Week 19 - Risk and Return Flashcards
What is the expected return on a security?
The expected return is the weighted average of all possible returns of a security, where the weights are the probabilities of each outcome occurring.
How can expected returns be interpreted in financial analysis?
Expected returns are often equated with the average return over multiple possible outcomes, assuming probabilities are accurately assigned.
What is the mathematical formula for expected return?
E(R) = nΣi=1 p_iR_i
p_i - probability of state i occurring
R_i - return of assets in state i
What are the key components needed to compute the expected return?
Possible states of the world (e.g., market conditions)
Returns associated with each state (R_i)
Probabilities of each state occurring (p_i)
Why do investors and analysts use expected returns?
Expected returns help investors assess the potential profitability of an asset and compare different investment options under uncertainty.
What are some limitations of expected return calculations?
It does not account for risk (volatility).
It is based on estimated probabilities, which may be inaccurate.
It assumes historical trends or rational expectations, which may not always hold.
What do variance and standard deviation measure in finance?
They measure the volatility (risk) of returns, showing how much actual returns deviate from the expected return.
What is the mathematical formula for variance?
σˆ2 = nΣi=1 p_i (R_i - E(R))ˆ2
p_i - probability of state i occurring
R_i - return in state i
E(R) - expected return
What does a high or low variance indicate?
High variance → Greater fluctuation in returns (more risk).
Low variance → More stable returns (less risk).
How is standard deviation (σ) related to variance?
σ = sqroot σˆ2
It is the square root of variance and provides a measure of dispersion in the same units as the returns.
Why do investors care about standard deviation?
It is widely used as a measure of risk.
Higher standard deviation means higher uncertainty in returns.
Used in risk-adjusted performance metrics like Sharpe Ratio.
How do variance and standard deviation differ?
Variance (𝜎ˆ2): Measures squared deviations from the mean (not in original units).
Standard deviation (σ): Expressed in the same units as returns, making it easier to interpret.
Stock A:
Variance: σˆ2_A = 0.03922
Standard deviation: σ_A = sqroot0.03922 = 0.198
Stock b:
Variance: σˆ2_B = 0.0206
Standard deviation: σ_A = sqroot0.0206 = 0.144
Which stock is riskier?
Stock A is riskier because it has a higher standard deviation (19.8%) compared to Stock B (14.4%). A higher standard deviation means greater volatility, implying more uncertainty in returns.
The decision depends on your risk tolerance and expected return:
If you prefer higher returns and can tolerate risk, you might choose Stock A.
If you prefer stability and lower risk, you might choose Stock B.
If both stocks have the same expected return, Stock B is the better risk-adjusted choice because it has lower volatility.
What is a portfolio in finance?
A portfolio is a collection of assets or securities, such as stocks, bonds, or mutual funds, held by an investor to achieve diversification and optimise risk and return.
How does adding an asset to a portfolio affect its overall risk and return?
The impact depends on:
The asset’s own risk and return characteristics.
Its correlation with existing portfolio assets.
If an asset has a low or negative correlation with the portfolio, it can help reduce overall risk.
If an asset has a high correlation, it may increase the portfolio’s risk.
How is the expected return of a portfolio calculated?
The portfolio’s expected return is the weighted average of the expected returns of the individual assets:
E(R_p) = nΣi=1 w_i E(R_i)
E(R_p) - Expected return of the portfolio
w_i - Weight (proportion) of asset i in the portfolio
E(R_i) = Expected return of asset i
How is portfolio risk measured?
Portfolio risk is measured using variance and standard deviation. Unlike expected return, portfolio variance considers asset correlations
What is the risk-return trade-off for a portfolio?
Higher expected return usually comes with higher risk (standard deviation).
Diversification can help reduce risk without necessarily reducing return.
The goal is to maximise return for a given level of risk (efficient frontier concept).
What is the formula for portfolio expected return?
E(R_p) = mΣj=1 w_j E(R_j)
E(R_p) - Expected return of the portfolio
w_j - Proportion (weight) of total portfolio invested in asset j
E(R_j) - Expected return of asset j
How do portfolio weights (w_j) affect expected return?
Higher weight on high-return assets → Increases portfolio return.
Higher weight on low-return assets → Lowers portfolio return.
Total portfolio weights must sum to 1 (Σw_j=1).
How else can the portfolio expected return be calculated?
By determining portfolio returns in each possible state (e.g., recession, normal, boom).
