Risk, Return and
The Capital Asset Pricing Model
(CAPM) Part 1
Dr. Humnath Panta
Lecture Outline
A.
B.
C.
D.
E.
F.
Return on Investment
Return and Risk: Individual Securities
Return and Risk: Portfolio
Types of Risk
The Efficient Portfolio
Portfolio Diversification and Risk
Learning Objectives
After watching this lecture video, you should be able to:
• Know how to calculate the return on an investment
• Explain and calculate the expected return and risk for an
individual security and a portfolio
• Explain and compute co-variance and correlation
• Explain the difference between stand-alone risk and risk
in a portfolio context.
• Understand the effects of diversification on portfolio risk
• Discuss the difference between diversifiable risk and
market risk, and explain how each type of risk affects
well-diversified investors.
• Understand the tradeoff between risk and return
Return on Investment
• The rate of return on a stock at any time t is a random
variable:
r 
D1  P1  P0
P  P0
D1

 1
P0
P0
P0
P0 is the price at the beginning of period
P1 is the price at the end of period (random variable)
D1 is the dividend at the end of period (random variable)
is the rate of return earned for investing in the stock over the period, or holding period return.
•
r
You bought a stock for $35 one-year ago and today you received dividends of
$1.25. The stock is now trading for $40. What is your percentage return?
• Dividend yield = 1.25 / 35 = 3.57%
• Capital gains yield = (40 – 35) / 35 = 14.29%
• Total percentage return = 3.57 + 14.29 = 17.86%
Return on Investment Cont.
•
•
The holding period return (HPR) is the total return from holding an investment for a specific time (its
holding period). This is different from an annualized return, which measures the return adjusted for a
one-year period, which may be more or less than the actual holding period.
Holding period return captures both the change in your investment's value
over time and any periodic benefits you receive from it.
Holding Period Return (HPR) 
D1  P1  P0 D1 P1  P0


P0
P0
P0
1
T
Annualized HPR  (1  HPR)  1
•
Holding period return is a very basic way to measure how much return you have obtained on a particular investment.
This calculation is on a per-dollar-invested basis, rather than a time basis, which makes it difficult to compare returns
on different investments with different time frames. When making comparisons such as this, the annualized
calculation shown above should be used.
Return on Investment Cont.
• Suppose your investment provides the following returns
over a four-year period:
Holding period return  (1  r1 )  (1  r2 )   (1  rn )  1
Year Return
1
10%
2
-5%
3
20%
4
15%
Your holding period return
 (1  r1 )  (1  r2 )  (1  r3 )  (1  r4 )  1
 (1.10)  (0.95)  (1.20)  (1.15)  1
 0.4421  44.21%
Return on Investment Cont.
•
Arithmetic average – is the sum of a series of numbers divided by the
count of that series of number.
n
Arithmetic average 
•
r
i
i
n

