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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: MV investors (in the case of risky assets and one riskfree asset) will chose to hold a portfolio that consists of a combination of the tangency portfolio (
l ) (risky!) and the riskfree asset (1  l ) So for the kth 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 kth 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 riskfree 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 MV 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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