Business Finance Investment
Fall 2016
Assignment #3
CAPM, Linear Factor Models, and Equity Returns
Due Date: November 10, 2016
The focus of this assignment is to better understand the CAPM and to implement asset
pricing models using actual data. You will
Sample Second Exam - Math 215 - Fall 2010
Write all your proofs with care, using full sentences and correct reasoning.
1. Prove
1
2
+
2
22
+
3
23
+ +
n
2n
=2
n+2
2n
for all n = 1, 2, 3, .
Solution. This is a straightforward induction argument. Solution om
Quiz 1 and Answers - Math 215 - Fall 2009
1. Give a complete and careful proof by induction that
20 + 21 + 22 + 23 + . + 2n = 2n+1 1
for n = 0, 1, 2, .
Answer. Let Pn for n = 0, 1, 2, . be the statement
20 + 21 + 22 + 23 + . + 2n = 2n+1 1.
Note that P0 st
Final Exam - Math 215 - Spring 2009 With Selected Solutions
Do problems 1, 2, 3, 4, 5 and choose one more problem from the remaining
seven problems on the exam. Write all your proofs with care, using full
sentences and correct reasoning.
+2
1
1. Prove 2 +
Exam2 - Math 215 - Fall 2010
Write all your proofs with care, using full sentences and correct reasoning.
1. Prove
1
2
+
2
22
+
3
23
+ +
n
2n
=2
n+2
2n
for all n = 1, 2, 3, .
Solution. This is a straightforward induction argument. Solution omitted.
2. (a)
Exam 2 - Math 215 - Fall 2009 With Solutions
Do problems 1, 2, 3, 4 and choose one more problem from problems 5, 6, 7, 8.
Write all your proofs with care, using full sentences and correct reasoning.
1
1. Prove 2 + 222 + 233 + +
Answer. Omitted.
n
2n
=2
n+
Exam1 With Solutions- Math 215 - Fall 2009
1. Let fn denote the n-th Fibonacci number where f0 = f1 = 1 and
fn+1 =
fn + fn1 for n = 1, 2, . Give a complete and careful proof by induction that
2
2
2
2
f0 + f1 + f2 + + fn = fn fn+1
for n = 0, 1, 2, .
Answer
Math 215
Written Homework 7 Solution
07/21/2009
1. (20 points total) The number of m-element subset of an n-element set, where 0
n!
n
m n, is
=
.
m
m!(n m)!
(a)
11
7
=
11!
111098
=
= 11103 = 330. (3 points)
7!4!
4321
(b) Since two particular individuals
MATH 215
Written Homework 6 Solution
07/21/2009
1. (20 points total) We are assuming (i) A B A C and (ii) A B A C .
(a) We wish to show B C . Let b B . Suppose that b A. Then b A B which
means b A C by (i). Therefore b C . (4) Now suppose b A. Since b B i
MATH 215
Written Homework 5 Solution
07/21/2009
1. (20 points total) This is very similar to WH4, Problem 1.
(a) When n = 1 the equation holds since both sides are A A1 in this case. (1)
Suppose n 1 and the equation holds for all sets A, A1 , . . . , An a
MATH 215
Written Homework 4 Solution
07/10/2009
1. (20 points total) If n = 1 the left hand side of the equation is A A1 which is the
right hand side. Therefore the equation is true for n = 1. (3)
Suppose n 1 and the equation holds for all sets A1 , . . .
MATH 215
Written Homework 3 Solution
07/01/2009
1. (30 points total) We rst construct a table for small values of n and for them the
values of the terms an of the Fibonacci sequence and the values of 2n3 .
n
1
2
3
4
5
6
7
an
1
1
2
3
5
8
13
2n3
1/4
1/2
1
2
Take Home Exam - Math 215 - Fall 2011
1. Prove by induction that
1 2 + 22 + 32 + . + n2 = n ( n + 1 ) ( 2 n + 1 ) / 6
for n = 1, 2, 3,. .
2. Let Un be the n-th Fibonacci number as defined by Eccles in
Definition 5.4.2. Show by induction that
U 1 2 + U2 2
Project Math 215 - Due Wednesday, April 29,2009.