10%  5%  20%  10%
 10%
4
Geometric average – average compound return per period over multiple
periods. The geometric average will be less than the arithmetic average
unless all the returns are equal.
1
n
Geometric average  [(1  r1 )  (1  r2 )  (1  r3 )  (1  r4 )]  1
1
4
 [(1.10)  (0.95)  (1.20)  (1.15)]  1
 0.4421.25  .0958  9.58%
Return on Investment Cont.
•
•
The main benefit to using the geometric mean is that the actual
amounts invested do not need to be known; the calculation
focuses entirely on the return figures themselves and presents an
"apples-to-apples" comparison when looking at two investment
options. You can measure an actual average return over multiple time
period.
Which is better?
•
•
The arithmetic average is overly optimistic for long horizons.
The geometric average is overly pessimistic for short horizons.
Return on Investment Cont.
Historical rates of return (1926-2007) of five important types of financial instruments in the United States.
Series
Average
Annual Return
Standard
Deviation
Large Company Stocks
12.3%
20.0%
Small Company Stocks
17.1
32.6
Long-Term Corporate Bonds
6.2
8.4
Long-Term Government Bonds
5.8
9.2
U.S. Treasury Bills
3.8
3.1
Inflation
3.1
4.2
Source: © Stocks, Bonds, Bills, and Inflation 2008 Yearbook™, Ibbotson Associates, Inc.,
Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights
reserved.
Distribution
– 90%
0%
+ 90%
Return on Investment Cont.
The trade-off between risk and return
18%
Small-Company Stocks
Annual Return Average
16%
14%
Large-Company Stocks
12%
10%
8%
6%
T-Bonds
4%
T-Bills
2%
0%
5%
10%
15%
20%
25%
Annual Return Standard Deviation
30%
35%
Return on Investment Cont.
•
Risk aversion: assumes investors dislike risk and require higher rates
of return to encourage them to hold riskier securities.
•
The Risk Premium is the added return (over and above the risk-free
rate) resulting from bearing risk or there should be a premium (higher
expected return) for bearing risk.
•
The difference between the expected return on a security or portfolio
and the "riskless rate of interest" is often termed its risk premium.
•
risk premium serves as compensation for investors to hold riskier
securities.
Return on Investment Cont.
•
An equivalent definition of a risk premium is: the expected excess
return on a security or portfolio, where excess return is the difference
between an actual return and that of a riskless security.
•
Average long-run excess return using stock market data:
•
•
•
The average excess return from large company common stocks for the
period 1926 through 2007 was: 8.5% = 12.3% – 3.8%
The average excess return from small company common stocks for the
period 1926 through 2007 was: 13.3% = 17.1% – 3.8%
The average excess return from long-term corporate bonds for the period
1926 through 2007 was: 2.4% = 6.2% – 3.8%
Risk and Return
.
Risk and Return Cont.
•
•
•
•
•
•
Expected return and risk
• Individual security and portfolio – expected return, variance and
standard deviation
The future is uncertain.
Investors do not know with certainty whether the economy will be growing
rapidly or be in recession.
Investors do not know what rate of return their investments will yield.
Therefore, they base their decisions on their expectations concerning the
future.
The expected rate of return on a stock represents the mean of a
probability distribution of possible future returns on the stock.
Risk and Return Cont.
• Statistics reminder:
• A random variable is a variable that can take
several possible values. We will denote random
variables by the superscript bar (e.g., r ).
•
For simplicity, we consider only discrete random variables
that may take on finite number of values.
• A probability distribution is a list of all possible
outcomes and the probability that each will occur.
Pr(r  rj )  p j , j  1,..., n
Risk and Return Cont.
The table below provides a probability distribution for the returns on stocks A and B
State
1
2
3
•
•
•
Probability
20%
50%
30%
Return On
Stock A
26%
18%
10%
Return On
Stock B
-4%
12%
20%
The state represents the state of the economy one period in the future i.e. 1 could
represent a recession and state 2 a normal and 3 a boom economy.
The probability reflects how likely it is that the state will occur. The sum of the probabilities
must equal 100%.
The last two columns present the returns or outcomes for stocks A and B that will occur in
each of the four states.
Risk and Return Cont.
•
•
Mean: the expected or forecasted value of a random variable. We will denote
the mean of a random variable by the superscript bar (e.g., r ).
Given a probability distribution of returns, the expected return can be
calculated using the following equation:
E ( r )  r  p1r1  p2 r2  ......  pn rn
E (r )  r 
Where:
n
pr
i 1
i
i
• E (r ) = the expected return on the stock
• n = the number of states
• pi = the probability of state i
• ri = the return on the stock in state i.
Risk and Return Cont.
• The expected return for stock A would be calculated as
follows:
E ( r )  r  p1r1  p2 r2  ......  pn rn
E (r )  r 
n
pr
i 1
i i
rA  (0.2  0.26)  (0.5  0.18)  (0.3  0.1)  0.172  17.2%
• Now you try calculating the expected return for stock B!
Risk and Return Cont.
•
•
Did you get 11.2% expected return for stock B? If so, you are
correct.
If not, here is how to get the correct answer:
E ( r )  r  p1r1  p2 r2  ......  pn rn
E (r )  r 
n
pr
i 1
i i
rB  (0.2  0.04)  (0.5  0.12)  (0.3  0.2)  0.112  11.2%
•
•
So we see that Stock A offers a higher expected return than Stock
B.
However, that is only part of the story; we haven't considered risk.
Risk and Return Cont.
•
•
•
•
•
What is Risk?
Risk reflects the chance that the actual return on an investment may be different
than the expected return - Risk is an uncertainty of an outcome
• As defined in Webster dictionary risk is a hazard, a peril, exposure to loss or
injury -Thus, risk refers to the chance that some unfavorable event will occur
Two types of investment risk
– Stand-alone risk- Individual stock risk
– Portfolio risk
Investment risk is related to the probability of earning a low or negative actual
return.
The greater the chance of lower than expected, or negative returns, the riskier the
investment.
Risk and Return Cont.
•
•
•
How to measure risk?
There is no universally agreed-upon definition of risk.
The measures of risk that we discuss are variance and standard
deviation.
•
•
•
•
•
The standard deviation is the standard statistical measure of the spread of
a sample, and it will be the measure we use most of this time.
Its interpretation is facilitated by a discussion of the normal distribution.
Standard deviation (σi) measures total, or stand-alone, risk.
The larger σi is, the lower the probability that actual returns will be
close to expected returns.
Larger σi is associated with a wider probability distribution of returns.
Risk and Return Cont.
•
A listing of all possible outcomes, and the probability of each occurrence. Can be shown graphically A
large enough sample drawn from a normal distribution looks like a bell-shaped curve.
The 20.0% standard deviation we
found for large stock returns from
1926 through 2007 can now be
interpreted in the following way: if
stock returns are approximately
normally distributed, the probability
that a yearly return will fall within 20.0
percent of the mean of 12.3% will be
approximately 2/3.
Probability
– 3s
– 47.7%
– 2s
– 27.7%
– 1s
– 7.7%
0
12.3%
68.26%
95.44%
99.74%
+ 1s
32.3%
+ 2s
52.3%
+ 3s
72.3%
Return on
large company
common stocks
Risk and Return Cont.
•
•
Dispersion of Returns (risk) can be measured using Variance and Standard
Deviation. Variance is a mathematical expectation of the average squared
deviation from the mean.
Variance of Return Formula:
E (Var )  s
2
E (Var )  s
2
Where:
 p1 ( r1  r ) 2  p2 ( r2  r ) 2  ......  pn ( rn  r ) 2

