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 September 29, 2015 Standard Deviation Mean

How Do You Create A Portfolio? Try It with Two Assets Mean Standard Deviation “X 1 ” “X 2 ” Where are the Portfolios That Can Be Created from Just These Two Assets ? September 29, 2015

Consider the ½, ½ Portfolio Mean of X 1 Mean of X 2 Mean of P Where P = ½ [X 1 ] + 1/2 [X 2 ] September 29, 2015

Variance of a Portfolio with two assets  P 2 =  (P -  P ) 2 n =  {  1 (X 1 -  1 ) +  2 (X 2 -  2 )} 2 n September 29, 2015

Variance with 2 Assets - Continued = (  1 ) 2  (  2 ) 2   { 2  1  2 (X 1 -  1 )(X 2 -  2 )} n = (  1 ) 2  (  2 ) 2   1  2 Cov (X 1,X 2 ) = (  1 ) 2  (  2 ) 2   1  2  1,2 September 29, 2015

Variance with 2 Assets - Continued = (  1 ) 2  (  2 ) 2   1  2  1,2 Recall the definition of the correlation coefficient:  1,2   1,2 1212 = (  1 ) 2  (  2 ) 2   1  2  1,2  1  2 September 29, 2015

Variance with 2 Assets - Continued  1,2   1,2 1212 where = (  1 ) 2  (  2 ) 2   1  2  1,2  1  2 What Happens if  = 1? September 29, 2015

Back to the ½, ½ Portfolio Mean of X 1 Mean of X 2 Mean of P Where P = ½ [X 1 ] + ½ [X 2 ] If   1  1,2  1/2  1 + 1/2  2 September 29, 2015

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

Today Tomorrow s1s1 s3s3 s2s2 And, we may not have any idea what the probabilities of s 1, s 2, s 3 may be!! September 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 September 29, 2015

How can you use “state prices?” To price any security – Price of a security j equals: P j = (p j,1 * q 1 ) + (p j,2 * q 2 ) + (p j,3 * q 3 ) This pricing formula is true if and only if the no- arbitrage assumptions is true September 29, 2015

Definition of an Asset in Modern Portfolio Theory: As a probability distribution of returns (usually a normal distribution) Instead of three (or any finite number of) states, there are an infinity of states possible with various probabilities of returns assigned -- Returns -- Probability Density Function September 29, 2015

Harry Markowitz’s Conclusion Standard Deviation Mean Maximizes Utility September 29, 2015

James Tobin’s Result Mean Standard Deviation Risk Free Asset Use of Leverage E September 29, 2015

CAPM – two conclusions M – the “efficient” basket The pricing rule based upon “beta” September 29, 2015 Bill Sharpe

Capital Market Line What is M ? RfRf M Mean STDD Answer: contains all “positively” priced assets, weighted by their “market” values. September 29, 2015

Security Market Line  i = R f +  i [  M – R f ] Rf Beta Mean MM 1 Security Market Line ii 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 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