Q5
How the CML is derived?
The line of possible portfolio risk and return combinations given the risk-free rate, risk and return of a portfolio of risky assets is referred to as the capital allocation line (CAL).

A simplifying assumption underlying modern portfolio theory is that investors have homogeneous expectations, i.e., they all have the same estimates of risk, return, and correlations with other risky assets for all risky assets. Under this assumption, all investors face the same efficient frontier of risky portfolios and will all have the same optimal risky portfolio and CAL.

Under the assumption of homogeneous expectations, this optimal CAL for all investors is termed the capital market line (CML).

A line used in the capital asset pricing model to illustrate the rates of return for efficient portfolios depending on the risk-free rate of return and the level of risk (standard deviation) for a particular portfolio.

The CAPM is a model for pricing an individual security or a portfolio

expected return = risk-free rate + portfolio beta* (the difference between the expected return on the market as a whole and the risk-free rate).

Efficient frontier:

Efficient frontier is the set of portfolios among all the possible portfolios of combinations of individual risky assets that offers the highest expected return for each level of risk (standard deviation

The concept of an efficient frontier can be used to illustrate the benefits of diversification. An undiversified portfolio can be moved closer to the efficient frontier by diversifying it. Diversification can, therefore, increase returns without increasing risk, or reduce risk without reducing expected returns. The assumptions:

The assumptions of Capital Market theory are primarily eightfold and I will attempt to explain them below. 1. Everyone is an Efficient Investor – It goes without saying that everyone wants to be a efficient investor. No investor wants economic loss and all...

...Q 1 - Two congruent squares overlap to form 3 congruent, non-overlapping rectangles, as shown. If the perimeter of each of these rectangles is 18, what is the area of each?
A 1 - We can look at this picture like this:
(This is one of the two squares that overlap)
And this:
Set the long side of the rectangle as X, and the short side as Y.
Now, we know that:
1. There are 2 long sides and 2 short sides in the rectangle
2. The perimeter is 18
3. The perimeter consists of all the sides of a shape
So, we can set:
18 = 2X + 2Y
Now, we look at the square.
We know that:
1. The square consists of 2 rectangles
2. Squares have 4 equal sides
3. The square we are looking at right now has two kinds of sides:
a. A side consisting of 1 long side. (X)
b. A side consisting of 2 short sides. (2Y)
Since we know that the sides of squares are equal, we can infer that X = 2Y
Then, we substitute it back in the first equation we got:
18 = 2X + 2Y
18 = 2X + X
18 = 3X
X = 6
Substitute X back in the equation again to find Y:
18 = 2(6) + 2Y
18 = 12 + 2Y
6 = 2Y
Y = 3
We want to know what the area of the rectangle is. The equation for that is multiplying the two different sides of the rectangle together. So:
X Y = Area
6 3 = 18
Answer: 18
Q 2 - What is the greatest possible sum of two multiples of 12, each less than 100, whose greatest common factor is 24?
A 2 - If the question says that the greatest common factor of the two numbers is 24, then...

...Difference Between CML AND SML
CML vs SML
CML stands for Capital Market Line, and SML stands for Security Market Line.
The CML is a line that is used to show the rates of return, which depends on risk-free rates of return and levels of risk for a specific portfolio. SML, which is also called a Characteristic Line, is a graphical representation of the market’s risk and return at a given time.
One of the differences betweenCML and SML, is how the risk factors are measured. While standard deviation is the measure of risk for CML, Beta coefficient determines the risk factors of the SML.
The CML measures the risk through standard deviation, or through a total risk factor. On the other hand, the SML measures the risk through beta, which helps to find the security’s risk contribution for the portfolio.
While the Capital Market Line graphs define efficient portfolios, the Security Market Line graphs define both efficient and non-efficient portfolios.
While calculating the returns, the expected return of the portfolio for CML is shown along the Y- axis. On the contrary, for SML, the return of the securities is shown along the Y-axis. The standard deviation of the portfolio is shown along the X-axis for CML, whereas, the Beta of security is shown along the X-axis for SML.
Where the market portfolio and risk free assets are determined by the...

...DERIVATION OF FORMULAS
constant acceleration
In order to be accurate, the title of this section should be "One Dimensional Equations of Motion for Constant Acceleration". Given that such a title would be a stylistic nightmare, let me begin this section with the following qualification. The equations of motion are valid only when acceleration is constant and motion is constrained to a straight line.
Given that we live in a three dimensional universe in which the only constant is change, you may be tempted to dismiss this section outright. It would be correct to say that no object has ever traveled in a straight line with constant acceleration anywhere in the universe at any time — not today, not yesterday, not tomorrow, not five billion years ago, not thirty billion years in the future, never. This I can say with absolute metaphysical certainty.
So what good is this section then? Well, in many instances, it is useful to assume that an object did or will travel along a path that is essentially straight and with an acceleration that is nearly constant. That is, any deviation from the ideal motion can be essentially ignored. Motion along a curved path may also be effectively one-dimensional if there is only one degree of freedom for the objects involved. A road might twist and turn and explore all sorts of directions, but the cars driving on it have only one degree of freedom — the freedom to drive in one direction or the opposite direction. (You can't drive...

