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CHAPTER SEVEN
Risk, Return, and Portfolio
Theory
J.D. Han
Learning Objectives
1.
2.
3.
Define the term “market risk” and explain how it
is related to expected return of a single financial
asset.
Measure the risk in portfolio of multiple assets
Identify the main aims of diversification, and
explain the principle benefits of international
versus domestic diversification.
Risk?
Default Risk; Credit Risk – all kinds of financial
instruments (bonds, loans, stocks)
 Inflation Risk – only bonds
 Market Risk – all kinds of financial instruments
“the chance that the actual outcome from an
investment will differ from the expected outcome”

How to measure Market Risk of
Individual Asset?
1. Variability= Deviation from its own
Average Rate of Return
“Mean Variance Approach”
2. Co-movement with the Market Index =
Relative Variability of Rate of Return to
the Market Index
“Capital Market Pricing Model”
Rate of Return

Recall that the rate of return is calculated by:
CFt  ( PE  PB ) CFt  PC
TR 

PB
PB
Where:
CFt = cash flows during the measurement period t
PE = Final price at the end of period t or sale price
PB = purchase price of the asset
PC = change in price during the period
Calculating Mean

Expected return – is the average of all possible
return outcomes, where each outcome is weighted
by the probability of its occurrence
E ( R) 
m
R
i 1
i
pri
Where:
E( R)= the expected return on a security
Ri = the ith possible return
pri = the probability of the ith return Ri
m = the # of possible returns
Variance- SD: Calculating Risk

Variance or standard deviation is typically used to
calculate the total risk associated with the
expected return
m
Variance =
   Ri  E ( R ) pri
2
2
i 1
Standard deviation =
  
2
•
Numerical Examples:
How to calculate the variance and the standard
deviation?
1) Data of r over 3 years: 4%, 6%, and 8%
E (r ) = (4 + 6 + 8)/3 = 6%
2  1/3(4 6)2  1/3(66)2  1/3(86)2  8/3
2) Data r: 3 times of 4, 5 times of 6, twice of 8
Now Mixing Multiple Assets in a
Portfolio

So far, we have examined Single Asset Case.

How about the return and risk of Multiple
Assets in an Investment Portfolio?
Portfolio’s Expected Return


The expected return is calculated as a weighted
average of the individual securities’ expected returns
The combination portfolio must add up to be 100
percent
n
E ( R p )   wi E ( Ri )
i 1
Where:
E(Rp) = the expected return on the portfolio
wi = the portfolio weight for the ith security
E(Ri) = the expected return on a single asset, or the ith
security
n = the # of different securities in the portfolio
Portfolio Risk

Portfolio risk is less than the weighted average of
the risk of the individual securities in a portfolio
of risky securities unless their correlation
coefficient is equal to one or they are perfectly
positively correlated

p

n
 w
i 1
i
i
,where p is risk of portfolio and i is risk of a single
asset i
Portfolio Risk p

Two factors must be considered in developing an
equation that will measure the risk of a portfolio
through variance and standard deviation
1. Weighted individual security risks 1., 2 …
2. Weighted co-movements between securities’
returns 1 2, 1 3,, 2 3 ….
- measured by the correlations between the
securities’ returns weighted again by the
percentage of investable funds placed in each
security
Covariance and Correlation Coefficient
 AB   RA,i  E ( RA )RB ,i  E ( RB )pri
m
i 1

 
, or 

AB
AB
A
AB 
 
AB
A


 = covariance between securities A and B
RA,I = one estimated possible return on security A
E(RA) =mean value; most likely result
m = the # of likely outcomes for a security for the period
pri = the probability of attaining a given return RA,i
Portfolio Risk
Correlation coefficient – is a statistical
measure of the relative co-movement
between the return on securities A and B
 The relative measure is bound between +1.0
and –1.0 with
AB = +1.0 = perfect positive correlation
AB = - 0.0 = zero correlation
AB = -1.0 = perfect negative correlation

*The expected rate of return and
standard deviation of the portfolio
should be:Two Asset Portfolio Case

Asset A ~(ErA, A) and Asset B ~ (ErB, B)
Suppose we mix A and B at ratio of w1 to w2 for a portfolio

P ~ (ErP, p)

Return:

Risk:
Erp = w1 ErA, + w2 ErB
 p  w1  A  w 2  B  2 w1 w2  AB  A  B
2
2
2
2
*AB is the correlation coefficient of rA and rB.
Diversification


Through diversification non-systematic risk can be
eliminated
Systematic risk cannot be eliminated
Total risk = non-systematic risk + systematic risk

Diversification can be performed:
1. Domestically
2. Internationally
Summary
1.
2.
3.
Uncertainly can be quantified in terms of probabilities,
and risk is commonly associated with the variance or
standard deviation of probability distributions.
When securities are combined, the combined risk of the
resulting portfolio depends not only on the individual
risk of the underlying securities but also on the statistical
correlation that exists between the individual returns.
Correlation coefficients of +1, 0, and –1 indicate perfect
positive correlation, statistical independence, and perfect
negative correlation respectively.
Summary
4.
5.
6.
Portfolio diversification reduces risk, and most investors
hold diversified portfolios.
Diversification enables the reduction of risk through the
elimination of company-specific or unique risk. Because
the returns of most securities are related to the general
state of the economy, they are positively correlated with
each other.
International diversification offers additional benefits in
terms of risk reduction.