Lecture 8

Risk and Return

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Learning Goals
1. Understand the meaning and fundamentals of risk,
return, and risk preferences.

2. Describe procedures for assessing and measuring the

risk of a single asset.

3. Discuss the measurement of return and standard

deviation for a portfolio and the concept of
correlation.

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2-2
Learning Goals (cont.)
4. Understand the risk and return characteristics of a
portfolio in terms of correlation and diversification, and
the impact of international assets on a portfolio.

5. Review the two types of risk and the derivation and role
of beta in measuring the relevant risk of both a security
and a portfolio.

6. Explain the capital asset pricing model (CAPM), its

relationship to the security market line (SML), and the
major forces causing shifts in the SML.

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2-3
Risk and Return Fundamentals
• In most important business decisions there are two key
financial considerations: risk and return.

• Each financial decision presents certain risk and return

characteristics, and the combination of these
characteristics can increase or decrease a firm’s share
price.

• Analysts use different methods to quantify risk

depending on whether they are looking at a single
asset or a portfolio—a collection, or group, of assets.

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Risk and Return Fundamentals (cont.)
• Risk is a measure of the uncertainty surrounding the
return that an investment will earn or, more formally,
the variability of returns associated with a given
asset.

• Return is the total gain or loss experienced on an

investment over a given period of time.
– calculated by dividing the asset’s cash distributions during
the period, plus change in value, by its beginning-of-period
investment value.

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Risk and Return Fundamentals (cont.)
• The expression for calculating the total rate of return
earned on any asset over period t, rt, is commonly
defined as

• where
rt = actual, expected, or required rate of return during period t
Ct = cash (flow) received from the asset investment in the time period t – 1 to t
Pt = price (value) of asset at time t
Pt – 1 = price (value) of asset at time t – 1

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Example
• At the beginning of the year, Apple stock traded for $90.75
per share, and Wal-Mart was valued at $55.33. During the
year, Apple paid no dividends, but Wal-Mart shareholders
received dividends of $1.09 per share. At the end of the year,
Apple stock was worth $210.73 and Wal-Mart sold for
$52.84.

• We can calculate the annual rate of return, r, for each stock.

Apple: ($0 + $210.73 – $90.75) ÷ $90.75 = 132.2%

Wal-Mart: ($1.09 + $52.84 – $55.33) ÷ $55.33 = –2.5%

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Historical Returns on Selected Investments
(1900–2011)

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Risk Preferences
• Economists use three categories to describe how
investors respond to risk.

– Risk averse is the attitude toward risk in which investors

would require an increased return as compensation for an
increase in risk.

– Risk-neutral is the attitude toward risk in which investors

choose the investment with the higher return regardless of its
risk.

– Risk-seeking is the attitude toward risk in which investors

prefer investments with greater risk even if they have lower
expected returns.
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Risk of a Single Asset: Risk Assessment

• Scenario analysis is an approach for assessing risk that

uses several possible alternative outcomes (scenarios) to
obtain a sense of the variability among returns.
– One common method involves considering pessimistic (worst),
most likely (expected), and optimistic (best) outcomes and
the returns associated with them for a given asset.

• Range is a measure of an asset’s risk, which is found by

subtracting the return associated with the pessimistic
(worst) outcome from the return associated with the
optimistic (best) outcome.

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Risk of a Single Asset: Risk Assessment (cont.)

• Norman Company wants to choose the better of two investments, A and B.

Each requires an initial outlay of $10,000 and each has a most likely annual
rate of return of 15%. Management has estimated the returns associated
with each investment. Asset A appears to be less risky than asset B. The risk
averse decision maker would prefer asset A over asset B, because A offers
the same most likely return with a lower range (risk).

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Risk of a Single Asset: Risk Assessment (cont.)

• Probability is the chance that a given outcome will

occur.

• A probability distribution is a model that relates

probabilities to the associated outcomes.

• A bar chart is the simplest type of probability

distribution; shows only a limited number of outcomes
and associated probabilities for a given event.

• A continuous probability distribution is a probability

distribution showing all the possible outcomes and
associated probabilities for a given event.
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Risk of a Single Asset: Risk Assessment (cont.)

• Norman Company’s past estimates indicate that the

probabilities of the pessimistic, most likely, and
optimistic outcomes are 25%, 50%, and 25%,
respectively. Note that the sum of these probabilities
must equal 100%; that is, they must be based on all
the alternatives considered.

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Risk of a Single Asset: Risk Assessment (cont.)

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Risk of a Single Asset: Risk Measurement

• Standard deviation (r) is the most common statistical indicator of an

asset’s risk; it measures the dispersion around the expected value.