...Derivation of the CAPM
We know from Markowtiz’ framework concerning two-fund separation that each investor will have a utility-maximizing portfolio that is a combination of the risk free asset and the tangency portfolio. If all investors see the same capital allocation line, they will all have the same linear efficient set called the Capital Market Line (CML). This forms a linear relationship between expected return of the portfolio and the standard deviation. If market equilibrium is to exist we know that the prices of all assets must adjust such that all assets are held by investors, there can be no excess demand. We get the market portfolio, M. Hence, in equilibrium the market portfolio will consist of all marketable assets held in proportion to their value weights.
If we invest a % in a risky asset, i, and (1-a) % in the market portfolio, we get the following mean and standard deviation:
Change in the mean and standard deviation with respect to the percentage of the portfolio, a, invested in asset i is a follows:
However we notice that by the definition of the market portfolio asset i is already hold in the market portfolio according to its market value weight. Therefore the percentage a in the equations is excess demand for i, which in equilibrium must be zero. We elaborate the new information in our equations:
The slope of the risk-return trade-off evaluated at point M in the graph, in market equilibrium, is:
This...

...Q1 The four major competitors in the computer work-station market are Sun Microsystems (29%), Hewlett-Packard (18.8%), IBM (16%),and Digital Equipment (11.6%) with other manufacturers holding 24.6% of the market. One year later a survey of computer workstations found 97 Sun, 86 HP, 70 IBM, 60 Digital and 82 other. Test at the 2.5% level of significance whether changes have occurred during the 1-year period.
1. State the critical value for the test.
2. Find the value of the test statistic (to 3 dec pl).
3. Can we conclude that the proportions have changed during the year? (yes/no)
OBSERVED
EXPECTED
(o-e)^2/e
97
115.42
2.939667
86
74.824
1.66929
73
63.68
1.364045
60
46.168
4.144087
82
97.908
2.584717
398
chi square value
12.70181
Critical value = 12.8
Test value = 12.701
No proportions have not changed.
-------------------------------------------------------
Q2
A film director wished to determine whether there existed a relationship between movie success and whether or not the movie was in 2D, 3D or Black and White.
Very unsuccessful Low success Medium success High success
2D 24 53 75 83
3D 29 56 93 109
Black and White 16 32 86 113
Assume that a 2.5% level of significance is to be used for the test.
1. Calculate the test statistic, reporting your answer correct to three decimal places.
Use the tables to determine the lower and upper bounds on the p-value for the test.
2. lower bound (if the lower bound is...

...Part 1 – Hypothesis: Sleep deprivation impairs task performance.
Part 2 – Experimental Design:
To begin the experiment, I would randomly assign 50 college students into two groups. The participants would not know the purpose of the experiment and they would not be able to make a choice about which group they are in. To begin, each person would be asked to study for a math exam, which will be taken the next day. Each person in the first group would be allowed to sleep for a maximum of 5 hours. The second group, on the other hand, would be asked to study for the test and to have enough sleep (about 8 hours). On the next day, they would have to answer 100 questions of a basic math test (addition, subtraction, multiplication, and division) within a time limit. Students would get one point for each correct answer, zero point for each blank question, and lose 0.5 point for each incorrect answer. The result of this test would be the measurement of each college student’s task performance.
Part 3
1. What is your independent variable? Be certain to identify your experimental and control conditions.
My independent variable would be the sleeping hours of only 5 hours maximum for my experimental group. For my control group, my independent variable would be more sleeping hours (about 8 hours).
2. What is your dependent variable? Provide the operational definition you will use.
My dependent variable is the completion and accuracy of the participant’s test result....

...exposure to very high doses of radiation and a high-dose radiation therapy used to treat other cancers. On the other hand, doctors have found that most people with CML do not develop it from high dose radiations. People with this cancer may not have any symptoms while being diagnosed. The most common way which detects CML is by having your yearly check-up or a medical examination. These signs and symptoms are likely to develop slowly. The most common signs and symptoms are: tiring more easily, shortness of breath doing usual day-to-day activities, pale skin color, enlarged sleep leading to a “dragging” feeling on the upper left side of the abdomen, night sweats, an ability to tolerate warm temperatures, and weight loss. It is hard to tell the difference in illnesses because these are very common to others. Most people with these familiar symptoms are not diagnosed with CML.
The treatments require some goals in order for this to become a curable illness. In the chronic phase of CML, which is when the symptoms are mild and white cells can still fight infection; you can return the blood cells to normal and kill all the viruses that have the cancer gene. The accelerated phase is when the patient can increase the chances of getting anemia and white blood cells wouldn’t be able to repair quite as easy. In this phase of CML the goal is to kill all the cells that contain the Brc-Abl gene. However, the...

...Carleton University
Honours Project
Final Draft Derivation and Application of the Black-Scholes Equation for Option Pricing
Author: Yeheng XU
Supervisor: Dr. David Amundsen
April 30, 2012
Abstract In this project, I will first study the concept of a stochastic process, and discuss some properties of Brownian Motion. Then I generalize Brownian Motion to what it called an Itˆ process. The above concepts will be used to derive the Black-Scholes Option Price o formula. Then an analytical solution for the equation will be provided by using mathematical tools such as Fourier Transformation and properties of the heat equation. Finally, I will implement a finite difference numerical scheme in MATLAB to simulate the original Black-Scholes equation for both European call and put options and compare to analytic solutions.
1
Contents
1 Introduction and Background 1.1 1.2 1.3 1.4 What is financial mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction of option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some economic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 5
2 Brownian Motion
7
3 Itˆ’s Lemma o
12
4 Black-Schloes Partial Differential Equation
15
5 Analytical Solution of the Black-Scholes Equation 5.1 5.2 The I.B.V.P. for the Black-Scholes Equation . . . . ....