Then, use the same probability-weighted expected return formula as for individual securities:
E(R_p) = nΣi=1 p_i R_p,i
p_i - probability of state i occurring
R_p,i - portfolio return in state i
How does diversification impact portfolio expected return?
Expected return is simply the weighted average (diversification does not increase it).
However, diversification helps reduce portfolio risk without necessarily lowering return.
What is portfolio risk?
Portfolio risk is the variability (volatility) of portfolio returns, measured using variance (σˆ2_p) and standard deviation (σ_p). It depends on individual asset risks and their correlations.
How do you calculate the portfolio return in each state?
R_p,st = w_1R_1 + w_2R_2 + … + w_mR_m
R_p,st - Portfolio return in state st
w_j - Portfolio weight of asset j
R_j - Return of asset j in that state
How do you calculate the expected return of a portfolio?
E(R_p) = nΣi=1 p_i R_p,st
p_i - Probability of state i occurring
R_p,st = Portfolio return in state i
How do you calculate portfolio variance?
σˆ2_p = nΣi=1 p_i (R_p,st - E(R_p))ˆ2
σˆ2_p - Portfolio variance
p_i - Probability of state i
R_p,st - Portfolio return in state i
E(R_p) - Expected portfolio return
How do you calculate portfolio standard deviation?
σ_p = sqroot σˆ2_p
σ_p - Portfolio standard deviation
σˆ2_p - Portfolion variance
How does diversification affect portfolio risk?
Diversification reduces portfolio risk if assets are not perfectly correlated.
Even if individual assets are volatile, combining them can lower overall risk.
The risk reduction depends on the correlation coefficient (ρ) between asset returns.
Example:
You own a portfolio that has £2,400 invested in stock A and £3,600
invested in stock B. The expected return on A is 8.6% and on B is 12.5%.
What is the expected return on the portfolio?
E(R_p) = nΣi=1 w_i E(R_i)
so E(R_p) = w_A E(R_A) + w_B E(R_B)
w_A and w_B are the weights of stocks A and B in the portfolio
E(R_A) = 0.086
E(R_B) = 0.125
Total investment:
Total portfolio value = 2400 + 3600 = 6000
w_A = 2400/6000 = 0.4, w_B = 3600/6000 = 0.6
E(R_p) = 0.4x0.086 + 0.6x0.125
= 0.1094
E(R_p) = 10.94%
The expected return on the portfolio is 10.94%.
What is diversification in investing?
Diversification is the practice of investing in multiple assets to reduce overall portfolio risk by spreading exposure across different securities, industries, and sectors.
The S&P 500 is composed of individual stocks, so why is its risk lower than most of them?
Diversification Effect: The S&P 500 includes 500 different companies, reducing the impact of any single stock’s volatility.
Unsystematic Risk is Reduced: Firm-specific risks (e.g., lawsuits, product failures) are diversified away.
Market-wide Risk Remains: The S&P 500 still has systematic risk (e.g., recessions, interest rate changes), but it’s less volatile than individual stocks.
How do the volatilities (σ) of individual stocks compare to the S&P 500?
Individual stocks have higher volatility:
AT&T: 22.7%
GE: 41.8%
Microsoft: 22.0%
S&P 500 volatility is lower: 18.6%
This shows how a diversified index smooths out individual stock fluctuations, leading to lower overall risk.
Why does the S&P 500 still have risk if it is diversified?
Diversification eliminates unsystematic risk (company-specific risks).
It does not eliminate systematic risk (market-wide risks like interest rates, inflation, economic downturns).
Even a broad index is exposed to macroeconomic conditions.
What are the main benefits of diversification?
Reduces total portfolio risk (by averaging out firm-specific risks).
Smooths returns over time (less extreme up/down swings).
Helps investors achieve more stable, risk-adjusted returns.
Cannot eliminate systematic risk, but minimises idiosyncratic (firm-specific) risk.
What does covariance of returns measure?
Covariance measures how much the returns on two risky assets move together. It shows whether the assets tend to move in the same direction (positive covariance) or in opposite directions (negative covariance).
What is the formula for covariance between two assets, A and B?
cov_ab = 1/(n-1) nΣi=1 (R_ai - E(R_a))(R_bi - E(R_b))
cov_ab - Covariance between asset A and asset B
R_ai, R_bi - Returns pf assets A and B for observation i
E(R_a), E(R_b) - Expected returns of assets A and B
n - total number of observations
How do you interpret the covariance value?
