Topic 2 - Portfolio Theory & CAPM Flashcards
What is meant by the term ‘Returns’?
Gains/Losses made on an investment over a period of time.
What is the formula to calculate total monetary returns?
Total monetary return = dividend income + capital gain (or loss)
When a shareholder receives the return they get on their investment, it comes in two forms:
1. cash called dividends
2. capital gain/capital loss
What is the formula to calculate total cash if the equity is sold?
Total cash if equity is sold = initial investment + total monetary return
What is meant by percentage return?
Percentage gain is a measure of the gain or loss on an investment expressed as a percentage of the initial amount
What is the formula to calculate percentage return?
percentage return = dividend yield / capital gains yield
What is the formula for dividend yield?
(dividend/beginning value) x 100
What is the formula for capital gains yield?
(difference in share price/beginning value) x 100
What is meant by holding returns period?
It is a measure of the total return an investor earns or loses on an investment during the time it is held
i.e.
- you buy a share today
- hold onto it for 5+ years
- you still get a return even though you arent selling the share
- you compare the price in the market to the price you originally paid
What is the formula for HPR?
(1+r1) x (1+r2) x ….. x (1+rt) -1
What is meant by average returns?
Average returns is a measure of the average gain or loss of an investment over a specified period
basically just calculating the mean
What is the formula for average return?
Average Return = sum of returns / number of periods
What are the two measures of risk?
Standard deviation and variance
Formula for standard deviation
Standard deviation = root(variance)
What is the symbol for standard deviation?
sigma
What is the symbol for variance?
sigma^2
What is the formula for variance?
1/T-1 * (r1-mean)^2 + (r2-mean)^2 ….
Which is more favourable: a higher SD or a lower SD?
Lower - because we dont want the return to be too different from the average.
Therefore, a lower SD indicates that your return is very similar to the average return
Give an example of low-volatility securities and why do they have low volatility
Government bonds have low volatility
- Governments borrow money by issuing bonds
- Once a week, the government sells some bills at an auction
- because the government can raise taxes to pay for the debt they incur, this debt is virtually free of the risk of default (called risk-free return)
What is risk premium?
the difference between the return on the risky asset and the risk-free return
What is the formula to calculate risk premium?
risk premium = returns - risk-free return
What is meant by expected return?
This is the return that an individual expects a security to earn over the next period
How to calculate the variance after calculating the expected returns?
Var(A) = Prob * (r(A) - Er(A))^2
What is meant by covariance?
Covariance is the statistical measure that measures the interrelationship between two securities
How do you interpret a positive and a negative covariance?
A positive covariance indicates that the two variables tend to move in the same direction
A negative covariance indicates that they move in opposite directions.
However, the magnitude of the covariance is not standardized, making it challenging to compare covariances between different pairs of variables.
What are the steps to calculating covariance?
- create a table with the headings: prob, Rx, Devx, Ry, Devy
- Dev x * Dev y
- Devxy * Probabibility
What is meant by correlation?
Correlation is a statistical measure that describes the strength and direction of a linear relationship between two securities.
Correlation ranges from -1 to 1.
How do you interpret correlation coefficients?
A correlation coefficient of 1 indicates a perfect positive linear relationship
A correlation coefficient of -1 indicates a perfect negative linear relationship
A correlation coefficient of 0 indicates no linear relationship.
The formula for correlation?
Cov(rL,rU) / SD(L)*SD(U)
What is meant by portfolio theory?
Portfolio theory is a way of managing investments to balance risk and return.
It suggests that by spreading your money across different types of investments (diversification), you can reduce the overall risk of your portfolio.
The goal is to create the most efficient mix of investments that offers the best possible return for a given level of risk, or the least amount of risk for a desired level of return.
According to the modern portfolio theory, what are rational investors considered to be?
“mean-variance” optimizers
- they choose an optimal portfolio based on the mean and variance (and covariance and correlation of return distributions)
Formula to calculate the weighted average? (Portfolio theory)
E(rP) = W(A)E(rA) + W(B)E(rB) …..
Formula to calculate the Var assets’ in a portfolio? (Portfolio theory)
Var(P) = Var(rL)W(L)^2 + Var(rU)W(U)^2 +2(WL)(WU)(CovL,U)
Formula to calculate the Cov of assets in a portfolio? (Portfolio theory) DOUBLE CHECK THIS FORMULA
Cov(rL,rU) = sum: prob[rL-E(rL)] * [rU-E(rU)]
*DOUBLE CHECK THIS FORMULA
What is the only circumstance for a diversified portfolio?
Diversification effect applies as long as there is less than perfect correlation
p<1
The formula for the coefficient of variation
Coefficient of Variation = SD / Er
Is a lower or higher coefficient of variation better?
lower
- there is a lower risk per unit of return
- basically means less risk
What is meant by efficient set?
It is a concept in modern portfolio theory that represents a set of optimal portfolios that offer the highest expected return for a defined level of risk for a given level of expected return
What is meant by opportunity set?
an investor can achieve any point on the curved line (between the plotted individual securities) by selecting different weights of the two assets in the portfolio.
i.e. they can change the proportions of the individual assets in the ‘pie’ to optimise their profits
These sets of investment opportunities are called opportunity sets
Where is the minimum variance on the curve?
