P.V. VISWANATH
FOR A FIRST COURSE IN INVESTMENTS
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 How do we characterize individuals’ preferences for
taking risk?
 How do we use utility functions over asset returns?
 How do we evaluate investors’ risk preferences?
 How do we allocate capital across risky and risk-free
portfolios?
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 Individuals have differing preferences for risk.
Therefore, what is a desirable set of investments for one
individual may not be desirable for another. A utility
function is a convenient way of describing individual’s
preferences for risk.
 In economics, a utility function is described over
quantities of goods. That works when we know the
consumption of a good with precision.
 However, if we are to describe preferences over what are
called lotteries, i.e. uncertain quantities of goods, then
we use a concept invented by von Neumann, called
Expected Utility, which can be used provided individuals’
preferences satisfy some basic conditions, such as
transitivity, i.e. if an individual prefers a choice A to B
and B to C, then he must prefer A to C.
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 In order not to have to worry about how individuals
choose combinations of goods, often in finance, we
use what is called a derived utility function.
 Even though individuals actually consume goods,
nevertheless because wealth can be used to acquire
goods, it is possible to define a utility function for
wealth, conditional on the prices of goods.
 Similarly, conditional on initial wealth, it is possible
to define a utility function on rates of return.
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 There are many such utility functions. Different utility functions have
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different properties. One such property is called constant relative risk
aversion, which establishes a certain stability of risk preferences.
This utility function is written as U(r) = E(r) – 0.005As2(r), where
returns are specified in percent per annum.
The constant risk aversion parameter is called A; higher values of A
represent a preference for less risk and r is measured in percentages.
The utility score can also be used as a certainty equivalent rate of
return.
That is, if the investor is indifferent between two portfolios, one with
an uncertain return of r0 and another with a certain return of r1, then
U(r0) = E(r0) – 0.005As2(r0) = U(r1) = E(r1) = r1
Thus, if a portfolio has an expected return of 15% and a std. dev. of 5%
and A=3 for a given individual, then he would be indifferent between
that portfolio and another portfolio with a certain return of 15(0.005)3(5)(5) or 14.625%
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 The utility function U is a good way of describing investors’
preferences under two conditions – one that return
distributions are normal (i.e. that only mean and standard
deviation of returns are necessary to describe the distribution)
– or – that investor preferences are quadratic.
 While some individuals might have approximately quadratic
utility functions, it still represents a strong restriction on
individual preferences.
 We know now (particularly post the 2008 mortgage crisis)
that return distributions probably are not normal. Hence we
must be careful in using our utility function.
 However, it still represents a very powerful way of describing
risk preferences and we can derive a lot of useful results with
that assumption.
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 Investment Advisors might find it useful to associate investors
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with utility functions.
If we decide that a mean-variance utility function is
appropriate, then what we need for this is to figure out the
investor’s A-number.
One way to do this is by using questionnaires, asking investors
which one of a pair of investment choices they would prefer.
This can then be used to construct indifference curves and
finally these indifference curves can be used to figure out the
investor’s A value.
Alternatively, we could observe individuals’ decisions when
confronted with risk.
 Finally, we could observe how much people are willing to
pay to avoid risk.
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 We start with a top-down allocation, first.
 Assuming we know the mean and variance of the risky
part of the investor’s portfolio, we compute the
proportion of the portfolio that would go into the risky
part and the riskless part, respectively.
 Let y=portion allocated to the risky portfolio, P
 (1-y)=portion to be invested in risk-free asset, F.
 Assume the following data:
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
 In the next slide, we see the investment opportunity set
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If borrowing
rates were
higher than
lending rates, as
if often the case,
the CAL is no
longer a straight
line.
 The investor must choose one optimal
portfolio, C, from the set of feasible choices, for
different values of y:
 Expected return of the complete portfolio:
 Variance:
E (rc )  rf  y  E (rP )  rf 
s ys
2
C
2
2
P
Note that there is an optimal value of y, which can be seen
graphically in the next slide.
We can also examine
the optimal portfolio in
the (E(r), s) plane.
The indifference curves
are straight lines in the
(E(r), s2) plane, but
convex in the (E(r), s)
plane.
We first view the shape
of the utility function in
this plane and then
superimpose it upon
the investment
opportunity set, viz. the
CAL.
 The investor must choose one optimal
portfolio, C, from the set of feasible choices
 Expected return of the complete portfolio:
E (rc )  rf  y  E (rP )  rf 
 Variance:
s C2  y 2s P2
 Substituting
these expressions into the utility
function and maximizing, we find
 In
𝐸 𝑟𝑃 − 𝑟𝑓
𝑦∗=
0.01𝐴𝜎𝑃 2
our example, this works out to
(15-7)/0.01(4*222)= 41.32%, for A=4
 How would we pick the risky portfolio?
 One option is to use a passive strategy. The passive
strategy avoids any direct or indirect security
analysis
 Supply and demand forces may make such a strategy
a reasonable choice for many investors
 A natural candidate for a passively held risky asset
would be a well-diversified portfolio of common stocks
such as the S&P 500.
 The capital market line (CML) is the capital allocation
line (CAL) formed from 1-month T-bills and a broad
index of common stocks (e.g. the S&P 500).
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 If we assume that the average investor has 85% in the
broad portfolio of risky assets in the marketplace
(based on data from 2009), and noting that the
average return on the S&P from 1926 to 2009 is about
7.9% with an annual std deviation of 20.8%, we can
reverse solve the equation for the optimal portfolio to
get an estimate of the risk aversion coefficient A for the
average investor.
 This turns out give us a value of A=2.15. Many studies
have also suggested values for A between 2 and 4.