Financial Management
4. Risk and Return. CAPM-model.
Liliya N. Zhilina, World Economy and Inrernational Relations
Department, Vladivostok State University of Economic and
Services (VSUES).
liliya.zhilina@vvsu.ru
4. Risk and Return.
CAPM-model.
Investment Return
 Investment returns measure the financial
results of an investment.
 Returns may be historical or prospective
(anticipated).
 Returns can be expressed in:

Money terms:
$ Received - $ Invested
$1,100
$1,000 = $100

Percentage terms:
($ Received – Invested) / $ Invested
$100 / $1,000 = 0.10 = 10%
• Cost (Price) of Debt Capital:
Interest Rate
• Cost of Equity:
Required
Dividend
Capital
Return = Return + Gain
Factors affect the cost of capital
• Production opportunities (returns, which a
producer expects to earn, makes the upper
limit of the payment for capital)
• Market conditions, especially interest rates and
tax rates.
• Time preferences for consumption / savings
• The firm’s investment policy. Firms with riskier
business-projects generally have a higher cost
of capital.
• The firm’s capital structure and dividend policy.
• Expected inflation (the higher risk - the
expensive capital)
Factors Influence Interest Rates
k = k* + IP + DRP + LP + MRP
= kRF +DRP + LP + MRP
Here:
k = Required rate of return on a debt security.
k* = Real risk-free rate.
kRF = Nominal risk-free rate.
IP = Inflation premium.
DRP = Default risk premium.
LP = Liquidity premium.
MRP = Maturity risk premium.
Premiums Added to k* for
Different Types of Debt
• Sort-Term Treasury (governmental):
only IP for ST inflation
• Long-Term Treasury (governmental):
IP for LT inflation, MRP
• Sort-Term corporate: ST IP, DRP, LP
• Long-Term corporate: IP, DRP, MRP, LP
More Factors Influence Interest
Rates
• The Central Bank Policy (money
supply – commercial banks crediting).
• Deficit (shortfall) / surplus of the
budget.
• Trade balance.
• Interest rates in other countries.
Risks Investing Abroad
Country Risk: depends on economy conditions,
political and social factors in a country.
Exchange Rate Risk: if investments (securities)
have been nominated in a national currency the
investments cost depends on exchange rate.
Exchange rate change depends on:
•
•
changing in a relative inflation;
increase in a country risk leads to drop of
national currency rate.
Risk
• Can be defined as the chance
that some unfavorable event will
occur.
What is investment risk?
Typically, investment returns are not
known with certainty.
Investment risk pertains to the
probability of earning a return less
than that expected.
The greater the chance of a return far
below the expected return, the greater
the risk.
How can be considered the risk of
investing in some asset?
1) on a stand-alone basis (each asset by itself);
2) In a portfolio context, where the investment is
combined with other assets and its risk is
reduced through diversification.
Stand-alone risks: compare risks of purchasing
а) short-term governmental bills with an
expected return of 5 %; and
б) shares of just being organized company that
plans to start drilling in a new oil field with an
expected return from +1000 % to –100 %.
Probability distribution of
returns of the stocks H and L
Return on the stocks,
k
H
L
Demand
Probability,
P
High
0,3
100%
20%
Average
0,4
15%
15%
Low
0,3
-70%
10%
Total
1,0
Average return (k):
gains matrix
Stock Н
Stock L
Demand
P
Return
Total
Return
Total
High
0,3
100%
0,3*100=30%
20%
0,3*20=60%
Average
0,4
15%
0,4*15=6%
15%
0,4*15=6%
Low
0,3
-70%
0,3*-70=-21%
10%
0,3*10=3%
Total
1,0
k=15%
k=15%
Probability distribution
Stock L
Stock H
-20
0
15
50
Rate of
return (%)
 Which stock is riskier? Why?
• Standard deviation measures the
stand-alone risk of an investment.
• The larger the standard deviation,
the higher the probability that
returns will be far below the
expected return.
