Economics 434
Financial Markets
Professor Burton
University of Virginia
Fall 2015
September 29, 2015
Clean up of 2 issues
• Definition of risk aversion
• Why the slope of the boundary of the feasible
set (why is the feasible set convex? ) is what it
is
September 29, 2015
Risk Aversion
• Anyone who is “risk averse” will always prefer
an expected return with zero variance to an
expected return with a positive variance
• In our presentation this means that increasing
variance will cause utility to fall
– Indifference curves slope up and to the right in
mean, standard deviation diagrams
– We usually assume more: that the curves are
convex to the origin (risk aversion is increasing as
wealth increases
September 29, 2015
Risk Aversion
implies negative slope but not the increasing slope shown below
Mean
Standard Deviation
September 29, 2015
How Do You Create A Portfolio?
Try It with Two Assets
Mean
“X1”
“X2”
Where are the Portfolios That Can
Be Created from Just These Two
Assets ?
Standard Deviation
September 29, 2015
Consider the ½, ½ Portfolio
Mean of
X1
Mean of P
Mean of
X2
September 29, 2015
Where P = ½ [X1] + 1/2 [X2]
Variance of a Portfolio
with two assets
 P2 =  (P - P)2
n
= {1(X1- 1) + 2(X2 - 2)}2
n
September 29, 2015
Variance with 2 Assets - Continued
= (1)212 + (2)222 + { 212(X1 - 1)(X2 - 2)}
n
= (1)212 + (2)222 + 212Cov (X1,X2)
= (1)212 + (2)222 + 2121,2
September 29, 2015
Variance with 2 Assets - Continued
= (1)212 + (2)222 + 2121,2
Recall the definition of the correlation coefficient:
1,2 
1,2
12
= (1)212 + (2)222 + 2121,212
September 29, 2015
Variance with 2 Assets - Continued
= (1)212 + (2)222 + 2121,212
where
1,2 
1,2
12
What Happens if  = 1?
September 29, 2015
Back to the ½, ½ Portfolio
If   1
1,2  1/21 + 1/22
Mean of
X1
Mean of P
Mean of
X2
September 29, 2015
Where P = ½ [X1] + ½ [X2]
If   1
Then all the portfolios are here
September 29, 2015
This Means the “boundary”of the possible portfolios
looks like this:
September 29, 2015
Main Topics Covered on Exam
• Bankruptcy
• No-Arbitrage and State Prices
• Capital Asset Pricing Model
(including Markowitz and Tobin)
September 29, 2015
Key takeaways about bankruptcy
• Equity becomes zero
• Owners of debt/liabilities take over
• Most importantly, assets are unchanged (they
don’t disappear or become worthless)
• Two outcomes:
– Either liquidation (Chapter 7) or
– Recapitalization (Chapter 11)
September 29, 2015
Tomorrow
s1
Today
s2
s3
And, we may not have any ideaSeptember
what the
probabilities of s1, s2, s3 may be!!
29, 2015
Fundamental Theorem of Finance
• The Assumption of No Arbitrage is True
• If and only if
• There exist positive state prices (one for each
state) that represent the price of a security
has a return of one dollar in that state and
zero for all other states
September 29, 2015
Interpreting the risk free rate
• What is the value of $ 1 tomorrow?
• What would you have to invest today to be
absolutely certain to receive $ 1 tomorrow?
• $ X (1 + r) = $ 1 which says: “if I invest $ x
today and earn the risk free rate, I will have $
1 tomorrow.
• Thus $ X =
$1
1+𝑟
September 29, 2015
How can you use “state prices?”
• To price any security
– Price of a security j equals:
Pj = (pj,1 * q1) + (pj,2 * q2) + (pj,3 * q3)
This pricing formula is true if and only if the noarbitrage assumptions is true
September 29, 2015
Definition of an Asset in Modern
Portfolio Theory:
As a probability distribution of returns (usually a normal distribution)
Probability Density Function
-- Returns -Instead of three (or any finite number of) states, there are
an infinity of states possible with various probabilities
of returns assigned
September 29, 2015
Harry Markowitz’s Conclusion
Mean
Maximizes
Utility
Standard Deviation
September 29, 2015
James Tobin’s Result
Mean
Use of Leverage
E
Risk Free
Asset
Standard Deviation
September 29, 2015
CAPM – two conclusions
Bill Sharpe
• M – the “efficient” basket
• The pricing rule based upon “beta”
September 29, 2015
Capital Market Line
Mean
M
Rf
What is M ?
Answer: contains all “positively”
priced assets, weighted by their
“market” values.
STDD
September 29, 2015
Security Market Line
i = Rf + i [M – Rf]
Mean
i
M
Rf
Security Market Line
Beta
1
September 29, 2015
Exam Procedures
• Starts Promptly after 9:30 AM
• Ends at 10:45 (graders will leave the room at 10:50,
no exams accepted after that they leave the room)
• Bring nothing to the exam but something to write
with; no use of cell phone or any other device with
information; no scratch paper, books, etc.
• No questions will be answered once the exam has
started (graders are instructed not to answer
questions once the exam has begun)
• A spillover room will be used. Look for a class email
that provides information about the room.
September 29, 2015
The Exam
• Covers reading
– Random Walk Down Wall Street
– Notes (all clickable from syllabus)
• Finite States
• Markowitz and Tobin
• CAPM
• Subjects
– Bankruptcy
– No-Arbitrage and State Prices
– CAPM
September 29, 2015
Exam Layout
• Five or six questions
• First question is always a set of definitions
– Usually 8 to 10 definitions
– Requiring short precise answers (no partial credit given on
these)
• Other questions tend to be short answer essay or short
problems
• No lengthy mathematical derivations, though math may be
used in definitions are in answering essay questions
September 29, 2015
September 29, 2015