Risk & Return

Question 1

How do you compare different investments based on their returns?

Answer 1

The realized return is the return that actually occurred over a previous time period

Ex: If we buy a stock on date t for Pt, receive a dividend on date t+1, and then sell the stock on that date for Pt+1, your realized return would be

Rt+1 = Divt+1 + Pt+1/Pt -1
= Divt+1/Pt + Pt+1 -Pt/Pt

Longer Periods: Assume all dividends are immediately reinvested in additional shares.

Question 2

What is the Realized Annual Returns Formula?

Answer 2

If a stock pays dividend at the end of each quarter, we can calculate the annual realized return (Rannual) from realized return on each quarter

Rannual = (1+RQ1)(1+RQ2)(1+RQ3)(1+RQ4) - 1

Question 3

Arithmetic average

Answer 3

Simple average from basic algebra

Problem:

$100 -> $200 -> $100

Avg = (100%-50%)/2 = 25%

100/200 = -50%

*doesn’t make any intuitive sense (didn’t make any money whatsoever

Question 4

What is Geometric average?

Answer 4

Compound annual growth rate (CAGR) measures the mean annual growth rate of an investment over a specified period, assuming consistent annual growth.

CAGR = (Ending value/Beginning Value)^1/n - 1

Significance: Smooths out volatility effects, providing a single annual growth rate that captures the effect of compounding, making it more realistic for long-term performance evaluation than simply averaging annual returns.

Question 5

What distinguishes a riskless investment from a risky investment?

Answer 5

Riskless Investment: The return in one year is certain. Example: Buying a Treasury security.

Risky Investment: There are different possible returns that it could earn. Example: Buying a share of AAPL.

Question 6

What is a probability distribution in the context of investment returns?

Answer 6

Probability Distribution: Assigns a probability 𝑃𝑟 that each possible return 𝑅
will occur. This helps investors understand the range of potential returns and the likelihood of each return happening.

Example: AAPL closed at $230 on Oct 30, 2024. In one year, the price will likely be around this level (+/- 20% return). There is still a non-zero probability that the price is zero (Apple Inc. bankrupted), though it is extremely unlikely.

Question 7

What is the expected return, and how is it calculated?

Answer 7

or mean return E[R] is a probability weighted average of the possible returns

𝐸[𝑅]=∑𝑃𝑟×𝑅

Theoretical Nature: In reality, we observe only one actual return. It represents the average amount one anticipates earning from an investment over a specified period, based on historical performance or the probability distribution of potential outcomes.

Purpose: Helps assess potential profitability and risk of investments, aiding informed decision-making and comparison of different options.

Long-Term Perspective: What we would earn on average if we could repeat the investment many times, drawing returns from the same distribution each time. (Typical performance expected)

Question 8

What is Variance?

Answer 8

Measures how “spread out” the distribution of the return is

*We assess risk by measuring how possible returns deviate from their mean

Recall that with risk-free investment, there is no deviation at all

Question 9

What is Standard Deviation?

Answer 9

is just the square root of the variance

also called volatility (sigma)

SD is in the same units as returns

Question 10

How can investors assess the return and risk of a potential investment?

Answer 10

Expected Return and Standard Deviation: To assess return and risk, one needs to determine the expected return and standard deviation.

Challenge: Requires knowledge of future returns distribution or repeated observations, both impractical.

Solution: Use historical data to estimate the future return distribution.

Question 11

How do historical returns help in estimating future return distributions?

Answer 11

Estimation: Using historical data, we estimate future return distributions by assuming future returns mimic past returns and each period’s realized return comes from the same probability distribution.

Empirical Distribution: By observing multiple periods, we form an empirical distribution of returns.

Variance and Volatility: Wider distributions indicate more variance, while narrow distributions, like T-bills, reflect minimal volatility. This helps in assessing investment risk and performance.

Question 12

How do you estimate the expected return and variance using historical returns?

Answer 12

Average Realized Returns: Provides an estimate of the expected return.

Average Annual Return Formula:

𝑅‾=1/𝑇∑𝑅𝑡

where 𝑅𝑡 is the realized return in year 𝑡.

Variance Estimation: Compute the average squared deviation of historical realized returns from the mean (using the average realized returns as an estimate of the mean).