n
p
i 1
i
( ri  r ) 2
n = the number of states
pi = the probability of state i
ri = the return on the stock in state i
= the expected return on the stock
•
•
r
Standard Deviation of Return: s  s 2
The larger the variance (or standard deviation), the lower the probability that
actual returns will be close to the expected return
Risk and Return Cont.
•
•
•
We use variance and standard deviation of the distribution of returns
as measure of risk.
We will once again use a probability distribution in our calculations.
The distribution used earlier is provided again for ease of use.
State
1
2
3
•
•
Probability
20%
50%
30%
E[rA] = 17.2%
E[rB] = 11.2%
Return On
Stock A
26%
18%
10%
Return On
Stock B
-4%
12%
20%
Risk and Return Cont.
•
The variance and standard deviation for stock A is calculated as follows:
E (Var )  s 2  p1 (r1  r ) 2  p2 (r2  r ) 2  ......  pn (rn  r ) 2
E (Var )  s
n
2
  pi (ri  r ) 2
i 1
s  (0.26  0.172) 2  0.2  (0.18  0.172) 2  0.5
2
A
 (0.10  0.172) 2  0.3  0.003136
sA 
•
•
0.003136  0.056  5.6%
Now you try the variance and standard deviation for stock B!
If you got .006976 and 8.35% respectively you are correct.
Risk and Return Cont.
•
•
If you didn’t get the correct answer, here is how to get it:
The variance and standard deviation for stock B is calculated as follows:
E (Var )  s 2  p1 ( r1  r ) 2  p2 ( r2  r ) 2  ......  pn ( rn  r ) 2
E (Var )  s
2

