1. Economics 434: The Theory of Financial Markets
professor Burton
Fall 2016
November 8, 2016

2. Second Mid-Term Coming Up
Thursday, November 8th
Will cover CAPM and APT (as well as everything else in class, powerpoints, readings)
November 8, 2016

3. Possible states in a two period economy
What can happen?
We can simplify and just think about these three possibilities S1
State 1 – Great Economy
Economy S2
Now
State 2 – Average Economy S3
State 3 – Financial & Economic Collapse
November 8, 2015

4. 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?
November 8, 2015

5. 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
November 8, 2015

6. X1 X2 X3 s1 s2 s3 Definition of Securities p1,1 p1,1 p1,2 p1,3 p2,3
November 8, 2015

7. 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
November 8, 2015

8. The following three conditions are not all true:
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.
November 8, 2015

9. 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
November 8, 2015

10. 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
November 8, 2016

11. q1, q2, q3 are the state prices for states 1, 2, 3
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
November 8, 2016

12. 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
November 8, 2015

13. 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)
November 8, 2016

14. 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) }
November 8, 2016

15. Pj = q { (pj,1 * π1) + (pj,2 * π2) + (pj,3 * π3) } But q = 1/(1+r)
price equals discounted expected value!
November 8, 2016

16. November 8, 2016