• Expected value of a return (r) is the average return that an

investment is expected to produce over time.

where
rj = return for the jth outcome
Prt = probability of occurrence of the jth outcome
n = number of outcomes considered
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Expected Values of Returns for Assets A and B

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Risk of a Single Asset: Standard Deviation

• The expression for the standard deviation of

returns, r, is

• In general, the higher the standard deviation,

the greater the risk.

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The Calculation of the Standard Deviation
of the Returns for Assets A and B

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The Calculation of the Standard Deviation
of the Returns for Assets A and B (cont.)

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Bell-Shaped Curve

Normal Distribution

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Matter of Fact
• All Stocks Are Not Created Equal
– Stocks are riskier than bonds, but are some stocks riskier
than others?
– A recent study examined the historical returns of large
stocks and small stocks and found that the average annual
return on large stocks from 1926-2009 was 11.8%, while
small stocks earned 16.7% per year on average.
– The higher returns on small stocks came with a cost, however.
– The standard deviation of small stock returns was a
whopping 32.8%, whereas the standard deviation on large
stocks was just 20.5%.

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Risk of a Single Asset: Coefficient of Variation

• The coefficient of variation, CV, is a measure of

relative dispersion that is useful in comparing the risks of
assets with differing expected returns.

• A higher coefficient of variation means that an

investment has more volatility relative to its expected
return.

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Risk of a Single Asset: Coefficient of
Variation (cont.)

• Using the standard deviations (from Table 8.4)

and the expected returns (from Table 8.3) for
assets A and B to calculate the coefficients of
variation yields the following:

CVA = 1.41% ÷ 15% = 0.094

CVB = 5.66% ÷ 15% = 0.377

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Personal Finance Example

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Personal Finance Example (cont.)

• Assuming that the returns are equally probable:

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Risk of a Portfolio
• In real-world situations, the risk of any single investment
would not be viewed independently of other assets.

• New investments must be considered in light of their

impact on the risk and return of an investor’s portfolio of
assets.

• The financial manager’s goal is to create an efficient

portfolio, a portfolio that maximum return for a given
level of risk.

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Portfolio Return & Standard Deviation

• The return on a portfolio is a weighted average of the

returns on the individual assets from which it is formed.

• where
wj = proportion of the portfolio’s total dollar
value represented by asset j
rj = return on asset j

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Portfolio Return & Standard Deviation (cont.)

• James purchases 100 shares of Wal-Mart at a price of

$55 per share, so his total investment in Wal-Mart is
$5,500. He also buys 100 shares of Cisco Systems at $25
per share, so the total investment in Cisco stock is $2,500.

– Combining these two holdings, James’ total portfolio is worth

$8,000.

– Of the total, 68.75% is invested in Wal-Mart ($5,500/$8,000)

and 31.25% is invested in Cisco Systems ($2,500/$8,000).

– Thus, w1 = 0.6875, w2 = 0.3125, and w1 + w2 = 1.0.

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Expected Return, Expected Value, and Standard
Deviation of Returns for Portfolio XY

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Expected Return, Expected Value, and Standard
Deviation of Returns for Portfolio XY (cont.)

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Risk of a Portfolio: Correlation
• Correlation is a statistical measure of the relationship between any
two series of numbers.
– Positively correlated describes two series that move in the same direction.
– Negatively correlated describes two series that move in opposite directions.

• The correlation coefficient is a measure of the degree of correlation

between two series.
– Perfectly positively correlated describes two positively correlated series
that have a correlation coefficient of +1.
– Perfectly negatively correlated describes two negatively correlated series
that have a correlation coefficient of –1.

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Risk of a Portfolio: Correlation (cont.)

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Risk of a Portfolio: Diversification
• To reduce overall risk, it is best to diversify by
combining, or adding to the portfolio, assets that have
the lowest possible correlation.

• Combining assets that have a low correlation with each

other can reduce the overall variability of a portfolio’s
returns.

• Uncorrelated describes two series that lack any

interaction and therefore have a correlation coefficient
close to zero.

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Risk of a Portfolio: Diversification (cont.)

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Forecasted Returns, Expected Values, and Standard
Deviations for Assets X, Y, and Z and Portfolios XY and XZ

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Risk of a Portfolio: Correlation,
Diversification, Risk, and Return

• Consider two assets—Lo and Hi—with the

characteristics described in the table below:

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Possible Correlations

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The Capital Asset Pricing Model (CAPM)

• The capital asset pricing model (CAPM) is the basic theory

that links risk and return for all assets.

• The CAPM quantifies the relationship between risk and

return.

• In other words, it measures how much additional return an

investor should expect from taking a little extra risk.

• The CAPM was introduced by Jack Treynor (1961), William

Sharpe (1964), John Lintner (1965) and Jan Mossin (1966)
independently, building on the earlier work of Harry
Markowitz (1952) on diversification and modern portfolio
theory.
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The CAPM: Types of Risk
• Total risk is the combination of a security’s nondiversifiable risk
and diversifiable risk.

