Topic 9: Risk & Return Flashcards
How do you compare different investments based on their returns?
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.
What is the Realized Annual Returns Formula?
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
Arithmetic average
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
What is Geometric average?
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.
What distinguishes a riskless investment from a risky investment?
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.
What is a probability distribution in the context of investment returns?
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.
What is the expected return, and how is it calculated?
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)
What is Variance?
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
What is Standard Deviation?
is just the square root of the variance
also called volatility (sigma)
SD is in the same units as returns
How can investors assess the return and risk of a potential investment?
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.
How do historical returns help in estimating future return distributions?
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.
How do you estimate the expected return and variance using historical returns?
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‾.
How can you convert monthly variance and volatility to annual variance and volatility?
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.
How can we measure and understand estimation error in average historical returns?
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.
What is the risk/return tradeoff for large portfolios?
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.
