Topic Outline

1. Relationship Between Risk and Return

2. Measures of Return

3. Risk and Return in a Portfolio Context

 

Definition of Terms

• Risk - is the variability of returns from those that are expected. Risk implies a degree of uncertainty. 

• Rate of Return - the amount received on an investment from holding that investment for a period of time relative to the amount of the initial investment. Returns indicate fluctuations in market prices for the investment along with any interest or dividends received, typically represented as a percentage relative to the initial market price of the investment.

 

Note: As a rule the more the risk, the higher the return; the lower the risk, the lower the return.

 

Relationship Between Risk and Return

1. Whether the expected return on an investment is sufficient to entice an investor depends on its risks and returns of alternative investments, and the investor’s attitude toward risk.

   • Most serious investors are risk averse. They have a diminishing marginal utility for wealth. Serious investors find less satisfaction in gaining an amount compared to the dissatisfaction they experience from losing the same amount. Due to this risk aversion, securities with higher risk must offer higher expected returns.

   • A risk-neutral investor uses an expected value approach because they view the utility of gaining as equivalent to the disutility of losing the same amount. Therefore, a risk-neutral investor maintains a purely rational stance toward risk.

   • A risk-seeking investor has an optimistic attitude towards risk. (S)he regards the utility of a gain as exceeding the disutility of a loss of the same amount.

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Measures of Return

 

Holding Period Return (HPR)

• Total return on an asset or portfolio over the period during which it was held.

 

Equation:

R = (Pt - Pt - 1) + DtPt - 1

 

where:

R = rate of return (holding period return)

Pt= stock price at the end of the period

t = time period

Pt-1 = stock price at the beginning of the period

Dt = cash dividend at the end of the time period

 

Annualized HPR

(1+ HPR) 1/n - 1

 

For Example:

What’s the one-year rate of return for an investor who bought a share of common stock for $20 one year ago, and the stock price increased to $22 during that period, with a $2 cash dividend paid per share?

• Pt (current stock price) = $22

• Pt-1 (previous stock price) = $20

• Dt (cash dividend) = $2

 

R = ($22 - $20) + $2$20 = $4$20 = 0.20 or 20%

 

* The time period (t) can be any length of time so the t represents the period for the common stock of one year. Hence, the rate of return on common stock is 20%.

 

Expected Returns

• The weighted average of potential returns, with the weights determined by the probabilities of each outcome, serves as a measure of the typical value within a probability distribution.

 

The formula for expected return is:

 

Expected rate of return = ∑ (Possible rate of return x Probability)

E(R) = i = 1n(Ri )(Pi)     or

 

where:

R = expected return

n = total number of possibilities

Ri = return for the ith possibility

Pi = probability of that return occurring

 

Example:

A company is considering investing in the common stock of one of two firms, ABC Corp. and XYZ Co. Below is calculated as follows:

 

ABC Corp. Stock

Rate of                                           Weighted

Return %         Probability %           Average

   80%        x          60%            =          48%

  (50)%       x         40%             =         (20)%

Expected rate of return (R)                 28%  

XYZ  Co. Stock

Rate of                                           Weighted

Return %         Probability %           Average

   30%        x          70%            =          21%

  (10)%       x         30%             =         (3)%

Expected rate of return (R)                 18%  

 

The expected rate of return on ABC Corp. stock is higher, but the risk of each investment also should be measured.

 

Standard Deviation

• Is a statistical measure of the variance or dispersion around the most likely expected return on an investment. It measures the variability of a distribution around the mean (average) and is computed as the square root of the variance.

 

The formula for standard deviation(o) is:

= i = 1n(Ri-R)2(Pi)

 

where:

= standard deviation

n = total number of possibilities

Ri = return for the ith possibility

Pi = probability of that return occurring

 

• In the equation, the deviations from the mean (Ri - R) are squared and then weighted by the probabilities of the returns occurring. Generally, a higher standard deviation indicates increased return variability and total risk.

• The standard deviation measures the tightness of the distribution and the riskiness of the investment.

 

Example:

The following measures the risk of the investments from the previous example using standard deviation.

ABC Corp. Stock

() = [(80%-28%)2 x 60%] + [(-50%-28%)2x 40%] = 4056 = 63.69%

 

XYZ Co. Stock

() = [(30%-18%)2 x 70%] + [(-10%-18%)2x 30%] = 336 = 18.33%

 

Although the investment in Xatalan stock has a higher expected return than the investment in Yarmouth stock (28% > 18%), it also is riskier than Yarmouth because its standard deviation is greater (63.69% > 18.33%). Therefore, to determine which investment is the better choice in terms of the risk-return tradeoff, we must measure the coefficient of variation (CV) of the expected returns on the two investments.