Positive covariance: Assets tend to move in the same direction (when one goes up, the other tends to go up).
Negative covariance: Assets tend to move in opposite directions (when one goes up, the other tends to go down).
Zero covariance: No predictable relationship between the movements of the assets.
How is covariance different from correlation?
Covariance measures the direction of the relationship but not the strength.
Correlation normalises covariance to provide a standardised measure (between -1 and 1) of both the strength and direction of the relationship.
Why is covariance important for portfolio risk?
Covariance is used in calculating the portfolio’s overall risk (variance and standard deviation). The correlation between asset returns affects how much diversification reduces the portfolio’s risk.
Positive covariance increases portfolio risk.
Negative covariance reduces portfolio risk by offsetting the price movements of the assets.
What is the formula for variance of returns for asset A?
var_a = σˆ2_a = cov_aa = 1/(n-1) nΣi=1 (R_ai - E(R_a))ˆ2
var_a - Variance of returns for asset A
R_ai - Returns of assets A for observation i
E(R_a) - Expected returns of assets A
n - total number of observations
What does variance measure in relation to an asset’s returns?
Variance measures the dispersion or spread of an asset’s returns around its expected return. A higher variance indicates more volatility or risk, meaning the asset’s returns fluctuate more widely from the expected return.
What is the key difference between covariance and variance?
Variance measures the variability of a single asset’s returns around its expected return.
Covariance measures how two assets move together relative to their expected returns.
If two assets have positive covariance, they tend to move in the same direction.
If the covariance is negative, the assets move in opposite directions.
If covariance is zero, the assets are unrelated in terms of their movements.
What is the covariance of an asset with itself?
The covariance of an asset with itself is its variance. This is because the asset’s returns move in relation to its own expected return, and it gives the same result as computing variance:
cov_aa = var_a
Thus, the formula for variance is simply the covariance of an asset with itself.
What does the correlation coefficient (ρ) measure?
The correlation coefficient measures the strength and direction of the linear relationship between two assets’ returns. It scales the covariance value to a range between [−1,+1], making it easier to understand the degree of association
What is the formula for calculating the correlation coefficient between two assets, A and B?
p_ab = cov_ab/ σ_a σ_b
p_ab - correlation coefficient between assets A and B
cov_ab - covariance between the returns of assets A and B
σ_a - standard deviation of assets A’s returns
σ_b - standard deviation of assets B’s returns
How do you interpret the values of the correlation coefficient?
p = +1: Perfect positive correlation (when one asset moves up, the other moves up by the same proportion).
p = -1: Perfect negative correlation (when one asset moves up, the other moves down by the same proportion).
p = 0: No correlation (the movements of the two assets are independent of each other).
p between 0 and 1: Positive correlation, but not perfect. The assets tend to move in the same direction, but not perfectly.
ρ between 0 and -1: Negative correlation, but not perfect. The assets tend to move in opposite directions, but not perfectly.
Why is the correlation coefficient useful in portfolio management?
The correlation coefficient helps to understand how assets move in relation to each other, which is crucial for:
Diversification: Choosing assets that are less correlated reduces portfolio risk.
Risk reduction: A mix of assets with low or negative correlation can reduce overall portfolio risk since they won’t move in the same direction at the same time.
How does the correlation coefficient affect portfolio risk?
If assets are perfectly positively correlated (ρ=+1), the portfolio’s risk will be the weighted sum of the individual risks.
If assets are perfectly negatively correlated (ρ=−1), the portfolio could have zero risk (if weights are adjusted correctly).
If assets are uncorrelated (ρ=0), diversification will still reduce risk, but not as effectively as when the assets are negatively correlated.
hat happens to the risk of a two-stock portfolio when p= -1?
When the correlation coefficient ρ=−1, the two stocks are perfectly negatively correlated, meaning they move in completely opposite directions. In this case, you can combine the two stocks to form a riskless portfolio, as the gains from one asset offset the losses of the other.
What happens to the risk of a two-stock portfolio when ρ=+1?
When the correlation coefficient ρ=+1, the two stocks are perfectly positively correlated, meaning they move in the same direction. In this case, there is no risk reduction through diversification, because both stocks will experience the same fluctuations. The portfolio risk will be the weighted sum of the individual risks.
What is the typical correlation (ρ) between stocks?
In practice, stocks generally have a correlation coefficient around ρ≈0.65, meaning they tend to move in the same direction but not perfectly. This provides some risk reduction, but not complete elimination of risk.
