day Octo ber 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 ID: 649901
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Slide1
Financial Market Theory
Thurs
day, October 19, 2017
Professor Edwin T BurtonSlide2
Finite State Version of MPT
October 19, 2017Slide3
Asset choice in a two period economySuppose 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 periodWhat should we own?
October 19, 2017Slide4
possible states in a two period economy
What can happen?We can simplify and just think about these three possibilities
NowS2
S1
S3
State 1 – Gets better
State 2 – Gets worse
State 3 – Muddles along
Economy
October 19, 2017Slide5
Three possible states and three available assetsThree states can occur – Good, bad, and mediocre (S
1, S2, S3)What are the available assets? X1, X2, X
3How will each asset perform in each state?October 19, 2017Slide6
The Definition of a “Real-World” SecurityGiven the states of the world: s1, s2, s3
A security is defined by its payoff in dollars in each state of the worldp1,i is the payoff for security i in state onep2,i is the payoff for security i in state twop3,i
is the payoff for security i in state three
September 1, 2015
October 19, 2017Slide7
s
1
s
3
s
2
Definition of Securities
X
1
X
2
X
3
X
4
X
5
p
1,1
p
1,2
p
2,1
p
1,1
p
3,1
p
1,2
p
2,2
p
3,2
p
1,3
p
2,3
p
3,3
p
1,4
p
1,5
p
2,4
p
2,5
p
3,4
p
3,5
September 1, 2015
October 19, 2017Slide8
What would constitute a riskless asset?Assume that owning one unit of X
r will return exactly 1 dollar regardless of stateReturn doesn’t have to be 1; could be anything. Easier to simply assume 1 unit of return in each stateXr is the “riskless asset”
X1$1
$1
$1State 1 – Economy gets better
State 2 – Economy gets worse
State 3 – Economy muddles along
Return
October 19, 2017Slide9
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 negativeYou may have sold security oneLong sale if you already owned it, but could be a short sale
September 3, 2015
October 19, 2017Slide10
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 threeƟ1P1 + Ɵ2P2
+ Ɵ3P3This could be positive or negative
September 3, 2015
October 19, 2017Slide11
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 + Ɵ3p2,3In state three: Ɵ1p3,1 + Ɵ2p
3,2 + Ɵ3p3,3
October 19, 2017Slide12
No Arbitrage MeansP1φ
1 + P2 φ2 + P3 φ3 ≤ 0 (Budget)ImpliesThe following three conditions are not all true:
p1,1φ1 + p1,2 φ2 + p1,3 φ3 ≥ 0P2,1φ1 + p2,2 φ2
+ p2,3 φ3 ≥ 0P3,1φ
1 + p3,2 φ2 + p3,3 φ
3
≥ 0
If the Budget holds exactly (equals zero), then at least one of the three conditions must
be strictly < 0.
October 19, 2017Slide13
Fundamental Theorem of FinanceThe Assumption of No Arbitrage is TrueIf 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 statesOctober 19, 2017Slide14
Diversification in a “Finite State” WorldMost 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 statesOctober 19, 2017Slide15
State PricesA state price is the price of a security that pays one unit in that state and zero in all other statesq1, q2, q3 are the state prices for states 1, 2, 3
q3 > q2 > q1October 19, 2017Slide16
Again: How can you use “state prices?”To price any securityPrice 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 truePrice of risk-free asset q = q1 + q2 + q3 October 19, 2017Slide17
Analyzing the risk free rateBuy the risk free asset, paying qInvest itNext period, you will have q (1+r)
We know that equals oneq (1+r) =1So q = 1/(1+r)October 19, 2017Slide18
Risk Adjusted ProbabilitiesPj = (pj,1
* q1) + (pj,2 * q2) + (pj,3 * q3)Define πi = q
i/qThese πi ‘s can be interpreted as probabilities since π1 + π2 + π3 = 1Substituting inPj = q { (pj,1 * π1) + (pj,2 * π
2) + (pj,3 * π3) }
October 19, 2017Slide19
Pj = q { (pj,1 * π
1) + (pj,2 * π2) + (pj,3 * π3) }But q = 1/(1+r)
Pj = =
price
equals discounted expected value!
October 19, 2017Slide20