n
 p (r
i 1
i
i
 r )2
s B2  ( 0.04  0.112) 2  0.2  (0.12  0.112) 2  0.5
 (0.2  0.112) 2  0.3  0.006976
sB 
•
•
0.006976  0.0835  8.35%
Stock A offers a higher expected return than Stock B, it is also less is riskier
since its variance and standard deviation are smaller than Stock B's.
This, however, is still only part of the picture because most investors choose to
hold securities as part of a diversified portfolio.
•
•
•
•
•
•
Risk and Return Cont.
Measuring stand – alone risk: the coefficient of variation - a standardized measure of
dispersion about the expected value, that shows the risk per unit of return.
If a choice to be made between two investments that have same expected returns but
different risk (std. deviation), most people would choose the one with lower risk.
In our example just illustrated, it is easy to make choice. A has higher expected return
and lower risk than security B. However, in real life you may have to make choice
between assets with different level risk and return.
To facilitate your choice we have another measure of risk: coefficient of variation (CV)
s
which can be defined as: Coefficien t of variation  CV 
r
s .056
s .0835
CV using our example, CVA 

 .33 x and CVB 

 .7457 x
r
.172
r
.112
The coefficient of variation shows the risk per unit of return, and it provides a more meaningful basis
for comparison than standard deviation when the expected returns on two alternatives are different.
Since the CV captures the effects of both risk and return, it is a better measure than the standard
deviation when evaluating stand-alone risk
Risk and Return Cont.
•
Practice Question: find expected return and risk for stock X and Y
State
Probability
Return On
Return On
Stock X
Stock Y
1
20%
5%
50%
2
30%
10%
30%
3
30%
15%
10%
4
20%
20%
-10%
Did you get E[RX]= 12.5% and E[RY] = 20%? If so, you are correct.
Similarly, if you get sX = 5.12% and sY = 20.49% you are correct.
• Although Stock Y offers a higher expected return than Stock X, it also is riskier since its
variance and standard deviation are greater than Stock X's.
• This, however, is still only part of the picture because most investors choose to hold
securities as part of a diversified portfolio.
Risk and Return Cont.
•
Most investors do not hold stocks in isolation. Instead, they choose to hold a
portfolio of several stocks.
• A portfolio is a group of assets. A portion of an individual stock's risk can be
eliminated, i.e., diversified away by creating a portfolio.
• A security’s portfolio weight is the percentage of the portfolio’s total value
invested in that particular asset – equal to 1.
• Portfolio weights can be positive (a “long” position) or negative (a “short”
position).
• Now, how would you measure expected return and risk on your portfolio?
•
The expected return and risk of a portfolio is also measured by the portfolio
expected return and standard deviation, just as with individual assets
Risk and Return Cont.
•
•
Assume we create a portfolio that is 50% invested in Stock A and 50%
invested in Stock B.
Let’s find portfolio rerun and risk using our pervious example
State
•
1
2
3
Probability
20%
50%
30%
Return On
Stock A
26%
18%
10%
From our previous calculations, we know that:
•
•
•
•
•
•
the expected return on Stock A is 17.2%
the expected return on Stock B is 11.2%
the variance on Stock A is .003136
the variance on Stock B is .006976
the standard deviation on Stock A is 5.6%
the standard deviation on Stock B is 8.35%
Return On
Stock B
-4%
12%
20%
Risk and Return Cont.

