Economics 434
Financial Markets
Professor Burton
University of Virginia
Fall 2015
September 10, 2015
Tomorrow
s1
Today
s2
s3
And, we may not have any ideaSeptember
what the
probabilities of s1, s2, s3 may be!!
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
2nd 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:
Pj = Price (of Security j) = (pj,1 * q1) + (pj,2 * q2) + (pj,3 * q3)
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
• Imagine a portfolio that consisted only of one of each of the state price
securities: Q1, Q2, Q3 with prices q1, q2, q3. (Call this portfolio, Q, which
consists of one unit of each state prices security).
• That portfolio, Q, would return exactly $ 1 regardless of which state
occurred – that means that portfolio would be the riskless asset.
• Price of Q, the riskless asset = q = q1 + q2 + q3
• Risk free rate = r =
1−𝑞
𝑞
=
1
𝑞
− 1 so that q =
September 10, 2015
1
1+𝑟
Interpreting the risk free rate
• What is the value today of $ 1 that I will receive tomorrow?
• Answer:
$1
1+𝑟
• This is called “discounting”. You are discounting $ 1 tomorrow back to it’s
present value. You use r because you want to be absolutely certain what a
dollar tomorrow is worth
• The more uncertain you are to get the $ 1 tomorrow the higher would be
the discount rate
• Lowest possible discount rate is r, for absolutely certain outcomes.
• Same logic applies to all future periods. For example, the value now of $ 1
that will be received two periods from now:
•
$1
1+𝑟 2
September 10, 2015
Create pseudo-probabilities (risk adjusted
probabilities)
• Define πi =
𝑞𝑖
𝑞
(recall that qi is the ith state
price and q is the sum of all state prices)
• Then:
 πi > 0 for all I
 π1 + π2 + π3 = 1
 This looks like probabilities for each state!
 In fact, these πi ‘s are called “risk-adjusted
probabilities”
September 10, 2015
The Pricing of Security j
Pj = =
1
1+𝑟
𝑝𝑗,1 ∗ 𝜋1 + 𝑝𝑗,2 ∗ 𝜋2 + 𝑝𝑗,3 ∗ 𝜋3
price equals discounted expected value!
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
September 10, 2015