 

Coefficient of Variation (CV)

• It is useful when the rates of return and standard deviations of two investments differ. It measures the risk per unit of return.

The formula is:

CV = R

* The CV is calculated by dividing the standard deviation by the mean of expected return.

 

• The lower the ratio, the better the risk-return tradeoff is.

• When assessing the risk of multiple investments, it's crucial to understand that relying solely on standard deviation can be misleading. Calculating the coefficient of variation provides a more effective basis for comparison because it quantifies the risk relative to the return.

 

Return in a Portfolio Context

Portfolio Return

• The weighted average of the expected returns of the individual assets within a portfolio. The weights correspond to the proportions of each asset in the portfolio, with the total weights summing up to 100%.

 

The general formula for the expected rate of return for a portfolio is:

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Rp = i = 1nWiRi

where:

Rp = expected return of a portfolio

n = number of different securities in the portfolio

Wi = proportion or weight of the total funds invested in security i

Ri = expected return for security i

 

• A common investment strategy involves building an efficient portfolio, aiming to maximize returns for a specific level of risk or minimize risk for a specific level of return.

• Assessing risk and return should focus on the overall portfolio of a firm, rather than on individual assets.

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QUESTIONS

  

1. The smaller the range of expected future returns, the greater the risk of a given investment as measured by its mean.

                     A. True

                     B. False

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2. If you are trying to determine whether to purchase Security A or Security B as the only holding in your portfolio, then you can consider the coefficient of variation in order to understand the risk-return relationship of the individual securities.

                    A. True

                    B. False

 

3. If the distribution of returns on an asset has a variance of zero, then the covariance of returns between that asset and the returns on any other asset must be equal to zero.

                    A. True

                    B. False

 

4. An investment security with high risk will have a(n)

                   A. Low expected return.                                                C. Increasing expected rate of return.

                   B. Lower price than an asset with low risk.                   D. High standard deviation of returns.

 

5. Catherine & Co. has extra cash at the end of the year and is analyzing the best way to invest the funds. The company should invest in a project only if the

                  A. Expected return on the project exceeds the return on investments of comparable risk.

                  B. Return on investments of comparable risk exceeds the expected return on the project.

                  C. Expected return on the project is equal to the return on investments of comparable risk. 

                  D. Return on investments of comparable risk equals the expected return on the project. 

 

6. City Development, Inc., is considering a new investment project that will involve building a large office block in Frankfurt-am-Main. The firm's financial analysis department has estimated that the proposed investment has the following estimated rate of return distributions.

Rate of Return        Probability

(5%)                          30%

10%                           50%

20%                           20%

 

 

Calculate the expected rate of return.

                   A. 5.5%                                          C. 10.5%

                   B. 7.5%                                          D. 11.7%

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7. George Wilson purchased Bright Light Industries common stock for $47.50 on January 31, 2016. The firm paid dividends of $1.10 during the last 12 months. George sold the stock today (January 30, 2017) for $54.00. What is George's holding period return?

                  A.16.00%                                        C.11.00%

                  B.14.00%                                        D.19.00%

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Questions 8 - 10.

The state of the economy has a strong effect on the expected returns for Techspace, Inc., as shown below:

 

State of the Economy                        Probability                         Techspace Returns

Recession                               .35                                  -10%

Stable                                     .40                                   10%

Expansion                              .25                                    30%

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8. What is the expected rate of return on Techspace, Inc., stock? 

                 A. 8%                                              C. 15%

                 B. 10%                                            D. 30%

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9. Given an expected rate of return on Techspace, Inc., stock of 8.0%, the standard deviation (ơ) is

                 A. 2.36%                                         C. 8.0%

                 B. 8.12%                                         D. 15.36%

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10. Given that the standard deviation () of Techspace returns is 15.36% and the expected rate of return is 8%, the coefficient of variation is

                 A. 1.0%                                          C. 1.92%

                 B. 1.23%                                        D. 2.36%

 

 

ANSWERS:

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1. B

2. A

3. A

4. D

5. A

6. B

7. A

8. A

9. D

10. C

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References:

Wiley. (2023). Wiley CMA Exam Review 2023 Study Guide Part 2: Strategic Financial Management. Hoboken NJ: John Wiley & Sons

Wiley. (2016). Wiley CMA Exam Review 2016 Study Guide Part 2: Strategic Financial Management. Hoboken NJ: John Wiley & Sons

Gleim, I.N & Flesher D.L. (2012). Gleim 16th Edition CMA Review Part 2: Financial Decision Making. Gleim Publications, Inc.

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