Instead of a third hour exam, I am asking you to write a paper
(hand-written or typed, as you wish). The paper will have as its title
"Logic, Sets and Mathematics" and you can use any materials or
problems
Discrete Calculus
Brian Hamrick
1
Introduction
n
How many times have you wanted to know a good reason that
n
i=
i=1
n(n + 1)
. Sure, its true by
2
n(n + 1)(2n + 1)
? Well, there
6
i=1
are several ways to arrive at these conclusions, but Discrete Calculus
Assignments. Math 215. Fall 2010.
1. Assignment Number One.
Due Monday, August 30, 2010.
Eccles. Read Chapters 1,2,3.
Read the article on Even/Odd by David Joyce
(available from our website).
Download the article on Peano Axioms from our website
(you do n
Math 215 - Ass ignment 5
Part 1.
Background for this problem is in Chapter 14 of Eccles, but it is
best for you to try the problem and then look at Chapter 14.
Consider the following discussion.
Suppose that F:X - P(X) is a well-defined function from a se
Math 215 -Fall 2010 - Assignment #6
1. Read Chapters 10, 12,13,14.
Read for the ideas, but particularly look at Chapter 14 in the light of
our work on Cantor and infinite sets. Chapter 10 you should read
now for the parts that you recognize. There is some
Math 215 -Fall 2010 - Assignment #7
Due Wednesday, November 24, 2010.
1. Read Chapters 15,16,17 (we have done all this in class).
Read Chapters 19,20,21 (this will be discussed in the week of
November 15 - 19).
2. page 225. Problems 2,3,5,6
3. For the pri
Ma t h 215 - Su p p lement on F init e a nd Infinit e Set s
by LK
We take as given (for this discussion) the natural numbers
N = cfw_1,2,3,4,5,.
and we let Nk = cfw_1,2,3,.,k be the set of the numbers from 1 to k.
It is assumed that the natural numbers ar
Euler's Formula for Plane Graphs
by LK
Leonhard Paul Euler (1707 - 1783) was a pioneering Swiss
mathematician who spent most of his life in Russia and
Germany. See <http:/en.wikipedia.org/wiki/Leonhard_Euler> for
information about his life and works.
This
Discrete Calculus
by Louis H. Kauffman
0. N o t a t i o n a l W a r n i n g . In these notes xk equals the k-th power of
x, but x(k) = x(x-1)(x-2).(x-k+1). Thus
x(0) = 1
x(1) = x
x(2) = x(x-1)
x(3) = x(x-1)(x-2)
and so on.
1. We are given a function f(n),
Boolean Algebra
Louis H. Kauffman
1 Introduction
The purpose of these notes is to introduce Boolean notation for elementary logic. In
this version of things we use 0 for F (False) and 1 for T (True). Negation is represented
by placing a bar (or overline)
Math 215
Problem Sampler
1. Construct truth tables for the statements (a) (not A) or B , (b) (not A) and B ,
(c) A B .
2. Construct truth tables for (a) not (A or B ), (b) (not A) and (not B ).
Deduce that the two statements are equivalent.
3. Find a stat
Math 215
Problem Sampler
1. Construct truth tables for the statements (a) (not A) or B , (b) (not A) and B ,
(c) A B .
2. Construct truth tables for (a) not (A or B ), (b) (not A) and (not B ).
Deduce that the two statements are equivalent.
3. Find a stat
MATH 215
Written Homework 2 Solution
06/26/2009
1. (25 points total) The assertion is 1 < n < 3 implies n3 < 2n2 + 15n for integers n.
a) Proof of the assertion by cases. There is but one case, n = 1. Since 13 = 1 < 17 =
212 + 151 the assertion is true. (
MATH 215
Written Homework 1 Solution
06/19/2009
1. (20 points total) We construct truth table for the statements of parts a) and b), adding
an extra column for convenience.
P Q not Q P and (not Q)
P Q not P (not P) or Q
TTF
T
TTF
F
and
.
TFF
F
TFT
T
FTT
T
Math 215, Fall 05 Homework #2
Solution
09/14/05
There are usually several valid proofs for a given proposition. Your proofs
do not necessarily have to match the ones below to be correct.
1. (20 points total) a) Let a R. By Axiom 6 there exists an element