# How to Calculate Rate of Return of Stock Portfolio

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```					Finance Homework                                                                          Julian Vu
p. 65 (3, 4), p. 66-69 (1, 2, 3, 4, 5, 12, 14), p. 107 (2), p. 109 (3,4)

2-3:
Given:
Security A                                       Security B
r = 7%                                           r = 12%
σ (standard deviation) = 35%                     σ (standard deviation) = 10%
rxy = -0.3                                       rxy = -0.7
ß = -1.5                                         ß =1
Which security is riskier? Why?

Use standard deviation. (Tighter probability distribution is less risky):
Because Security B has a smaller standard deviation ( 10% > 35%), meaning that Security B has a
smaller range of variation, then security B is less risky.

2-4:
You own a portfolio with \$250,000 worth of long-term U.S. gov. bonds.
a. Would your portfolio be riskless?

In this case, we are examining default-risk. Because U.S. gov. bonds are backed by the word of the
U.S. Government, they are deemed to be riskless, regardless of whether it is a 90-day T-Bill or a
long-term T-Bond.

b. Now suppose you hold a portfolio consisting of \$250,000 worth of 30-day Treasury bills. Every
30 days your bills mature, and you reinvest the principal in a new batch of bills. 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?

If it is a Treasury bill, then the amount received will not variate from amount expected. Because you
do not expect more every time you reinvest (ﬁxed-income), then you theoretically do not face risk.
Only in the case of treasury notes, however.

c. Can you think of an asset that would be completely riskless? Could someone develop such an
asset? Explain.

I thought that treasury notes were completely riskless. I understand that if the U.S. Government
went for broke and could not repay their debts, then the risk would be existent, however such is not
generally the case.
2-1:
A stock’s return has the following distribution:
Demand for the                           Proﬁtability of this                   Rate of Return
Company’s Products                       Demand Occurring                       if this demand occurs
Weak                                     0.1                                    (50%)
Below Average                            0.2                                    (5%)
Average                                  0.4                                    16%
Above Average                            0.2                                    25%
Strong                                   0.2                                    60%

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

r^m = 11.40%, σ = 26.69%, CV = 2.34

2-2
An individual has \$35,000 invested in a stock which has a ß = 0.8, and \$40,000 invested in a stock
with a ß of 1.4. If these are the only two investments in her portfolio, what is her portfolio’s beta?

2-3
Assume that the risk-free rate (RF) 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 ß of 1.2?

Return on market:

Return on stock:
2-4
Assume that the risk-free rate is 6%, and 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%?

2-5
e market and Stock J have the following probability distributions:

Probability     rM       rJ
0.3             15%      20%
0.4             9%       5%
0.3             18%      12%

a. Calculate the expected rates of return for the market and Stock J

b. Calculate the standard deviations for the market and Stock J.

Stock:
probability                  return on stockri
return on market         pi *        ri - r^     (ri - r^)^2 rm - ri    rm - ri^2 (ri - r^)^2 * Pi
0.3          0.15        0.2        0.06       0.084 0.007056         -0.05    0.0025 0.002117
0.4          0.09       0.05        0.02      -0.066 0.004356          0.04    0.0016 0.001742
0.3          0.18       0.12      0.036        0.004 0.000016          0.06    0.0036 0.000005

Expected return: (r^) 0.116                   Sum:       0.022                         Variance: 0.003864
Standard dev: 0.062161

Market:
probability                                      p
return on market return on stocki * rm       rm - r^       (rm - r^)^2 (rm - r^)^2) * Pi
0.3             0.15           0.2         0.045         0.015    0.000225 0.0000675
0.4             0.09          0.05         0.036        -0.045    0.002025      0.00081
0.3             0.18          0.12         0.054         0.045    0.002025 0.0006075

Expected rate:         0.135                   Variance:      0.001485
Standard dev:     0.0385357
σm = 3.85%, σj = 6.22%

c. Calculate the coeﬃcients of variation for the market and Stock J.

CVM = 0.3319, CVi = 0.5362

===========================================

2-12
Stocks A and B have the following historical returns:
Stock A Returns        Stock B Returns
Year          rA              rB
2002          (18%)           (14.5%)
2003          33%              21.80%
2004          15%             30.50%
2005          (0.50%)         (7.60%)
2006          27.00%          26.30%

a. Calculate the average rate of return for each stock during the 5-year period.

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? What would have
been the average rate of returno n 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 A, Stock B, or the portfolio? Why?
2-14

3-2
Security A has an expected rate of return of 6 percent, a standard deviation of returns of 30%, a
correlation coeﬃcient with the market of -0.25, and a beta coeﬃcient of -0.5 Security B has an
expected return of 11%, a standard deviation of returns of 10%, a correlation with the market of
0.75, and a ß of 0.5. Which security is more risky, why?

3-3
e beta coeﬃcient of an asset can be expressed as a function of the asset’s correlation with the
market as follows:

a. Substitute this expression for beta into the Security Market Line (SML), Equation 3-9. is results
in an alternative form of the SML.

b. Compare your answer to part a with the Capital Market Line (CML), Equation 3-6. What
similarities are observed? What conclusions can be drawn?

3-4
Suppose you are given the following information. e beta of company i, bi, is 1.1, the risk-free rate,
rRF, is 7%, and the expected market premium, rM - rRF, is 6.5%. (Assume that ai - 0.0).
a. Use the Security Market Line (SML) of CAPM to ﬁnd the required return for this company.

b. Because your company is smaller than average and more successful than average (that is, it has a
low book-to-market ratio), you think the Fama-French three-factor model might be more
appropriate than the CAPM. You estimate the additional coeﬃcients from the Fama-French three-
factor model: e coeﬃcient for the size eﬀect, ci, is 0.7, and the coeﬃcient for the book-to-market
eﬀect, di, is -0.3. If the expected value of the size factor is 5%, and the expected value of the book-
to-market factor is 4%, what is the required return using the Fama-French three-factor model?

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