Portfolio Theory Flashcards
efficient frontier
optimal portfolios offering the maximum possible expected return for a given level of risk
modern portfolio theory
theory on how risk-averse investors can construct portfolios to optimize or maximize expected return based on a given level of market risk
–> risk is an inherent part of higher reward
assumptions of portfolio theory
- investors want to maximize returns for a given level of risk –> will choose the asset with the higher expected rate of return
- investors are generally risk averse –> will choose the asset with the lower perceived level of risk
- risk is measured by the volatility of expected returns
- investors base decisions only on expected return and risk
portfolio weights
xi = value of investment i / total value of portfolio
–> portfolio weights must add up to 1 or 100%
expected return of a portfolio
weighted average of the returns on the investments in the portfolio
R(p) = x1R1 + x2R2 + … + xnRn
mean return of a portfolio
simple average on each periods returns
–> adding all returns in each period and then dividing this figure by the total number of returns (periods)
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R = (Ri1 + Ri2 + Ri3 … + Ri) / T
variance of a portfolio
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Var(R) = 1/T x (sum of) (Rt - R)²
standard deviation of a portfolio
SD(R) = sqrt(Var(R))
correlation
how the movement of stocks or the returns of two stocks influcence each other
important measure for diversification because you do not want a portfolio of stocks that are completely correlated with each other
positive / negative / no correlation
positive correlation: movement in the same direction; return of both stocks is higher or lower than the average at the same time
perfect positive correlation: linear relationship, for every unit one stock’s return increases, the other stock’s return increases one unit as well
negative correlation: movement in different / opposite directions; one stock has a higher-than-average return when the other has a lower-than-average return
perfect negative correlation: linear relationship, for every one unit one stock’s return increases, the other stock’s return decreases one unit
no correlation / uncorrelated: one cannot make any assumptions on the movements of the returns of two stocks, they do not influence each other and the returns are completely unrelated to each other
diversification and correlation
if two stocks are perfectly positively correlated, then there is a risk-return trade-off between the two securities
–> benefits to diversification whenever the correlation between two stocks is less than perfect
the effect of diversification on portfolio risk
with an increase in number of stocks of a portfolio, the portfolio’s volatility will decrease if it is diversified well
the decrease of volatility will go towards a certain volatility that is non-diversifiable / systematic (market risk)
diversification can decrease the company risk / unsystematic / diversifiable volatility
covariance
measure of how two assets relate / move together relative to their individual mean values over time
positive covariance: two returns tend to move together
negative covariance: two returns tend to move in opposite directions
can range from - infinity to + infinity
covariance formula
cov(R1R2) = [r1 - E(r1)] x [r2 - E(r2)]
–> return 1 - expected return 1 multiplied by return 2 - expected return 2
covariance of historical data is divided by (T - 1)
correlation
relative measure of the relationship between two assets
a measure of the common risk shared by stocks that does not depend on their volatility
can range from -1 to +1
correlation formula
cr = cov / (standard deviation 1 x standard deviation 2)
benefits of diversification based on portfolio risk calculations
if portfolio risk is less than the risk of each repsective asset, the invesrtor should invest in both assets instead of one or the other
effect of a portfolio’s risk of investing in assets that are less than perfectly correlated
when correlation = +1 –> portfolio risk is unaffected, it is simply the weigthed average of the sd of the two assets
when correlation = 0 –> portfolio risk is reduced witht he addition of two assets that are not perfectly correlated
when correlation = -1 –> maximum risk reduction occurs with the addition of two assets that are perfectly negatively correlated
maximum risk reduction of portfolio
maximum risk reduction can be achieved with the addtition of two assets that are perfectly negatively correlated
steps of calculating covariance of a two asset portfolio with historical data
- calculate the mean / average return of each individual asset
