Economics 434 Financial Markets Professor Burton University of Virginia Fall 2015 September 10, 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 10, 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 10, 2015

2 nd Fundamental Theorem of Finance State prices exist (not necessarily in the real world) If and only if Every security’s price is equal to its returns in each state “priced” by the state price for each state: P j = Price (of Security j) = (p j,1 * q 1 ) + (p j,2 * q 2 ) + (p j,3 * q 3 ) September 10, 2015

What are the takeaways? That state prices are not necessarily the same for each state. – The available securities and their prices securities determine state prices. If there were a different set of securities, then state prices will generally be different – Determining what is a “good” state and what is a “bad state” will depend upon the available securities. States where virtually all the securities have very positive payoffs is a “good” state, whereas states for which there are few, if any, securities with positive payoffs is a “bad” state. (This is all relative). – State prices for bad states will exceed state prices for good states. (to convince yourself of this, imagine you have a portfolio of securities that do very poorly in the bad state but great in the good state. If someone offered you your choice of state securities, which would you choose?) Diversification is about finding securities with positive payoffs in bad states (not simply about lack of correlation of returns in good states). – Two benefits of diversification Lowers risk for a given return Increases potential return for a given level of risk All of this follows from the simple assumption of “no-arbitrage” September 10, 2015

Interpreting the “risk-free” rate and the riskless security September 10, 2015

The Risk Free Security September 10, 2015

Interpreting the risk free rate September 10, 2015

Create pseudo-probabilities (risk adjusted probabilities) September 10, 2015

The Pricing of Security j September 10, 2015

Do We Really Not Need Actual Probabilities of Each State? The “state price” view requires knowledge of existing security prices (required to verify the no-arbitrage condition) How are these security prices determined? We don’t really know that until we decide on a “market model” that determines security prices September 10, 2015

Two Main “Market Model” Candidates Capital Asset Pricing Model (CAPM) – William Sharpe etal, 1962 – Two parameter theory Mean, variances of returns But, concludes that “covariances” drive pricing (through betas) – Terminology widely used in all of finance Arbitrage Pricing Theory (APT) – Steve Ross, 1982 – Father of “multi-factor” models Widely used in Asset Management industry Mainly regressions of returns on explanatory variables – Special case of APT is CAPM September 10, 2015