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FINANCE Lecture Notes - Lecture 55: Expected Return, Weighted Arithmetic Mean, Capital Asset Pricing Model

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Fordham University
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Finance
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FINANCE
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Finance chapter 11: optimal portfolio choice & capm: the expected return of a portfolio, to find optimal portfolio: need to define portfolio and analyse return. Portfolio weights: fraction of total investment in portfolio held in each individual investment in pf. Portfolio weight x(i) = value of investment (i) / total value of portfolio. Return on portfolio: the weighted average returns on the investments in the portfolio. Expected return of portfolio: the weighted average of expected return of the investments within in: the volatility of a two-stock portfolio. Covariance: expected product of deviations of two return from their means. If two stocks move together, their returns tend to be above or below average at same time = covariance is positive. If two stocks move in opposite directions, one tends to be above average when the other is below average = covariance is negative. Correlation: measures how returns move in relation to each other; quantifies strength of relationship: similar interpretation as covariance.

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Related Questions

Hi guys could someone help me and explain in detail the two fundamental principles of investments – the risk/return tradeoff and diversification. Also, describe how asset allocation and the correlation of various securities within asset classes impact the riskiness of an investor’s portfolio.

I found an answer but it seems to me very long and not sure about it so can you help me out guys

ANS:

Risk /Return Tradeoff

Risk, return and liquidity are three dimensions of investment decision. Risk/Return trade off considers two dimension ie risk and return. As risk of an investment increases , its expected return will decrease. Risk is measured by volatility of return which can be statistically obtained by standard deviation of return. Higher the standard deviation, higher will be volatility of return and higher the risk

Risk can be also measured as a relative risk compared to the market. This is measured by stock beta with reference to a market index. Higher the beta, higher the value of stock Beta, higher will be the risk .

Investors seeking higher return will need to take higher risks.

Treasury bonds are almost risk free assets. This has the lowest rate of return. Stocks have higher risk , hence have higher expected return. High beta stocks will have higher risk and higher returns.

An investor should understand his risk taking ability and will have to make a trade off between risk and return depending on his risk apetite.

Diversification:

There is a saying, do not put all your eggs in one basket. The reason is that it has higher risk if all eggs are in one basket. If the eggs are distributed in different basket, risk of loss decreases.

Same principle is applied in investment. Diversification reduces risk.

If one invests in different classes of assets like bonds, stock, commodities and properties –his risk is likely to reduce. Reduction in return in one asset is likely to be compensated by return in other asset.

Within one class of asset also, there should be diversification to reduce risk.

For example one stock “A “may have expected return of 10% and standard deviation of 5%.This risk can be reduced by diversification in number of assets and the ratio of Risk/return can be decreased.

If we diversify and instead of investing 100% in stock A, we invest 60% in “A” and 40% in another stock “B” having expected return15% and standard deviation of 8%.assume the correlation between the two assets are 0.5

Portfolio return=wa*Ra+wb*Rb

Wa=Weight of asset A=0.6

Wb=Weight of asset B=0.4

Ra=Return of A=10%

Rb=Return of B=15%

Sa=Standard deviation of A=5%

Sb=Standard deviation of B=8%

Corr(a,b)=Correlation of A & B=0.2

Portfolio return=0.6*10+0.4*15=12%

Portfolio Variance=(wa^2)*(Sa^2)+(wb^2)*(Sb^2)+2wa*wb*Cov(a,b)

Cov(a,b)=Corr(a,b)*Sa*Sb=0.1*5*8=4

Portfolio Variance=(0.6^2)*(5^2)+(0.4^2)*(8^2)+2*0.6*0.8*4=23.08

Portfolio standard deviation=Square root of Variance=(23.08^0.5)=4.8%

The diversification has increased return and reduced risk

Stock A was giving 10% return with 5% standard deviation. With diversification of portfolio we have 12% return and 4.8% standard deviation. Risk return ratio is decreased.

It may be noted that, higher the correlation between assets in the portfolio, higher will be risk in terms of standard deviation. Hence it is better to diversify with uncorrelated assets.

