Portfolio theory Flashcards
(28 cards)
Expected return and Portfolio return
∑pi*ri
rp =∑ wi *ri
Define diversification
Diversification is the process of increasing the number of securities within in a portfolio in order to reduce total risk (specifically unsystematic) without lowering expected return.
-Diversification is effective when assets don’t move perfectly together—some rise while others fall—so their risks offset each other.
relation between diversification and correlation coefficient
Diversification most efficient when the correlation coefficient p between assets is low or negative because this means the assets don’t move together, helping to reduce overall risk.
Perfect positive correlation means no risk reduction.
Perfect negative correlation allows for complete risk elimination
=Pab = 1, no unsystematic risk can be diversfied.
-Pab = -1, all risk can be diversified away, move it exact opposite,
-Pab = 0 zero correlation.
-1+1 = unlileky to occur in real world situation.
describe covariance
-Covariance, cov, refers to an absolute measure of the extent to which two sets of numbers, assets, move together over time (directional relationshiop of two assets over time)
COVab =∑(Rai - e(Ra)) * (Rbi - e(Rb)) / n
COVav = pab * σa * σa
-Sample covariance
e(ra/b) is the mean/average return of assets a and b over the time period.
describe correlation coefficinet
-refers to a relative measure of both strength and direction of a relationship between two variables, assets. Value between -1-1.
ρab = COVab /σa*σb,
given bycovariance between 2 assets divided by product of the standard deviations
define systematic and unsystematic risk + relation to correlation coeffieints + draw on graph
-sytematic risk refers to overall market risk which is undiversfiable and effects returns of all assets. Eg) policy, econmic business cycles, finaltion etc.
-Unsystematic risk, also known as diversifiable risk, is risk that is industry/asset specific and can be reduced/eliminated through diversification. Unsystematic risk is decreasing in number of portfolio assets.
Describe risk aversion + eg\
-RA refers to the preference to avoid risks especiaclly. More specifically its a Preference for certainty over a gamble with equal or higher expected value
-risk-averse people prefer a guaranteed outcome (e.g., $100) over a risky alternative with the same expected value (e.g., 50% chance of $200 or $0)
-Risk averse indivudals have a concave utility function.
Formula for the variance (risk) of a two asset portfolio:
2 asset portfolio variance:
-σp^2 = w1^2σ1^2 + w2^2σ2^2 + 2(w1w2p12σ1σ2)
-SD of port with 2 assets is just the SWRT of the variance.
-The variance of a two-asset portfolio is derived by combining the individual variances of each asset weighted by their weight, plus two times the product of their weights and the covariance between the assets.
What is the correlation coeffient between between the market return and a risk-free asset + why
0
-The return of the risk free asset is constant and doesnt change relative to market changes.
The most important criteria when adding new investments to a portfolio is..
correlation of the new investment with the rest of the portfolio
if an individual owns only one security the most appropriate measure of risk is + why
-Standard deviation, as it captures the total volatility of that security’s returns, reflecting both systematic and unsystematic risks.
REMINDER Variance=σ^2
SD = sqrt(variance)
formula for expected standard deviation and expected variance
Variance σ^2 = ∑(pi * ( ri - E(R) )^2 )
-sum of prob state pccuring * (return in that state - baseline/expected return)squared
-Standard deviation σ is square root of the variance sqrt(variance)
Variance (risk measure) for historical returns
σ^2 = ∑( Ri - E(R) )^2 / N
-n is population when toy have every retur
what does the steepness of utility curves indicate?
An investor’s preferences for risk and return are shown by their utility curve.
Steep utility curve: Indicates high risk aversion; the investor requires a higher reward for taking on additional risk.
Flat utility curve: Indicates low risk aversion; the investor is more willing to risk for smaller increases in expected return.
Utility curve slope: Represents the trade-off between risk and return—steeper means more risk-averse, flatter means more risk-tolerant.
