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Chapter 6: Problem 1

A stock's expected return has the following distribution: $$\begin{array}{lcc} \begin{array}{l} \text { DEMAND FOR THE } \\ \text { COMPANY'S PRODUCTS } \end{array} & \begin{array}{c} \text { PROBABILITY OF THIS } \\ \text { DEMAND OCCURRING } \end{array} & \begin{array}{c} \text { RATE OF RETURN } \\ \text { IF THIS DEMAND } \\ \text { OCCURS } \end{array} \\ \hline \text { Weak } & 0.1 & (50 \%) \\ \text { Below average } & 0.2 & (5) \\ \text { Average } & 0.4 & 16 \\ \text { Above average } & 0.2 & 25 \\ \text { Strong } & 0.1 & 60 \\ & 1.0 & \\ \hline \end{array}$$ Calculate the stock's expected return, standard deviation, and coefficient of variation.

Short Answer

Expert verified
Expected return is 11.4%, standard deviation is 29.58%, and CV is 2.593.

Step by step solution

01

Understand the Problem

We need to calculate the stock's expected return, standard deviation, and coefficient of variation based on the provided probabilities and rates of return. Understanding the connection between probability, rate of return, and these measures is crucial for solving this problem.
02

Calculate the Expected Return

The expected return is calculated as the sum of the product of each rate of return and its probability. This can be expressed as \( E(R) = \sum p_i \times R_i \), where \( p_i \) is the probability and \( R_i \) is the rate of return.\[E(R) = (0.1 \times -0.5) + (0.2 \times -0.05) + (0.4 \times 0.16) + (0.2 \times 0.25) + (0.1 \times 0.6)\]\[E(R) = -0.05 - 0.01 + 0.064 + 0.05 + 0.06 = 0.114\] Therefore, the expected return is 11.4%.
03

Calculate the Variance and Standard Deviation

First, calculate the variance using the formula \( \sigma^2 = \sum p_i (R_i - E(R))^2 \).\[\sigma^2 = (0.1 \times (-0.5 - 0.114)^2) + (0.2 \times (-0.05 - 0.114)^2) + (0.4 \times (0.16 - 0.114)^2) + (0.2 \times (0.25 - 0.114)^2) + (0.1 \times (0.6 - 0.114)^2) \]\[\sigma^2 = 0.037849 + 0.006688 + 0.0009216 + 0.018496 + 0.023524\]\[\sigma^2 = 0.0874786\] The standard deviation is \( \sigma = \sqrt{0.0874786} = 0.2958 \) or 29.58%.
04

Calculate the Coefficient of Variation

The coefficient of variation (CV) is calculated using \( CV = \frac{\sigma}{E(R)} \), where \( \sigma \) is the standard deviation and \( E(R) \) is the expected return.\[CV = \frac{0.2958}{0.114} = 2.593\] This means the coefficient of variation is 2.593.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution
Probability distribution is a core concept in statistics and finance that shows us how likely different outcomes are. Think of it as a map of various future scenarios. In the context of investments, it tells us the probability that a stock will yield certain rates of return.
- A probability distribution assigns a probability to all possible outcomes (in this case, different demand levels) that always total to 1. - For example, the given probabilities for the stock's returns based on demand are 0.1, 0.2, 0.4, 0.2, and 0.1 for "Weak," "Below average," "Average," "Above average," and "Strong" demand, respectively, summing up to 1.0.
Understanding probability distributions allows investors to predict likely scenarios and make informed decisions.
This distribution provides insights into the riskiness of an investment by showing all possible returns the investment might yield.
Standard Deviation
Standard deviation helps us understand risk by measuring the variation or spread of possible investment returns around the expected return. The larger the standard deviation, the more varied the returns are likely to be, indicating a riskier investment.
- It calculates how much the return values deviate from the expected return. So, it tells us how much the actual returns might differ from what we are expecting. - In the exercise, once the expected return is determined to be 11.4%, the standard deviation is calculated using each probability-weighted deviation from this expected return.
Mathematically, the standard deviation is the square root of the variance, which is the average of squared deviations from the expected return.
The given standard deviation of 29.58% implies that the actual returns could vary quite a bit from the expected return of 11.4%.
Coefficient of Variation
The coefficient of variation (CV) is a relative measure of variability. It provides context to the standard deviation, allowing us to understand the risk of an investment relative to its expected return.
- CV is especially useful when comparing the risks of two or more investments with different expected returns. It tells us how much risk one bears per unit of return.- Calculated as CV = \(\frac{\sigma}{E(R)}\), where \(\sigma\) is the standard deviation and \(E(R)\) is the expected return.- In the exercise, with a standard deviation of 29.58% and an expected return of 11.4%, the CV is 2.593.
A CV of 2.593 suggests a relatively high level of risk compared to the expected return. The higher the CV, the greater the relative risk, helping investors assess potential return variation when making choices.

