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Bodie's Notes Qn 7

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7. Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%. Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of 12%, and the other half an alpha of 22%. Suppose the analyst buys $1 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks. a) What is the expected profit (in dollars) and standard deviation of the analyst’s profit? b) How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?
The Answer that is given is:

a) Shorting an equally-weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the ten positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the systematic market factor as RM, the expected dollar return is (noting that the expectation of non-systematic risk, e, is zero):
$1,000,000 × [0.02 + (1.0 × Rm)] − $1,000,000 × [(−0.02) + (1.0 × Rm)] = $1,000,000 × 0.04 = $40,000

The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving RM sum to zero.) Thus, the systematic component of total risk is also zero. The variance of the analyst’s profit is not zero, however, since this portfolio is not well diversified.

Conceptually I can make sense but I am not sure of the return/profit calculation
I am not able to understand the calculation of [0.02 + (1.0 × Rm)] and what happens to the returns calculated by the analyst? Is it just a sampling bias?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
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#2
Hi @Srilakshmi I apologize but we copied the question incorrectly. It should read (emphasis mine):
7. Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%. Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of +2%, and the other half an alpha of -2%. Suppose the analyst buys $1 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks.
a) What is the expected profit (in dollars) and standard deviation of the analyst’s profit?
b) How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?
In this way, the +0.02 and -0.02 are just the alphas, α. Although I think the formula might be slightly better to include the risk-free rates, Rf, which cancel. Sorry for the confusion created by the typo. :(
 

jshi

New Member
#4
Hi @Srilakshmi I apologize but we copied the question incorrectly. It should read (emphasis mine):

In this way, the +0.02 and -0.02 are just the alphas, α. Although I think the formula might be slightly better to include the risk-free rates, Rf, which cancel. Sorry for the confusion created by the typo. :(
Hi @David Harper CFA FRM , I am also stuck on this question and have been searching the forum for answers. Could you explain how you reached $1,000,000 × [0.02 + (1.0 × Rm)] − $1,000,000 × [(−0.02) + (1.0 × Rm)] = $1,000,000 × 0.04 = $40,000 without skipping steps? How are we using alpha to calculate the expected return and where does the risk free rate go?

On another note, I also found this chapter a little bit hard to grasp as there were numerous applications and assumptions regarding the multifactor model - I know we have the learning outcomes, but what would you say are the key things principle-wise and formula-wise to pay attention to?

Also nice to meet you, this is my first post, but as you can see I am only on T1... so you shall see more of me as we got onto the more difficult topics!
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
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#5
Hi @jshi

welcome! This is Bodie's question, btw. He is using the single-factor formula for the excess return of the portfolio given by R(p) = α(p) + β(p)*R(m) so that where beta, β(p) = 1.0 ...
  • if alpha, α(p) = 2.0% then for the long component we have R(p) = α(p) + β(p)*R(m) = +0.020 + 1.0*R(m)
  • if alpha, α(p) = -2.0% then for the short component we have R(p) = α(p) + β(p)*R(m) = -0.020 + 1.0*R(m)
In this way, the return is really just given by: [+0.020 + R(m)] - [-0.02 + R(m)] = +0.02 -(-0.02) = +0.040 which can be multiplied by the $1.0 mm for the dollar return, although as shown might be easier until/unless you are comfortable with how the short position manifests as negative, such that this is a $1.0 million portfolio that is long 100% and short 100%.

By "excess" return he (and we too in the FRM) mean "in excess of the risk-free rate" such that we could say:
  • Excess return R(p) = α(p) + β(p)*R(m) and because R(p) = gross return - risk-free rate, where Rf is the risk-free rate, we could say
  • Gross return RG(p) = α(p) + Rf + β(p)*R(m)
And we could insert the Rf into the first step but it would just get eliminated as the gross return would be [+0.020 + Rf + R(m)] - [-0.02 + Rf R(m)]; i.e., the short position subtracts the same risk-free rate that is added to the long. Is why we don't need to know it.

Re: I also found this chapter a little bit hard to grasp as there were numerous applications and assumptions regarding the multifactor model - I know we have the learning outcomes, but what would you say are the key things principle-wise and formula-wise to pay attention to?

I agree with your assessment. For me, the key reason that this chapter is "a little bit hard to grasp ..." is simply that Bodie Kane is a thorough, rigorous, logical text and we are assigned Chapter 10 but none of the earlier chapters that provide the scaffolding! This is a weakness of GARP's anthology approach, in my estimation. Re: what are they key things to pay attention to? It's always hard for me to say because we are always trying to help improve the exam with our feedback so it's hard for me to disassociate from my teacher/advocate role frankly. Exam-wise, i think most important is probably
  • LO: Calculate the expected return of an asset using a single-factor and a multifactor model.
  • The meaning and utility of well-diversified; i.e., eliminates idiosyncratic risk
  • LO: Describe and apply the Fama-French three factor model in estimating asset returns.
I hope that's helpful!
 

akrushn2

Member
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#6
Little confused by this question.

The variance of dollar returns from the positions in the 20 stocks is: 20 × [(100,000 × 0.30)] = 18,000,000,000. If we are taking $100,000 for each stock whether long or short we shouldn't be multiplying by 20. it should be multiplied by 10 right?

Also I don't get sqrt of 18 million as 134164. I get 42426.40.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#7
Hi @akrushn2 I actually think the setup is slightly confusing in a way that could lend to your interpretation, but the gross exposure is $2.0 million = $1.0 mm long plus $1.0 mm short; $100,00 each over 20 positions. The firm-specific returns are uncorrelated but there are 20 of them; notice how they are presumed uncorrelated per the 20*(position_variance) = 18,000,000,000 without any covariance returns; this is because they are uncorrelated. Your 42426.40 is the square root of $1.8 billion but note the dollar variance is $18.0 billion.
 
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