Diversification reduces unsystematic risk,but it cannot reduce systematic risk of overall market or economy

Consider the following statement to answer Questions i-iv: You manage a risky portfolio with an expected rate of return of 17% and a standard deviation of27%. The Treasury-bill rate is 7%. (i) One of your clients chooses to invest 70% of a portfolio in your fund and 30% in a T-bill moneymarket fund. What is the expected value and standard deviation of the rate of return on yourclient’s portfolio?(2 marks) (ii) Suppose that your client decides to invest in your portfolio a proportion y of the total investmentbudget so that the overall portfolio will have an expected rate of return of 15%. (a) What is the proportion y? (Hint: Use CAPM formula) (b) Further suppose that your risky portfolio includes the following investments in the givenproportions: Stock A (27%), Stock B (33%), and Stock C (40%). What are your client’sinvestment proportions in your three stocks and the T-bill fund?(2 marks) (iii) Now suppose that your client prefers to invest in your fund a proportion ythat maximizes theexpected return on the overall portfolio subject to the constraint that the overall portfolio’sstandard deviation will not exceed 20%. (a) What is the investment proportion, y? (b) What is the expected rate of return on the overall portfolio?(2 marks) (iv) Suppose that your client’s degree of risk aversion is A = 3.5. What proportion, y, of the totalinvestment should you suggest that he invest in your fund?(2 marks) (v) The market price of a security is $40. Its expected rate of return is 13%. The risk- free rate is 7%and the market risk premium is 8%. What will be the market price of the security if itscovariance with the market portfolio doubles (and all other variables remain unchanged)?Assume that the stock is expected to pay a constant dividend in perpetuity.(2 marks) QUESTION 2: OMEGA Currently Omega Corporation’s shares are selling at $60 per share and company is paying $5 per share dividend. Dividends are expected to grow at an annual rate of 3% for foreseeable future. Required rate of return for investors is 12% At the same time, Heller Corporation’s shares are selling at $58 per share and company is paying $4 per share dividend. Dividends are expected to grow at an annual rate of 5% for foreseeable future. Required rate of return for investors is 12% a) Calculate the current value of each stock on the basis of Dividend Discount Model. b) On the basis of above calculation, determine either each stock is overvalued or undervalued. (5 marks)

What is the taxable equivalent yield on a municipal bond with a yield to maturity of 4.5 percent for an investor in the 39 percent marginal tax bracket?

1.76 percent

4.50 percent

7.38 percent

11.54 percent

Rank the following bonds in order from lowest credit risk to highest risk all with the same time to maturity, by their yield to maturity: JM Corporate bond with yield of 12.25 percent, IB Corporate bond with yield of 4.49 percent, TC Corporate bond with yield of 8.76 percent, and B&O Corporate bond with a yield of 5.99 percent.

JM bond, TC bond, B&O bond, IB bond

IB bond, B&O bond, TC bond, JM bond

TC bond, B&O bond, IB bond, JM bond

JM bond, IB bond, B&O bond, TC bond

A 5.75 percent coupon bond with 12 years left to maturity is offered for sale at $978.83. What yield to maturity is the bond offering? (Assume interest payments are paid semi-annually and par value is $1,000.)

3.00 percent

3.09 percent

5.75 percent

6.00 percent

A client in the 28 percent marginal tax bracket is comparing a municipal bond that offers a 3.25 percent yield to maturity and a similar-risk corporate bond that offers a 4.10 percent yield. Which bond will give the client more profit after taxes?

The municipal bond.

The corporate bond.

Both give the client equal profits after taxes.

There is not enough information given to determine answer.

Portfolio Weights An investor owns $15,000 of Adobe Systems stock, $15,500 of Dow Chemical, and $17,000 of Office Depot. What are the portfolio weights of each stock?

Adobe System = 0.3333, Dow Chemical = 0.3333, Office Depot = 0.3333

Adobe System = 0.3158, Dow Chemical = 0.3263, Office Depot = 0.3579

Adobe System = 0.3263, Dow Chemical = 0.3158, Office Depot = 0.3579

Adobe System = 0.2667, Dow Chemical = 0.3333, Office Depot = 0.4000

Portfolio Return Year-to-date, Company O had earned a -7.4 percent return. During the same time period, Company V earned 9.65 percent and Company M earned 2.68 percent. If you have a portfolio made up of 20 percent Company O, 40 percent Company V, and 40 percent Company M, what is your portfolio return?

4.93%

6.41%

3.45%

19.73%

Average Return The past five monthly returns for K and Company are 4.55 percent, 4.72 percent, -.65 percent, -.15 percent, and 9.30 percent. What is the average monthly return?

3.874%

1.614%

1.481%

3.554%

Portfolio Weights If you own 270 shares of Air Line Inc at $18.95, 170 shares of BuyRite at $9.9, and 370 shares of Motor City at $45.95, what are the portfolio weights of each stock?

