# Finance 660 by qfc86623

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```									Chapter 7 – Introduction to Risk, Return and the Opportunity Cost of Capital
Chapter 8 – Risk and Return (section 8-2 and 8-3, skim)

These chapters describe how risk is measured and is part of a three-chapter sequence describing how the risk of a
project‟s cash flows determines the discount rate (the opportunity cost of capital) for these cash flows. We then use
the discount rate to calculate the present value of the project‟s future expected cash flows. Subtracting the initial
investment gives us the project NPV.

You will notice a slight difference in notation between these notes and the text. Please take this into account in your
studying.

************************************************************

Through most of our discussion of Chapter 6, we assumed that the future cash flows of a project are known
with certainty. However, in most circumstances future cash flows are uncertain. When a corporation‟s future cash
flows are uncertain:

   Step one - Calculate expected future cash flows.

Calculation of project cash flows (under certainty) was discussed in Chapter 6. This involved determining
project revenues, expenses, tax depreciation, corporate income taxes, working capital adjustments, inflation, etc.

To calculate expected cash flows for risky projects, the types of calculations done in Chapter 6 may have to be
performed many times resulting in something like this:

Project A‟s time one cash flow in a booming economy = \$155
Project A‟s time one cash flow in a normal economy = \$135
Project A‟s time one cash flow in a recession economy = \$40

To calculate expected cash flows, you need to determine the probabilities:

Probability of a booming economy = 20%
Probability of a normal economy = 60%
Probability of a recession economy = 20%

What is the expected project cash flow for time one in the above example?
If the initial investment is \$100, what is the IRR for this project?

In the above example, I have assumed that project cash flows depend only on the type of economy. This is
obviously a simplification. Several other factors can affect project cash flows, each with their own cash flows
and probabilities. For example, consider consumer demand:

Project A‟s time one cash flow in a booming economy and high consumer demand = \$200 (10% prob.)
Project A‟s time one cash flow in a booming economy and low consumer demand = \$110 (10% prob.)

Project A‟s time one cash flow in a normal economy and high consumer demand = \$165 (30% prob.)
Project A‟s time one cash flow in a normal economy and low consumer demand = \$105 (30% prob.)

Project A‟s time one cash flow in a recession economy and high consumer demand = \$60 (10% prob.)
Project A‟s time one cash flow in a recession economy and low consumer demand = \$20 (10% prob.)

   Step two - Discount expected cash flows at the opportunity cost of capital to determine their present value.

The opportunity cost of capital is based on the cash flow‟s risk (higher risk, higher opportunity cost of capital)

Notice that the above example gives project cash flows and probabilities. We use these to calculate
expected project cash flows (and the present value of these cash flows).

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Sometimes you might want to calculate present values of individual parts of the project. For example:

Economy                 Probability            Project            Product 1            Product 2
Boom                       0.20                 \$155                \$115                  \$40
Normal                     0.60                 \$135                 \$95                  \$40
Recession                  0.20                  \$40                  \$0                  \$40
Expected                                        \$120                 \$80                  \$40

PV of cash flows from product 1 + PV of cash flows from product 2 = PV of project cash flows

Separate present value calculations (of the two parts) will give you the same answer as you would get
by calculating the PV of the project cash flows.

However, separate calculations may allow you to focus on the risks of the different parts of the project.

How could you use this concept in the above example?

The opportunity cost of capital for cash flows is the expected return for financial assets with the exact same
amount of risk as these cash flows.

   Risk-free cash flows should use a discount rate equal to the expected return for risk-free securities (e.g., the
one-month Treasury-Bill interest rate). We are using 5% as the risk-free rate in this class.

   Higher risk cash flows should be discounted at a higher discount rate to reflect their higher level of risk. For
example, consider a project with cash flows that are just as risky as large U.S. firm common stocks.

Examine Table 7.1 (page 149 of the textbook). Then describe how the opportunity cost of capital would be
determined for the risky cash flows of this particular project.

Relevant Table 7.1 information
Average annual return for the large U.S. firm common stocks (1900 – 2003) = 11.7%
Average annual return for Treasury Bills (1900 – 2003) = 4.1%

Source: Yahoo Finance
S&P 500 for 2004: 10.7%               Treasury Bills for 2004: 1.5% (approximate)
S&P 500 for 2005: 3.0%                Treasury Bills for 2005: 3.0% (approximate)
S&P 500 for 2006: 15.2%               Treasury Bills for 2006: 5.0% (approximate)

(Note: For purposes of class discussion, ignore what has happened so far in 2007, i.e., assume we are at the
beginning of the year.)