Variance
=1/𝑇−1∑(𝑅𝑡−𝑅‾)^2

Note: Divide by 𝑇−1 because we lose one degree of freedom by calculating 𝑅‾.

Question 13

How can you convert monthly variance and volatility to annual variance and volatility?

Answer 13

Conversion: If we calculated variance and volatility (standard deviation) of realized monthly returns, we can convert them to annual values.

Assumption: Monthly returns are independently and identically distributed over time.

Question 14

How can we measure and understand estimation error in average historical returns?

Answer 14

Estimation Error: The average historical return is an estimate of the true expected return; estimation errors occur.

Standard Error (SE): Statistically measures estimation error.

SE = SD(𝑅)/square root 𝑁

Use of SE: Determines a reasonable range for the true expected return.

95% Confidence Interval:

HistoricalAverageReturn
±(2×StandardError)

Confidence: We are 95% confident that the true expected return will fall within two standard deviations of the average return.

Question 15

What is the risk/return tradeoff for large portfolios?

Answer 15

Positive Relationship: Plot of the average return vs. the volatility shows a positive relationship.

Risk Aversion: Consistent with the view that investors are risk-averse; riskier investments (high volatility) must offer higher average returns to compensate for the actual risk.

Higher Volatility: Investments with higher volatility should have a higher risk premium and, therefore, higher returns.

Question 16

What is the risk/return tradeoff for individual stocks?

Answer 16

Graph Analysis: The average return vs. volatility graph shows a positive relationship but no clear linear function.

Individual Stocks: Returns are scattered and hard to predict based on volatility, generally below the dotted line.

Systematic Risk: Market-wide/systematic risk is the only factor affecting expected return. Use the risk vs. return relationship to estimate expected return.

Diversification: Individual stocks offer less return for risk compared to the index due to lack of diversification.

Question 17

What are common and independent risks, and how does diversification work?

Answer 17

Common Risk: Risk that affects all securities.

Independent Risk: Risk that affects a particular security.

Diversification: Averaging out of independent risks in a large portfolio.

Examples:

Theft Insurance: Thefts in different houses are largely unrelated; independent risk.

Earthquake Insurance: Earthquakes affect all houses in a region; common risk.

Question 18

What are the main types of risks in holding a stock?

Answer 18

Independent Risk: Arises from firm-specific news.

Also known as firm-specific, idiosyncratic, unique, or diversifiable risk.

Common Risk: Arises from market-wide news.

Also known as systematic, undiversifiable, or market risk.

Diversification: In a large stock portfolio, firm-specific risk will tend to cancel out across firms (diversified away).

Systematic risk will not be diversified.

Question 19

Firm Specific or Systematic Risk:

Apple recalls iPhones due to battery overheating issues

Answer 19

Firm specific

Question 20

Firm Specific or Systematic Risk:

Starbucks CEO steps down, causing strategic uncertainity

Answer 20

Firm specific

Question 21

Firm Specific or Systematic Risk:

Inflation rises, reducing common purchasing power and impacting all retail stocks

Answer 21

Common risk

Question 22

Firm Specific or Systematic Risk: Boeing faces grounding of its fleet after safety concerns with a specific aircraft model

Answer 22

Firm specific risk

Question 23

Firm Specific or Systematic Risk: Oil prices surge due to geopolitical tensions in the Middle East impacting energy costs worldwide

Answer 23

Common risk

Question 24

How does diversification affect different types of risk in stock portfolios?

Answer 24

Actual Firms: Have both firm-specific and systematic risk exposures.

Hypothetical Firms:

Type S Firms: Only affected by systematic risk.

No risk reduction through diversification.

Type I Firms: Only affected by idiosyncratic risk.

All risk diversified away with a sufficient number of firms

Diversification Effect: Idiosyncratic risk is eliminated through diversification, leaving just systematic risk.

Question 25

What is the risk premium/excess return?

Answer 25

The additional expected return over the risk-free rate that investors require to compensate for taking on risk. The return of a security over the risk-free rate

Question 26

How is systematic risk measured?

Answer 26

Beta: Quantifies the sensitivity of an investment's returns relative to the overall market returns. Beta of 1: The investment's price moves with the market. Beta > 1: The investment is more volatile than the market. Beta < 1: The investment is less volatile than the market. Purpose: Helps investors understand the extent to which an investment is exposed to market-wide risk factors, which are beyond the control of individual companies or sectors. Steps to Estimate Expected Return: Measure the investment's systematic risk (Beta). Determine the risk premium required. Estimate the expected return of an investment (ONLY systematic risks are compensated).

