Hence, i = bi1f1 + + binfn , fi := Cov(RM, fi)/2 M. That is to say,
the overall beta of the asset can be considered to be made up from
underlying factor betas that do not depend on the particular asse
+ 2 parameters, those of ai s bi s, 2 ei s, and f and 2 f . Now
suppose a portfolio has weight w = (w1, , wm) where Pwi = 1. Then
its return can be calculated by R = XwiRi = Xwibif + Xwiei = bf + e wh
1. SingleFactor model Single-factor model assumes that there is a
single factor that affects all assetss performance and all assets are
correlated to each other through this single factor. Though simp
reasonable under risk 2 M. (2) For a particular asset ak with k < 1, its
systematic risk 2 k 2 M is smaller than the risk of the market
portfolio, so its expected return k is smaller than the expect m
market system consists of a risk-free asset with return rate 3%, and
three risky assets with the following parameters: Mean Return Cov(Ri ,
Rj ) Asset a1 a2 a3 a1 0.1 0.04 -0.006 0.016 a2 0.2 -0.006 0
here we provide a rigorous analysis, showing that the solution we
obtained is indeed the unique solution to the conditional minimization
problem. We use the same notation 1, u, C, w1, w2, e1, e2 as be
, Rj ) = 0.3 2 for all i 6= j and Var(Ri) = 2 . Calculate the risk of the
following portfolios: (i) wi = 1/m for all i = 1, , m; (ii) w = 3/m for all i
= 1, , m/2 and wi = 1/m for i = m/2 + 1, , m. (A
not contain any non-system risk. Finally, we introduce two important
indexes used in finance community: Jensen Index Jk = k 0 k(M
0), Sharp index k = k 0 k Theoretically, Jk = 0. The real data Jk
thu
random variables representing the returns of all risky assets (at end
time). As the market portfolio has weight wM, its return is the random
variable RM = Xm i=1 wM iRi = (R, wM). Hence, the market po
argument. If, based on (1.10), there is a large demand of one particular
asset thereby causing short supply, its price will arise, thereby
decreasing its rate of return. Similarly, assets under light
a simplified approach. It is believed that one can sort out a few factors
so that the returns of all assets can be traced back to these factors. A
factor model that represents this connection between
have already seen that the CAMP model ends up Ri = iRM + ei where
RM is the return of the market portfolio. Thus, CAMP model can be
regarded as a factor model. 2. Arbitrage Pricing Theory Now assume
t
driven by demand according to which assets share price changes. Take
an extreme example. Suppose the weight of a stock is negative in the
market portfolio; then, according to the CAPM theory, everybod
constants 0, 1, , n such that the well-diversified portfolio having
a rate R = bifi + + bnfn + e with the expected return = 0 + b11 +
+ bnn. Here the simple APT theorem applies since the mutual fund
given by (1.10). This weight is universal in the sense that it is
independent of any individual investor. That everybody invest according
to the CAPM theory has profound consequences. (a) The market h
depend on its unique risk Var(k). (5) That Cov(RM, k) = 0 for all k = 1,
, n states the following: The market portfolio has no unsystematic
risk, i.e., its expected return does not depend on each in
hyperbola is the Markowitz frontier, where dashed thick curve is the
remaining part of the Markowitz curve. The thick tangent line is the
capital market line when risk-free rate 0 is less that . The t
assets with parameters given in Exercise 1.12. (a) Assume the risk-free
rate is 0.2. Plot the Markowitz curve and the Capital Market line. (b)
Assume the risk-free rate is 0.1. Plot the Markowitz curv
. .
() h=91
=91/365 () * (100 95) / 95 5.2632% i .
, 222 1 0.9091% 220 f .
real *
* 1 4.3148%, i f i f
1/ real 1 1 18.4639%. * h i i Problem. (CT1, September
2006, Problem 3). An individual ha
deviation is on the Markowitz curve. The unbounded region on the
right-hand side of the hyperbola is called the Markowitz bullet or
attainable region; the top half of the hyperbola is called the Marko
PORTFOLIO THEORY Now consider the important constant k := Cov(Rk,
RM) 2 M . (1) From the formulas we just derived, k = (k0)/d
(M0)/d . Thus, k = k 0 M 0 , k 0 = k(M 0), k =
0 + k(M 0). (1.13) This lin
efficient frontier, that is, for efficient portfolios, whose weight are given
by the minimum risk weight line, being a linear combinations of two
special weights. Theorem 1.2 (Two-Fund Theorem) Two ef
Under certain assumption, we may reasonably believe that Q is
independent of the price P. The price is in certain way artificial (driven
by demand). The fair price of an asset should be judged by its
Given R, find a portfolio w = (w1, , wm) R m that minimizes
Var[R] = Xm i=1 Xm j=1 wiwjij subject to Xm i=1 wi = 1, E[R] = Xm i=1
iwi = . The solution. This problem can be solved by using the
Lagrange
two and three assets respectively, find the portfolios that have
minimum risk under condition (i) shorting selling is allowed (ii) short
selling is forbidden. Assume the covariance matrix (ij ) is kno
M . Substitute this in the pricing formula we then obtain the following
certainty equivalent pricing formula: P = 1 1 + 0 n E(Q) (M
0)Cov(Q, RM) 2 M o . Exercise 1.19. Suppose the risk-free rate is 3
non-trivial since all constants e1, , em reduce to a single constant
0. Proof. Set 1 = (1, , 1), u = (1, , m) and bk = (b1k, , bmk),
k = 1, , n Suppose (w, 1) = 0 and (w, bk) = 0 for all k = 0, , n.
C
R m dot product, we then have w = 11C1 2 + 2uC1 2 = w1 + (1
)w2, where = 1(1, 1C1 )/2 and w1, w2 are weights of two
portfolios given by w1 := 1C1 (1, 1C1) , w2 := uC1 (1, uC1) . Here
the proportion (
. i=1% , 2%
, f (
) 0.01 0.02 0.030612 3.0612%. 1 1 0.02 i f f
Problem (CT1, September 2008, Problem 1) A 91-day
government bill is purchased for 95 at the time of issue and is
redeemed at the ma