# Required Rate of Return, Expected Return, Discrete Distribution

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Please show all the work.

Questions:

1) What types of risk are measured by standard deviation and beta? Which type of risk is more important? Where does correlation fit in? Assume the market risk premium is 5%. Research a current stock and determine its required rate of return using the SML.

#6-3

Security A has an expected return of 7%, a standard deviation of returns of 35%, a correlation coefficient with the market of -0.3, and a beta coefficient of -1.5.

Security B has an expected return of 12%, a standard deviation of returns of 10%, a correlation with the market of 0.7, and a beta coefficient of 1.0. Which security is riskier?

Why?

#6-4

Suppose you owned a portfolio consisting of $250,000 of government bonds with a maturity of 30 years.

a. Would your portfolio be riskless?

b. Now suppose you hold a portfolio consisting of $250,000 of 30-day Treasure bills. Every 30 days your bills mature, and you reinvest the principal ($250,000) in a new batch of bill. Assume that you live on the investment income from your portfolio and that you want to maintain a constant standard of living. Is your portfolio truly riskless?

c. Can you think of any asset that would be completely riskless? What security comes closest to being riskless? Explain.

#6-5

If investors' aversion to risk increased, would the risk premium on a high-beta stock increase by more or less than that on a low-beta stock? Explain.

#6-6

If a company's beta were to double, would its expected return double?

Problems:

#6-1

Portfolio Beta

An individual has $35,000 invested in a stock with a beta of 0.8 and another $40,000 invested in a stock with a beta of 1.4. If these are the only two investments in her portfolio, what is her portfolio's beta?

#6-2

Required Rate

of Return

Assume that the risk-free rate is 6% and that the expected return on the market is 13%. What is the required rate of return on a stock that has a beta of 0.7?

#6-4

Expected Return:

Discrete Distribution

A stock's return has the following distribution:

Demand for the Probability of This Rate of Return if this

Company's products Demand Occurring Demand Occurs (%)

Weak 0.1 -50%

Below average 0.2 (5)

Average 0.4 16

Above average 0.2 25

Strong 0.1 60

1.0

Calculate the stock's expected return, standard deviation, and coefficient of variation.

#6-7

Required Rate of

Return

Suppose rRF = 9%, rM = 14%, bi = 1.3.

a. What is ri, the required rate of return on stock i?

b. Now suppose rRF (1) increases to 10% or (2) decreases to 8%. The slope of the SML remains constant. How would this affect rM and ri?

c. Now Assume rRF remains at 9% but rM (1) increases to 16% or (2) falls to 13%. The slope of the SML does not remain constant. How would these changes affect ri?

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#### Solution Preview

Please find attached file.

Questions:

1) What types of risk are measured by standard deviation and beta? Which type of risk is more important? Where does correlation fit in? Assume the market risk premium is 5%. Research a current stock and determine its required rate of return using the SML.

Since the standard deviation is basically a measure of an investment's volatility, the type of risk that is most widely measured using it is the market risk. This is so because one of the most unpredictable elements within investments is Market movement which is known to get volatile any time. The beta on the other hand, being a measure of volatility but only in comparison to a market index or benchmark, can be used to measure the market risk as well. However with the standard deviation not tied down to any specific benchmark, it can also be used to measure both default risk and foreign exchange risks.

The type of risk that is more important is the standard deviation. This is because the standard deviation gives an actual return on the fund, irrespective of whether it is in line with any set benchmark. The beta on the other hand may indicate an investment as performing well relative to the market, but in retrospect be underperforming given its own internal characteristics which state that it ought to be giving the shareholders a greater return and not simply beating the market.

Correlation in this instance fits in when comparing how two or more investments may be performing or behaving in respect to each other.

Market risk premium can be defined as the difference between an investment's risk-free rate and its expected returns within the market. The required rate of return however is the minimum rate of return that investors anticipate that their investment should return to them. The required rate of return is generally derived from the formula:

Ke= Krf + (Km-Krf)ß

In this instance, the risk free rate is usually the three month U.S. Treasury bill rate, which here will be set at 0.025%.

