Dr. Jiang Investment: Risk and Return

Summary

This lecture introduces key concepts in risk and return for investment, including methods for calculating holding period returns, average returns (arithmetic and geometric), and internal rate of return (IRR). It explains how historical return distributions and volatility measures like standard deviation help assess financial risks. The Sharpe ratio is used to evaluate risk-adjusted returns, while asset allocation between risky and risk-free assets is explored. The lecture also covers optimal portfolio allocation based on an investor’s risk aversion coefficient, using practical examples and calculations to enhance understanding of portfolio management strategies.

Takeaways

• 😀 Holding period return (HPR) is calculated by adding capital gains and dividends, then dividing by the initial price to determine the investment's return over a period.
• 😀 Arithmetic average return ignores compounding, while geometric average return accounts for it, providing a more accurate measure of real-world investment growth.
• 😀 Geometric average is especially useful for mutual funds as it reflects the compounded growth rate over time, unlike arithmetic average which can overestimate returns.
• 😀 Risk measures such as standard deviation and variance quantify the volatility of returns, with standard deviation being the most widely used metric for risk.
• 😀 A normal distribution allows us to estimate the probability of outcomes within one, two, and three standard deviations from the mean (68%, 95%, and 99.7%, respectively).
• 😀 The Sharpe ratio is a key metric for risk-adjusted returns, calculated as the excess return over the risk-free rate divided by the standard deviation of the return.
• 😀 A higher Sharpe ratio indicates a better risk-return tradeoff, making it a crucial tool for comparing portfolios with varying levels of risk.
• 😀 Portfolio allocation between risky and risk-free assets depends on the proportion of funds invested in each, with the total return and risk being a weighted average.
• 😀 The Capital Allocation Line (CAL) illustrates the relationship between expected return and risk for a portfolio that combines risky and risk-free assets, with its slope representing the Sharpe ratio.
• 😀 The optimal allocation to risky assets is determined by an investor's risk aversion coefficient, which measures how much risk an investor is willing to take for a given return.
• 😀 A questionnaire can help estimate an individual's risk aversion coefficient, categorizing investors based on their willingness to take on risk and guiding portfolio allocation decisions.

Q & A

What is the Holding Period Return (HPR) and how is it calculated?

The Holding Period Return (HPR) measures the total return on an investment over a specific period. It is calculated as the sum of capital gains and dividend distribution divided by the initial price. For example, if an asset starts at $20, ends at $24, and provides a dividend of $1, the HPR would be (24 - 20 + 1) / 20 = 25%.

What is the difference between arithmetic and geometric average returns?

Arithmetic average return is simply the sum of returns over a period divided by the number of periods, ignoring compounding. In contrast, the geometric average return accounts for compounding over time and represents the actual growth rate of the investment.

Why is geometric average return generally considered more accurate than arithmetic average?

Geometric average return is more accurate because it considers the compounding effect of returns over time, which is how investments grow in reality. Arithmetic average, on the other hand, may overestimate the true growth as it ignores volatility and compounding.

What is the formula for the Sharpe ratio, and how does it help in evaluating investments?

The Sharpe ratio is calculated as (Return of the portfolio - Risk-free rate) / Standard deviation of the portfolio. It helps evaluate the risk-adjusted return of an investment, showing how much excess return an investor is receiving per unit of risk.

How can historical return distribution help in measuring risk and return?

Historical return distribution, typically modeled using a normal distribution, helps in understanding the range of possible returns for an asset. It allows investors to estimate the mean return and volatility (standard deviation or variance) to quantify risk and return.

What is the relationship between risk (standard deviation) and normal distribution?

In a normal distribution, the standard deviation indicates the probability of observing a return within a certain range of the mean. For example, one standard deviation above or below the mean covers about 68% of possible outcomes, two standard deviations cover 95%, and three standard deviations cover 99.74%.

How do you calculate the expected return and risk for a portfolio combining a risky asset and a risk-free asset?

The expected return for a portfolio combining a risky asset and a risk-free asset is a weighted average of the individual returns. The standard deviation of the portfolio is calculated using the standard deviation of the risky asset, weighted by the proportion of the portfolio allocated to it. The risk-free asset has zero standard deviation.

What is the Capital Allocation Line (CAL), and what does it represent?

The Capital Allocation Line (CAL) represents the risk-return trade-off of a portfolio that combines a risky asset and a risk-free asset. The slope of the CAL is determined by the Sharpe ratio of the risky asset, and it shows how the portfolio's expected return increases with more risk (higher standard deviation).

How does the risk aversion coefficient affect the optimal asset allocation?

The risk aversion coefficient (denoted as 'a') reflects an investor’s preference for risk. A higher coefficient indicates a lower tolerance for risk, leading to a higher allocation in risk-free assets and a lower allocation in risky assets. The optimal allocation is determined by balancing the investor's risk aversion with the risk-return trade-off of the assets.

What is the purpose of using the internal rate of return (IRR) in investment analysis?

The internal rate of return (IRR) is used to evaluate the profitability of an investment. It is the discount rate that makes the present value of cash inflows equal to the initial investment. IRR accounts for the timing and amount of cash flows, providing a measure of return that adjusts for the changing investment amounts over time.