# Does the multi-index model predict returns better than the single-index model? Explain

Home, - Derive the standard version of CAPM

**Question 1 - (i) Derive the standard version of CAPM.**

Answer - Use standard arbitrage conditions to derive CAPM by considering a generic equation R ¯_{i} = a + bβ_{i} of a straight line in expected return and beta space. Set beta equals to zero and beta equals to one to obtain R ¯_{I }= R_{F}, + β_{i} (R ¯_{M} - R ¯_{F}). Note that this equation describes any security or portfolios in an economy with riskless lending or borrowing.

**(ii) Does the multi-index model predict returns better than the single-index model? Explain.**

Answer - That is not correct. The more indexes are added, the more complex things become and the more accurately the historical correlation matrix is reproduced. However, this does not imply that future correlation matrices will be forecast more accurately because, although more complex models better describe the historical correlations, they also introduce more noise when it comes to prediction.

**Question 2 - (i) Show that, in an equally weighted portfolio, if the returns on different assets are uncorrelated an increase in the number of assets may bring the variance of the portfolio close to zero. What happens if the assets are correlated?**

Answer - Use the equation of an equally weighted portfolio _{P}σ^{2} = 1/n(_{j} σ ¯^{2} - σ ¯j_{k}) + σ ¯_{jk} to show the results, in particular when σ ¯_{jk} is equal to zero and when σ ¯_{jk} is not equal to zero.

**(ii) Assume that the average variance of return for an individual security is 20 and that the average covariance is 10. How many securities need to be held before the risk of an equally weighed portfolio is only 10% more than the minimum? **

Answer - Consider the equation of an equally weighted portfolio _{P}σ^{2} = 1/n(_{j}σ ¯^{2} - σ ¯_{jk}) + σ ¯_{jk}. As the number of securities (n) approaches infinity, an equally weighted portfolio's variance (total risk) approaches a minimum equal to the average covariance of the pairs of securities in the portfolio, which is given as 10. Therefore, 10% above the minimum risk level would result in the portfolio's variance being equal to 11. Setting the formula of an equally weighted portfolio equal to 11 and using _{j}σ ¯^{2} = 20 and σ ¯_{jk} = 10 we have:

_{P}σ^{2} = 1/n (20 - 10) + 10 = 11

Solving the above equation for N gives N = 10 securities.