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Most popular questions from this chapter

You have a \(\$ 2\) million portfolio consisting of a \(\$ 100,000\) investment in each of 20 different stocks. The portfolio has a beta equal to \(1.1 .\) You are considering selling \(\$ 100,000\) worth of one stock that has a beta equal to 0.9 and using the proceeds to purchase another stock that has a beta equal to \(1.4 .\) What will be the new beta of your portfolio following this transaction?

Suppose \(\mathrm{k}_{\mathrm{RF}}=9 \%, \mathrm{k}_{\mathrm{M}}=14 \%,\) and \(\mathrm{b}_{\mathrm{i}}=1.3\) a. What is \(\mathrm{k}_{\mathrm{i}}\), the required rate of return on Stock i? b. Now suppose \(k_{R F}(1)\) increases to 10 percent or (2) decreases to 8 percent. The slope of the SML remains constant. How would this affect \(\mathrm{k}_{\mathrm{M}}\) and \(\mathrm{k}_{\mathrm{i}}\) c. Now assume \(\mathrm{k}_{\mathrm{RF}}\) remains at 9 percent but \(\mathrm{k}_{\mathrm{M}}(1)\) increases to 16 percent or (2) falls to 13 percent. The slope of the SML does not remain constant. How would these changes affect \(\mathrm{k}_{\mathrm{i}}\)

Stocks \(X\) and \(Y\) have the following probability distributions of expected future returns: $$\begin{array}{ccc} \text { PROBABILITY } & \mathbf{X} & \mathbf{Y} \\ \hline 0.1 & (10 \%) & (35 \%) \\ 0.2 & 2 & 0 \\ 0.4 & 12 & 20 \\ 0.2 & 20 & 25 \\ 0.1 & 38 & 45 \end{array}$$ a. Calculate the expected rate of return, \(\hat{\mathrm{k}},\) for Stock \(\mathrm{Y}\). ( \(\hat{\mathrm{k}}_{\mathrm{X}}=12 \%\).) b. Calculate the standard deviation of expected returns for Stock X. (That for Stock \(\mathrm{Y}\) is 20.35 percent.) Now calculate the coefficient of variation for Stock \(\mathrm{Y}\). Is it possible that most investors might regard Stock \(Y\) as being less risky than Stock \(X\) ? Explain.

Stocks A and B have the following historical returns: $$\begin{array}{lcc} \text { YEAR } & \text { STOCK A'S RETURNS, } \mathbf{k}_{\mathrm{A}} & \text { STOCK B'S RETURNS, } \mathbf{k}_{\mathbf{B}} \\ \hline 1997 & (18.00 \%) & (14.50 \%) \\ 1998 & 33.00 & 21.80 \\ 1999 & 15.00 & 30.50 \\ 2000 & (0.50) & (7.60) \\ 2001 & 27.00 & 26.30 \end{array}$$ a. Calculate the average rate of return for each stock during the period 1997 through 2001 b. Assume that someone held a portfolio consisting of 50 percent of Stock \(A\) and 50 percent of Stock B. What would have been the realized rate of return on the portfolio in each year from 1997 through 2001 ? What would have been the average return on the portfolio during this period? c. Calculate the standard deviation of returns for each stock and for the portfolio. d. Calculate the coefficient of variation for each stock and for the portfolio. e. If you are a risk-averse investor, would you prefer to hold Stock \(\mathrm{A}\), Stock \(\mathrm{B}\), or the portfolio? Why?

Suppose you won the Florida lottery and were offered (1) \(\$ 0.5\) million or (2) a gamble in which you would receive \(\$ 1\) million if a head were flipped but zero if a tail came up. a. What is the expected value of the gamble? b. Would you take the sure \(\$ 0.5\) million or the gamble? c. If you choose the sure \(\$ 0.5\) million, are you a risk averter or a risk seeker? d. Suppose you actually take the sure \(\$ 0.5\) million. You can invest it in either a U.S. Treasury bond that will return \(\$ 537,500\) at the end of a year or a common stock that has a \(50-50\) chance of being either worthless or worth \(\$ 1,150,000\) at the end of the year. (1) What is the expected dollar profit on the stock investment? (The expected profit on the T-bond investment is \(\$ 37,500\).) (2) What is the expected rate of return on the stock investment? (The expected rate of return on the T-bond investment is 7.5 percent. (3) Would you invest in the bond or the stock? (4) Exactly how large would the expected profit (or the expected rate of return) have to be on the stock investment to make \(y\) ou invest in the stock, given the 7.5 percent return on the bond? (5) How might your decision be affected if, rather than buying one stock for \(\$ 0.5\) million, you could construct a portfolio consisting of 100 stocks with \(\$ 5,000\) invested in each? Each of these stocks has the same return characteristics as the one stock - that is, a \(50-50\) chance of being worth either zero or \(\$ 11,500\) at year-end. Would the correlation between returns on these stocks matter?
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