Updated (approximate) averages including information from 2004 - 2006
Average annual return for large U.S. firm common stocks (1900 – 2006) = 11.6%
Average annual return for Treasury Bills (1900 – 2006) = 4.1%
Average difference (1900 – 2006) = 7.5%

   Arithmetic versus geometric averages

Assume you buy a stock for \$100 (at t = 0). The stock falls 50% in year 1, but increases 50% in year 2. What is
your stock selling for at t = 2?

What do you need to earn in the second year to get you back to \$100?

We should use arithmetic averages for estimating risk premiums (discussed below)

We should use geometric averages for computing growth in investment

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   Example (stock without dividends)

Time        Price                     Dividend                 Return
0           \$100
1           \$110                      \$0
2           \$132                      \$0
3           \$125.4                    \$0

Arithmetic average =

Geometric average =

   Example (stock with dividends)

Time        Price                     Dividend                 Return
0           \$50
1           \$55                       \$0
2           \$57                       \$6.25
3           \$50                       \$1.30

Arithmetic average =

Geometric average =

   Compounding and the growth in investment values

Assume your great-great grandparents put \$1000 in the bank for you in 1900, earning 4.0% per year (geometric
average for T-Bills). What is the value of this \$1000 investment today (107 years later)?

Assume your great-great grandparents put \$1000 in the bank for you in 1900, earning 9.7% per year (geometric
average for large U.S. firm common stocks). What is the value of this \$1000 investment today (107 years
later)?

1.   What is the market risk premium? The extra return investors expect for the stock market (e.g., large
U.S. firm common stocks) over the risk-free rate.

Based on the law of one price, what investors “expect” for the stock market should (in equilibrium) be
equal to what investors “require” for the stock market.

In general, what investors “require” for a particular investment should be based on the
investment‟s opportunity cost of capital (i.e., the expected rate of return on other investments with
the same risk level)

Now, applied to the stock market (e.g., large U.S. firm common stocks)

If investors‟ expected return for the stock market is higher than what they require (i.e., the stock
market‟s opportunity cost of capital), then:

If investors‟ expected return for the stock market is lower than what they require (i.e., the stock
market‟s opportunity cost of capital), then:

So, to have both buyers and sellers in the market, expected = required.

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2.   What is the expected return (over the next year) for an investment in large U.S. firm common stocks?

   Current short-term Treasury Bill rate
   Market risk premium for large U.S. firm common stocks (estimated as the avg. historical

3.   In the previous equation, we assumed two things:

   The average historical premium is a good approximation of the unobservable market risk
   The market risk premium is constant across time

4.   Based on this, what would happen if Treasury Bill interest rates increased from 5% to 10%?

   Expected large U.S. firm common stock return with a 5% risk-free interest rate =
   Expected large U.S. firm common stock return with a 10% risk-free interest rate =

5.   Problems with using historical data (for the period 1900-2006) to approximate the market risk

Is the average historical premium a good approximation for what investor‟s expect for the future?

   Perhaps events have occurred that have caused previous market returns to be different than what
investor expected.

What are some surprise events that could have caused the actual market return to differ from
expected?

   Maybe the high average historical return for large U.S. firm common stocks is due to a decrease in

Price for a portfolio of large U.S. firm common stocks = \$1000
Expected dividend in one year = \$100
Opportunity cost of capital = 13.4%, Growth rate in dividend (in perpetuity) = 3.4%

What is the expected return for large U.S. firm common stocks using the above information?

What happens if, over the next year, the opportunity cost of capital for large U.S. firm common
stocks declines to 9.4%? (Keep everything else the same.)

New large U.S. firm common stock price =
New expected return for large U.S. firm common stocks =

What happens to the average annual return for large U.S. firm common stock when this most
recent year is included in the average?

   Results from recent surveys of financial economists and corporate CFOs imply a market risk
premium between 5.5% and 7% (see footnote 11, page 152).
   Using analysis from dividend yields and dividend growth rates results in a risk premium of 5.3%
   Brealey, Myers, and Allen suggest the use of a market risk premium in the range of a 5% - 8%.
(Note – this is lower than the range of 6% - 8.5% given in the previous edition of the text.)

In class, we will use 5% as the risk-free interest rate and 8.4% as the market risk premium.

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Computing the opportunity cost of capital for risky cash flows of a project

From the above, we know how to calculate the opportunity cost of capital for risk free cash flows and for cash
flows that are just as risky as large U.S. firm common stocks. What about cash flows of different amounts of
risk?

