In other words we cannot replace mp with mm without

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ossibly not for the market as a whole. In other words, we cannot replace mp with mM without being careful. (For you: Why?) Next, we will show that it is valid for the entire market! RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/10/2012. © P. KOLM. 6 Recall that in the previous lecture we showed: M-V investors (in the case of risky assets and one risk-free asset) will chose to hold a portfolio that consists of a combination of the tangency portfolio ( l ) (risky!) and the risk-free asset (1 - l ) So for the k-th investor, we have mp,k = rf + lk (mTang - rf ) = lk w Tang¢m + (1 - lk )rf How much they choose to hold in each one (i.e. the size of lk ) is determined by their risk aversion RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/10/2012. © P. KOLM. 7 Let us calculate the market portfolio in this situation! Let us assume that the wealth of the k-th investor is Wk Then the market portfolio (the aggregate investments in the risky portfolio)1 is wM = 1 W åWklkw Tang = k 1 w Tang åWk lk W k where W = åWk is the total aggregate wealth in the economy k The amount allocated to the risk-free asset is åW (1 - l ) = 0 k k k because “borrowing = lending” in equilibrium, otherwise markets will not clear. Therefore, åW l kk k = åWk = W k RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/10/2012. © P. KOLM. 8 Using this, we obtain 1 w Tang åWk lk W k 1 w Tang åWk = W k = w Tang wM = We have shown that in the M-V economy the market and tangency portfolios are the same! (For you: Make a list of all the assumptions needed for this economy!) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/10/2012. © P. KOLM. 9 So what about CAPM? Let us now see what happens on the aggregate market level For each investor k, we have Swk = gk (m - rf ⋅ i ) Aggregating over all investors, we get Sw Agg = L (m - rf ⋅ i ) where w Agg = 1 W å wkWk and L = k 1 W ågW k k k But we know w Agg º wM , therefore SwM = L (m - rf ⋅ i ) RISK AND PORTFOLI...
View Full Document

Ask a homework question - tutors are online