Required rate of return for Citigroup in this case would be:

Ke= Krf + (Km-Krf) ß

Ke= 2.5 + (5-2.5)2.56

Ke= 2.5 + 6.4

Ke= 8.9%

#6-3

Security A has an expected return of 7%, a standard deviation of returns of 35%, a correlation coefficient with the market of -0.3, and a beta coefficient of -1.5.

Security B has an ...

#### Solution Summary

A required rate of return, expected return and discrete distribution is examined.

Finance: Bond valuation, YTM, beta, Portfolio Required Return, DPS, constant growth

Please see attached file for problems.

Week 3 Problem Set Solutions

5-1: Bond Valuation with Annual Payments.

Jackson Corporations bonds have 12 years remaining to maturity. Interest is paid annually, the bonds have a $ 1,000 par value, and the coupon interest rate is 8%. The bonds have a yield to maturity of 9%. What is the current market price of these bonds?

5-2: Yield to Maturity for Annual Payments.

Wilson Wonders bonds have 12 years remaining to maturity. Interest is paid annually, the bonds have a $ 1,000 par value, and the coupon interest rate is 10%. The bonds sell at a price of $ 850. What is their yield to maturity?

5-3: Current Yield for Annual Payments.

Heath Foods bonds have 7 years remaining to maturity. The bonds have a face value of $ 1,000 and a yield to maturity of 8%. They pay interest annually and have a 9% coupon rate. What is their current yield?

5-7: Bond Valuation with Semiannual Payments.

Renfro Rentals has issued bonds that have a 10% coupon rate, payable semiannually. The bonds mature in 8 years, have a face value of $ 1,000, and a yield to maturity of 8.5%. What is the price of the bonds?

5-8: Yield to Maturity and Call with Semiannual Payments.

Thatcher Corporations bonds will mature in 10 years. The bonds have a face value of $ 1,000 and an 8% coupon rate, paid semiannually. The price of the bonds is $ 1,100. The bonds are callable in 5 years at a call price of $ 1,050. What is their yield to maturity? What is their yield to call?

6-1: Portfolio Beta.

An individual has $ 35,000 invested in a stock which has a beta of 0.8 and $ 40,000 invested in a stock with a beta of 1.4. If these are the only two investments in her portfolio, what is her portfolios beta?

6-3: Expected and Required Rates of Return.

Assume that the risk- free rate is 5% and the market risk premium is 6%. What is the expected return for the overall stock market? What is the required rate of return on a stock that has a beta of 1.2?

6-4: Expected Return Discrete Distribution.

A stocks return has the following distribution:

Calculate the stocks expected return, standard deviation, and coefficient of variation.

6-9: Portfolio Required Return

Suppose you are the money manager of a $ 4 million investment fund. The fund consists of four stocks with the following investments and betas:

6-10: Portfolio Beta

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 which has a beta equal to 0.9 and using the proceeds to purchase another stock which has a beta equal to 1.4. What will be the new beta of your portfolio following this transaction?

6-11 Required Rate of Return

Stock R has a beta of 1.5, Stock S has a beta of 0.75, the expected rate of return on an average stock is 13%, and the risk- free rate of return is 7%. By how much does the required return on the riskier stock exceed the required return on the less risky stock?

Time to Reflect

Assume that the U.S. financial market consists of two firms, Food Mart and Tech Mart. Use the following stock return data which include U.S. T-Bills to calculate the expected return, beta of market for each security, and the required rate of return of all securities in the market including the market itself.

8-1: DPS Calculation

Thress Industries just paid a dividend of $ 1.50 a share ( i. e., D0 $ 1.50). The dividend is expected to grow 5% a year for the next 3 years, and then 10% a year thereafter. What is the expected dividend per share for each of the next 5 years?