How does holding multiple stocks in a portfolio impact risk?
Holding a diversified portfolio of multiple stocks can lower risk, especially when stocks have lower correlation with each other. By combining assets with different correlations, the portfolio’s overall risk is reduced compared to holding individual assets, though it is rarely eliminated entirely. Diversification works best when assets are not perfectly correlated.
How does diversification affect portfolio risk?
Diversification lowers risk by combining assets whose returns are not perfectly correlated. When stocks in a portfolio have a correlation less than +1, diversification can help reduce the portfolio’s overall risk because the assets don’t move exactly together. However, diversification does not eliminate risk entirely, especially if stocks have positive correlation.
Why do investors typically hold many stocks in their portfolios?
Investors hold many stocks to diversify risk. By holding a variety of stocks, investors can reduce the impact of any single stock’s poor performance on the overall portfolio. The more diversified the portfolio, the less risk it carries, as long as the correlation between assets is not too high.
Let’s break down these graphs, which are all about how investments in stocks can work together.
Imagine you have two stocks, let’s call them Stock X and Stock Y. These graphs show how their returns (how much money they make) relate to each other.
The First Set: Correlation of +1
Correlation: This is a fancy word for how closely two things move together. A correlation of +1 means they move perfectly in the same direction.
The Graphs:
X and Y: Notice how the graphs for Stock X and Stock Y look exactly the same? They both go up and down at the same time.
X + Y: The combined graph shows what happens when you put your money into both Stock X and Stock Y. Because they move together, the combined return is just a bigger version of the individual returns.
If you have two stocks with a correlation of +1, it’s like having two copies of the same thing. You don’t get any extra benefit or reduce any risk by having both.
The Second Set: Correlation of -1
Correlation: A correlation of -1 means the stocks move perfectly in opposite directions. When one goes up, the other goes down by the same amount.
The Graphs:
X and Y: Notice how the graphs are mirror images of each other.
X + Y: The combined graph is a flat line. This means the overall return is always the same, no matter what the individual stocks do.
This is really interesting! If you have two stocks with a correlation of -1, you’ve completely eliminated risk. When one stock does poorly, the other does well, balancing it out.
What is the difference between realised and expected returns?
Realised returns are the actual returns that an investor receives from an investment over a specific period of time.
Expected returns are the anticipated returns based on the investor’s predictions or market forecasts.
What are the components of realised returns?
Realised returns consist of two components:
Expected component: The return that investors expect based on available information.
Unexpected component: The return that differs from the expected return, caused by unforeseen events or surprises in the market.
What is the unexpected component of returns?
The unexpected component is the part of the return that deviates from what was expected.
It arises due to new information, surprises, or events that affect the market or investment.
Can the unexpected component be positive or negative?
Yes, the unexpected return can either be positive (when returns exceed expectations) or negative (when returns fall short of expectations).
What happens to the average of the unexpected return component over time?
Over time, the average of the unexpected return component tends to be zero.
This means that while there can be both positive and negative surprises, they balance out on average.
Why is the unexpected return important for investors?
Unexpected returns reflect the uncertainty and risks inherent in investing.
They help investors understand that returns are not always predictable and can vary due to unforeseen factors.
What do announcements and news contain in relation to stock prices?
Announcements and news typically contain both:
Expected component: The part that investors anticipate based on prior information.
Surprise component: The new, unforeseen information that may affect stock prices.
How does the surprise component of news affect stock prices?
The surprise component has a direct impact on stock prices.
Unanticipated news or earnings reports that differ from expectations lead to price movements, either up or down, depending on the nature of the surprise.
What is the evidence for the effect of unexpected news on stock prices?
It’s obvious from watching stock price movements following unexpected news.
When earnings or news differ from expectations, stock prices often change in response, demonstrating the market’s reaction to surprises.
What is the relationship between efficient markets and trading on unexpected news?
Efficient markets are markets in which investors trade on unexpected news.
In such markets, stock prices quickly reflect the new information as investors act on surprises and adjust prices accordingly.
How does the ease of trading on surprises affect market efficiency?
The easier it is to trade on surprises, the more efficient the market becomes.
If information can be quickly acted upon, prices adjust rapidly, making the market more efficient in reflecting new data.
Why do price changes appear random in efficient markets?
In efficient markets, price changes are random because we cannot predict when or what kind of surprises will occur.
Prices fluctuate due to the continuous arrival of new, unexpected information, and these movements are unpredictable.
What is the formula for total return?
