Assume that stock market returns do follow a single-index structure. An investment fund analyses 125 stocks in order to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate ________ estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor

Question

BlackTom AI · Accepted Answer

Each stock has one expected return and one sensitivity (beta) to the single macroeconomic factor, so you need 125 estimates of expected returns and 125 estimates of sensitivity coefficients.

1) Relevant knowledge points and concepts
- In a single-index (factor) model each asset’s return is described by its expected return and its sensitivity (beta) to the common factor plus an idiosyncratic error.  
- A full mean–variance model without factor structure would require N expected returns and N(N+1)/2 unique elements of the covariance matrix. A single-index model replaces the many pairwise covariances with betas and the factor variance.

2) Step-by-step reasoning
- Number of stocks N = 125.  
- Expected returns: one per stock ⇒ 125 estimates.  
- Sensitivity coefficients (betas): one per stock in a single-index model ⇒ 125 estimates.

3) Intermediate calculations (to show why alternative counts are incorrect)
- Full covariance unique elements would be N(N+1)/2 = 125×126/2 = 15750/2 = 7,875 (not 7,750).  
- All matrix entries (if you wrongly counted every cell) would be N^2 = 125^2 = 15,625.  
These larger numbers correspond to covariance entries in unrestricted models, not to betas in a single-index model.

4) Why the correct answer is correct
- The single-index structure means pairwise covariances are determined by product of betas and the factor variance; you only need to estimate each asset’s beta (one per asset) rather than each pairwise covariance. Thus both required counts are simply N = 125.

5) Why other options are incorrect (brief)
- Options citing 15,625 or similar are counting every covariance cell (N^2) rather than the single beta per asset.  
- Options citing 7,750 or 7,875 confuse unique covariance entries; those are irrelevant because the single-index model replaces them with betas.  
- The 25/15,625 option is unrelated to N = 125.

6) Conclusion
Therefore the fund must compute 125 estimates of expected returns and 125 estimates of sensitivity coefficients, confirming the given answer: 125; 125.

Similar Questions

Consider the single-index model. The alpha of a stock is 2%. The return on the market index is 10% and the risk-free rate of return is 3%. The stock earns a return 14%, and there are no firm-specific events affecting the stock performance. The β of the stock is

The fugitive slave law

All of the following were provisions within the Comprimise of 1850 EXCEPT:

Opponents to slavery's expansion organized what political party in 1848?

The Wilmot Proviso proposed a resolution

Freedom in California, after the war, was generally

Who spoke out against the US-Mexican War?

"Fifty-four forty or fight" referred to

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler