Initiate data input function Flashcards
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
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
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.
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
When calculating portfolio returns through Linear combinations of two random variables what exactly are we doing?
- 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)
- 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
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
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
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.
How do we measure the risk of a two asset portfolio?
At first this may seem rather complex, in fact it is not.
- Understand the that risk is measured by variance
- weight each of your assets
- Measure the covariance of your assets
- 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.
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
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)
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)
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
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)
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.