• Coefficient of variation is an
alternative measure of stand-alone
risk.
Measures of Risk Formulas:
Expected rate of return, k^ :
^
k
The standard deviation, s :
s =
Variance =
s=
 k
n
i =1

s
2
2
i
 k Pi .
Coefficient of variation (risk per unit of
^
return). CV = s / k
Risks and Returns of the stocks H & L
sН= ((100 - 15)20.3 + (15-15)20.4 + (-70 -15)20.3)1/2
= 65,84%.
sL= ((20 - 15)20.3 + (15-15)20.4 + (10-15)20.3)1/2
=3,87%.
Expected
Stock
rate of return
H
L
15%
15%
Standard
deviation
Coefficient
of variation
65,84%
3,87%
4,39
0,26
The stock Н is 17 times risky than the stock L
Probability Ranges for a Normal
(Gaussian ) Distribution
68,26%
95,46%
99,74%
-3σ
-2σ
-1σ
k
+1σ
+2σ
+3σ
Probabilities that returns would be in the intervals
±1 SD, ±2 SD, ±3SD
Risk-averse investors
• The average investor is risk-averse
(dislikes risk), which means that he
or she must be compensated for
holding risky assets.
• At the market, where most investors
are risk-averse, riskier assets have
higher required returns than less
risky assets.
Portfolio risk
• An asset is stored in a stock portfolio could be
considered less risky than stored alone.
• The portfolio (diversification) provides average
return but much lower risk. The key here is
negative correlation.
• Correlation is the tendency of two variables to
move together.
• Expected return of a portfolio is the weighted
average of expected returns of separate assets:
n
kp = w1k1 + w2k2 + …. wnkn = ∑wiki
i=1
Two-Stock Portfolios
• Two stocks can be combined to form a riskless
portfolio if r = -1.0.
• Risk is not reduced at all if the two stocks have r =
+1.0.
• In general, stocks have r
but not eliminated.
0.65, so risk is lowered
• Investors typically hold many stocks.
• What happens when?
• r = 0 suggests that the two variables are not related
to one another; that is, they are independent.
r – correlation coefficient
What would happen to the
risk of an average few-stock
portfolio as more randomly
selected stocks were added?
sp would decrease because the added
stocks would not be perfectly correlated,
but kp would remain relatively constant.
Amount of a Portfolio Stock and its Risk
sp (%)
Company Specific
(Diversifiable) Risk
35
Stand-Alone Risk, sp
20,1
Market Risk
0
10
20
30
40
The market portfolio is a portfolio
consisting of all stocks. Its SD ≈ 20,1%
2 000+
# Stocks in Portfolio
Stand-alone =
risk
Market +
risk
Diversifiable
risk
Market risk is that part of a security’s
stand-alone risk that cannot be
eliminated by diversification.
Firm-specific, or diversifiable, risk is
that part of a security’s stand-alone risk
that can be eliminated by diversification.
•
As more stocks are added, each new stock has a
smaller risk-reducing impact on the portfolio.
• sp falls very slowly after about 40 stocks are
included. The lowest limit for sp is about
20,1% =
sM.
• Rational investors will
minimize risk by holding
portfolios. They bear only market risk, so prices and
returns reflect this lower risk.
• By forming well-diversified portfolios, investors can
eliminate about half the riskiness of owning a single
stock.
How should the risk of an individual stock
be measured?
• Сapital Аsset Рricing Мodel (CAPM). By William
Sharp and othrers.
• CAPM is the equilibrium model, which sets up
dependence between risk and required return for
assets (shares) in the well diversified portfolio.
• The assumption of CAPM. Risk has one influence
factor: a stock volatility relative to the market.
• The relevant risk of an individual stock, which is
called its beta coefficient, is defined under the
CAPM as the amount of risk that the stock
contributes to the market (well-diversified) portfolio.
How are betas calculated?
• Run a regression with returns on the stock in
question plotted on the Y axis and returns on the
market portfolio plotted on the X axis.
