Financial Market Theory Thursday, October 19, 2017 Professor Edwin T Burton

Finite State Version of MPT October 19, 2017

Asset choice in a two period economy Suppose that the world only has two periods; there is only one more period after today Suppose we want to buy assets now (in this period) that will do well by the end of this upcoming single period What should we own? October 19, 2017

possible states in a two period economy What can happen? We can simplify and just think about these three possibilities State 1 – Gets better Economy S2 Now State 2 – Gets worse S3 State 3 – Muddles along October 19, 2017

Three possible states and three available assets Three states can occur – Good, bad, and mediocre (S1, S2, S3) What are the available assets? X1, X2, X3 How will each asset perform in each state? October 19, 2017

The Definition of a “Real-World” Security Given the states of the world: s1, s2, s3 A security is defined by its payoff in dollars in each state of the world p1,i is the payoff for security i in state one p2,i is the payoff for security i in state two p3,i is the payoff for security i in state three October 19, 2017 September 1, 2015

X1 X2 X3 X4 X5 s1 s2 s3 Definition of Securities p1,1 p1,1 p1,2 p1,3 October 19, 2017 September 1, 2015

What would constitute a riskless asset? Assume that owning one unit of Xr will return exactly 1 dollar regardless of state Return doesn’t have to be 1; could be anything. Easier to simply assume 1 unit of return in each state Xr is the “riskless asset” Return $1 State 1 – Economy gets better X1 $1 State 2 – Economy gets worse $1 State 3 – Economy muddles along October 19, 2017

What Does a Security Cost Today? P1 times Ɵ1 is what it costs to buy a quantity Ɵ1 of security one at price P1. Or simply: P1 Ɵ1 Similarly for 2, 3, etc. P1. is always a positive number, but what about Ɵ1. That might be negative You may have sold security one Long sale if you already owned it, but could be a short sale October 19, 2017 September 3, 2015

So, What Does a Portfolio of Securities Cost? A portfolio is three numbers in a world of three securities: Ɵ1, Ɵ2, Ɵ3 where the Ɵ’s are the amounts purchased or sold of securities one, two and three Ɵ1P1 + Ɵ2P2 + Ɵ3P3 This could be positive or negative October 19, 2017 September 3, 2015

What does this security pay What does this security pay? (these can be negative as well as positive) In state one: Ɵ1p1,1 + Ɵ2p1,2 + Ɵ3p1,3 In state two: Ɵ1p2,1 + Ɵ2p2,2 + Ɵ3p2,3 In state three: Ɵ1p3,1 + Ɵ2p3,2 + Ɵ3p3,3 October 19, 2017

No Arbitrage Means P1φ1 + P2 φ2 + P3 φ3 ≤ 0 (Budget) Implies The following three conditions are not all true: p1,1φ1 + p1,2 φ2 + p1,3 φ3 ≥ 0 P2,1φ1 + p2,2 φ2 + p2,3 φ3 ≥ 0 P3,1φ1 + p3,2 φ2 + p3,3 φ3 ≥ 0 If the Budget holds exactly (equals zero), then at least one of the three conditions must be strictly < 0. October 19, 2017

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 that has a return of one dollar in that state and zero for all other states October 19, 2017

Diversification in a “Finite State” World Most assets perform well in good state –that’s the definition of a “good state” Most assets do terribly in the bad state – that’s the definition of a “bad state” Diversification in the sense of protection against downside losses – finding assets that pay off in bad states October 19, 2017

State Prices A state price is the price of a security that pays one unit in that state and zero in all other states q1, q2, q3 are the state prices for states 1, 2, 3 q3 > q2 > q1 October 19, 2017

Again: 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 no-arbitrage assumptions is true Price of risk-free asset q = q1 + q2 + q3 October 19, 2017

Analyzing the risk free rate Buy the risk free asset, paying q Invest it Next period, you will have q (1+r) We know that equals one q (1+r) =1 So q = 1/(1+r) October 19, 2017

Risk Adjusted Probabilities Pj = (pj,1 * q1) + (pj,2 * q2) + (pj,3 * q3) Define πi = qi/q These πi ‘s can be interpreted as probabilities since π1 + π2 + π3 = 1 Substituting in Pj = q { (pj,1 * π1) + (pj,2 * π2) + (pj,3 * π3) } October 19, 2017

Pj = q { (pj,1 * π1) + (pj,2 * π2) + (pj,3 * π3) } But q = 1/(1+r) price equals discounted expected value! October 19, 2017