Portfolio Theory Flashcards
What is a portfolio?
A collection of securities with different risks and returns associated.
What is the realised return?
A geometric average of past returns.
What are the three characteristics of portfolios?
First: Expected Return
Second: Variance
Third: Covariance
What is the expected return?
Expected return is the best guess for the return that will be achieved the next period.
What are the methodologies we can use to find the expected return?
Distribution of Returns and Sample of Past Returns
What is the formula for the expected return when we use the methodology of distribution of returns?
E(ri) = ∑piri
What is the excel formula we should use to find the expected return (using distribution of returns)?
=sumproduct()
When can we use the mean return to find the expected return?
When observations are independent and identical distributed, the Law of Large Numbers states that the mean return converges in probability to the true expected return,
What is the excel formula we should use to find the expected return (using a sample of past returns)?
=average
What is the variance of the return?
The best guess for the measurement error. It tells us how much should an investor expect to fail prediction. It is the measure of risk.
What are the methodologies we can use to find the variance?
Distribution of Returns and Sample of Past Returns
What is the formula to find the variance using the distribution of returns?
∑ pi [ri - E(ri)]^2
True or false: The variance of the mean return is the true variance.
False
What is the volatitility?
The risk, it’s the standard deviation
In excel, what formula should we use to find the variance?
=var.s()
In excel, what formula should we use to find the standard deviation?
=stdev.s()
What does the covariance measure?
The relationship between two random variables.
What does it mean when a covariance is positive?
On expectation, the returns of the two securities move in the same direction.
What does it mean when a covariance is negative?
On expectation, the returns of the two securities move in opposite directions.
What does it mean when a covariance is zero?
We suspect the securities are independent.
Why do we compute the coefficient of correlation?
Because we can only interpret the sign of the covariance.
What is the interval of values the coefficient of correlation can take?
Between-1 to 1
How do you calculate the coefficient of correlation?
COV/σaσb
What does it mean if the coefficient of correlation is 1?
The two securities move by the same amount in % points in the same direction.
What does it mean if the coefficient of correlation is -1?
The two securities move by the same amount in % points in opposite directions.
In Excel, how do you compute the covariance?
=covariance.s()
In Excel, how do you compute the coefficient of correlation?
=correl()
How can risks be grouped?
Firm Specific/Idiosyncratic Risk and Systematic Risk
What is Idiosyncratic Risk?
Types of risk that share no correlation.
What is Systematic Risk?
Risk that affects all firms, but not all in the same way.
How do we compute the weight?
Value allocated in a specific security/Wealth
What is a long position?
When an investor invests in a security, the return will be received. w > 0
What is a short position?
When an investor short sells a security, the return will be paid.
w < 0
What is the expected return of a portfolio?
The weighted average of the expected return of its components.
True or false: when focusing on diversification, we focus on long positions and short positions.
False, only long positions
Why do investors create portfolio?
Gains from diversification
True or false: there is an added advantage of creating portfolios in terms of returns.
False, the expected return of a portfolio is given by the weighted average of the individual expected returns.
True or false: the portfolio’s volatility is the weighted average of the individual volatilities.
False, due to the covariance term.
What are the gains from diversification?
The difference between the portfolio’s volatility and the average volatility of the individual securities in the portfolio.
When do we have gains from diversification?
When the standard deviation of the portfolio is lower than the individual standard deviations.
What happens the more we diversify?
The more we diversify, the further the portfolio’s risk is from the average of individual risks.
True of false: the more we diversify the less risk the portfolio has.
False, diversification doesn’t mean less risk, but even if the portfolio’s risk increases the weighted average has to increase more.
How can we eliminate firm specific risk?
By diversifying.
What is the minimum variance portfolio?
The minimum variance portfolio is the combination of weights that combines two securities that presents the lowest volatility.
What is the formula for the MVP?
wa = (σb^2 - COV) / (σa^2 + σb^2 - 2COV)
True or false: the coefficient of correlation has no effect on the expected return of a portfolio.
True
What happens to the variance when the correlation increases and both securities are long positions?
Increases
What happens to the variance when the correlation increases and both securities are short positions?
Increases
What happens to the variance when the correlation increases and one security is a short position and the other a long position?
Decreases
When can we create portfolios with zero risk?
When the correlation is either -1 (positive weights) or 1 (negative weights).
What are efficient portfolios?
For a given risk, it’s the one with the highest expected return. For a given expected return, it’s the one with the lowest risk.
What does the Sharpe Ratio tell us?
For one unit of risk how many units of excess return the portfolio returns.
If over time the risk free changes, how can we say there’s no risk?
Over time it won’t be zero but when we invest the volatility of the investment is zero.
What is the Market Portfolio?
The Market Portfolio combines all risky securities (not the risk free). When we’re maximising sharpe ratio, we’re maximising gains from diversification which eliminates risks. The market portfolio only includes systematic risks The market portfolio selects its weights for all securities in order to maximise the sharpe ratio.
True or false: the market portfolio includes idiosyncratic risk.
False, only systematic risk.
True or false: the market portfolio includes the risk free security.
False, only risky assets
What does the Capital Market Line tell us?
The CML tells us all the different efficient portfolios.
Can we use the Capital Market Line to combine two risky securities?
No, only the risk free with the market portfolio.
How do we find the optimal portfolio?
Since all portfolios along the CML, we also need to maximise the utility function of the investor.
True or false: Assume that the covariance between two stocks is negative. When the return of one stock is higher than its own expectation the return of the other stock is always below its own expectation
False
True or false: Consider that the coefficient of correlation between two stocks is +1. This implies that when the return of one stock increases by 5pp (pp - percentage points), the return of the other increases by 5pp (pp – percentage points).