The expected return of a portfolio is simply the weighted-average of the
expected returns of assets in the portfolio.
rp  w1r1  w2 r2  w3 r3  .....  wn rn
rp 
n
wr
i 1
i
i
rp  .5 x.172  .5 x.112  .142  14.20%
•
The rate of return on the portfolio is a weighted average of the returns on
the stocks in the portfolio.
Risk and Return Cont.
•
We use standard deviation to measure portfolio risk. Standard deviation is
a little more tricky and requires that a new probability distribution for the
portfolio returns be constructed.
•
The Variance of the return on a portfolio Does Not Equal the Weighted
Average of the variances of the returns on individual stocks!
•
The variance of the rate of return on the two risky assets portfolio is:
σ P2  w A2 σ A2  w B2 σ B2  2w A w B σ A σ B ρ A , B
where A,B is the correlation coefficient between the returns on the two assets.
where A,B is the correlation coefficient between the returns on the two assets.
Risk and Return Cont.
•
The variance/standard deviation of a portfolio reflects not only the
variance/standard deviation of the stocks that make up the portfolio but
also how the returns on the stocks in the portfolio move together or related.
•
Two measures of how the returns on a pair of stocks vary together are the
covariance and the correlation coefficient.
•
Covariance is a measure that combines the variance of a stock’s returns with
the tendency of those returns to move up or down at the same time other stocks
move up or down.
•
Since it is difficult to interpret the magnitude of the covariance terms, a related
statistic, the correlation coefficient, is often used to measure the degree of comovement between two variables. The correlation coefficient simply
standardizes the covariance.
Risk and Return Cont.
• Covariance measures how two random variables are
related. In this case, how are the returns of stock X and
stock Y related? Do they move together or not?
n
s A, B  Cov[ A, B]   Pi  [(rAi  rA )  (rBi  rB )]
i 1
s AB  0.3  [(0.10  0.172)  (0.20  0.112)]
 0.5  [(0.18  0.172)  (0.12  0.112)]
 0.2  [(0.26  0.172)  (0.04  0.112)]
 0.00190  0.00003  0.00268  0.00454
Risk and Return Cont.
• Correlation is a “standardized” measure of how two
random variables are related. Correlation is always
between -1 and +1. If two variables move together, they
are positively correlated. If they move opposite of each
other they are negatively correlated.
 A, B  Corr ( A, B) 
 AB
Cov( A, B)
s As B
 0.00454

 0.972
(0.056)(0.084)
Risk and Return Cont.
•
The variance of the rate of return on the two risky assets
portfolio is:
σ 2  w2 σ 2  w2 σ 2  2w w σ σ ρ
P
A A
B B
A B A B A, B
 .52  .056 2  .52  .084 2  2  .5  .5  .056  .084  .972
 .00078  .00174  .00227
 .00026
σ
•
P

σ2 
P
.00026  .0161  1.61%
Now you compute variance and standard deviation if correlation is +1.
Did you get variance =.00487 and Std.dev =6.98%? You got right
answers.
Risk and Return Cont.
•
How to find minimum variance portfolio?
σ 2  Cov ( A, B)
08352  (0.00454)
B
Minimum Variance Portfolio 

 .60
2
2
2
2
σ  σ  2Cov ( A, B) .056  0835  2 x(0.00454)
A B
•
If you put 60% in stock A your portfolio will have the lowest risk
σ 2  w2 σ 2  w2 σ 2  2w w σ σ ρ
P
A A
B B
A B A B A, B
 .6 2  .056 2  .4 2  .084 2  2  .6  .4  .056  .084  .972
 .00113  .00113  .002195
 .00006
σ
P

σ2 
P
.00006  .008  .8%
Risk and Return Cont.
• Return and risk using historical data
n
r
r  r2  ....  rn
ri  1

