Chapter 8: The capital Asset pricing model Flashcards
Additional assumptions about markets and investors:
A4: Markets are in equilibrium: The total demand for any financial
instrument equals its total supply.
A5: Uniform horizon: All investors are investing for the same period of time.
A6: Homogeneity: All investors agree on the expected returns of investments, their standard deviations of returns and the
correlations between these returns. All investors can borrow and lend unlimited amounts of money at the same uniform risk-free rate.
- same feasible set and same risk free return and same market line if satisfy A5 and A6.
- concentrating on risky? then same proportions as same feasible sets
A7: No friction: There are no transaction costs and no taxes.
The market portfolio is the only efficient portfolio consisting entirely of risky investments.
What is this portfolio?
Theorem 8.1:
Let I_1, I_2, . . . , I_n be all risky investments,
and assume their total market value is w1,w2, . . . ,wn.
Let W = w1 + w2 + · · · + wn.
The market portfolio consists of a portfolio of investments of wj/W in I_j for each 1 ≤ j ≤ n
(finding the market value for each investment w’s)
as all investors have the same proportion of each risky investments and axioms A1,2,3,4,5,6. FInd the total value of investing in each and spread investments.
stock market indices
tracker funds:
You may have heard of stock market indices (e.g., FTS100, the Dow Jones Industrial Index, S&P 500, Nikkei, CAC40, DAX, etc.) The values of these indices are the weighted average price of a set
of stocks, where the weights are proportional to the proportion of the total value of a stock as part of the total value of the whole set
of stocks.
You might have also heard of tracker funds: these are investments that hold shares in the same proportion as a given index, e.g.,
FTSE 100 trackers.
The previous theorem says roughly that the only risky investments in the portfolio of an investor who assumes axioms A1-A7 must be tracker funds.
Actively managed funds
There are other types of investment funds, actively managed funds.
These funds invest money in stocks carefully chosen by
spectacularly highly paid fund managers.
These funds demand high fees from investors in return for applying their talents in choosing the way in which the fund’s assets will be
invested. So you are asked to pay large fees to have axiom A6 broken: these
fund managers claim to possess knowledge which is not apparent to lesser investors. There is a ongoing debate on whether these fund managers are worth these high fees
The market price of risk
For any efficient investment A lying on the market line we have
rA − rB = ((rM − rB)/σM)σA
where rB is the risk-free interest rate and M is the market portfolio.
We interpret (rM − rB)/σM
as the market price of risk:
this slope measures how much more return investors demand for an increase of one unit in the volatility of their returns.
We want a similar expression for the excess return above the
risk-free interest rate for non-efficient portfolios, e.g., individual stocks.
Theorem 8.2:
Theorem 8.2:
For any portfolio A we have
rA − rB =
[(rM − rB)/σ^2_M ]*Cov(A, M)
where rB is the risk-free interest rate, M is the market portfolio and Cov(A, M) is the covariance between the return of A and the return of M
For any 0 ≤ t ≤ 1, let portfolio Πt consist of an investment of t in A and an investment of 1 − t in M.
The curve c given by (σ_Π_t,r_Π_t) in the σ-r
plane joins points A and M.
points of our portfolio
c intersects the capital market line at M, and the capital market
line must be tangent to c at M:
Otherwise c would cross the capital market line and we would have
a portfolio above the capital market line, contradicting the fact that the capital market line is the efficient frontier.
diagram:
curve c is like the efficient frontier, under capital market line from risk free r_B has intersection point M
Point A is the bottom of the efficient frontier, under r_A and c is this curve.
DEFINITION:
given any investment A what is the beta coefficient
Definition: The beta coefficient of a portfolio A is defined as
β = [Cov(A, M) ]/σ^2_M
cover and var in market portfolio
along with the assumptions of the world this tells us all investments lie on a line…
consider beta r plane not sigma r plane
theorem
The security market line is the linear relation between expected returns r and beta coefficients β given by
r = rB + (rM − rB )β.
We can restate the previous theorem:
for any investment with expected return r and beta coefficient β the point (β,r) lies on the security market line#
consider beta r plane not sigma r plane
must lie on the line! , since the result in Theorem 8.2 is not based on no-arbitrage arguments, and so there is no apparent way to
translate discrepancies between real-life data and CAPM
predictions into arbitrage strategies.