Total Return (R) = Expected Return (E(R)) + Unexpected Return (U)
This formula shows that the realized return is composed of both expected and unexpected components.
What are the components of the unexpected return (U)?
The unexpected return (U) can be further broken down into two parts:
Systematic portion (m): The part of the return driven by broad market movements and factors that affect all securities (e.g., interest rates, economic events).
Unsystematic portion (e): The part of the return specific to an individual asset, such as company performance or unique news about a particular stock.
unexpected return (U) = systematic portion (m) + unsystematic portion (e)
What is the breakdown of total return?
Total Return (R) = Expected Return (E(R)) + Systematic Portion (m) + Unsystematic Portion (e)
This shows how total return is made up of:
Expected return (E(R)): The anticipated return based on available information.
Systematic return (m): The portion of the unexpected return due to market-wide factors.
Unsystematic return (e): The portion of the unexpected return due to firm-specific factors.
What is the difference between systematic and unsystematic returns?
Systematic Return (m): The return due to market-wide factors that affect all assets, such as interest rate changes or economic events.
Unsystematic Return (e): The return due to firm-specific factors, like company earnings reports or management changes, which affect only individual assets.
What role do systematic and unsystematic risks play in total return?
Systematic risk affects the overall market and is non-diversifiable (investors cannot eliminate it by holding a diversified portfolio).
Unsystematic risk is unique to individual companies or sectors and can be mitigated by diversifying the investment portfolio.
What is systematic risk?
Systematic risk refers to risk factors that affect a large number of assets simultaneously.
It is also known as non-diversifiable risk, market risk, or macro risk because it impacts the entire market or economy.
What are the key characteristics of systematic risk?
Systematic risk is non-diversifiable, meaning it cannot be eliminated through diversification.
It is associated with broad market or economic factors that affect most assets, not just individual securities.
What are some examples of factors that cause systematic risk?
Examples of factors that cause systematic risk include:
Changes in GDP: Economic growth or recession.
Inflation: Rising or falling price levels.
Interest rates: Central bank decisions affecting borrowing costs.
Wars: Geopolitical instability affecting global markets.
Oil price shocks: Sudden changes in oil prices impacting various industries.
How does systematic risk differ from unsystematic risk?
Systematic risk affects the entire market and cannot be diversified away (e.g., interest rate changes, economic recessions).
Unsystematic risk is specific to individual companies or sectors (e.g., management changes, company earnings), and can be reduced by diversification.
How does systematic risk impact investment portfolios?
Since systematic risk affects all assets, it cannot be avoided through diversification.
Even a well-diversified portfolio will be exposed to market-wide risks like economic downturns or global crises.
What is unsystematic risk?
Unsystematic risk refers to risk factors that affect a limited number of assets rather than the entire market.
It is also called diversifiable risk, unique risk, asset-specific risk, or micro risk.
What are the key characteristics of unsystematic risk?
Unsystematic risk is specific to individual companies, industries, or sectors.
It can be eliminated by holding a diversified portfolio of assets.
It is not related to broader market movements or macroeconomic factors.
What are some examples of factors that cause unsystematic risk?
Examples of factors that cause unsystematic risk include:
Labor strikes: Work stoppages disrupting company operations.
Part shortages: Supply chain disruptions affecting production.
Dividend cuts: Reduction in expected dividends, affecting stock prices.
Management changes: Changes in leadership impacting company strategy.
How does unsystematic risk differ from systematic risk?
Systematic risk affects the entire market and cannot be diversified away (e.g., interest rate changes, economic recessions).
Unsystematic risk is specific to individual assets or sectors and can be eliminated through diversification.
Systematic or unsystematic?
1. Inflation increases by 2%
2. Discovery of a major gas field
3. A firm loses a law case
4. Oil prices increase
5. Government subsidies for solar panels are withdrawn
6. Economic growth is lower than expected
7. Decrease in textile imports
- Systematic Risk. Inflation affects the entire economy and all assets, making it a market-wide factor.
- Unsystematic Risk. This affects specific companies or industries (e.g., energy sector), not the whole market.
- Unsystematic Risk. This is specific to that firm, impacting its stock and operations, not the entire market.
- Systematic Risk. Changes in oil prices affect a broad range of industries globally, including transportation and manufacturing.
- Unsystematic Risk. This primarily impacts companies or industries involved in solar energy and alternative energy, not the overall market.
- Systematic Risk. Economic growth impacts the entire economy, affecting most assets and industries.
