[ Pobierz całość w formacie PDF ]
individual investors hold stocks with lottery-like characteristics. Kumar
notes that lotteries offer a low probability of a high payoff (positive
skewness), feature low prices per ticket, and have negative expected
returns. Similarly, lottery-like stocks feature positively skewed return
distributions, low prices per share, and possibly negative expected
returns.
After classifying stocks into lottery stocks and non-lottery stocks,
Kumar compares the portfolio weights individual investors assign to lot-
tery stocks to both a random assignment as well as the portfolio weights
institutional investors assign to lottery stocks. A random assignment
would allocate 0.74 percent of the portfolio to lottery stocks. On aver-
age, individual investors allocate 8.3 percent of their portfolios to lot-
tery stocks. Kumar also reports that even the most wealthy individual
investors allocate 7.7 percent of their portfolios to lottery stocks. In
contrast, institutional investors allocate 0.28 percent of their portfolios
to lottery stocks.
At the opposite end of the spectrum lie stocks that are the mirror
images of lottery stocks. These stocks feature low volatility, are less
positively skewed, have higher prices per share, and higher expected
returns. The random assignment would allocate 54 percent of a port-
folio to stocks that are diametrically opposite to lottery stocks. The
weight assigned by individual investors is 33 percent, and the weight
assigned by institutional investors is 59 percent.
64 Behavioralizing Portfolio Selection Theory
Polkovnichenko (2002) reports that the wealthiest group of investors
hold almost the same amount in direct equity as indirect equity. Simi-
larly, Kumar reports that investors who hold larger mutual fund port-
folios invest more in lottery stocks. Kumar also finds that the highest
turnover is among buyers of lottery stocks.
In line with risk-seeking behavior in the domain of losses, Kumar
finds that when economic conditions deteriorate, people increase their
allocations to lottery stocks. In particular, he finds that on a risk-
adjusted basis, those who invest in lottery stocks earn 5.9 percent a
year less than those who do not. Those who invest heavily in lottery
stocks earn 8.9 percent less than those who invest moderately in lottery
stocks. The 8.9 percent actually corresponds to 13.1 percent on a risk-
adjusted basis.
4.1.2 Theoretical Issues
Consider the traditional Markowitz portfolio selection problem when
security returns are normally distributed. Formally, let w is a vector
of portfolio weights, be a return covariance matrix, R be a vector of
expected returns, and " [0, ") measure the degree of risk tolerance.
Then the mean-variance efficient frontier is the family of solutions to the
w-minimization of wT w - RT w, for varying . Associated with each
mean-variance portfolio is a normally distributed return distribution.
A key feature of the mean-variance efficient portfolios is the two-
fund separation property. In this case, any mean-variance efficient port-
folio is a convex combination of a unique risky portfolio and the risk-free
portfolio.
4.1.2.1 Full Maximization with Behavioral Preferences
The starting point for behavioralizing portfolio theory involves the
replacement of a mean-variance objective function with a behavioral
objective function in the optimization procedure. The two main objec-
tive functions used in the literature are the prospect theory weighting
function Wiv(xi) and the SP/A function V (SP,A).
Consider how different an optimized behavioral portfolio return dis-
tribution is from a mean-variance return distribution. To describe the
4.1 Preference for Positively Skewed Returns: Full Maximization 65
Fig. 4.1 Illustration of the payoff pattern to the risk-free security and a combination of the
risk-free security and unique mean-variance risky security. The x-axis is the gross return to
the unique risky mean-variance security.
contrast, consider a series of graphs with the return to the unique risky
mean-variance portfolio on the x-axis. Figure 4.1 illustrates the returns
to two portfolios relative to the unique risky portfolio. One of the two
portfolios is risk-free, returning the risk-free rate, and the other com-
bines the risk-free security and the unique risky security.
Figure 4.2 illustrates the return to a behavioral portfolio associ-
ated with prospect theory preferences when the weighting function is
the identity function, meaning the weights are themselves probabili-
ties. Notice that the returns to loss states are very low, and in fact
are associated with a lower bound on achievable return. This bound-
ary property stems from the fact that the value function is convex in
the domain of losses. The kink at the origin of the value function, in
combination with subcertainty, can produce a flat region with neither
gain nor loss. In the domain of gains, the behavioral return is similar
to a neoclassical return, as the value function is concave in gains. See
also Barberis and Huang (2008).
