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Risk and Return Exercises in
Corporate Finance
Risk and Return Exercises
1. Stock Returns
Suppose a stock had an initial price of $75 per share, paid a dividend of $1.20 per share during the year, and had an ending share price of $86.
To calculate the total return on this stock, we can use the formula:
Total Return = (Ending Price - Beginning Price + Dividends) / Beginning Price
Plugging in the values: Total Return = ($86 - $75 + $1.20) / $75 = 0.1600 or 16.00%
The total return on this stock over the year was 16.00%.
2. Bond Returns
Suppose you bought a 6% coupon bond one year ago for $1,040. The bond sells for $1,063 today.
a. Assuming a $1,000 face value, what was your total dollar return on this investment over the past year? The total dollar return is the sum of the interest received and the change in the bond's price. Interest received = 6% of $1,000 = $60 Change in bond price = $1,063 - $1,040 = $23 Total dollar return = $60 + $23 = $
b. What was your total nominal rate of return on this investment over the past year? The total nominal rate of return is calculated as: Nominal Rate of Return = (Total Dollar Return) / (Initial Investment) Nominal Rate of Return = $83 / $1,040 = 0.0798 or 7.98%
c. If the inflation rate last year was 3%, what was your total real rate of return on this investment? The real rate of return is calculated as: Real Rate of Return = (1 + Nominal Rate of Return) / (1 + Inflation Rate) - 1 Real Rate of Return = (1 + 0.0798) / (1 + 0.03) - 1 = 0.0478 or 4.78%
3. Calculating Returns and Variability
Using the following returns, calculate the average returns, the variances, and the standard deviations for X & Y:
The average return for X and Y can be calculated by summing the returns and dividing by the number of observations.
The variance for X and Y can be calculated by squaring the difference between each return and the average return, then summing the squared differences and dividing by the number of observations.
The standard deviation is the square root of the variance.
4. Calculating Returns
You bought a stock 3 months ago for $43.18 per share. The stock paid no dividends. The current price is $46.21.
To calculate the annualized percentage return (APR) of your investment, we can use the formula:
APR = [(Ending Price - Beginning Price) / Beginning Price] * (365 / 90)
Plugging in the values: APR = [($46.21 - $43.18) / $43.18] * (365 / 90) = 0.0700 or 7.00%
The APR of your investment is 7.00%.
5. Systematic versus Unsystematic Risk
Classify the following events as mostly systematic or mostly unsystematic:
a. Short-term interest rates increase unexpectedly - Systematic b. The interest rate a company pays on its short-term debt borrowing is increased by its bank - Unsystematic c. Oil prices unexpectedly decline - Systematic d. An oil tanker ruptures, creating a large oil spill - Unsystematic e. A manufacturer loses a multimillion-dollar product liability suit - Unsystematic f. A Supreme Court decision substantially broadens producer liability for injuries suffered by product users - Systematic
6. Beta
Consider the following quotation from a leading investment manager: "The share of Southern Co. have traded close to $12 for most of the past 3 years. Since Southern's stock has demonstrated very little price movement, the stock has low beta. Texas Instruments, on the other hand, has traded as high as $150 and as low as its current $75. Since the TI's stock has demonstrated a large amount of price movement, the stock has a very high beta".
Do you agree with this analysis? Explain.
No, I do not agree with this analysis. The beta of a stock is not solely determined by the magnitude of its price movements. Beta measures the sensitivity of a stock's returns to the returns of the overall market. A stock with low price volatility can still have a high beta if its returns are highly correlated with the market. Conversely, a stock with high price volatility can have a low beta if its returns are not strongly correlated with the market. The analysis provided oversimplifies the relationship between price volatility and beta.
