The solution to the Math Review (I) problem set
Q1:
E(X) = (a + b)/2 = 5.5
P(2< X < 6) =
6
V(X) = (b - a)2/12 = 25/12 = 2.083
61
2 f ( x)dx = 3 5dx =
63 3
=
5
5
Q2: The cumulative distn function for X is:
0,
1
,
F ( x ) = Pr ( X x ) = 3
1
,
2
1,
if _ x
Figures for Ch 1
Fig 1
u
u(co , .)
co
Fig 2
u(co , c1)
c1
co
Fig 3
c1
co
1
Fig 4
c1
co
Fig 5
c1
co
Fig 6
y1
x1
x1
x2
2
Fig 7: The combination of co and c1 can be changed along the IC to keep utility
constant, equal to u
c1
: u (co , c1 ) u
co
Fig 8
c1
c1
Problems for Ch 1
1. Graphically demonstrate the Fisher separation theorem for the case where an
individual ends up lending in financial markets. Label the following points on the
graph: initial endowment, Wo; optimal production investment, (Po, P1); opti
Problems for Ch 4
1. You have a logarithmic utility function, U(W) = ln (W), and you current level of
wealth is $5000.
(a) Suppose that you are exposed to a situation which results in a 50/50 chance
of winning or losing $1000. If you can buy insurance whi
Problems for Ch 5
1. Security A pays $30 if state 1 occurs and $10 if state 2 occurs. Security B pays
$20 if state 1 occurs and $40 if state 2 occurs. The price of security A is $5 and the
price of security B is $10.
(a) Set up the payoff table for securi
Probability and Statistics Review ( I )
Random variable (denoted by a bold capital-case letter)
A random variable (RV) is not a sure thing but takes on more than one value with a
probability distribution. For a discrete RV denoted by X, it has n possible
A supplementary problem for Ch 7
(This is optional, and you may not do it)
Q. Given the following var-cov matrix and expected returns vector (for assets X and
Y) for a two-asset world:
0
0.01
=
0.0064
0
X 1 0.2
X = 0.1
2
(a) What is the expected r
Ch 6. Objects of Choice
Using mean and variance as choice criteria in EU
EU for a continuous prob. distn. can be constructed in a manner similar to that for a
discrete case, which is introduced in the Review. The key point is on the equivalence
between a
Ch 10. Efficient Capital Markets: Theory
A. Definition of Efficiency
The capital market is a place in which funds are transferred from lenders who have
excess funds to borrowers who have productive opportunities.
A market is allocationally efficient when
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Problems for the Review (I)
1. A uniform RV is denoted by X with a mass (3, 8). Draw its p.d.f and c.p.f. on two
separate graphs. Calculate E(X), V(X) and the prob. of X falling between (2, 6).
2. A RV, Y, has the following frequency distn:
Y
0
1
2
q
1/4
Solution to the supplement Q for Ch7
(a) A zero-beta portfolio has zero covariance with the market portfolio, by definition.
Also, using matrix notation, the covariance between two portfolios is
cov = W1W2
where
W1 = the row vector of weights in the zero-
Ch 1 Introduction
We compare a world without capital markets to one with them in order to show that
no one is worse off while at least one individual is better off in a world with capital
markets.
Two periods: period o (today),
period 1 (tomorrow)
Endowme
Ch 4.
Utility Theory under Uncertainty
The theory of choice
Ch 1 deals with choices of consumption and investment with capital markets under no
uncertainty. This chapter is concerned mainly with the foundation for the theory of
investor choice with uncert
Ch 6. Objects of Choice
Using mean and variance as choice criteria in EU
EU for a continuous prob. distn. can be constructed in a manner similar to that for a
discrete case, which is introduced in the Review. The key point is on the equivalence
between a
Solution to the Ch 4 assignments
Q1: (a) E[U(W)] = .5 ln 4,000 + .5 ln 6,000 = 8.4967825
Let U(W) = E[U(W)], i.e., ln (W) = 8.4967825. Then, W = $4,898.98
Thus, M RP = 5,000 - 4,898.98 = $101.02 ( <$125).
So, he would not buy the insurance for $125
(b) Gi
Solution to Ch 7 problem set
Q1:
Yes. Given a riskless asset, two-fund separation obtains, and if you can observe any one
of the following: 1) the percent of the investors portfolio held in the market portfolio,
or 2) the of the investors portfolio, you c
Solution to the Ch 5 assignments
Q1: (a)
Security A
Security B
Prob. distn.
State 1
30
20
/
State 2
10
40
/
prices
5
10
(b) pj denotes the price of pure security in state j. Solve
30 p1 + 10 p 2 = 5
20 p1 + 40 p 2 = 10
p1 = $0.1
p 2 = $0.2
Q2: (a) pi d
Ch 10. Efficient Capital Markets: Theory
A. Definition of Efficiency
The capital market is a place in which funds are transferred from lenders who have
excess funds to borrowers who have productive opportunities.
A market is allocationally efficient when
Solution to Ch 6 problem set
Q1:
a) E(X) = 9.0,
var(x) = 28,
E(Y) = 5.0,
Var(Y) = 50.8,
Cov(X,Y) = -1.2
b) E(R p ) = aE(X) + (1-a)E(Y)
2
2
2
Var(R p ) = a 2 x + 2ab cov( X , Y ) + (1 a ) y
% in X % in Y E(R p )
125
-25
10.0
100
0
9.0
75
25
8.0
50
50
7.0
2
Ch 7 CAPM and APT
The derivation of the CAPM
The market portfolio M must be an efficient portfolio in equil, i.e., it must lie on the
upper half of the mean-var investment opportunity set. Assume that asset returns
follow a normal distn., which is complet