chapter 8 book: valuing portfolios and stuff Flashcards
Risk
the possibility of incurring harm
ex post returns
past or historical returns
ex ante returns
expected returns
the return on an investment consists of which two components
the income yield
the capital gain (or loss) yield
The income yield
the return earned in the form of a periodic cash flow received by the investors
the income yield formula
Income yield = CF1 / P0
CF1 = the expected cash flows to be received
P0 = the purchase price (or beginning market price)
The capital gain (or capital loss)
measures the appreciation (or depreciation) in the price of the asset from some starting price, usually the purchase price or the price at the start of the year
capital gain yield formula
Capital gain yield = (P1 - P0) / P0
capital gain
the appreciation in the price of an asset from some starting price
usually the purchase price or the price at the start of the year
capital loss
the depreciation in the price of an asset from some starting price
usually the purchase price or the price at the start of the year
total return formula
income yield plus the capital gain (or loss) yield
(CF1 + P1 - P0) / P0
paper losses
capital losses that people do not accept as losses until they actually sell and realize them
our decision to acknowledge paper gains and losses depends on what?
our investment horizon
A day trader
someone who buys and sells based on intraday price movements
what do we mean when we say that investors have to mark to market the prices of all financial securities over the relevant investment horizon?
investors always carry securities at the current market value regardless of whether they sell them
–> the total return includes the effect of paper gains and losses on securities not yet sold
How can we measure the ex post or historical returns?
The arithmetic mean or geometric average
The arithmetic mean or arithmetic average
the most commonly used value in statistics
the sum of all of the returns divided by the total number of observations
The arithmetic mean or arithmetic average formula
Arithmetic mean (AM) = (Eri / n)
ri = the individual returns
n = the total number of observations
The geometric mean
measures the compound growth rate over multiple periods
this is the growth rate in the value invested or, equivalently, the compound rate of return
The geometric mean formula
(1 + rn) ^(1/n) - 1
the standard deviation
a measure of risk over all the observations
A more accurate measure of risk
the square root of the variance, denoted as σ
The Arithmetic Mean is appropriate to use when?
when we are trying to estimate the typical return for a given period, such as a year
what do we use when we are interested in determining the “true” average rate of return over multiple periods?
the Geometric Mean
–> We use the GM because it measures the compound rate of growth in our investment value over multiple periods
the better way to estimate the average return when we are interested in the performance of an investment over time
the geometric mean
gives us a better insight
expected returns
estimated future returns
often estimated based on historical averages
problem with expected earnings
there is no guarantee the past will repeat itself
expected returns formula
ER = E(ri · Probi)
ER = the expected return on an investment
ri = the estimated return in scenario i
Probi = the probability of state i occurring
difference between arithmetic mean and expected returns
they both look at past data
the difference is that expected returns consider different probabilities while arithmetic mean weights the probabilities equally
which is better between the scenario based and historic approach in the short term future? why?
scenario based approach
because where we are today has a huge bearing on what is likely to happen over a short period
which is better between the scenario based and historic approach in the long term future? why?
historic based approach
tends to be better because it reflects what actually happens, even if it was not expected
the range
the difference between the maximum and minimum values
why is the standard deviation a more accurate measure of risk than the range?
because the range uses only two observations, the maximum and minimum, whereas the standard deviation uses all the observations
The formula of the standard deviation for a series of historical or ex post returns is
historical approach
ex post σ = ((E(ri - r_)^2) / (n - 1))^(1/2)
ri = the return in year i
r_ = the average return
n = number of observations
basically, whats inside the square root is the variance
the variance
denoted as σ^2
square of the standard deviation
expressed in units of %2
The formula of the standard deviation for a scenario based ex ante returns
scenario based approach
ex ante σ = (E(Probi)(ri - ER)^2)^(1/2)
ex ante is scenario based or historic approach
scenario based approach
ex post is scenario based or historic approach
historic approach
The coefficient of variation (CV)
an alternative measure of risk compared to the variance or standard deviation
a standardized measure of dispersion that can be used to compare the risks of two investments with different arithmetic means
not expressed as a percentage
The coefficient of variation (CV) formula
Standard deviation of returns / Arithmetic mean return
a higher risk is a higher or lower coefficient of variation (CV)?
a lower coefficient of variation (CV)
A portfolio
a collection of securities, such as stocks and bonds, that are combined and considered a single asset
may refer to the holdings of a single investor or to holdings that are managed as a unit by one or more portfolio managers on behalf of their clients
modern portfolio theory (MPT)
The study of portfolios and the potential gains related to them
the theory that securities should be managed within a portfolio, rather than individually, to create risk-reduction gains
also stipulates that investors should diversify their investments so as not to be unnecessarily exposed to a single negative event
why investors should diversify their investments?