8-2: Constant Growth Valuation

Boehm Incorporated is expected to pay a $ 1.50 per share dividend at the end of the year ( i. e., D1 =$ 1.50). The dividend is expected to grow at a constant rate of 7% a year. The required rate of return on the stock, rs, is 15%. What is the value per share of the company's stock?

8-3: Constant Growth Valuation

Woidtke Manufacturings stock currently sells for $ 20 a share. The stock just paid a dividend of $ 1.00 a share ( i. e., D0 =$ 1.00). The dividend is expected to grow at a constant rate of 10% a year. What stock price is expected 1 year from now? What is the required rate of return on the company's stock?

8-4: Preferred Stock Valuation

Basil Pet Products has preferred stock outstanding which pays a dividend of $ 5 at the end of each year. The preferred stock sells for $ 50 a share. What is the preferred stocks required rate of return?

8-11: Nonconstant Growth Stock Valuation

Assume that the average firm in your companys industry is expected to grow at a constant rate of 6% and its dividend yield is 7%. Your company is about as risky as the average firm in the industry, but it has just successfully completed some R& D work that leads you to expect that its earnings and dividends will grow at a rate of 50% [ D1= D0( 1 g) D0=( 1.50)] this year and 25% the following year, after which growth should match the 6% industry average rate. The last dividend paid ( D0) was $ 1. What is the value per share of your firms stock?

8-12: Nonconstant Growth Stock Valuation

Simpkins Corporation is expanding rapidly, and it currently needs to retain all of its earnings; hence it does not pay any dividends. However, investors expect Simpkins to begin paying dividends, with the first dividend of $ 1.00 coming 3 years from today. The dividend should grow rapidly at a rate of 50% per year during Years 4 and 5. After Year 5, the company should grow at a constant rate of 8% per year. If the required return on the stock is 15%, what is the value of the stock today?

8-13: Preferred Stock Valuation

Rolen Riders issued preferred stock with a stated dividend of 10% of par. Preferred stock of this type currently yields 8%, and the par value is $ 100. Assume dividends are paid annually. a. What is the value of Rolens preferred stock? b. Suppose interest rate levels rise to the point where the preferred stock now yields 12%. What would be the value of Rolens preferred stock?

8-14: Return on Common Stock

You buy a share of The Ludwig Corporation stock for $ 21.40. You expect it to pay dividends of $ 1.07, $ 1.1449, and $ 1.2250 in Years 1, 2, and 3, respectively, and you expect to sell it at a price of $ 26.22 at the end of 3 years.

a. Calculate the growth rate in dividends.

b. Calculate the expected dividend yield.

c. Assuming that the calculated growth rate is expected to continue, you can add the dividend yield to the expected growth rate to get the expected total rate of return. What is this stocks expected total rate of return?

8-17: Constant Growth Stock Valuation

Suppose a firms common stock paid a dividend of $ 2 yesterday. You expect the dividend to grow at the rate of 5% per year for the next 3 years, and, if you buy the stock, you plan to hold it for 3 years and then sell it.

a. Find the expected dividend for each of the next 3 years; that is, calculate D1, D2, and D3. Note that D0 $ 2.

b. Given that the appropriate discount rate is 12% and that the first of these dividend payments will occur 1 year from now, find the present value of the dividend stream; that is, calculate the PV of D1, D2, and D3, and then sum these PVs.

c. You expect the price of the stock 3 years from now to be $ 34.73; that is, you expect P 3 to equal $ 34.73. Discounted at a 12% rate, what is the present value of this expected future stock price? In other words, calculate the PV of $ 34.73.

d. If you plan to buy the stock, hold it for 3 years, and then sell it for $ 34.73, what is the most you should pay for it?

e. Use Equation 8- 2 to calculate the present value of this stock. Assume that g 5%, and it is constant.

f. Is the value of this stock dependent on how long you plan to hold it? In other words, if your planned holding period were 2 years or 5 years rather than 3 years, would this affect the value of the stock today, P 0? Reizenstein Trucking (RT) has just developed a solar