Air Line = .2700, BuyRite = .1700, MotorCity = .3700

Air Line = .3333, BuyRite = .3333, MotorCity = .3333

Air Line = .3333, BuyRite = .2099, MotorCity = .4568

Air Line = .2150, BuyRite = .0707, MotorCity = .7143

Portfolio Return At the beginning of the month, you owned $6,500 of Company G, $8,900 of Company S, and $2,800 of Company N. The monthly returns for Company G, Company S, and Company N were 8.15 percent, -1.59 percent, and -.14 percent. What is your portfolio return?

2.14%

6.42%

3.29%

2.13%

The standard deviation of the past five monthly returns for PG Company are 2.75 percent, -0.75 percent, 4.15 percent, 6.29 percent, and 3.84 percent. What is the standard deviation?

2.309 percent

2.581 percent

3.256 percent

3.406 percent

Rank the following three stocks by their risk-return relationship, best to worst. Rail Haul has an average return of 10 percent and standard deviation of 19 percent. The average return and standard deviation of Idol Staff are 12 percent and 22 percent; and of Poker-R-Us are 11 percent and 25 percent.

Idol Staff, Rail Haul, Poker-R-Us

Rail Haul, Idol Staff, Poker-R-Us

Idol Staff, Poker-R-Us, Rail Haul

Poker-R-Us, Rail Haul, Idol Staff

Which of the following statements is correct?

A single stock has a lot of diversifiable risk.

A single stock has more market risk than a diversified portfolio of stocks.

Bonds and stocks have a high correlation because they are both financial assets.

None of these statements is correct.

Which of the following statements is correct with regards to diversification?

Diversifying reduces the return of the portfolio.

Diversifying reduces the market risk of the portfolio.

Diversifying reduces the dollar return of the portfolio.

None of these statements is correct.

Expected Return Compute the expected return given these three economic states, their likelihoods, and the potential returns:

Economic State

Probability

Return

Fast Growth

.1

29%

Slow Growth

.8

14%

Recession

.1

-29%

14.0%

14.3%

11.2%

17.0%

Risk Premium If the annual return on the S&P 500 Index was 14.00 percent. The annual T-bill yield during the same period was 6.50 percent. What was the market risk premium during that year?

20.50%

14.00%

7.50%

6.50%

CAPM Required Return A company has a beta of .69. If the market return is expected to be 13.9 percent and the risk-free rate is 5.95 percent, what is the company's required return?

11.44%

15.54%

17.39%

9.59%

Portfolio Beta You have a portfolio with a beta of .94. What will be the new portfolio beta if you keep 40 percent of your money in the old portfolio and 60 percent in a stock with a beta of 1.54?

1.24

1.00

1.30

2.48

Under/Over Valued Stock A manager believes his firm will earn a 17.8 percent return next year. His firm has a beta of 1.68, the expected return on the market is 15.8 percent, and the risk-free rate is 5.8 percent. Compute the return the firm should earn given its level of risk and determine whether the manager is saying the firm is under-valued or over-valued.

22.6%, under-valued

27.544%, under-valued

27.544%, over-valued

22.6%, over-valued

What is the taxable equivalent yield on a municipal bond with a yield to maturity of 4.5 percent for an investor in the 39 percent marginal tax bracket?

1.76 percent

4.50 percent

7.38 percent

11.54 percent

Rank the following bonds in order from lowest credit risk to highest risk all with the same time to maturity, by their yield to maturity: JM Corporate bond with yield of 12.25 percent, IB Corporate bond with yield of 4.49 percent, TC Corporate bond with yield of 8.76 percent, and B&O Corporate bond with a yield of 5.99 percent.

JM bond, TC bond, B&O bond, IB bond

IB bond, B&O bond, TC bond, JM bond

TC bond, B&O bond, IB bond, JM bond

JM bond, IB bond, B&O bond, TC bond

A 5.75 percent coupon bond with 12 years left to maturity is offered for sale at $978.83. What yield to maturity is the bond offering? (Assume interest payments are paid semi-annually and par value is $1,000.)

3.00 percent

3.09 percent

5.75 percent

6.00 percent

A client in the 28 percent marginal tax bracket is comparing a municipal bond that offers a 3.25 percent yield to maturity and a similar-risk corporate bond that offers a 4.10 percent yield. Which bond will give the client more profit after taxes?

The municipal bond.

The corporate bond.

Both give the client equal profits after taxes.