so that they are not unnecessarily exposed to a single negative event
The expected return on a portfolio
the weighted average of the expected returns on the individual securities in the portfolio
The “portfolio weight” of a particular security is the percentage of the portfolio’s total value that is invested in that security
The expected return on a portfolio formula
ERp = E(wi · ERi)
ERp = the expected return on the portfolio
ERi = the expected return on security i
wi = the portfolio weight of security i
The standard deviation of a two-security portfolio (first formula excluding the correlation coefficient)
σp = (((WA)^2)((σA)^2) + (WB)^2)((σB)^2) + 2(WA)(WB)(COVAB))^(1/2)
σp = the portfolio standard deviation
COVAB = the covariance of the returns on security A and security B
covariance
a statistical measure of the correlation of the fluctuations of the annual rates of return of different investments
related to the correlation coefficient
covariance first formula formula (excluding the correlation coefficient)
COVAB = EProbi · (rA,i - r_A)(rB,i - r_B)
rA,i = the ith return on security A
rB,i = the ith return on security B
the correlation coefficient
ρAB
related to covariance and individual standard deviations
a statistical measure that identifies how security returns move in relation to one another
has a maximum value of +1.0, which denotes perfect positive correlation, and a minimum value of −1.0, which denotes perfect negative correlation
the correlation coefficient formula
ρAB = COVAB / (σA · σB)
covariance new formula with the correlation coefficient
COVAB = ρAB · σA · σB
the formula to calculate the portfolio standard deviation with the correlation coefficient
σp = (((wA)^2)((σA)^2) + (wB)^2)((σB)^2) + 2(wA)(wB)(ρAB · σA · σB))^(1/2)
what does a positive correlation coefficients imply?
that the returns on security A tend to move in the same direction as those on security B
what does a negative correlation coefficients imply?
the returns on security A tend to move in the opposite direction to those on security B
what does the covariance measure when we compare it to the correlation coefficient? (it always measures it anyways)
measures the strength, or magnitude, of the relationship between two variables
the lower the correlation coefficient, the lower or higher the standard deviation involved?
the lower
the secret of MPT
by combining securities in a portfolio, we can reduce risk
risk reduction increases as we combine securities that are less than perfectly correlated
what does it mean when we say that the relationship between the correlation coefficient and the standard deviation of our pertfolio is bowed?
because, as it decreases towards -1, it directly touches the horizontal line perpendiculairement
on the other hand, as it is at 1, it becomes on straight line (does not curve anymore)
how to calculate the standard deviation of a portfolio that removes all risk?
σp = wσA - (1 - w) · σB
when p = -1
where, the singular w = σB / (σA + σB)
the basis of hedging
The perfect negative correlation case
taking an offsetting position so as to minimize risk
three securities portfolios standard deviation
just write it on your cheat sheet if you need it bro
the father of modern portfolio theory
Harry Markowitz
Harry Markowitz’s three main assumptions for investors
- Investors are rational decision-makers.
- Investors are risk averse
- Investor preferences are based on a portfolio’s expected return and risk (as measured by variance or standard deviation)
risk averse
to dislike risk and require compensation to assume additional risk
efficient portfolios
those that offer the highest expected return for a given level of risk or offer the lowest risk for a given expected return
The first step in the Markowitz analysis
to determine the expected return-risk combinations available to investors from a given set of securities
we allow the portfolio weights to vary
the minimum variance frontier
the curve produced when determining the expected return-risk combinations available to investors from a given set of securities by allowing the portfolio weights to vary
the attainable portfolios
portfolios that may be constructed by combining the underlying securities
those that lie on the minimum variance frontier
the minimum variance portfolio (MVP)
a portfolio that lies on the efficient frontier and has the minimum amount of portfolio risk available from any possible combination of available securities
The importance of the minimum variance portfolio MVP
portfolios lying below it, on the bottom segment of the minimum variance frontier, are dominated by portfolios on the upper segment
the efficient frontier
the segment above the inefficient frontier in the minimum variance frontier
Rational, risk-averse investors will be interested in holding only those portfolios (which offer the highest expected return for their given level of risk)
diversification
the process of investing funds across several securities, which results in reduced risk
Random diversification or naïve diversification
the act of randomly diversifying without regard to relevant investment characteristics, such as company size, industry classification, and so on
basically, what I’ve been doing for 90% of the time
unique (non-systematic) risk or diversifiable risk
The part of the total risk that is eliminated by diversification
the market (systematic) risk or non-diversifiable risk
The part that is not eliminated by diversification
This portion of the risk cannot be eliminated because all the securities in the portfolio will be directly influenced by overall movements in the general market or economy
formula for total risk
market (systematic) risk + unique (non-systematic) risk
non-diversifiable risk + diversifiable risk
why evidence suggests the benefits of international diversification have been declining?
because global equity markets become more integrated