True
True or false: The covariance between a stock and itself is +1.
False
Stocks X and Y expected returns are 20% and 10%. Stock X’s volatility is two-times higher than stock Y’s volatility. The returns on stocks X and Y are independent. The risk-free rate is 2%.
True or false: Consider that you want to create a portfolio W that combines stocks X and Y, which has an expected return of 12%. If the coefficient of correlation between X and Y increases, the standard deviation of portfolio W increases.
True
Stocks X and Y expected returns are 20% and 10%. Stock X’s volatility is two-times higher than stock Y’s volatility. The returns on stocks X and Y are independent. The risk-free rate is 2%.
True or false: Consider a portfolio “P” that is invested 50% in X and 50% in Y. The higher the stock’s Y volatility, the lower will be the Sharpe Ratio of portfolio P.
True
Stocks X and Y expected returns are 20% and 10%. Stock X’s volatility is two-times higher than stock Y’s volatility. The returns on stocks X and Y are independent. The risk-free rate is 2%.
True or false: A portfolio that is invested 20% in X and 80% in Y has a zero volatility.
False
True or false: The Sharpe Ratio of a portfolio that combines the riskless asset and portfolio X always has the same Sharpe Ratio as portfolio X.
True
True or false: The CML can give you the expected return of a portfolio that combines three stocks and has a given standard deviation.
False
True or false: The SML only provides equilibrium returns of efficient portfolios.
False
True or false: Portfolio’s diversification decreases the portfolio’s total risk.
False
True or false: Two stock that are in equilibrium and have the same exposure to systematic risk, should present the same equilibrium expected return.
True
True or false: Consider that you created a portfolio with equal weights in 5 stocks. If the coefficient of correlation for each pair of stocks is +1 you can compute the portfolio’s standard deviation using the following expression: σp = ∑ wiσi
True
True or false: As an investor diversifies her portfolio, the portfolio’s standard deviation decreases.
False
True or false: The correlation of a stock with itself is the variance.
False
True or false: Consider an economy with N stocks. All stocks have the same expected return, the same standard deviation, and the covariance for every different pair of stocks is zero. As any investor diversifies his portfolio, the portfolio’s Sharpe ratio increases.
True
True or false: Consider a portfolio with 40% invested in security X, 80% in security Y, and a short position in security Z with a weight of -20%. The lower the coefficient of correlation between securities X and Z the lower will be the portfolio’s expected return.
False
True or false: The higher the beta for an efficient portfolio, the lower will be the weight in the riskfree asset.
True
True or false: Assume that all securities are in equilibrium. The beta on all efficient portfolios is the same as the beta for the market portfolio.
False
True or false: The higher the coefficient of risk-aversion for a given investor, the lower will be the Sharpe ratio for the investor’s optimal efficient portfolio.
False
True or false: The beta of a portfolio is given by the weighted average of the individual betas from the portfolio’s individual securities.
True
True or false: For an efficient portfolio, the higher the weight on the market portfolio, the higher will be its Sharpe ratio.
False
True or false: All efficient portfolios present the same percentage of systematic risk.
False
True or false: The more an agent benefits from diversification gains, the lower will be the portfolio’s standard deviation.
False
True or false: When all coefficients of correlation are lower than 1, any portfolio with more than two securities will satisfy the following condition: σp < ∑ wiσi
False
True or false: When the correlation between two stocks is +1 it is impossible to have gains from diversification.
False
True or false: The higher the coefficient of correlation between two stocks the lower will be the portfolio’s volatility
False
True or false: For an equally weighted portfolio the lower the coefficients of correlation of its components the higher will be its Sharpe Ratio.
True
True or false: Two securities with the same beta have the same exposure to systematic risk
True
True or false: A portfolio is efficient if for a given volatility it is the portfolio with the highest expected return.
True
True or false: A portfolio is efficient if it has the same beta as the Market portfolio.
False
True or false: The higher the weight on the market portfolio, on an efficient portfolio, the higher will be its beta.
True
True or false: All portfolios with a beta of 1 (same as the market portfolio) are efficient portfolios.
False
True or false: The more an investor diversifies her portfolio, the lower will be the following difference: |𝜎𝑝 − ∑𝑖 𝜔𝑖𝜎𝑖|.
False
True or false: The more an agent benefits from diversification gains, the lower will the portfolio’s average standard deviation.
False
True or false: Assume that the covariance between two stocks is negative. When the return of one stock is higher than its own expectation the return of the other stock is on average below its own expectation.
True
True or false: The minimum variance portfolio with two stocks presents a standard deviation equal to zero.
False
True or false: Consider a portfolio with 40% invested in security A, 80% in security B, and a short position in security C with a weight of -20%. The lower the coefficient of correlation between securities A and B the higher will be the portfolio’s Sharpe ratio.
True
True or false: The Sharpe Ratio of a portfolio that combines the riskless asset and portfolio W always has a higher Sharpe Ratio than portfolio W.
False
True or false: When using the CML, if you insert the standard deviation of security A you get the expected return of security A.
False
True or false: The SML provides equilibrium returns for any security, any portfolio, and any efficient portfolio.
True
True or false: Efficient portfolios combine the risk-free asset with the portfolio with the lowest Sharpe ratio.
False
True or false: The beta on all efficient portfolios is the same as the market portfolio’s beta.
False
True or false: The more risk averse the agent is the lower will be the Sharpe ratio on her optimal efficient portfolio.
False
