# In other words we cannot replace mp with mm without

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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...
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