Download Returns and Risk

Returns and Risk

Realized Returns—expect (after the fact) return

Expected Return- estimated return

Investment Decision—trade off between risk (expected) and return (expected)
Trade-Off Between Risk and Return


Required Return = Risk free rate + Risk premium
Required Return = Cost of Capital to Firm
 Two components of Required Return:
 Risk-free rate
 Risk premium
First step in the investment decision process
 Knowledge of long-term return (actual) and risk (actual)
The past is a good guidepost but
Don’t use it as a hitching post
 Historical returns play a large part in estimating future unknown returns
Return
 Objective of investor: Maximize Wealth
 Micro-basis—maximize expected returns given a level of risk
 Components of Return
 Yield—income component
 Dividend Yield
 Current Yield
 Capital gain (loss) [ Price Change]
 Change on price of a security over some period
 Total Return for period = Yield over period + Price Change over period
5-1
Risk
 the chance that the actual outcome from an investment will differ from the expected
outcome
 Greater the variability----the greater the risk
Sources of Risk:
1. Interest Rate Risk
 security prices move inversely to interest rates
 bonds more directly affected
2. Market Risk
 variability in returns resulting from fluctuations in the overall market
 common stock affected more than bonds
 Market risk components:
 Recessions, wars, structural change in the economy, and changes in consumer
preferences
3. Inflation Risk
 purchasing power risk
 with unexpected inflation, the real return (inflation-adjusted) involves risk even when
the nominal return is certain (fixed)
 Fisher equation
4. Business Risk
 risk associated with a particular industry or environment
 risks associated with the business cycle
 factors that affect the business cycle
 monetary policy, changes in technology, changes in supply of raw materials
5. Financial Risk
 debt financing
 financial leverage
6. Liquidity Risk
5-2
7. Exchange Rate Risk (Currency Risk)
 variability in returns on securities caused by currency fluctuations
8. Country Risk (Political Risk)
Types of Risk
 Systematic (Market)
 Nonsystematic (Company)
 Total Risk= Systematic Risk + Nonsystematic Risk
Holding Period Return
(Total Return)
TR 
(Any Cash pymts received over the period)  (Price change over the period)
Price at beginning of the period
TR 
Any Cash pymts received over the period
Price change over period

Price at beginning of the period
Pr ice at beginning of the period
TR  Period' s Yield  Period' s Price Change
5-3
Return Relative

Total return for an investment for a given time period stated on the basis of
RR  1  TR
Return Relative 
CFt  PE
PB
Cumulative Wealth Index


Measures the cumulative effect of returns over time, typically on the basis pf a $1 invested
Measures the level rather than the changes in wealth
CWI n  WI 0 [(1  TR 1 )(1  TR 2 )...(1  TR n )]
CWI n  Yield X Price Change
TR n 
5-4
CWI n
1
CWI n 1
International Returns

Convert cashflows into US dollars and therefore experience currency or exchange rate risk
Total Return in Domestic Terms  [ RR *

Ending Value of foreign currency
] 1
Beginning Value of foreign currency
Example: Buy foreign company stock @ 175.86 pesos (1 peso=.29) and sell after 2 years @ 250 pesos (
1 peso=.27) No dividends
Geometric Mean = 19.23%
Geometric Mean in $ = [1.1923 * (.27/.29)] –1 =11.01%
TR= ( 250/175.86 ) - 1 = 42.2%
TR in $ = [ 1.422 * (.27/.29)] –1 = 32.4%
Annual Returns

Arithmetic Mean
 Measures the central tendency of a distribution
X

X
n
Geometric Mean (compounded return)
G  [(1  TR 1 )(1  TR 2 )...(1  TR n )]1 / n  1
G  (CWI n )1 / n  1
where WI 0  $1
5-5
 measures the annual compound rate of growth over time
 uniform rate at which the $ actually grows over time (per period)
 measures the realized change in wealth over multiple periods
 reflects the variability of returns
 always < arithmetic mean
When Should Use Arithmetic vs Geometric