In this class, we will:
 Use the CAPM model.
 In the CAPM, risk is measured by beta (the portion of total risk that cannot be diversified away).
 We will initially use a variation of the CAPM to solve for the PV of the project‟s cash flows. With this
number we will also calculate the NPV.
 With the PV, we will calculate the project‟s risk and opportunity cost of capital and use the opportunity
cost of capital to verify our calculation of the project PV and NPV.

Statistical and other tools used in the calculation of the beta of a project’s risky cash flows
 Calculation of expected cash flows
 Calculation of returns and expected returns
 Calculation of the variance (and standard deviation) of returns
 Calculation of the covariance (and correlation) of returns

This next section of the notes is meant to build up your intuition of the determinants of beta and project
value. A detailed example follows. You are expected to review most of this detailed example on your own.

ABC Corporation is considering an investment of \$100 (at t = 0) in either project A or project B. Each project will
last only one year. Time 1 cash flows from these projects depend on the state of the economy. Cash flows for a
\$100 risk-free investment and for a \$100 investment in the “market” are provided for comparison purposes.

State                                          1                       2                           3
Economy                                      Boom                    Normal                    Recession
Probability                                  20%                      60%                        20%
Risk-Free                                     \$105                     \$105                      \$105
Market                                        \$143                     \$116                       \$76
Project A                                     \$155                     \$135                       \$40
Project B                                      \$15                     \$105                      \$136

1.    What is the expected time one cash flow from the four investments?

     E(CFrf) = (\$105)(0.2) + (\$105)(0.6) + (\$105)(0.2) = \$105
     E(CFM) = (\$143)(0.2) + (\$116)(0.6) + (\$76)(0.2) = \$113.4
     E(CFA) = (\$155)(0.2) + (\$135)(0.6) + (\$40)(0.2) = \$120
     E(CFB) = (\$15)(0.2) + (\$105)(0.6) + (\$136)(0.2) = \$93.2

Based on the initial investment of \$100, we can calculate a one-period rate of return.

State                                          1                       2                           3
Economy                                      Boom                    Normal                    Recession
Probability                                  20%                      60%                         20%
Risk-Free                                     5%                      5%                          5%
Market                                       43%                      16%                        -24%
Project A                                    55%                      35%                        -60%
Project B                                     -85%                      5%                        36%

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2.   What is the expected return – based on the initial investment

E(rrf) using initial investment = (5%)(0.2) + (5%)(0.6) + (5%)(0.2) = (\$105 / \$100) – 1 = 5%
E(rM) using initial investment = (43%)(0.2) + (16%)(0.6) + (-24%)(0.2) = (\$113.4 / \$100) – 1 = 13.4%
E(rA) using initial investment = (55%)(0.2) + (35%)(0.6) + (-60%)(0.2) = (\$120 / \$100) – 1 = 20%
E(rB) using initial investment = (-85%)(0.2) + (5%)(0.6) + (36%)(0.2) = (\$93.2 / \$100) – 1 = -6.8%

Calculation of the PV of the project’s time one cash flow

Step one – calculate the variance of the market‟s returns
Step two – calculate the covariance between the project‟s cash flows and the market‟s returns
Step three – use the above two numbers to calculate the PV

3.   What is the variance of the market’s returns

   rM i  E (rM )  pi
N
2                      2
M
i 1

2M = [(0.43 - 0.134)2 0.2 + (0.16 - 0.134)2 0.6 + (-0.24 - 0.134)2 0.2] = 0.045904

4.   What is the covariance between the project’s cash flow and the market’s returns?

Cov (CFA , rM )   CFAi  E (CFA ) rM i  E (rM )  pi
N

i 1
Cov(CFA ,rM) =[(\$155-\$120)(0.43-0.134)(0.2) + (\$135-\$120)(0.16-0.134)(0.6) +
(\$40-\$120)(-0.24-0.134)(0.2)] = \$8.290000

Cov(CFB ,rM) =[(\$15-\$93.2)(0.43-0.134)(0.2) + (\$105-\$93.2)(0.16-0.134)(0.6) +
(\$136-\$93.2)(-0.24-0.134)(0.2)] = -\$7.646800

5.   What is the PV of the projects’ time one cash flows? (See footnote 16, page 227 of the textbook.)

            Cov(CFA , rM )(E (rM )  rf ) 
PVA   E (CFA )                                 (1  rf )
                       M2

