Matakuliah
Tahun
: F0892 - Analisis Kuantitatif
: 2009
EXPECTED RETURN PORTFOLIO
Pertemuan 8
7.1 What is a “Portfolio?”
• A portfolio is a list of securities indicating how
much is (or will be) invested in each one;
• The question is: which portfolio should be
selected?
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7.2 The Return on a Portfolio
• You have funds W0 to invest in a portfolio at the
beginning of the period;
• By definition we have W1 = W0(1 + rp), where rp is the
return on your portfolio;
• But at time 0, you do not know what W1 will be for any
given portfolio because you do not know what rp will be
for that portfolio;
• At time 1 you know what is W1;
• Hence, rp is a random variable ex ante but is known ex
post.
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7.3 How Should You Evaluate Alternative Portfolios?
• What is the “good thing” you want from your portfolio? Return.
You want “return to be high;”
• What is the “bad thing” you want to avoid in your your portfolio?
Risk. You want “risk to be low;”
• Hence, how should you evaluate alternative portfolios? By
examining their expected returns and their standard deviations.
• Observe, therefore, that you have two conflicting objectives:
(1) maximize the expected return on your portfolio; and
(2) minimize the standard deviation of your portfolio.
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• Harry Markowitz and his paper “Portfolio Selection” [Journal of
Finance (1952)] are generally viewed as the origin of modern
portfolio theory. He was awarded the 1990 Nobel Prize with
William Sharpe and Merton Miller for their pioneering work in
the theory of financial economics.
7.4 Calculating the Expected Return of a Portfolio
• The expected return on a portfolio is the weighted average of
the expected returns on the securities included in that portfolio;
• What is the weight of a security in a portfolio? It is the
percentage of wealth invested in that security;
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• The formula to compute the expected return on a
portfolio of N securities is
N
r p  X i r i
,
i1
where
X i is the weight of security i; and
ri
is the expected return on security i.
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Example
On January 25, 1999, you bought the following stocks:
Table 7.1
Quantity
Price
Stock
200
171
Microsoft
500
185 5/8
IBM
1,000
62 1/2 Coca-Cola
You expect to earn the following (annual) returns on the
stocks you bought: 20% on Microsoft, 12% on IBM, and
15% on Coca-Cola. What is the expected return on
your portfolio?
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First, we need to compute the weight of each security.
The total wealth invested is equal to
W0 =($171*200) + ($185.625*500)+($62.50*1,000)
=$34,200+$92,812.50+$62,500
=$189,512.50.
Hence,
X1=18.05%[=$34,200/$189,512.50];
X2=48.97%[= $92,812.50 /$189,512.50];
X3=32.98%[= $62,500 /$189,512.50].
[Note that X3= 1-X1-X2]
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EXPECTED RETURN OF PORTFOLIO
Most investors do not hold stocks in isolation.
Instead, they choose to hold a portfolio of several
stocks. When this is the case, a portion of an
individual stock's risk can be eliminated, i.e.,
diversified away. This principle is presented on the
Diversification page. First, the computation of the
expected return, variance, and standard deviation of
a portfolio must be illustrated.
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Once again, we will be using the probability distribution for the returns on stocks A and B.
Once again, we will be using the probability distribution for the returns on stocks A
and B.
State
1
2
3
3
Return on
Probability Stock A
20%
5%
30%
10%
30%
15%
20%
20%
Return on
Stock B
50%
30%
10%
-10%
From the Expected Return and Measures of Risk pages we know that
the expected return on Stock A is 12.5%, the expected return on Stock B
is 20%, the variance on Stock A is .00263, the variance on Stock B is
.04200, the standard deviation on Stock S is 5.12%, and the standard
deviation on Stock B is 20.49%.
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Portfolio Expected Return
The Expected Return on a Portfolio is computed as the weighted
average of the expected returns on the stocks which comprise the portfolio.
The weights reflect the proportion of the portfolio invested in the stocks.
This can be expressed as follows:
where
E[Rp] = the expected return on the portfolio,
N = the number of stocks in the portfolio,
wi = the proportion of the portfolio invested in stock i, and
E[Ri] = the expected return on stock i.
For a portfolio consisting of two assets, the above equation can be
expressed as
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For a portfolio consisting of two assets, the above
equation can be expressed as
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Expected Return on a Portfolio of Stocks A and B
Note: E[RA] = 12.5% and E[RB] = 20%
Portfolio consisting of 50% Stock A and 50% Stock B
Portfolio consisting of 75% Stock A and 25% Stock B
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