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1

Two types of securities we are concerned with
Performance of the two?

Fixed income securities: Government bonds, treasury bills, treasury notes, convertible bonds, commercial paper.

Variable income securities: Preferred stock, common stock, investment companies, mutual funds.

Performance = Fixed income have low average return and low standard deviation. Variable have higher average return but also higher standard deviation

2

Calculating final value and annual return on simple investments

Calculating final value and annual return on compounding investments

Effective annual rate of return and a short understanding of how to calculate effective annual return

What happens to the annual rate as the compounding period becomes smaller?

Final value = initial value x (1+annual rate of return x time period)

Final value = initial value x (1 + (nominal annual rate of return / number of compounds per year)) ^number of compounds per year x time period

Effective annual rate of return = the return that would have been observed if compounding had occurred annually.

Calculating effective annual return = actually very simple = simply use the compounding formula and replace n with 1

Annual rate decreases as compounding period becomes smaller

3

Discrete random variable vs continuous random variable?

What does a probability density function do?

Discrete = can take only countably many values = tossing a tie or flipping a coin has set and specific outcomes

continuous = takes values in an interval or over the whole real line.

Probability density function = gives the probability that our discrete random variable X will take the value x.

4

Define the following terms:
Mean
variance
Covariance
Correlation coefficient

Mean = average
variance = how far a set of (random) numbers are spread out from their average value
Covariance = a measure of how much two random variables vary together
Correlation coefficient = a statistical measure of the degree to which changes to the value of one variable predict change to the value of another

5

When calculating portfolio returns through Linear combinations of two random variables what exactly are we doing?

1. We are first identifying the weight of each asset in our portfolio i.e. in an equally weighted portfolio with 2 assets A and B a = 1/2 and b = 1/2 as they each constitute a half (note the capital A = the asset and the little a = the weighting (for use in the formulas)
2. We are then determining the sample variance and covariance of our portfolio based upon the sample variance, covariance, and weighting of our Assets A and B

6

What is a matrix
What are the components and Dimensions of a Matrix
What is a column vector and a row vector?
How do we tell if two matrices are well defined?

Matrix = a rectangular array of numbers
Components = All of the values within the matrix
Dimensions = Rows x Columns (note: equal rows and columns means we have a Square matrix)
Column vector = One specific column ( m x 1)
Row vector = one specific row (1 x n)

Well defined = the first matrices must have the same number of columns as the second matrices rows

7

How do we perform matrix transposition

How do we perform matrix addition and subtraction

How do we perform matrix multiplication?

How do we perform matrix inversion?

Matrix transposition: First row of first matrix becomes first column of the transposed first matrix

Firstly perform matrix transposition Then simply add a1,1 to b1,1 etc.

Matrix multiplication is slightly tricky. For our product we draw our rows from our first matrix and our columns from our second matrix. That is to say we multiply each element from our first row in matrix 1 with each element in column 1 of matrix 2 and then add the numbers. We then multiply each element from row 1 of matrix 1 with each element from column 2 of matrix 2. Then repeat using row 2 of matrix 1

Inverse = we will not have to invert complex matrices, only diagonal ones with 0s around. Simply find the inverse of each diagonal element... super easy

8

What is a diagonal matrix?
What is an identity matrix? (In) (capital I small n)
What is an invertible matrix? How do we denote Y as the inverse matrix of X?
How do we demonstrate that Y = X^-1?

Diagonal matrix = an n x n matrix with zeros everywhere, except on its diagonal.
Identity matrix = a diagonal matrix with 1s and 0s only
Invertible matrix = where XY = In = YX

Y as the inverse of X = X^-1

Y = X^-1: Here we would simply want to show that XY = In... That is that Matrix X multipled by matrix Y is an identity matrix.

9

How do we measure the risk of a two asset portfolio?

At first this may seem rather complex, in fact it is not.
1. Understand the that risk is measured by variance
2. weight each of your assets
3. Measure the covariance of your assets
4. Simply utilise the variance formula for your portfolio.
- This formula simply adds the squared weight of asset one multiplied by its squared standard deviation to the squared weight of asset two multiplied by its squared deviation... and then adds 2 times the weight of asset one times the weight of asset two times the co-variance of the weights.

10

What is the budget constraint?
When does a portfolio contain a long position?
When does a portfolio contain a short position?
What do we mean by short selling?
How could we benefit from short selling?

Budget constraint = Because portfolio weights define the fractional amount invested in each asset,
they must sum to 1.
Long position = Xi > 0
Short position = Xi < 0
Short selling = achieved by borrowing the security from someone and then selling it in the market. The short seller is obliged to return the security to its rightful owner after a certain time, together with any cashflows (such as dividends) payed by the security in the interim. When short selling is allowed, some portfolio weights can be either negative or greater than one.