Arithmetic Mean—better measure of average (typical) performance over a single period
Geometric Mean—better measure of change in wealth over time
 Better measure of typical performance over multiple periods
Example:
Stock Starting Price
A
10
B
10
Period 1
20
8
Period 2 Arith Mean Geo Mean
10
25%
0%
12
15%
9.54%
Inflation Adjusted Returns
Inflation Adjusted Holding Period 
1  TR
1
1  IF
Inflation Adjusted Compound Yield (annual yield) 
Inflation Adjusted Cumulative Wealth Index 
5-6
1 G
1
1  IF
CWI n
Inflation Index
Risk Measurements
Risk

possibility that actual cash flows (returns) will be less than forecasted cash flows
(returns).
 Since we have uncertainty, we have returns that are variable and each potential
outcome can be assigned a probability.
Return
 Expected value of return on a firm is the weighted average of all possible return
outcomes where each outcome is weighted by its respective probability of
occurrence.
n
R̂ =  R j P j
j=1
i.e.:
2 companies
CPI - utility company
IFP - farm equipment company
Rate of Return Anticipated
State of
Economy
Probability
CPI
IFP
Recession
.2
10%
-4%
Normal
.6
18%
18%
Boom
.2
26%
40%
1.0


RCPI = .1(.2)+.18(.6)+.26(.2) = .18

RIFP = -.04(.2)+.18(.6)+.40(.2) = .18
5-7
Have same expected return. Does this imply same risk?
 If you have investments that are approximately equal in size and the
expected returns are the same then the standard deviation is an
appropriate measure of risk. It is an absolute measure of variability. It
measures the dispersion of a variable about its mean.
n
=
 (R j - R̂ ) 2 P j
j=1
2
2
2
CPI = (.10 - .18 ) .2 + (.18 - .18 ) .6 + (.26 - .18 ) .2 =
25.6 = 5.06%
2
2
2
IFP = (-.04 - .18 ) .2 + (.18 - .18 ) .6 + (.4 - .18 ) .2
193.6 = 13.91%
Standard Deviation for Sample:
n
s =
 (R j - R̂ ) 2
j=1
n 1
5-8
Probability of Occurrence
 If assume a normal probability distribution, can use standard deviation to
assess the probability of an event occurring

Number of standard deviations, Z, that a particular value of R is from
expected value R can be computed:
Z=

R-R

Note: If the calculated standard deviation is negative, you are
finding the probability to the left of the point in interest
i.e. IFP what is probability of a return of 0%?
Z=
1.
0 - .18
= - 1.29
.1391
Calculate Z
Return 0% is 1.29 standard deviation below the mean.
2.
CPI
1.
Nearest standard deviation in Table: Area to left .0985.
Probability below a return of 0% = 9.85%
Calculate Z
Z=
0 - .18
= - 3.5573
.0506
Return of 0% is 3.5573 standard deviations below the mean.
2.
Nearest standard deviation in Table is 3.5 - Area to left is .00023
 Probability below 0% = .023%
5-9

If comparing investments with different expected returns, the coefficient of
variation provides a better measure of risk.

Coefficient of variation considers relative variation & is defined as ratio of σ to
the expected return, R
V=

R̂
i.e. 2 assets
T
S
R
25%
10%
σ
20%
18%
S may appear to be less risky
VT =
VS =
T
=
R̂ T
S
R̂ S
=
.2
= .8
.25
.18
= 1.8
.10
Therefore S is more risky than T
5-10
EXPECTED RETURNS ON PORTFOLIO
E(Rp) = weighted average of the expected returns from individual securities in
the portfolio.
n
R̂ p =  W i R̂ i
i =1
where  Wi = 1 0  Wi  1
RISK PREMIUMS


Risk premiums—additional return investors expect to receive
Equity Risk Premium
 Stock return –risk free rate
(1  TR cs )
1
(1  rf )

Bond Horizon Premium

Long term government bond rate - risk free rate
(1  TR GB )
1
(1  rf )

Bond Default Premium

Long-term Corporate rate-Long-term government rate
(1  TR Corp )
(1  TR GB )
5-11
1
PORTFOLIO RISK

Portfolio risk - risk associated with collections of assets or securities.
 Generally, a security by itself has more risk then when held in a portfolio.
 Diversification --- reduce variability of return