The minimum variance is the highest point on the curve, right at the bend
When is an investor likely to avoid investing in a portfolio?
When the expected return of the portfolio is less than the minimum variance (mv)
How do you know if the portfolios are diversified, just by looking at the graph?
Diversification is shown because the curved line is always to the left of the straight line
What is one thing you can deduce from the efficient set graph?
The lower the correlation, the more bend there is on the curve as it goes closer to p=-1
This indicates that the diversification effect rises as p declines
What is the formula for total risk?
total risk = systematic risk + unsystematic risk
Describe the graph of a portfolio as the number of securities increases
As the number of securities increases, the portfolio risk decreases
However, the portfolio variance can never drop to zero, it reaches a floor of cov, which is the covariance of each pair of securities
The more securities that are added, the further risk is reduced. Therefore, in an ideal world without transaction costs, it can be argued that we never achieve ‘too much’ diversification.
However, in the real world there are additional costs for more diversification
As more securities are added to a portfolio, what happens?
As more securities are added to a portfolio, the variance of the portfolio decreases, which indicates diversification effect
reduced variance/SD = reduced risk = inc. diversification
What can you do if you dont have capital but still want to invest?
You can still invest by borrowing, but this is VERY risky because the returns must be high enough to pay for the interest (leverage).
Instead, an investor might want to combine a risky investment with a risk-free investment (e.g. government treasury bills)
Suppose you want to invest (w) in the risky asset r, and (1-w) in the risk-free asset f.
What will be the expected return?
E(rP) = WE(rR) + (1-w)rF
simplifying and rearranging this, we get
E(rP) =rF + w*[E(rR) - rF]
Suppose you want to invest (w) in the risky asset r, and (1-w) in the risk-free asset f.
What will be the variance?
Var(p) = (w^2 * Var(R)) + ((1-w)^2 * Var(F)) + 2(w)(1-w)Cov(r,f)
however, since risk free assest have no variability, Var(F) and Cov(r,f) = 0
so we’re left with: Var(p) = w^2 * Var(R)
Describe the capital market line
on the x-axis is the Expected return on the portfolio
on the y-axis is the SD of the portfolio
The y-intercept (where portfolio standard deviation = 0), is when your portfolio only consists of a risk-free rate
As you invest more, the SD increases as well as the expected return of the portfolio.
There is a linear relationship between the portfolio’s SD and expected return.
What is meant by the separation principal?
The investors investment decision consists of two separate steps:
1) The investor will first estimate the optimal portfolio
2) The investor must then decide how much to invest in the optimal portfolio and how much to borrow in the risk-free asset depending on their risk preferences.
What is the best measure of risk of a security in a large portfolio?
the beta of the security
What does the CAPM model descibe?
The CAPM model describes the relationship between expected return and risk
What is the formula for CAPM?
E(ri) = rF + Bi * (E(rM) - rF)
E(ri) = the expected return on security i
rF = the risk-free rate
Bi = the beta of i
E(rM) = Expected return on the market portfolio
(E(rM) - rF) = the expected market risk premium
What is the formula for Beta?
Bi = (Cov(r,m)) / Var(m)
What does beta measure?
Beta measure the sensitivity of an asset’s return to movements in the market portfolio
i.e. beta measure how a security responds to movements in the market portfolio
What is the formula for beta that includes correlation?
Bi = Corr(i,m) * SD(i) / SD(m)
How do you interpret a beta with a value <0
opposite direction as the market index
How do you interpret a beta with a value =0
movement of the security is uncorrelated with that of the market
How do you interpret a beta with a value 0<b<1
same direction but less than the movement of the market
How do you interpret a beta with a value =1
same direction and equal to the movement of the market
How do you interpret a beta with a value >1
same direction but greater than the movement of the market
List the three assumptions of CAPM
1) investors hold diversified portfolios (investors are well-diversified)
2) Perfect capital market assumptions e.g. no individual dominated the market, no transaction costs, investors are rational and risk-averse
3) Single time horizon: This means that all investors have the same investment horizon and evaluate the expected returns and risks over the same period.
List the 4 limitations of CAPM
1) No Consideration of Behavioral Factors:
CAPM does not account for behavioural factors, such as investor sentiment, market psychology, or irrational behaviour, which can impact asset prices in real-world situations.
2) Perfect capital market assumptions are unrealistic e.g. No Transaction Costs or Taxes:
The model assumes no transaction costs or taxes, which may not reflect the actual costs associated with buying and selling assets. In reality, transaction costs and taxes can significantly affect investment returns.
3) Single-Period Model: CAPM is a single-period model that assumes investors evaluate risk and return over a specific time horizon. In reality, investors often have multiple time horizons, and the single-period assumption may not capture the complexity of long-term investment strategies.
4) Constant Beta:
CAPM assumes that an asset’s beta remains constant over time. In practice, betas can change due to shifts in a company’s business operations, industry dynamics, or other factors, making the assumption of a constant beta questionable.