Figure 4.3 illustrates the possible impact of the weighting func-
tion in the domain of gains. With overweighting of probabilities at the
extremes, the behavioral return distribution corresponds to an inverse-S
66 Behavioralizing Portfolio Selection Theory [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl akte20.pev.pl
individual investors hold stocks with lottery-like characteristics. Kumar
notes that lotteries offer a low probability of a high payoff (positive
skewness), feature low prices per ticket, and have negative expected
returns. Similarly, lottery-like stocks feature positively skewed return
distributions, low prices per share, and possibly negative expected
returns.
After classifying stocks into lottery stocks and non-lottery stocks,
Kumar compares the portfolio weights individual investors assign to lot-
tery stocks to both a random assignment as well as the portfolio weights
institutional investors assign to lottery stocks. A random assignment
would allocate 0.74 percent of the portfolio to lottery stocks. On aver-
age, individual investors allocate 8.3 percent of their portfolios to lot-
tery stocks. Kumar also reports that even the most wealthy individual
investors allocate 7.7 percent of their portfolios to lottery stocks. In
contrast, institutional investors allocate 0.28 percent of their portfolios
to lottery stocks.
At the opposite end of the spectrum lie stocks that are the mirror
images of lottery stocks. These stocks feature low volatility, are less
positively skewed, have higher prices per share, and higher expected
returns. The random assignment would allocate 54 percent of a port-
folio to stocks that are diametrically opposite to lottery stocks. The
weight assigned by individual investors is 33 percent, and the weight
assigned by institutional investors is 59 percent.
64 Behavioralizing Portfolio Selection Theory
Polkovnichenko (2002) reports that the wealthiest group of investors
hold almost the same amount in direct equity as indirect equity. Simi-
larly, Kumar reports that investors who hold larger mutual fund port-
folios invest more in lottery stocks. Kumar also finds that the highest
turnover is among buyers of lottery stocks.
In line with risk-seeking behavior in the domain of losses, Kumar
finds that when economic conditions deteriorate, people increase their
allocations to lottery stocks. In particular, he finds that on a risk-
adjusted basis, those who invest in lottery stocks earn 5.9 percent a
year less than those who do not. Those who invest heavily in lottery
stocks earn 8.9 percent less than those who invest moderately in lottery
stocks. The 8.9 percent actually corresponds to 13.1 percent on a risk-
adjusted basis.
4.1.2 Theoretical Issues
Consider the traditional Markowitz portfolio selection problem when
security returns are normally distributed. Formally, let w is a vector
of portfolio weights, be a return covariance matrix, R be a vector of
expected returns, and " [0, ") measure the degree of risk tolerance.
Then the mean-variance efficient frontier is the family of solutions to the
w-minimization of wT w - RT w, for varying . Associated with each
mean-variance portfolio is a normally distributed return distribution.
A key feature of the mean-variance efficient portfolios is the two-
fund separation property. In this case, any mean-variance efficient port-
folio is a convex combination of a unique risky portfolio and the risk-free
portfolio.
4.1.2.1 Full Maximization with Behavioral Preferences
The starting point for behavioralizing portfolio theory involves the
replacement of a mean-variance objective function with a behavioral
objective function in the optimization procedure. The two main objec-
tive functions used in the literature are the prospect theory weighting
function Wiv(xi) and the SP/A function V (SP,A).
Consider how different an optimized behavioral portfolio return dis-
tribution is from a mean-variance return distribution. To describe the
4.1 Preference for Positively Skewed Returns: Full Maximization 65
Fig. 4.1 Illustration of the payoff pattern to the risk-free security and a combination of the
risk-free security and unique mean-variance risky security. The x-axis is the gross return to
the unique risky mean-variance security.
contrast, consider a series of graphs with the return to the unique risky
mean-variance portfolio on the x-axis. Figure 4.1 illustrates the returns
to two portfolios relative to the unique risky portfolio. One of the two
portfolios is risk-free, returning the risk-free rate, and the other com-
bines the risk-free security and the unique risky security.
Figure 4.2 illustrates the return to a behavioral portfolio associ-
ated with prospect theory preferences when the weighting function is
the identity function, meaning the weights are themselves probabili-
ties. Notice that the returns to loss states are very low, and in fact
are associated with a lower bound on achievable return. This bound-
ary property stems from the fact that the value function is convex in
the domain of losses. The kink at the origin of the value function, in
combination with subcertainty, can produce a flat region with neither
gain nor loss. In the domain of gains, the behavioral return is similar
to a neoclassical return, as the value function is concave in gains. See
also Barberis and Huang (2008).
Figure 4.3 illustrates the possible impact of the weighting func-
tion in the domain of gains. With overweighting of probabilities at the
extremes, the behavioral return distribution corresponds to an inverse-S
66 Behavioralizing Portfolio Selection Theory [ Pobierz całość w formacie PDF ]