PVA = [\$120 – [(\$8.29)(0.084)] / 0.045904] / (1.05) = \$99.8382
NPVA = -\$100 + \$99.8382 = -\$0.1618

PVB = [\$93.2 – [(-\$7.6468)(0.084)] / 0.045904] / (1.05) = \$102.0885
NPVB = -\$100 + \$102.0885 = \$2.0885

Calculation of beta, the CAPM opportunity cost of capital, and the verification of the project PV and NPV

Step one - calculate the project returns (using the PV) and the expected return (using the PV)
Step two – calculate the variance and standard deviation of the market‟s returns
Step three – calculate the variance and standard deviation of the project‟s returns (using the PV)
Step four - calculate the correlation coefficient between the project‟s returns (using the PV) and the market‟s returns
Step five – use the above three numbers to calculate the beta
Step six – use the beta to calculate the CAPM opportunity cost of capital and the PV and NPV

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6.    What are the returns for the two projects (using the PV)?

State                                             1                        2                           3
Economy                                         Boom                    Normal                     Recession
Probability                                     20%                      60%                         20%
Project A                                     55.2512%                 35.2188%                   -59.9352%
Project B                                    -85.3069%                 2.8519%                     33.2177%

7.    What is the expected return for the two projects (using the PV)?

E(rA) using PV = (55.2512%)(0.2) + (35.2188%)(0.6) + (-59.9352%)(0.2) = (\$120 / \$99.8382) – 1 = 20.1945%
E(rB) using PV = (-85.3069%)(0.2) + (2.8519%)(0.6) + (33.2177%)(0.2) = (\$93.2 / \$102.0885) – 1 = -8.7067%

8.    What is the variance of returns for the market, and for the two projects (using the PV)?

2M = 0.045904 (calculated above – variance is the same as based on the initial investment)
2A = [(0.552512 - 0.201945)2 0.2 + (0.352188 – 0.201945)2 0.6 + (-0.599352 - 0.201945)2 0.2] = 0.166539
2B = [(-0.853069 - (-0.087067))2 0.2 + (0.028519 - (-0.087067))2 0.6 + (0.332177 - (-0.087067))2 0.2] =
0.160521

9.    What is the standard deviation of returns for the market, and for the two projects (using the PV)?

M = 0.214252
A = 0.408091
B = 0.400651

Aside - What is the variance and standard deviation for the risk-free asset?

10. What is the correlation coefficient between the project’s returns (using the PV) and the market’s return?

Note -The correlation coefficient measures the tendency of two asset's returns to move in the same direction.
The maximum correlation coefficient is 1.0 (perfect positive correlation) and the minimum is -1.0 (perfect
negative correlation). A correlation coefficient of 0 means that the two asset's returns are uncorrelated.

     AM = Cov (rA ,rM,) / (A M ) = [(0.552512 - 0.201945) (0.43 - 0.134) (0.2) + (0.352188 - 0.201945) (0.16 -
0.134) (0.6) + (-0.599352 - 0.201945) (-0.24 - 0.134) (0.2)] / [(0.408091) (0.214252)] = 0.949675
     BM = Cov (rB,rM,) / (B M ) = [(-0.853069 - (-0.087067)) (0.43 - 0.134) (0.2) + (0.028519 - (-0.087067)) (0.16
- 0.134) (0.6) + (0.332177 - (-0.087067)) (-0.24 - 0.134) (0.2)] / [(0.400651) (0.214252)] = -0.872593

The maximum correlation coefficient is 1.0 (perfect positive correlation) and the minimum is -1.0 (perfect
negative correlation). A correlation coefficient of 0 means that the two asset's returns are uncorrelated.

Aside – What is the correlation coefficient between the risk-free asset and the market?

The „market‟ investment and project A have a very high positive correlation. This indicates that when the
„market‟ has a return that is higher than its expected return, project A should also have a return that is higher
than its expected return (and vice versa).

The opposite relationship (a negative correlation) is found between the „market‟ investment and project B. The
returns of the „market‟ investment and the risk-free investment are uncorrelated.

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11. What is the beta for the cash flows of Project A and Project B?

Investors in ABC Corp. want the corporation‟s managers to only invest in positive NPV projects. To calculate
the NPV, we need to discount the expected t = 1 cash flow at the opportunity cost of capital for each of the
projects.

NPVA = -\$100 + \$120 / (1 + opportunity cost of capital for project A)
NPVB = -\$100 + \$93.2 / (1 + opportunity cost of capital for project B)

As stated above, the opportunity cost of capital depends on the riskiness of the projects. How risky are these
two projects?

Well-diversified investors in ABC Corp. are concerned with how these projects will affect their investment
portfolio‟s risk. They are not concerned with risk that will be diversified away.

