University of California, Los Angeles
Department of Statistics
Statistics C183/C283
Instructor: Nicolas Christou
The efficient frontier with short sales allowed and risk free lending and borrowing
Suppose riskless lending and borrowing exists. Let
R
f
be the return of the riskless asset (savings
account, treasury bills, government bonds, etc.). We will examine the geometric pattern of combi
nations of the riskless asset and a risky portfolio.
Consider the portfolio possibilities curve below constructed using different combinations of two
stocks. The efficient frontier is the concave part of the curve that begins with the minimum risk
portfolio and extends to the maximum expected return portfolio.
Risk (standard deviation)
Expected return
Suppose now that the investor wants to invest a portion of her wealth on portfolio
A
(a point on
the efficient frontier) and her remaining wealth on the riskless asset. Let
x
be the portion invested
in portfolio
A
and 1

x
the portion invested in the riskeless asset.
This combination is a new
portfolio. It has the following expected return and standard deviation:
1
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 Spring '11
 Nicolas
 Statistics, Standard Deviation, EFFICIENT FRONTIER, σp

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