expected return of 12 percent and a risk level of 12 percent,
while Stock C has an expected return of 16 percent and a risk level of 16 percent. If we
assume that the risk level given for each stock is equal to the standard deviation of its
return, we can find the coefficients of variation for the stocks as follows:
Since these values are equal, the coefficient of variation measure suggests that these two
investments are equally attractive on a risk-adjusted basis.
While this analysis appears to make sense, there is a conceptual problem with using the
coefficient of variation to compute the amount of risk an investor can expect to realize for
each 1 percent of expected return. This problem arises because investors expect to earn a
positive return even when assets are completely risk free. For example, as shown in Exhibit
7.3, from 1926 to 2009 investors earned an average return of 3.71 percent each year on 30day Treasury bills, which are considered to be risk free.3 If investors can earn a positive
risk-free rate without bearing any risk, then it really only makes sense to compare the risk
of the investment, sRi, with the return that investors expect to earn over and above the riskfree rate. As we will discuss in detail in Section 7.6, the expected return over and above the
risk-free rate is a measure of the return that investors expect to earn for bearing risk.
This suggests that we should use the difference between the expected return, E (Ri), and
the risk-free rate, Rrf, instead of E (Ri) alone in the coefficient of variation calculation. With
this change, Equation 7.4 would be written as:
where CVi* is a modified coefficient of variation that is computed by subtracting the riskfree rate from the expected return.
Let’s compute this modified coefficient of variation for the AMD and Intel example. If the
risk-free rate equals 0.03, or 3 percent, the modified coefficients of variation for the two
stocks are:
We can see that the modified coefficient of variation for AMD is smaller than the modified
coefficient of variation for Intel. This tells us that an investment in AMD stock is expected to
have less risk for each 1 percent of return. Since investors prefer less risk for a given level
of return, the AMD stock is a more attractive investment.
A popular version of this modified coefficient of variation calculation is known as the
Sharpe Ratio. This ratio is named after 1990 Nobel Prize Laureate William Sharpe who
developed the concept and was one of the originators of the capital asset pricing model
which is discussed in Section 7.7. The Sharpe Ratio is simply the inverse of the modified
coefficient of variation:
Sharpe Ratio
A measure of the return per unit of risk for an investment
For the stocks of AMD and Intel, the Sharpe Ratios are:
You can read more about the Sharpe Ratio and other ratios that are used to measure riskadjusted returns for investments at the following Web site: http://en.wikipedia.org/wiki/sharperatio.
This tells us that investors in AMD stock can expect to earn 0.524 percent for each one
standard deviation of return while …