Returns and Standard Deviations Stock A: Expected return = 12%, Standard deviation = 20% Stock B: Expected return = 15%, Standard deviation = 25% Stock C: Expected return = 18%, Standard deviation = 30%
a. What is the expected return on an equally weighted portfolio of these 3 stocks? The expected return on an equally weighted portfolio is the average of the expected returns of the individual stocks: Expected Return = (12% + 15% + 18%) / 3 = 15%
b. What is the variance of a portfolio invested 20% each in A and B, and 60% in C? The variance of a portfolio is calculated using the formula: Var(P) = w_A^2 * Var(A) + w_B^2 * Var(B) + w_C^2 * Var(C) + 2 * w_A * w_B * Cov(A,B) + 2 * w_A * w_C * Cov(A,C) + 2 * w_B * w_C * Cov(B,C)
Assuming the covariances are unknown, we can simplify the calculation: Var(P) = 0.2^2 * 0.2^2 + 0.2^2 * 0.25^2 + 0.6^2 * 0.3^2 = 0.0400 + 0.0625 + 0.1620 = 0.
The standard deviation of the portfolio is the square root of the variance: Std Dev(P) = √0.2645 = 0.5145 or 51.45%
11. Calculating Portfolio Betas
You own a stock portfolio invested 10% in stock Q, 35% in Stock R, 20% in Stock S, and 35% in Stock T. The betas for these 4 stocks are 0.75, 1.90, 1.38 and 1.16, respectively.
To calculate the beta of the overall portfolio, we can use the formula:
β_P = Σ w_i * β_i
Where: β_P = Portfolio beta w_i = Weight of stock i in the portfolio β_i = Beta of stock i
Plugging in the values: β_P = 0.10 * 0.75 + 0.35 * 1.90 + 0.20 * 1.38 + 0.
- 1.16 β_P = 0.075 + 0.665 + 0.276 + 0.406 = 1.
The beta of the overall portfolio is 1.422.
12. Calculating Portfolio Betas
You own a portfolio equally invested in a risk-free asset and 2 stocks. If one of the stocks has a beta of 1.65 and the total portfolio is equally as risky as the market, what must the beta be for the other stock in your portfolio?
Let's call the beta of the other stock β.
Since the portfolio is equally invested in the risk-free asset and the two stocks, the portfolio weight of each stock is 0.5.
The portfolio beta is calculated as: β_P = 0.5 * 1.65 + 0.5 * β = 0.825 + 0.5β
Since the portfolio is equally as risky as the market, the portfolio beta must be 1.0.
Substituting this in the equation: 1.0 = 0.825 + 0.5β 0.175 = 0.5β β = 0.
The beta of the other stock in the portfolio must be 0.35.
13. Using CAPM
A stock has a beta of 1.15, the expected return on the market is 11%, and the risk-free rate is 5%. What must the expected return on this stock be?
We can use the Capital Asset Pricing Model (CAPM) to calculate the expected return on the stock:
E(R_i) = R_f + β_i * (E(R_m) - R_f)
Where: E(R_i) = Expected return on the stock R_f = Risk-free rate β_i = Beta of the stock E(R_m) = Expected return on the market
Plugging in the values: E(R_i) = 5% + 1.15 * (11% - 5%) = 5% + 1.15 * 6% = 12.90%
The expected return on this stock must be 12.90%.
14. Using CAPM
A stock has an expected return of 10.2%, the risk-free rate is 4% and the market risk premium is 7%. What must the beta of this stock be?
We can use the Capital Asset Pricing Model (CAPM) to solve for the beta of the stock:
E(R_i) = R_f + β_i * (E(R_m) - R_f) 10.2% = 4% + β_i * (7%) β_i = (10.2% - 4%) / 7% = 0.
The beta of this stock must be 0.8857.
15. Security Market Line (SML)
Suppose you observe the following situation:
Security | Beta | Expected Return --- | --- | --- A | 0.80 | 9.0% B | 1.20 | 12.0% C | 1.50 | 14.0%
Assume these securities are correctly priced. Based on the CAPM, what is the expected return on the market?
We can use the CAPM equation to solve for the expected return on the market:
E(R_i) = R_f + β_i * (E(R_m) - R_f)
