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Unformatted text preview: Outlines of Lecture 1 Objectives
1. Define riskaverse investors and determine
personal required risk premium on a risky asset
2. Discuss in which conditions the meanvariance analysis is consistent with the expected
utility maximization References
UtilityFunction.pdf; Quant_Review.pdf; Chapter
6 & Appendix A&B , BKM
Hsiulang Chen Portfolio Analysis 1 Introduction Asset Allocation or “Portfolio Theory” Meanvariance portfolio analysis was developed
by Harry Markowitz in the early 1960’s. This
work won the Nobel prize in economics in 1990. This work is widely regarded as the first step in
modern finance. An Asset Allocation Problem:
– How much of your wealth should you invest
in each security?
Hsiulang Chen Portfolio Analysis 2 Introduction Before MeanVariance Analysis A typical investment advice:
– If you are young, you should be putting money into a
couple of good growth stocks, maybe even into a few
small stocks. Now is the time to take risks.
– If you are close to retirement, you should be putting all
of your money into bonds and safe stocks, and
nothing into the risky stocks. Don’t takes risks with
your portfolio at this stage in your life. Though this advice was intuitively compelling, it
was, of course, all wrong.
Hsiulang Chen Portfolio Analysis 3 Introduction After MeanVariance Analysis We now know that the optimal portfolio of risky
assets is exactly the same for everyone, no
matter what their tolerance for risk. Investors should control risk not by reallocating
among risky assets, but through the split
between risky and riskfree assets. The portfolio of risky assets that investors
should hold should contain a large number of
assets. It should be welldiversified.
Hsiulang Chen Portfolio Analysis 4 Introduction Hsiulang Chen Portfolio Analysis 5 Introduction Hsiulang Chen Portfolio Analysis 6 Introduction Hsiulang Chen Portfolio Analysis 7 Introduction Hsiulang Chen Portfolio Analysis 8 Why should an individual utility function be
U(r)=E(r)½Aσ2(r)?
In choices under uncertainty, individuals
maximize his or her expected utility U, which is a
function of wealth W.
Is the concept of expected utility maximization
consistent with the concept of meanvariance
analysis? Hsiulang Chen Portfolio Analysis 9 Concepts of Risk Aversion
An individual is said to be risk averse if and only if his utility function is
concave (i.e. U’’(W)<0). The individual will not accept “fair” gambles
(i.e. E(ε̃)=0)[BKM (8th), p. 188] ][BKM (9th), p. 191]
$150
P=0.5 $100 U(W)
U(150)
U(100)
E(U) $50 U(50) Will you accept this fair gamble ε ̃?
E(payoff) = 0.5 ×$50 + 0.5 ×$150
= $100 What is your expected utility level
from playing the gamble? E[U($100+ε̃)] =
0.5×U($50)+0.5×U($150) W
50 WC 100 150 U($100) > E[U($100+ε̃)] No!
⇒∃ WC ∋ U(WC)= E[U($100+ε̃)]
(Certainty Equivalent)
⇒ ∃ π ∋ U(Wπ)= E[U(W+ε̃)] 10 WC is called the certainty equivalent of the gamble W+ε̃. The
difference between 100 and WC is called the (insurance) risk
premium π.
To avoid a present gamble a riskaverse individual would be
willing to pay an insurance risk premium.
Q. What is π for an investor with log utility in this risky game?
Q. Will risk premium π depend on an individual risk preference only? U ′′(W )
≡ 1 σ ε2 A(W )
π =− σ
~
2
U ′(W )
1
2 2
~
ε This is derived by taking a Taylor series approximation of the
equation U(Wπ)= E[U(W+ε̃)] around the point ε̃=0 and π=0
[UtilityFunction.pdf]. A(W) is the Pratt(1964)Arrow(1970) measure
of absolute risk aversion. Note that the risk premium π depends on
the uncertainty of the risky asset and on the individual’s coefficient
of absolute risk aversion. Since both σε̃2 and U’(W) are > 0,
concavity of the utility function ensures that π>0.
Q. Can you use this approximation to figure out the insurance risk
premium π for an investor with log utility?
Hsiulang Chen Portfolio Analysis 11 In finance, however, the compensatory risk premium is more useful. To
induce a riskaverse individual to undertake a fair gamble, a compensatory
risk premium πC would have to be offered, making the package actuarially
favorable. It corresponds to the “extra” return expected on riskier assets. ∃
πC ∋ U(W)= E[U(W+ε̃+πC]
In the previous example, U(100)=0.5×U(50+πC)+0.5×U(150+πC)
U(W) $150 U(150)
U(100) P=0.5 $100
U(50) $50
+πC +πC +πC
50 Hsiulang Chen 100 150 What is the compensatory risk
premium required for accepting this
fair gamble for an investor with log
utility? Portfolio Analysis 12 Example 1
Suppose the utility function is U(W)=(W)0.5. Consider the
following simple game:
p $150 1p $50 $100 1. What is the utility level at wealth levels $150 and $50?
2. What is the expected utility if p still equals to 0.5?
3. What is the certainty equivalent of the risky prospect?
4. Does this utility function also display risk aversion?
5. Does this utility function display more or less risk
aversion than the log utility function? Hsiulang Chen Portfolio Analysis 13 Example 2 You and your friend Michael just have a trip to Las Vegas. Suppose
your utility function is U(W)= – 1/W and the utility function of your
friend Michael is U(W)=lnW, where W is the terminal wealth.
A casino company offers a simple game described as follows:
The prize of the game depends on a coin you toss. If the head
appears, you get $200. If the tail appears, you get $100. To enter
the game one pays an entry fee $135.
1. Who is willing to participate this game?
2. If the prize of this game changes to $250 versus $50, instead of
original $200 versus $100, will your answer in part (a) change?