Short selling benefit = if we invest the proceeds from the short sell into an investment with higher expected rate of return

11

What is a combination line?
What does a combination line tell us?
How do we find the equation for a hyperbola in mean-standard deviation space?
What is the Global MVP (G)?
How can we find the required weight of asset 1 in a two asset portfolio which satisfies the point G on a hyperbola in mean-standard deviation space?

Combination line = the set of points in mean-standard deviation space that are achievable by combining together two assets into one portfolio

Combination line = A combination line shows the relationship between the expected return and
the standard deviation for a portfolio of two assets. It indicates, for differing portfolio weights for each of the two assets, how the expected rate of return and standard deviation change.
Hyperbola equation = simply rearrange the formula for expected return of a two asset portfolio in terms of X1 and substitute into the formula for variance of returns for a two asset portfolio
Global MVP = the portfolio with the least possible variance of returns is known as the global minimum variance portfolio = the vertex of the hyperbola in mean-standard deviation space
Finding Asset 1 weight for G = take the hyperbola equation in terms of X1 and find the first derivative and equate it to 0. (This equation is given to us)

12

How do we denote perfect positive and perfect negative correlation in regards to a two asset portfolio?
What type of combination lines will we have in these special cases?
In the case of a risk free asset in our portfolio when are we lending and when are we leveraging at the risk free rate?

Perfect positive: p1,2 = +1
Perfect negative p1,2 = -1

Combination lines = linear and G has zero variance (is riskless) (this is true for both perfect positive and perfect negative. This is helpful to remember when sketching the combination line in these instances as we know our two combination lines meet somewhere on our Y axis where our sigma is 0)

lending = X1 < 1
Leveraging = X1 > 1 (borrowing)

13

What do we mean by Utility in regards to investment decisions?
What do we mean by Marginal utility is diminishing?
How can it be shown that an investor is risk averse?

Utility = positive outcome = (happiness) = we always attempt to maximise our Utility when investing (maximise our happiness), our positive outcome.

Diminishing = We value each additional unit of wealth less than the previous unit (if you have $100 an extra $100 is amazing, a further extra $100 is somewhat marginally less amazing and so on and so on.

Risk averse = utility function is concave

14

When is an asset said to be efficient, inefficient?

You have 3 assets. A1 and A2 have no assets in their upper left quadrant, A3 has A2 in its upper left quadrant? Which asset do you choose?

What is an indifference curve?

Efficient = when no other assets exist to the upper left of our asset in question in mean-standard deviation space (imagine a plane with x = mean and y = std)
Inefficient = when above does not hold true

Choose = either A1 or A2 as they are efficient, which of the two you choose in the end depends on your utility function (risk averse or not?)

Indifference curve = collection of points over which expected utility is the same i.e. as risk increase our rise increases proportionally such that our utility is always the same.... a riskier investment along this curve will bring about a level of utility proportionally equivalent to that of a less risky investment along this curve. (However with multiple curves we always choose the upper left one)

15

We want to choose a combination of two assets which when plotted in mean-standard deviation space are both efficient, how do we choose our weighting of each asset?

We must first establish our indifference curve utilising our level of utility. We then combine our two assets to produce a hyperbola in mean-standard deviation space. The optimal portfolio the investor should choose is the point on this hyperbola that is just tangent to the investor’s indifference curve.

16

What is the Markowitz problem?
What is the MVS? What does it look like in mean-std space vs mean-variance space?
What is the Two fun theorem?

Markowitz problem = finding the vector of portfolio weights which minimize the variance of our portfolio

MVS = Minimum Variance Set (MVS) = All portfolios (combinations of asset weights) that have the minimum variance for a certain target portfolio returns.

Mean-std = hyperbola
Mean-variance = parabola

two fun theorem = states that the entire MVS can be created by combining together any two distinct portfolios that already exist on the MVS = suggests an investment management company only needs to create two funds for all their clients

17

When defining the lagrangian for a minimization problem including risky assets and one risk free asset (using matrix notation) what is our target return constraint?
Note: our lagrangian removes the budget constraint when we include a risk free asset and as such we only have our target return constraint

What are our two first order conditions when including a risky asset?

When including a risky asset does the variance equation of the portfolio change? explain

target return = λ[μP − {xμ + rF (1 − xT 1)}] (here the right hand side expression denotes the risk free return multiplied by the weighting of the risk free asset in the portfolio.

First order conditions:
1: Ωx∗ = λ(μ − rF 1)
2: (μ − rF 1)Tx∗ = μP − rF .

Variance equation = σ^2P = xTΩx = does not change because the risk-free asset does not add risk to the portfolio

18

What are our first order conditions with a multiple asset portfolio?