If an investor is invested in the market (the market portfolio), then beta is the relevant risk measure (not
variance or standard deviation). Beta measures the contribution of an asset to the riskiness of the „market
portfolio.‟ The market portfolio has a beta of one. Betas greater than one are more risky than the market (and
vice versa).

Calculation of beta

A = AM (σA / σM)

A = 0.949675 * (0.408091 / 0.214252) = 1.80887
B = -0.872593 * (0.400651 / 0.214252) = -1.63175

rf = 0.0 * (0.0 / 0.214252) = 0.0
M =1.0 * (0.214252 / 0.214252) = 1.0

   Project A is more risky than the market portfolio.
   Project B is less risky than the market portfolio.

12. A quick look at the Capital Asset Pricing Model (CAPM)

Using certain semi-reasonable assumptions, it can be shown that all risk-averse investors will want to invest in
two assets: i) the market portfolio, and ii) a risk-free asset.

The market portfolio is a portfolio of all risky assets (stocks, bonds, real estate, fine art, human capital, etc.)
held in proportion to their market value. The risk-free asset could be Treasury Bills or some other risk-free
asset (bank saving's account, bank CD's, money-market mutual funds, etc.). Example:

\$10000 total investment
\$2000 in a bank certificate of deposit - risk-free investment
\$8000 in a global mutual fund - market investment

Based on the discussion in the notes so far, we learned that beta is the appropriate measure of risk for investors
that own a portion of the market portfolio. This also applies to an investment such as the one above (i.e., part
investment in the market portfolio and part in a risk-free investment).

Remember: Beta measures the contribution of an asset to the riskiness of the market portfolio (after taking into
account the reduction in risk from diversification).

Betas greater than one are more risky than the market portfolio.
Betas less than one are less risky than the market portfolio.

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Risk-averse investors will not want to own high-risk assets unless they are compensated with a low price (or
equivalently a high expected return).

Example:
Price              Expected T = 1 Cash Flow                 Beta
Asset I                            \$100                         \$110                           High
Asset J                            \$100                         \$110                           Low

Notice that both assets have the same expected return (10%), but asset I has more risk. This is not an
equilibrium relationship. Investors will not want to own asset I if they can own asset J that has the same
expected return, but lower risk. What will happen to asset I's price and expected return?

Based on this, if an asset has above-average risk, then investors will require an above-average expected
return (and vice versa).

13. The Capital Asset Pricing Model (CAPM) is one finance model that takes into account this risk/return
tradeoff. According to this model, the required return for asset a is: ra = rf + a (E(rm) - rf) = rf + a
(MRP)

Notice that the CAPM uses beta for a measure of risk.

In a CAPM setting, the required return means the expected return required by investors based on the asset's
risk (beta) and market-wide expected returns [rf and E(rm)]. Graphically, the CAPM equation gives the formula
for the Security Market Line (SML), see Figure 8-7 in the text, page 195. For example, assume that rf = 5%
and E(rM) - rf = 8.4%. The intercept of the SML is the risk free rate (5%). The slope is the market risk premium
(8.4%).

What is the required return for the following investments / projects? Also, how are these required returns
related to expected returns? Assuming the CAPM is correct:

rf = 0            rrf = 5% + 0 (8.4%) = 5%
M = 1             rM = 5% + 1 (8.4%) = 13.4%
A = 1.80887       rA = 5% + 1.80887 (8.4%) = 20.1945%
B = -1.63175      rB = 5% + -1.63175 (8.4%) = -8.7067%

Expected Return – (calculated using the initial investment)

E(rrf) = (\$105 / \$100) – 1 = 5%
E(rM) = (\$113.4 / \$100) – 1 = 13.4%
E(rA) = (\$120 / \$100) – 1 = 20%
E(rB) = (\$93.2 / \$100) – 1 = -6.8%

Notice that for financial assets:
Required return (using CAPM) = expected return (based on the initial investment)
Required return (using CAPM) = expected return (based on the PV of the future expected cash flows)

Notice that for the two projects:
Required return (using CAPM)  expected return (based on the initial investment)
Required return (using CAPM) = expected return (based on the PV of the future expected cash flows)

As discussed earlier, competition in the financial markets (plus market perfection and efficiency) causes:

The cost of the financial asset (reflected in the initial investment) to equal the present value of the future
expected cash flows (NPV = \$0)

Resulting in required returns = expected returns (based on initial investment)

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However, this should also happen for any asset that trades in an active and competitive market place
(land, fine art, coins)

14. Graphical presentation of the CAPM (the SML)

15. Expected rates of return can differ from required rates of return for:

Assets that do not trade on an active and competitive secondary market (greater probability of mis-
pricing).