n
n
n
Variance(s
2
) 
 (r
i 1
i
 r )2
n 1
n
COV ( R A , RB ) 
i
i 1
 (r
i 1
i
 rA ) ( ri  rB )
n 1
Types of Risk
Individual assets have two types of risk:
1.
Market Risk - Economy-wide sources of risk that affect a large number
of assets (e.g., the overall stock market). Also called “non-diversifiable
risk” and/or “systematic risk.”
2.
Unique Risk - Risk that affects at most a small number of assets. Also
called “diversifiable risk,” “unsystematic risk,” and/or “idiosyncratic risk.”
•
•
•
•
Total risk = systematic risk + unsystematic risk
The standard deviation of returns is a measure of total risk.
For well-diversified portfolios, unsystematic risk is very small.
Consequently, the total risk for a diversified portfolio is essentially equivalent
to the systematic risk.
Types of Risk Cont.
•
•
•
•
Systematic risk is the risk associated with the market
Risk factors that affect a large number of assets
Also known as non-diversifiable risk or market risk
Includes such things as changes in GDP, war, inflation, interest rates,
etc.
• An extra-ordinarily hot summer creates a spike in energy prices nationwide.
• Companies are discovered to be systematically abusing GAAP rules.
• A global recession occurs.
=> Systematic risk cannot be diversified away.
Types of Risk Cont.
•
•
•
•
•
Unsystematic risk is the risk associated with individual asset which is
diversifiable which is also known as a unique or firm specific or idiosyncratic risk
Risk factors that affect a limited number of assets
Includes such things as labor strikes, part shortages, etc.
The risk that can be eliminated by combining assets into a portfolio
If we hold only one asset, or assets in the same industry, then we are exposing
ourselves to risk that we could diversify away.
•
•
A rogue currency trader racks up $750M in losses before being discovered.
•
A company’s main manufacturing facility burns to the ground.
A company is sued because the tires on a particular make of SUV tend to de-tread,
resulting in fatalities.
The Efficient Portfolio
According to the “Modern Portfolio Theory-MPT” a risk averse investor can construct
a portfolio to optimize or maximize expected return for the given level of risk. The
MPT was introduced by Harry Markowitz (1952) in his paper “Portfolio Selection”
published in the Journal of Finance.
According to the MPT, it’s possible to construct an “efficient frontier” of optimal
portfolios offering the maximum possible expected return for a given level of risk.
The efficient portfolio (optimal portfolio) is a portfolio that provides the greatest
expected return for a given level of risk, or equivalently, the lowest risk for a given
expected return.
There are four basic steps involved in portfolio constructions:
A. Security selection
B. Asset allocation
C. Portfolio Optimization and D. Performance measurement
The Efficient Portfolio Cont.
Stock A
Stock B
The Efficient Set for Two Assets
17.2%
11.2%
5.6%
8.4%
-0.972
Port Std
Dev
5.60%
4.92%
4.23%
3.55%
2.88%
2.23%
1.60%
1.05%
0.79%
1.05%
1.60%
2.22%
2.88%
3.55%
4.23%
4.91%
5.60%
6.29%
6.97%
7.66%
8.35%
Port Exp
Ret
17.2%
16.9%
16.6%
16.3%
16.0%
15.7%
15.4%
15.1%
14.8%
14.5%
14.2%
13.9%
13.6%
13.3%
13.0%
12.7%
12.4%
12.1%
11.8%
11.5%
11.2%
Portfolio Expected Return
20.0%
Wt
Stock A
100.0%
95.0%
90.0%
85.0%
80.0%
75.0%
70.0%
65.0%
60.0%
55.0%
50.0%
45.0%
40.0%
35.0%
30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
100% STOCK A
15.0%
100% STOCK B
10.0%
5.0%
0.0%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
Portfolio Std Deviation
We can consider other portfolio weights
besides 50% in Stock A and 50% in Stock B…
return
The Efficient Portfolio Cont.
•
•
•
The Efficient Set with
Various Correlations
 = -1.0
 = 1.0
 = 0.2
s
Relationship depends on correlation coefficient: -1.0 < r < +1.0
If r = +1.0, no risk reduction is possible
If r = –1.0, complete risk reduction is possible
The Efficient Portfolio Cont.
return
The Efficient Set for
Many Securities
Individual Assets
sP
Consider a world with many risky assets; we can still identify
the opportunity set of risk-return combinations of various
portfolios.
return
The Efficient Portfolio Cont.
The Efficient Set for
Many Securities
minimum
variance
portfolio
Individual Assets
sP
The section of the opportunity set above the minimum
variance portfolio is the efficient frontier.
return
The Efficient Portfolio Cont.
rf
The Efficient Set with
Riskless Borrowing and
Lending
Balanced
fund
100%
bonds
100%
stocks
In addition to stocks