Assumptions of Markowitz Portfolio theory 1952
- Investors are rational and risk-averse
- Portfolio decisions based on mean and variance
- Returns are normally distributed
- No transaction costs or taxes
Describe the envelope curve + how its dervied
-The envelope curve shows all possible combinations of N stocks (all possible portfolios) in a return/risk space (Rp, σp). Plots all potential portfolios in a risk return space.
-The envelope curve is derived by calculating all possible combinations of N risky assets (with different weights), plotting their expected returns against portfolio risk (standard deviation), and tracing the outer boundary of these points to form the curve.
Define the efficinet frontier and what it shows
The efficient frontier shows the set of portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given level of return. portfolios considered “optimal” as they represent the best possible trade-offs between risk and return.
-Any portfolio/point on the efficient frontier maximises expected return for a given risk level, or minimises risk for a given return level, any point/portfolio on frontier is better than any portfolio below it, as it provides more return for the same or less risk.
describe the process to get from envelope curve to the efficient frontier + describe a point on the efficient frontier
Start with the envelope curve and identify the minimum var portfolio, portfolio with the lowest possible risk (Sd).
Eliminate the lower portion of the curve below the minimum var point, as these portfolios are dominated (provide lower returns for the same or higher risk).
Or, iteratively remove all dominated portfolios until you are left with the efficient frontier.
The remaining upper portion of the curve is called the efficient frontier.
Portfolios on the efficient frontier are not dominated — they are optimal choices for investors, offering the highest expected return for a given level of risk or the lowest risk for a given expected return.
This set of portfolios maximizes expected return for a given risk level or minimizes risk for a given return level.
Any portfolio on the efficient frontier is prefferd to any portfolio below it, as it provides more return for the same or less risk
Define the capital market line and how its dervived + formula
-introduce a risk free asset
CML refers to a line showing risk-return trade-off for possible portfolios combining a risk-free asset and the market portfolio.
-CML given by Rp = Rf + σp* (Rm - Rf)/σm
Rm market port return, σm: sd of market portfolio.
-Slope is the shrape ratio, derive the fomrula by
-CML shows Shows all possible efficient portfolios formed by mixing the risk-free asset and the single risky stock.
-shows that investors can both lend and borrow at a risk-free rate of return
-all points on the CML (except the market portfolio itself) dominate portfolios on the efficient frontier of risky assets alone, as you can lend/borrow
define the market portfolio + market portfolio beta + interpet
The market portfolio M is A portfolio that includes all risky assets in the economy, not just stocks, weighted by their market values. M lies on the eff frontier providing the highest return for any given level of risk
beta of mkt port is 1.0, serving as a benchmark for assessing the systematic risk of individual assets relative to the market.
Beta > 1.0: Indicates that the security or portfolio is more volatile than the market. For example, a beta of 1.2 suggests the asset is 20% more volatile than the market.
Beta < 1.0: Signifies that the security or portfolio is less volatile than the market.
Beta = 1.0: Implies that the security or portfolio’s volatility matches that of the market.
Describe tangecy between Cml and efficient frontier
The CML is a straight line starting at the risk-free rate, representing portfolios combining the risk-free asset and the market portfolio.
The Efficient Frontier is a curve showing the best risky portfolios (without the risk-free asset).
This tangency point represents the optimal combination of risky assets and the risk-free asset that provides the highest expected return for a given level of risk. Also represents the market portolio.
At this point, the Sharpe Ratio (excess return per unit of risk) is msximised.
Portfolios on the CML dominate those on the efficient frontier alone because mixing with the risk-free asset improves the risk-return trade-off.
define market portfolio and the sharp ratio
Market Portfolio:
The market portfolio is the tangency point/portfoli on between the efficient frontier and the market porfolio
-*The market portfolio is the portfolio of all stocks in the economy
with weights equal to their relative market values over total market value
Sharpe Ratio:
The Sharpe Ratio measures the risk-adjusted return of a portfolio. It is calculated as the excess return of the portfolio per unit of risk.
What does Markowitz Portfolio Theory explain?
It explains how investors choose portfolios to maximize their expected return for a given level of risk (or minimize risk for a given return), based on their preferences
