Intro to Finance: Risk and Return - Expected Return Probability-Weighted average

What This Is

Expected Return is a probability-weighted average of possible returns on an investment. It's a crucial concept in finance as it helps investors and analysts evaluate the potential return on a security or portfolio. For example, consider a portfolio with two possible outcomes: a 10% return with a 60% probability and a 20% return with a 40% probability. The expected return would be (0.6 x 10%) + (0.4 x 20%) = 12%.

Key Formulas & Symbols

• Expected Return (ER) =? (Probability × Return) where ER = expected return, Probability = probability of each outcome, Return = return on each outcome.
• Probability = P(A) = Number of favorable outcomes / Total number of outcomes where P(A) = probability of event A, Number of favorable outcomes = number of outcomes that meet the condition, Total number of outcomes = total number of possible outcomes.
• Return = R = (FV - PV) / PV where R = return, FV = future value, PV = present value.
• Discounted Expected Return (DER) = ER / (1 + r) where DER = discounted expected return, ER = expected return, r = discount rate.
• Risk-Adjusted Return = RAR = ER - (Risk Premium) where RAR = risk-adjusted return, ER = expected return, Risk Premium = additional return required for taking on risk.
• Sharpe Ratio = SR = (R - RF) / ? where SR = Sharpe ratio, R = expected return, RF = risk-free rate,-= standard deviation of returns.
• Treynor Ratio = TR = (R - RF) / ? where TR = Treynor ratio, R = expected return, RF = risk-free rate,-= beta of the asset.

Step-by-Step Calculation

1. Identify all possible outcomes and their associated probabilities.
2. Calculate the return for each outcome using the formula Return = (FV - PV) / PV.
3. Multiply each return by its associated probability to get the weighted return.
4. Sum up the weighted returns to get the expected return.
5. If using a discount rate, calculate the discounted expected return using the formula DER = ER / (1 + r).

Common Mistakes

• Mistake: Using the arithmetic mean instead of the geometric mean for calculating expected return.
• Correction: The geometric mean is used because it takes into account the compounding effect of returns.
• Mistake: Confusing expected return with actual return.
• Correction: Expected return is a probability-weighted average of possible returns, while actual return is the actual return on an investment.
• Mistake: Not considering the time value of money when calculating expected return.
• Correction: The time value of money should be considered by using a discount rate to calculate the discounted expected return.

Exam / CFA Tips

• Tip: Be careful with question wording, as it may imply a specific calculation or assumption.
• Tip: Use the correct formula for calculating expected return, and make sure to consider all possible outcomes.
• Tip: Be aware of the difference between expected return and actual return.

Quick Practice Problem

A portfolio has two possible outcomes: a 10% return with a 60% probability and a 20% return with a 40% probability. What is the expected return on this portfolio?

Answer: 12% Explanation: (0.6 x 10%) + (0.4 x 20%) = 12%

Last-Minute Cram Sheet

• The expected return is a probability-weighted average of possible returns.
• The expected return is calculated using the formula ER =? (Probability × Return).
• The geometric mean is used to calculate expected return.
• The time value of money should be considered when calculating expected return.
• The actual return is not the same as the expected return.
• The Sharpe ratio is used to measure risk-adjusted return.
• The Treynor ratio is used to measure risk-adjusted return.
• The risk-free rate is used as a benchmark for expected return.
• The beta of an asset is used to measure its systematic risk.
• The expected return is not the same as the required return.