- calculate the deviation from the mean / average return for each asset and year
- multiply the deviation of the assets with the corresponding ones from that year
- add all years together
- for historical data, divide the sum by (T-1)
formula variance of a two-stock portfolio
Var = (weight asset 1)² x (sd asset 1)² + (weight asset 2)² x (sd asset 2)² x 2 x (weight asset 1) x (weight asset 2) x (sd asset 1) x (sd asset 2) x (correlation of the two assets)
or
Var = (weight asset 1)² x (sd asset 1)² + (weight asset 2)² x (sd asset 2)² x 2 x (weight asset 1) x (weight asset 2) x (covariance)
effects of correlation on diversification
a lower correlation can lead to greater diversification effects in a portfolio
even though the return of stock A is more volatile than the return of stock B, a lower correlation of A and C can lead to better diversification and therefore a lower portfolio volatility
formula variance of a three asset portfolio
Var = (weight asset 1)² x (sd asset 1)² + (weight asset 2)² x (sd asset 2)² + (weight asset 3)² x (sd asset 3)² x 2 x (weight asset 1) x (weight asset 2) x (sd asset 1) x (sd asset 2) x (correlation of 1 and 2) + 2 x (weight asset 1) x (weight asset 3) x (sd asset 1) x (sd asset 3) x (correlation of 1 and 3) + 2 x (weight asset 2) x (weight asset 3) x (sd asset 2) x (sd asset 3) x (correlation of 2 and 3)
inefficient and efficient portfolio
inefficient portfolio: it is possible to find another portfolio that is better in terms of expected return and volatility
efficient portfolio: there is no way to reduce the volatility of the portfolio without lowering its expected return
What is a dominant portfolio?
- there is no other equally risky portfolio that has a higher expected return
- there is no other portfolio with the same expected return that has less risk
What is the efficient frontier?
reducing total risk through diversification because almost all assets are less than perfectly correlated –> considering many assets at various weights
–> efficient frontier represents all the dominant portfolios in risk/return space
–> plotting all assets and portfolios in a return and standard deviation, it will form a broken eggshell shape
–> portfolios along efficient frontier have the highest return for the quantity of risk they are willing to assume
Why would an investor always choose a portfolio along the efficient frontier?
You would never choose an asset or portfolio beneath the efficient frontier because they have a lower return with similar risk
Minimum Variance Portfolio
- the possible portfolio with the least amount of risk
- portfolio composed of risky assets with the smallest standards deviation
efficient portfolio
a portfolio that contains only systematic risk
–> no way to reduce the volatility of the portfolio without lowering its expected return
market portfolio
an efficient portfolio that contains all shares and securities in the market (often the S&P 500 is used)
calculating the weights of the MVP
w(A) = [(sd B)² - cov(AB)] / [(sd A)² + (sd B)² - 2 x cov(AB)]
w(B) = 1 - w(A)
long position vs short position
long position = a positive investment in a security
short position = a negative investment in a security
–> short sale means selling a stock you do not own and then buying the stock back in the future
—> used when you expect a stock price to decline in the future
value of a short sale
in calculations we think of short sale as a negative investment, so we use e.g. - 10.000€ for the calculation of the portfolio weight, giving it a negative weight in the portfolio
capital market line (CML)
shows the risk and return of all possible portfolios created from some combination of treasury bills (risk-free asset) and the market portfolio
–> runs through y-Achse and the market portfolio in the graph
adding a risk-free asset
risk can be reduced by investing a portio of the portfolio into a risk-free investment
–> will likely reduce the expected return
tangent portfolio
- where the line with the risk-free investment is tangent to the efficient frontier of risky investments
- portfolio with the highest sharpe ratio
- combinations of the risk-free asset and the tangent portfolio provide the best risk and return trade-off available to an investor
- tangent portfolio is efficient
- all efficient portfolios are combinations of the risk-free investment and the tanget portfolio
sharpe ratio
measures the ratio of reward-to-volatility provided by a portfolio
share ratio = portfolio excess return / portfolio volatility
excess return = expected return - return of risk free investment
optimal risky portfolio
the combination of risky assets that maximizes the sharpe ratio
–> maximizing the slope of the line