Expected standard deviation of portfolio
n
p   W j  j
j =1

Can reduce variability of return of a portfolio by adding negatively
correlated securities. The correlation between two variables is a relative
statistical measure of the degree to which two variables move together.
The correlation coefficient is the extent to which high (low) values of one
variable is associated with high (low) values of another. (Markowitz)
Perfectly positively correlated +1
Perfectly negatively correlated -1
Two variables unrelated
0



o
Correlation coefficient =
ij =
Covariance i, j
ij
=

Standard deviation  i  j
COV (R i , R j ) 

p ij
( R i  R i )( R j  R j )
2
 (R i  R i )
 (R j  R j )
2
1
( R i  R i )( R j  R j )
n 1
Usually, returns from most investments, a firm or individual considers are
positively correlated with other investments held by the firm.
5-12

Most returns from stock are positively correlated.
5-13

However, whenever individual securities in a portfolio are less than
perfectly correlated, diversification can reduce the portfolio's risk below
the weighted average of the total risk of individual sectors.
2 security portfolio
p = W A A + W B B + 2 W A W B P AB A B
2
2
2
2
PAB: correlation coefficient
N security portfolio
n
n
n
p =  w  +   w i w j ij
i
1 j1
i 1
2
i
2
i
i j

Portfolio Standard Deviation
n
p =
 (R jp - R̂ p ) 2 P j
j=1
where
Rjp is the actual return on the portfolio during state j
^
R p is the expected return on the portfolio
5-14
SECURITY MARKET LINE

Required return of any risky asset is determined by the prevailing r f and risk
premium.
SML : E(K j) = R f +
(E(K m) - R f )
m
2
Cov( K j , K m)
K j = R f + j

Graphically the SML indicates the current required rate of return in the
markets for a given amount of systematic risk.

Systematic risk of a security is a function of total risk of security as measured
by σj of security jth returns and the σm of returns from market portfolio and the
correlation of jth security returns with all other securities in the market p(KjKm).
 β=
measure of volatility of a security's returns relative to the returns of a
market index.
j =


Covariance j, m P jm  j m
=
2
Variance m
m
measure of systematic risk
Beta is the slope of the regression line between the periodic returns on
the market and periodic returns for sec j.
K j = a j + bjRm + ej


Security j's Characteristic Line:
bj: Historical Beta
5-15


Market
β=1
β j > 1 more risky than market
β j < 1 less risky than market
Portfolio Beta:
n
p =  W j B j
j=1
SML Devlopment

Risk premium for security j, Θj, =
difference between the required return, Kj, and the risk-free rate, rj.
Θj=Kj-rf


Market risk premium:


Θm=Km- rf

if B j = 1 then Θ j = Θ m
Building on this
B j = 2 then
Θ j = 2Θm
= B jΘ m
= B j (K m - r f)
and if the return on a security is the rf + risk premium then
Kj=rf+Θj
= r f + B j (K m - r f)
5-16

Graphically SML

Slope of SML (Market risk premium):
(Km - rf)/ (1-0)
5-17
Which securities are good investments given a level of risk?
i.e.
Returns:
rf = .05
km = .25
Bj = .5
Kx = .05 + .5(.25-.05) = .15
if E(Rx) > 15% then undervalued - good buy
Prices:
D1 = 5
P1 = 50
Pe =
Kj = .15
If actual market price
Pm > 47.83 Over Valued (sell)
Pm < 47.83 Under Valued (buy)
5-18
5 + 50
= 47.83
(1 + .15)



The SML can shift in a parallel fashion due to changes in expected
inflation.
The SML can change its slope due to a change in consumer
confidence.
The SML can shift and change its slope simultaneously.
CAPITAL ASSET MARKET LINE
 Recall SML:
E(K j) = R f +
(E(K m) - R f )
m
2
Cov: (ρj m σj σm)

Capital Market Line considers Total Risk

If ρ(Kj, Km) = 1 then SML reduces to CML
j =
 jm  j m
m
2
if ρj m = 1 then
j =
i
m
5-19
Cov( K j , K m)
and
CML
E(K j) = R f +
(E(K m) - R f )
m
E(K j) = R f +  j (E(Km) - R f )
5-20
j