• Diversifiable risk is the portion of an asset’s risk that is

attributable to firm-specific, random causes; can be eliminated
through diversification. Also called unsystematic risk.

• Nondiversifiable risk is the relevant portion of an asset’s risk

attributable to market factors that affect all firms; cannot be
eliminated through diversification. Also called systematic risk.

• Because any investor can create a portfolio of assets that will

eliminate virtually all diversifiable risk, the only relevant risk is
nondiversifiable risk.
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Risk Reduction

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Risk and Return: The CAPM
• The beta coefficient (b) is a relative measure of
nondiversifiable risk. An index of the degree of
movement of an asset’s return in response to a change
in the market return.
– An asset’s historical returns are used in finding the asset’s
beta coefficient.
– The beta coefficient for the entire market equals 1.0. All
other betas are viewed in relation to this value.

• The market return is the return on the market portfolio

of all traded securities.
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 Beta Derivation

• The green line y=2x above is

steeper than the line y=x because
each value of y on the line
is twice that of x.
• The blue line y=0.5x above is
shallower than the line y=x because
each value of y on the line
is half that of x.
• The number in front of x in the
formula tells us how steep the line
will be. We call this the gradient. 42
Selected Beta Coefficients and Their
Interpretations

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Risk and Return: The CAPM (cont.)
• The beta of a portfolio can be estimated by using the
betas of the individual assets it includes.

• Letting wj represent the proportion of the portfolio’s total

dollar value represented by asset j, and letting bj equal
the beta of asset j, we can use the following equation to
find the portfolio beta, bp:

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Example: Mario Austino’s Portfolios V and W

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Risk and Return: The CAPM (cont.)
• The betas for the two portfolios, bv and bw, can be
calculated as follows:

bv = (0.10  1.65) + (0.30  1.00) + (0.20  1.30) +

(0.20  1.10) + (0.20  1.25)
= 0.165 + 0.300 +0 .260 + 0.220 + 0.250 = 1.195 ≈ 1.20

bw = (0.10  .80) + (0.10  1.00) + (0.20  .65) + (0.10  .75) +

(0.50  1.05)
= 0.080 + 0.100 + 0.130 +0 .075 + 0.525 = 0.91

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Risk and Return: The CAPM (cont.)
• Using the beta coefficient to measure nondiversifiable
risk, the capital asset pricing model (CAPM) is given in
the following equation:

rj = RF + [bj  (rm – RF)]

where
rt = required return on asset j
RF = risk-free rate of return, commonly measured by the return on a U.S.
Treasury bill
bj = beta coefficient or index of nondiversifiable risk for asset j
rm = market return; return on the market portfolio of assets
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Risk and Return: The CAPM (cont.)
• The CAPM can be divided into two parts:
1. The risk-free rate of return, (RF) which is the required
return on a risk-free asset, typically a 3-month U.S.
Treasury bill.

2. The risk premium.

– The (rm – RF) portion of the risk premium is called the market risk
premium, because it represents the premium the investor must
receive for taking the average amount of risk associated with
holding the market portfolio of assets.

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Historical Risk Premium
Treasury bills

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Example: CAPM
Benjamin Corporation, a growing computer software
developer, wishes to determine the required return
on asset Z, which has a beta of 1.5. The risk-free
rate of return is 7%; the return on the market
portfolio of assets is 11%. Substituting bZ = 1.5, RF =
7%, and rm = 11% into the CAPM yields a return of:

rZ = 7% + [1.5  (11% – 7%)] = 7% + 6% = 13%

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Security market line (SML)
• The security market line (SML) is the depiction of the
capital asset pricing model (CAPM) as a graph that
reflects the required return in the marketplace for
each level of nondiversifiable risk (beta).

• It reflects the required return in the marketplace for

each level of nondiversifiable risk (beta).

• In the graph, risk as measured by beta, b, is plotted

on the x axis, and required returns, r, are plotted on
the y axis.

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Security Market Line (cont.)

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Shifts in the SML
• The security market line (SML) is not stable over
time, and shifts in the SML can result in a
change in required return.

• The position and slope of the SML are affected

by two major forces:
– Inflationary expectation
– Risk aversion

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Inflation Shifts SML

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Risk Aversion Shifts SML

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Risk and Return: The CAPM (cont.)
• The CAPM relies on historical data which means
the betas may or may not actually reflect the
future variability of returns.
– Therefore, the required returns specified by the model
should be used only as rough approximations.

• The CAPM assumes markets are efficient.

– Although the perfect world of efficient markets
appears to be unrealistic, studies have provided
support for the existence of the expectational
relationship described by the CAPM in active markets
such as the NYSE.
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