Projects: the expected return (based on the initial investment) for a project can be more or less than the
required return implied by the CAPM.

16. As an estimate of the opportunity cost of capital, we will use the required return based on the CAPM. In
other words, to determine the appropriate discount rate for a project:

Calculate the project's beta.

Use the CAPM equation to determine the expected return (and required return) for financial assets with the
same amount of beta risk.

Use this expected return (required return) as the discount rate for calculating the NPV of the project (or as
the cutoff rate used with the IRR method).

`For projects A and B

NPVA = PV of expected (time 1) cash flow – initial investment = \$120 / (1 + 0.201945) -\$100 = \$99.8382 -
\$100 = -\$0.1618

NPVB = PV of expected (time 1) cash flow – initial investment = \$93.20 / (1 – 0.087067) - \$100 =
\$102.0885 - \$100 = \$2.0885

We have verified our earlier calculations of the PV and NPV!

Plot the SML and projects A and B on the same graph

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Selected quiz questions from the textbook for Chapters 7 and 8

7-1, 7-2 (see footnote 17, page 156), 7-4, 7-5, 7-8, 8-5, 8-6, 8-8

Chapter 7 and 8 Review Questions

1.   What is the opportunity cost of capital? What is typically used as the opportunity cost of capital for projects
with risk free cash flows? What is meant by a constant market risk premium? Assuming a constant market risk
premium, what happens to the expected return for large U.S. firm common stocks if there is an increase in the
risk free interest rate?

2.   Historically, what has been the average difference between the returns on large U.S. firm common stocks and
Treasury Bills? Is this a good estimate for the market risk premium? According to Brealey, Myers, and Allen
what is a good estimate for the market risk premium?

3.   Know how to calculate:

A)   Expected cash flows
B)   Expected returns
C)   Variance of returns
D)   Standard deviation of returns
E)   Covariance and correlation coefficient
F)   Beta
G)   The CAPM discount rate
H)   The PV of cash flows

4.   What is the maximum and minimum correlation coefficient? How is the maximum and minimum correlation
coefficient related to the maximum and minimum possible beta?

5.   What are "real life" examples of the risk-free security and the market portfolio?

6.   What is the beta risk of the market portfolio? What is the beta of the risk free asset? What does it mean (in
terms of risk) if the beta is (a) greater than one, (b) less than one, or (c) less than 0?

7.   How is the beta riskiness of an asset related to its market value and expected return? Be able to use the CAPM
equation to calculate required rates of return for assets based on their beta risk. (This graphs the security market
line.) What are examples of assets that fall on the SML? How do market forces ensure that these assets locate
on the SML?

8.   What does it mean if a project locates above (or below) the SML with respect to the project's NPV?

Chapter 7 and 8 Practice Problems - Plenty of practice problems are given in the notes. Here are a couple more
practice problems.

1.   Using the following annual returns over a 4 year period, what is the geometric average return? -7.17%

Year 1        50%
Year 2        10%
Year 3        -10%
Year 4        -50%

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2.   The following are stock prices for a non-dividend paying stock. Using these prices, what is the geometric
average return over this 3-year period? 13.80%

Time         Price
0           \$95
1           \$105
2           \$165
3           \$140

3.   The following are stock prices for a dividend paying stock. Using these prices and dividends, what is the
geometric average return over this 3-year period? 8.74%

Time         Stock Price   Dividend
0             \$45
1             \$50           \$0
2             \$60           \$2
3             \$54           \$2

4.   The following are stock prices for a non-dividend paying stock. Using these prices, what is the geometric
average return over this 3-year period? 8.58%

Time         Stock Price
0             \$50
1             \$55
2             \$65
3             \$64

5.   The following are stock prices and dividends for a stock. Using these prices and dividends, what is the
geometric average return over this 3-year period? 21.67%

Time         Stock Price   Dividend
0             \$50           \$0
1             \$55           \$0
2             \$70          \$5.5
3             \$80          \$3.5

6.   The following are stock prices and dividends for a stock. Using these prices and dividends, what is the
geometric average return over this 3-year period? -0.84%

Time         Stock Price   Dividend
0             \$100          \$0
1              \$80          \$0
2              \$70         \$8.0
3              \$84         \$3.5

7.   The following are stock prices and dividends for a stock. Using these prices and dividends, what is the
geometric average return over this 3-year period? 2.42%

Time         Stock Price   Dividend
0             \$100          \$0
1             \$110          \$0
2             \$121          \$4
3              \$95          \$9