• The stock’s beta coefficient (bi or β) is equal to
the slope angle v of the regression line, which
measures relative volatility:
βi = bi = (σi / σM)* riM
• here: riM – the correlation between the i’th stock’s
expected return and the expected return on the market;
σi, σM – the standard deviations of the i’th stock’s
expected return and the expected return on the market
Use the historical stock returns to
calculate the beta for the stock of Y (MD).
Years
1
2
3
4
5
6
7
8
9
10
Market
25,7%
13,0%
-8,2%
25,0%
22,5%
15,7%
40,0%
12,0%
-5,8%
-9,1%
Y
40,0%
-35,0%
-25,0%
45,0%
10,0%
15,0%
46,0%
-25,0%
25,0%
-24,0%
Calculating Beta for Y (MicroDrive)
40%
KY
20%
0%
-40%
-20%
0%
20%
kM
40%
-20%
ky = 1,25 kM - 0,0918
-40%
R² = 0,412
How is beta calculated?
• The regression line, and hence beta, can be
found using a spreadsheet program.
In this example, b = 1.25.
• Analysts typically use four or five years’ of
monthly returns to establish the regression line.
Some use 52 weeks of weekly returns.
• The R2 measures the percent of a stock’s
variance that is explained by the market. The
typical R2 is:
– 0.3 for an individual stock
– over 0.9 for a well diversified portfolio
How is beta interpreted?
• If b = 1.0, stock has average (of the broader
market) risk.
• If b > 1.0, stock is riskier than average.
• If b < 1.0, stock is less risky than average.
• Most stocks have betas in the range of 0.5 to
1.5.
• Theoretically, it is possible for a stock to have
a negative beta: when the stock’s return would
tend to rise whenever the returns on other
stocks fall.
• Value Line (valueline.com)
Expected Return versus Market Risk
Expected
Security
return
Risk, b
HighTech.
17.4%
1.29
Market
15.0
1.00
Utilities
13.8
0.68
T-bills
8.0
0.00
Collections
1.7
13.4
 Which of the alternatives is best?
Use the SML to calculate each
alternative’s required return
• The Security Market Line (SML) is part
of the Capital Asset Pricing Model
(CAPM).
• SML: ki = kRF + (RPM)bi
• kRF – risk free return, kM – market return.
^
• Assume kRF = 8%; kM = kM = 15%.
• RPM = (kM - kRF) = risk premium
•
= 15% - 8% = 7%.
Required Rates of Return
kHT =
=
kM =
kUt =
kT-bill =
kColl =
8.0% + (7%)(1.29)
8.0% + 9.0%
8.0% + (7%)(1.00)
8.0% + (7%)(0.68)
8.0% + (7%)(0.00)
8.0% + (7%)(-0.86)
= 17.0%.
= 15.0%.
= 12.8%.
= 8.0%.
= 2.0%.
Expected versus Required
Returns
^
HT
k(exp.)
17.4%
k (req.)
17.0% Undervalued
Market
15.0
15.0
Fairly valued
Ut
13.8
12.8
Undervalued
T-bills
8.0
8.0
Fairly valued
Coll
1.7
2.0
Overvalued
The slope of the SML equation is (kM - kRF), the market
risk premium.
ki (%) SML: ki = kRF + (RPM) bi
ki = 8% + (7%) bi
.
HT
kM = 15
kRF = 8
.
. .
. T-bills
Market
Ut
Coll.
-1
0
1
2
Risk, bi
SML and Investment Alternatives
Limitation of the CAPM
• The CAPM is a single factor model.
• Richard Roll showed that it is virtually impossible to
prove investors behave in accordance with CAPM
theory.
• CAPM/SML concepts are based on expectations,
yet betas are calculated using historical data. A
company’s historical data may not reflect investors’
expectations about future riskiness.
• Betas of individual securities are not good
estimators of future risk.
• Betas of portfolios of 10 or more randomly selected
stocks are reasonably stable.
• Past portfolio betas are good estimates of future
portfolio volatility.