There is not enough information given to determine answer.

Portfolio Weights An investor owns $15,000 of Adobe Systems stock, $15,500 of Dow Chemical, and $17,000 of Office Depot. What are the portfolio weights of each stock?

Adobe System = 0.3333, Dow Chemical = 0.3333, Office Depot = 0.3333

Adobe System = 0.3158, Dow Chemical = 0.3263, Office Depot = 0.3579

Adobe System = 0.3263, Dow Chemical = 0.3158, Office Depot = 0.3579

Adobe System = 0.2667, Dow Chemical = 0.3333, Office Depot = 0.4000

Portfolio Return Year-to-date, Company O had earned a -7.4 percent return. During the same time period, Company V earned 9.65 percent and Company M earned 2.68 percent. If you have a portfolio made up of 20 percent Company O, 40 percent Company V, and 40 percent Company M, what is your portfolio return?

4.93%

6.41%

3.45%

19.73%

Average Return The past five monthly returns for K and Company are 4.55 percent, 4.72 percent, -.65 percent, -.15 percent, and 9.30 percent. What is the average monthly return?

3.874%

1.614%

1.481%

3.554%

Portfolio Weights If you own 270 shares of Air Line Inc at $18.95, 170 shares of BuyRite at $9.9, and 370 shares of Motor City at $45.95, what are the portfolio weights of each stock?

Air Line = .2700, BuyRite = .1700, MotorCity = .3700

Air Line = .3333, BuyRite = .3333, MotorCity = .3333

Air Line = .3333, BuyRite = .2099, MotorCity = .4568

Air Line = .2150, BuyRite = .0707, MotorCity = .7143

Portfolio Return At the beginning of the month, you owned $6,500 of Company G, $8,900 of Company S, and $2,800 of Company N. The monthly returns for Company G, Company S, and Company N were 8.15 percent, -1.59 percent, and -.14 percent. What is your portfolio return?

2.14%

6.42%

3.29%

2.13%

The standard deviation of the past five monthly returns for PG Company are 2.75 percent, -0.75 percent, 4.15 percent, 6.29 percent, and 3.84 percent. What is the standard deviation?

2.309 percent

2.581 percent

3.256 percent

3.406 percent

Rank the following three stocks by their risk-return relationship, best to worst. Rail Haul has an average return of 10 percent and standard deviation of 19 percent. The average return and standard deviation of Idol Staff are 12 percent and 22 percent; and of Poker-R-Us are 11 percent and 25 percent.

Idol Staff, Rail Haul, Poker-R-Us

Rail Haul, Idol Staff, Poker-R-Us

Idol Staff, Poker-R-Us, Rail Haul

Poker-R-Us, Rail Haul, Idol Staff

Which of the following statements is correct?

A single stock has a lot of diversifiable risk.

A single stock has more market risk than a diversified portfolio of stocks.

Bonds and stocks have a high correlation because they are both financial assets.

None of these statements is correct.

Which of the following statements is correct with regards to diversification?

Diversifying reduces the return of the portfolio.

Diversifying reduces the market risk of the portfolio.

Diversifying reduces the dollar return of the portfolio.

None of these statements is correct.

Expected Return Compute the expected return given these three economic states, their likelihoods, and the potential returns:

Economic State

Probability

Return

Fast Growth

.1

29%

Slow Growth

.8

14%

Recession

.1

-29%

14.0%

14.3%

11.2%

17.0%

Risk Premium If the annual return on the S&P 500 Index was 14.00 percent. The annual T-bill yield during the same period was 6.50 percent. What was the market risk premium during that year?

20.50%

14.00%

7.50%

6.50%

CAPM Required Return A company has a beta of .69. If the market return is expected to be 13.9 percent and the risk-free rate is 5.95 percent, what is the company's required return?

11.44%

15.54%

17.39%

9.59%

Portfolio Beta You have a portfolio with a beta of .94. What will be the new portfolio beta if you keep 40 percent of your money in the old portfolio and 60 percent in a stock with a beta of 1.54?

1.24

1.00

1.30

2.48

Under/Over Valued Stock A manager believes his firm will earn a 17.8 percent return next year. His firm has a beta of 1.68, the expected return on the market is 15.8 percent, and the risk-free rate is 5.8 percent. Compute the return the firm should earn given its level of risk and determine whether the manager is saying the firm is under-valued or over-valued.

22.6%, under-valued

27.544%, under-valued

27.544%, over-valued

22.6%, over-valued