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8.   The following are stock prices and dividends for a stock. Using these prices and dividends, what is the
geometric average return over this 3-year period? 13.68%

Time      Stock Price    Dividend
0          \$80            \$0
1          \$100           \$2
2          \$105           \$4
3          \$102           \$9

9.   Project X is a one-year project. The standard deviation of Project X‟s returns (based on its present value) is
20%. The standard deviation of the market‟s returns is 15%. What is the lowest possible beta for Project X?
(Hint: think about the formula for beta.) -1.33

10. Project X is a one-year project. The standard deviation of Project X‟s returns (based on its present value) is
15%. The standard deviation of the market‟s returns is 25%. What is the highest possible beta for Project X?
(Hint: think about the formula for beta.) 0.6

11. If an asset‟s standard deviation of returns is 20% and the market‟s standard deviation of returns is 25%, what is
the maximum and minimum possible beta for the asset? Maximum = 0.80, minimum = -0.80.

12. Consider the following returns (based on present values of future cash flows) for the risk-free asset, the market,
and Project X. Based on this information, Project X‟s returns have a __________ correlation with the market
returns.

State                                             1                       2                           3
Economy                                         Boom                    Normal                    Recession
Probability                                     20%                      60%                        20%
Risk-Free                                        5%                      5%                          5%
Market                                           43%                      16%                        -24%
Project X                                        20%                      5%                         -15%

A. Negative
C. Zero

13. Which of the following two assets has the highest beta? The difference between the two assets is bolded.
Standard deviations and correlations are calculated based on the present value of future cash flows.

Asset A
Expected cash flow = \$130
Initial investment = \$100
Standard deviation of Asset A‟s returns = 30%
Standard deviation of the Market‟s returns = 20%
Correlation between Asset A‟s returns and the market‟s returns = -0.5
Asset B
Expected cash flow = \$130
Initial investment = \$100
Standard deviation of Asset B‟s returns = 40%
Standard deviation of the Market‟s returns = 20%
Correlation between Asset B‟s returns and the market‟s returns = -0.5

A. Asset A has the highest beta. (Correct Answer)
B. Asset B has the highest beta.
C. Both assets have the same beta.

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14. Which of the following two assets has the highest beta? The difference between the two assets is bolded.
Standard deviations and correlations are calculated based on the present value of future cash flows.

Asset AA
Expected cash flow = \$130
Initial investment = \$100
Standard deviation of Asset AA‟s returns = 30%
Standard deviation of the Market‟s returns = 20%
Correlation between Asset AA‟s returns and the market‟s returns = 0.0
Asset BB
Expected cash flow = \$130
Initial investment = \$100
Standard deviation of Asset BB‟s returns = 40%
Standard deviation of the Market‟s returns = 20%
Correlation between Asset BB‟s returns and the market‟s returns = 0.0

A. Asset AA has the highest beta.
B. Asset BB has the highest beta.
C. Both assets have the same beta. (Correct Answer)

15. Which of the following two assets has the highest beta? The difference between the two assets is bolded.
Standard deviations and correlations are calculated based on the present value of future cash flows.

Asset A
Expected cash flow = \$130
Initial investment = \$100
Standard deviation of Asset A‟s returns = 30%
Standard deviation of the Market‟s returns = 20%
Correlation between Asset A‟s returns and the market‟s returns = 0.0

Asset B
Expected cash flow = \$130
Initial investment = \$100
Standard deviation of Asset B‟s returns = 40%
Standard deviation of the Market‟s returns = 20%
Correlation between Asset B‟s returns and the market‟s returns = 0.0

A. Asset A has the highest beta.
B. Asset B has the highest beta.
C. Both assets have the same beta. (Correct Answer)

16. Using the following information, what is the beta for project X? 0.9170

Initial investment cash flow (at time 0) for the project = -\$100
Time 1 expected cash flow for the project = \$126
Standard deviation of the project‟s returns (using the PV) = 20.862355%
Standard deviation of the market‟s returns = 21.425219%
Expected return for the market = 13.4%
Risk free interest rate = 5%
Correlation between the market‟s returns and the project‟s returns (using the PV) = 0.941731739
Correlation between the market‟s returns and the risk free asset‟s returns = 0

17. Project X is a one-year project. Using the following information, what is the PV of the project‟s time one cash
flow? \$114.57

Initial investment cash flow (at time 0) for the project = -\$100
Time 1 expected cash flow for the project = \$127