Why? (Can you answer this without recalculation?)
3. (Advanced) What is the maximum entry fee you are willing to pay
for this fair coin game? Given a biased coin game and the entry
fee of $135, what is the minimum probability of the coin head
appearing so that you will play it?
Hsiulang Chen Portfolio Analysis 14 Example 3 Suppose that your wealth is $250,000. You buy a
$200,000 house and invest the remainder in a riskfree
asset paying an annual interest rate of 6%. There is a
probability of .001 that your house will burn to the
ground and its value will be reduced to zero.
1. With a log utility of endofyear wealth, how much would
you be willing to pay for insurance (at the beginning of
the year)? Assume that its endofyear value will be
$200,000 if the house does not burn down.
2. If the cost of insuring your house is $1 per $1,000 of
value, what will be the certainty equivalent of your endofyear wealth if you insure your house at:
i. ½ its value
ii. Its full value
iii. 1.5 times its value
Hsiulang Chen Portfolio Analysis 15 MeanVariance Analysis
Consider an investor’s utility from an investment in a given portfolio of
assets. Let denote the rate of return on the portfolio, so that the end
of period wealth of the investor is W=W0 (1+r̃P). For convenience, we
write U(W=W0(1+r̃P)) simply as U( r̃P).
Take a Taylor series expansion of U( rP) around the mean of rP, E(rP):
̃
̃
̃ ~
U (r ) = U ( E (~ )) + (~ − E (~ ))U ′( E (~ )) + (~ − E (~ )) U ′′( E (~ )) + ⋅ ⋅ ⋅
r
r
r
r
r
r
r
p p p p p 1
2! 2 p p p If we ignore thirdorder and higher terms because:
i. The utility function is quadratic (U’’’=0), or
ii. Portfolio rates of return are normally distributed, so that third,
fourth, and all higher central moments are all a function of the
mean or variance, or
iii. The actual utility function and actual portfolio returns are close
enough to those assumed in (i) and (ii) such that higher order
terms are small and can be safely neglected
Hsiulang Chen Portfolio Analysis 16 Then taking the expectation of both sides of the
equation, we obtain E[U (~ )] = U ( E (~ )) + σ U ′′( E (~ ))
r
r
r
p p 1
2 2 p The investor’s expected utility is a function of only
the mean and variance of the rate of return on the
portfolio.
In the real world, people’s utility functions may
take a variety of forms. For example, E(U) = E(r)λσ2, E(U) = E(r)λσ2λTE2, U = E(r)0.5Aσ2 [BKM,
p.159 (8th) or 163 (9th)], and Utility = alphaλσ2transaction costpenalties in [BARRA, p.173]. An utility function is used to quantify the
personal riskreturn tradeoff.
Hsiulang Chen Portfolio Analysis 17 Investor preferences can be depicted as indifference
curves. Each curve traces out the combinations of
{E(r),σ(r)} yielding the same level of utility. In an example of
U(r)=rCE=E(r)½A σ2(r), Hsiulang Chen Portfolio Analysis 18 Investors with various degrees of risk aversion in a
given utility function U(r)=rCE=E(r)½A σ2(r).
Q1. What will be the certainty equivalent
rate of return for this risky investment ?
Q2. What will be another equivalent risky
investment to this investor? Q3. Who is indifferent between this
investment and an investment of (.6, .15)? How can we see that rCE=0.03 ? What is 0.25 ½ 0.78 (0.75)2 ?
Hsiulang Chen Portfolio Analysis 19 Concept Checks
Given utility function U(r)=E(r)½A σ2(r) and the market riskfree rate of 3%
Consider a risky investment (σ(r), E(r))=(75%,25%). It offers the risk premium of 22%.
Will an investor with A=1.00 take this risky investment?
Sol1: If investing in this, the utility level is 0.25½ * 1.00* (0.75)2 = 0.03125
This is also the certainty equivalent rate of return for this risky investment. This is his
personal riskfree rate assigned to this risky investment. The investor thinks this risky
investment is worth a certain rate of 3.125%. Since his personal riskfree rate on this
is below the market riskfree rate, he prefers the market riskfree rate to the risky
investment. Alternatively, you might think the investor has a personal required risk
premium of (25%(3.125%)=28.125%) for this risky investment, which is higher than
what it is offered at 22%.
Sol 2: If investing in the market riskfree rate, the utility level is 0.03. The investor
prefers the market riskfree rate because it has higher utility level. Hsiulang Chen Portfolio Analysis 20 Hsiulang Chen Portfolio Analysis 21 As an investment advisor, how do you gauge your
clients willingness to take risk?
How is the utility function of an individual specified in reality? Check
https://swww.canada.etrade.com/cwMFQuestionnaireIntro.shtml website. Hsiulang Chen Portfolio Analysis 22 Hsiulang Chen Portfolio Analysis 23 Hsiulang Chen Portfolio Analysis 24 Deal or No Deal (a TV show): 26
briefcases that contains money, varying
from one cent to $1 million. The
contestant picks one briefcase as his own
and then begins to open the other 25,
each time, by process of elimination,
revealing a little more about what his own
case might hold. At the end, the
contestant can also trade his brief case
for the last unopened one. Periodically, a
“banker” offers a deal to the contestant:
Stop playing now and take the money
offered.
An offer of $37,000 to quit winning
$200,000 in a 1/5 chance. NO!
An offer of $67,000 to quit winning
$200,000 in a 1/3 chance. NO!
An offer of $99,000 to quit winning
$200,000 or $50 (1/2 chance). YES! Hsiulang Chen Q. Is Mr. Johnson less risk averse than
an investor with log utility? Portfolio Analysis 25 ...
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 Fall '11
 HengChen

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