1: Ωx∗ = λ1 + γμ
2: 1Tx∗ = 1
3: μTx∗ = μP

19

Determining the MVS graphically with multiple assets and one risk free asset:

What are the two steps involved here?

What is the Tangency portfolio?

As per our formula sheet how do we go about producing the MVS equation?

Step 1: determine the set of possible portfolios that can be constructed using only the risky assets. This will generally be a shaded hyperbolic region in mean-standard deviation space

Step 2: draw the combination lines that connect the risk-free security to each of the possible risky portfolios. When aggregated together, these
lines form a solid triangular region in mean-standard deviation space

Tangency portfolio = The aforementioned triangular formation will have an upper boundary (the combination line with the highest slope) and this is the MVS of the triangular formation. This MVS will touch the hyperbola of the MVS for only the risky assets at one point. This point is denoted T and is the Tangency portfolio

MVS equation = rearrange the equation for σ^2P in terms of μP = the equation produces two symmetric lines in mean-standard deviation space that intersect the y-axis at rf

20

What does the one fund theorem state?
What does the one fund theorem suggest?

One fund theorem = When a risk-free asset exists, the entire set of efficient portfolios can be created by combing together one fund of risky assets, T, with the risk-free asset. The fund T is the tangency portfolio

One fund theorem suggests = an investment management company only needs to create one fund of risky assets for all of their clients

21

What does Tobins seperation Theorem state?

an individuals investment decision can be separated into two parts

First: the investor determines the tangency portfolio T, i.e., the portfolio where the combination line that connects the risk-free asset with the risky asset MVS is just tangent to the MVS.

Second: the investor combines this portfolio T with the risk-free asset in such a way that his/her utility is maximised.

22

What are the two restrictions we consider on the risk-free asset?

1. When risk-free borrowing is not possible
2. When risk-free borrowing is available but at a higher rate than is available for lending

23

What are the implications where we have No Risk-free borrowing (2)?

1. The MVS is no longer a straight line as we exceed T. This point of tangency connects to the MVS with risky assets only. Hence we have a straight line between F and T and then curvature equal to that of the MVS with risky assets only beyond point T

2. the one fund theorem no longer holds

24

If an asset is completely uncorrelated with the market what will its Beta and Mu equal?

under the CAPM what symbol do we use to denote portfolio risk and individual asset risk?

What does our Epsilon term measure?

Why does Epsilon = 0 when it comes to the SML?

Interesting Note: If i have a portfolio of 1 asset which has high diversifiable risk then i wont be paid for holding that risk. i.e. it is better to hold an asset with high systematic risk

Beta uncorrelated = 0 = Beta measures the sensitivity of an assets return to the return of the market portfolio
Mu = Risk free

Individual asset risk = Beta
Portfolio risk = Sigma

Epsilon = Error term = Diversifiable (idiosyncratic) risk
Epsilon at 0 = Because the SML measures systematic risk (beta) and hence there is 0 un-systematic risk. This is also why we use Mu (Mu is an average) and when we average out Epsilon we get 0 because (essentially we diversify it)

25

Isobeta lines:
What is an Isobeta line
What does the line imply?

Isobeta line = a line drawn in mean-standard deviation space starting on the CML and extending to the right.

Isobeta line implies = that all assets along the line have the same Beta, however as we move to the right our Variance increases... because we are adding more diversifiable risk. So although all assets along the line have the same Beta and hence the same return... assets to the right have greater total variance due to increased diversifiable risk.

26

What are risk factors?
What is a factor model?
What type of model is the CAPM?
What is the SFM?
If we rearrange the CAPM what can we find alpha to be?

Risk factors = underlying sources of randomness which effect the returns of an asset: Market index, GDP, employment rate, interest rate, etc.

Factor model = expresses the connection between the returns of individual assets and a few risk factors

CAPM = is a single factor model (however, it is not The Single Factor model)

SFM = The single factor model

Rearrange CAPM: alpha = rf (1 - B)

27

What is the characteristic line?
Explain the notion that the characteristic lines rise over run equates to beta?

Under the SFM how do we calculate covariance between two assets?

Under the SFM how do we calculate R square? How do we interpret this result of our R square in terms of diversification?

Characteristic line = similar to the SML or CMl, they characteristic line is a line of best fit drawn through the plot of return on i Vs return on market.

Rise over run = Beta measures the sensitivity of an assets return to the return of the market portfolio. We are analysing the returns of asset i vs returns of the market and as such rise over run gives us the Beta ratio

Covariance = Bi x Bj x market sigma^2

R^2 = systematic risk / total risk = simply calculate systematic risk using our factor model formula provided to us and divide by the total risk

R^2 and diversification = well obviously if our R^2 is 1 then our systematic risk must make up our total risk and henceforth there is no unsystematic risk = no diversifiable risk...