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Variance of the project‟s returns (using the PV) = 0.02635916
Standard deviation of the project‟s returns (using the PV) = 0.16235503
Variance of the market‟s returns = 0.04590400
Standard deviation of the market‟s returns = 0.21425219
Expected return for the market = 0.134
Risk free interest rate = 0.05
Covariance between the project‟s cash flows and the market‟s returns = \$3.6620
Correlation between the market‟s returns and the project‟s returns (using the PV) = 0.91887202
Correlation between the market‟s returns and the risk free asset‟s returns = 0

18. You are calculating the beta for a project‟s time one cash flow. (This is a one-year project, and the time one
cash flow is expected to be \$150.) Assume that the correlation between project returns and market returns is
equal to -0.50. Both the standard deviation of the project‟s returns and the standard deviation of the market‟s
returns are positive. (The correlation and standard deviations are correctly calculated using the present value of
the project‟s cash flows.) Holding other things constant, what is the effect of a higher standard deviation of
project returns on the beta of project‟s time one cash flow?

A. The beta will become higher.
B. The beta will become lower. (Correct answer)
C. This change will have no affect on the beta.

19. Assume that the risk-free interest rate is 5% and the market risk premium is 8.4%. What is the required rate of
return for the following financial assets?

Beta for financial asset AAA = -1.5. Answer = -7.6%
Beta for financial asset BBB = 1.5. Answer =17.6

20. What is the expected return (based on the initial investment) for the two financial assets described above if the
market is in equilibrium (and perfect and efficient)? -7.6% and 17.6% respectively.

21. Project Z requires an initial investment of \$100 and has a time one expected cash flow of \$95. The beta of
Project Z's time one cash flow is -1.5. The risk free rate is 5%, the market risk premium 8.4%. Using the
CAPM, what is the NPV of Project Z? \$2.81

22. Both Project X and Y require an initial investment of \$1000 and both have an expected one-year return of 10%.
Using this information and the other information provided below, which project has the higher NPV?

0                                   1
Project X expected cash flows                   -\$1000                               \$1100
Project Y expected cash flows                   -\$1000                               \$1100

Risk-free interest rate = 5%
Discount rates determined by the CAPM equation

Correlation between the market returns and Project X returns = -0.2
Correlation between the market returns and Project Y returns = +0.2

Annual standard deviation of returns for the market = 20%
Annual standard deviation of returns for the Project X = 40%
Annual standard deviation of returns for the Project Y = 20%

Hint: You have enough information to calculate the NPV of the two projects. But you can also answer the
problem without calculations if you correctly use some finance intuition. Solution:
The beta for Project X = [(-0.2)(0.40)] / 0.20 = -0.4
The beta for Project Y = [(+0.2)(0.20)] / 0.20 = +0.2
With a lower beta, Project X will have a lower discount rate and a higher NPV

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23. You are considering the following two mutually projects (Projects X and Y):

Expected cash flows

0                  1
Project X                 -\$100               \$130
Project Y                 -\$100               \$130

Standard deviation of project returns and correlation of project returns with market returns (both using the PV
of cash flows)

Std. Deviation        Correlation
Project X                 40%                 -0.6
Project Y                 30%                  0.4

Other information:

     The standard deviation of the market = 20%
     The risk free rate = 5%
     The market risk premium = 8.4%
     Use the CAPM to calculate the opportunity cost of capital

Using the information above, which of these two mutually exclusive projects should the firm select?

B.   Project Y
C.   Either Project X or Project Y (they both have the same positive NPV)
D.   Neither Project X or Project Y (they both have the a negative NPV)

24. A project will produce one of two possible cash flows in one year (cash flows and probabilities given below).
The beta for these cash flows is 1.2. The risk free rate is 5% and the market risk premium is 8.4%. Using the
CAPM, what is the present value of the project‟s time one cash flow? \$92.11

Probability                                             60%                                      40%
Project Cash Flow                                       \$130                                     \$70

Hint: Calculate the expected time one cash flow. Then calculate the discount rate using the CAPM. Use the
discount rate to calculate the present value.

25. A project has the following cash flows:

0                1
-\$100            \$150

Assume that the correlation between project returns and the market returns is equal to 0.50. Both the standard
deviation of the project‟s returns and the standard deviation of the market‟s returns are positive. (The
correlation and standard deviations are correctly calculated using the present value.) Holding other things
constant, what is the effect of a higher correlation on the NPV of the project? (Hint, assume the correlation is
increased to 0.60. Use the CAPM to calculate the discount rate.)

A. This will increase the NPV of the project.
B. This will decrease the NPV of the project. (Correct Answer)
C. This will have no affect on the NPV of the project.

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