and bonds, consider
a world that also has
risk-free securities
like T-bills.
s
• Now investors can allocate their money across the T-bills and a
balanced mutual fund. Note: CML = Capital Market Line, which is
the relationship between total risk and return for efficient portfolios.
The Efficient Portfolio Cont.
return
Market Equilibrium
M
rf
sP
With the capital allocation line identified, all investors choose a
point along the line—some combination of the risk-free asset and
the market portfolio M. In a world with homogeneous
expectations, M is the same for all investors.
return
The Efficient Portfolio Cont.
rf
Balanced
fund
100%
bonds
100%
stocks
Market Equilibrium
s
Where the investor chooses along the Capital Market Line
depends on his/her risk tolerance. The main point is that all
investors have the same CML.
Portfolio Diversification and Risk
•
Diversification - reduce risk without an equivalent reduction in expected returns by
spreading the portfolio across many asset classes.
•
The reduction in risk is achieved by offsetting the worse-than-expected returns from one
asset by the better-than-expected returns from another.
•
•
•
•
Diversification is easier to achieve with less correlated assets (negative correlation is the best).
Diversification is easier to achieve with a larger number of assets.
Anticipated and Unanticipated Components of News. So the return on a security consists of
two parts:
• First, the expected returns
• Second, the unexpected or risky returns
A way to write the return on a stock in the coming month is: R  r  U  r  m  
Portfolio Diversification and Risk Cont.
Assume asset 1 and 2 has expected return of 18% and
12%, and volatility of 30% and 20%, respectively. What
are the expected return and volatility of a portfolio that
invests 50% in each asset, if the correlation between
the two assets is
(i) 0
(ii) –1
(iii) 1?
Portfolio Diversification and Risk Cont.
(i)
w 1  w 2  .5, 12  0  s12  12s1s 2  0
rp  (.5)(.18)  (.5)(.12)  .15
s p  (.5) 2 (.3) 2  (.5) 2 (.2) 2  0  .0325  s p  18%
2
(ii) expected return is still .15 and
12  1  s 12  12s 1s 2  (1)(.3)(.2)  .06
s p 2  (.5) 2 (.3) 2  (.5) 2 (.2) 2  2(.5)(.5)(.06)  .0025  s p  5%
(iii) expected return is still .15 and
12  1  s 12  12s 1s 2  (1)(.3)(.2)  .06
s p 2  (.5) 2 (.3) 2  (.5) 2 (.2) 2  2(.5)(.5)(.06)  .0625  s p  25%
Portfolio Diversification and Risk Cont.
• Portfolio return and risk with N assets
• Portfolio Mean
rp
 E[ ~
rp ] 
n
w r
i 1
i
• Portfolio Variance
N
N
i 1
j 1
s p2    wi w js ij
i
Portfolio Diversification and Risk Cont.
Calculating Portfolio Variance (3 Assets)
σ P2  w A2 σ A2  w B2 σ B2  2w A w B σ A σ B ρ A , B
• Step 1: Set up the following table
Portfolio
=
w1 Stock1
+
=
w1 Stock 1 + w2 Stock 2
+
w3 Stock 3
w12s12
w1w2s12
w1w3s13
w2w1s21
w22s22
w2w3s23
w3w1s31
w3w2s32
w32s32
w2 Stock2
+
w3 Stock3
•
Step 2: Adding up the boxes to calculate the portfolio variance.
Portfolio Diversification and Risk Cont.
Stock
1
2
3
4
5
6
N
1
2
3
4
5 6
Stock
N
Calculating Portfolio
Variance (N Assets)
• Just add up all the
boxes!!
• The shaded boxes
(on the diagonal)
contain variance
terms.
• All the rest contain
covariance terms
Portfolio Diversification and Risk Cont.
Components of portfolio variance: For a portfolio with N
securities, there are N variance terms (the diagonal) and N 2-N covariance
terms (everything else). The covariance terms dominate as N gets large:
N=2:
N=10:
N=100:
N=1000:
2 variance terms; 2 covariance terms (50%)
10 variance terms; 90 covariance terms (10%)
100 variance terms; 9900 covariance terms (1%)
1000 variance terms; 999,000 covariance terms (.1%)
=> The variance of very large portfolios comes almost entirely from the
covariance among the securities in the portfolio. The variance of individual
securities does not matter much!
Portfolio Diversification and Risk Cont.
• Each Asset’s contribution to total portfolio variance
equals to the sum of terms in row i
• An asset’s contribution to the risk of a well-diversified
portfolio is determined by its average covariance with
other assets, not by its own variance.
• An asset’s average covariance with other assets is
called the systematic (or market) risk.
Portfolio Diversification and Risk Cont.
s
In a large portfolio the variance terms are
effectively diversified away, but the covariance
terms are not.
Diversifiable Risk;
Nonsystematic Risk; Firm
Specific Risk; Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk; Market Risk
n
Portfolio Diversification and Risk Cont.