Question 27

What is the natural candidate for an efficient portfolio?

Answer 27

Market Portfolio: A portfolio of all stocks and securities in the market. Use large market indices as proxies for the market portfolio, e.g., S&P 500 Index (US) and S&P/TSX Composite Index (Canada). If the market portfolio is efficient, any changes in its value arise from systematic/common risk.

Question 28

What is Beta and Systematic Risk?

Answer 28

The beta of a security is the expected percent change in the excess return of that security given a 1% change in the excess return of the market portfolio A.k.a sensitivity to systematic or market risk! - Market portfolio has beta equal 1 - Beta > 1, more sensitive than the market - Beta < 1, less sensitive than market

Question 29

What is the market risk premium?

Answer 29

The expected excess return for the market portfolio (Expected market return - risk-free rate).

Question 30

What is an efficient portfolio?

Answer 30

Any portfolio that contains only systematic risk and all unsystematic risk has been diversified away (no diversifiable risk remains)

Question 31

Why is a security with only diversifiable risk having an expected return that exceeds the risk-free rate inconsistent with both an efficient capital market and the CAPM?

Answer 31

CAPM: The expected return of a security depends on its systematic risk (beta). Diversifiable risk is an unsystematic risk that does not affect expected return as it can be diversified away. Hence, a security with only diversifiable risk should not exceed the risk-free rate. Efficient Market: In an efficient market, stock prices reflect all information. Securities should not offer higher returns for diversifiable risk, which can be eliminated through diversification. If a stock provides additional returns for diversifiable risk, it signals market inefficiency and creates an arbitrage opportunity for risk-free profit.

Question 32

Why is a security with a beta of 1 having a return of 15% last year when the market had a return of 9% consistent with both an efficient capital market and the CAPM?

Answer 32

CAPM does not specify that expected returns should always match actual returns. It only specifies that the expected return should be based on systematic risk (beta). A security with a beta of 1 is expected to have the same return as the market on average, but deviations in any single period are consistent with CAPM due to unexpected events or news. Efficient Market Hypothesis (EMH): Stock prices reflect all available information, making it impossible to consistently achieve above-market returns without taking on additional risk. Deviations from expected returns can occur due to unexpected news or events, but these are quickly incorporated into stock prices.

Question 33

Why is the statement "Small stocks with a beta of 1.5 tend to have higher returns on average than large stocks with a beta of 1.5" inconsistent with CAPM but not necessarily EMH?

Answer 33

CAPM: Securities with the same beta should have the same expected return, regardless of size. Efficient Market Hypothesis (EMH): Suggests that smaller stocks may have higher returns due to the additional risk associated with their size. This risk, along with the potential size premium, is publicly available information and should be reflected in stock prices.

Question 34

What is the relationship between diversifiable risk and risk premium?

Answer 34

Diversifiable Risk: Does not offer an additional risk premium. If diversifiable risk did offer a premium, investors could buy stocks, earn the higher premium, and diversify to eliminate risk, an opportunity that would quickly be exploited. Investors can eliminate firm-specific risk by diversifying their portfolios, so they do not require compensation (i.e., risk premium). Investors can costlessly remove diversifiable risk from their portfolio by diversifying so they do not demand a risk premium for it

Question 35

What are the key principles regarding risk premium?

Answer 35

The risk premium for diversifiable risk is zero, so investors are not compensated for holding firm-specific risk. The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk.

Question 36

How can we estimate expected returns for securities with different systematic risk (beta)?

Answer 36

Capital Asset Pricing Model (CAPM): 𝐸[𝑅]=𝑟𝑓+𝛽×(Expected market return−𝑟𝑓) Principle: Higher beta investments should offer higher expected returns to compensate investors for taking on additional risk.

Question 37

What does beta measure in the context of CAPM?

Answer 37

Beta: Measures how sensitive an investment's excess return is to the market portfolio's excess return. Beta of 1.5: If the market's excess return changes by 1%, the investment's excess return is expected to change by 1.5%, aligning with CAPM. Higher beta indicates more sensitivity and thus higher expected returns to compensate for increased systematic risk.