By Petr Adamek, John Y. Campbell, Andrew W. Lo, A. Craig Mackinlay, Luis M. Viceira

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5) 2 m 0 m 2 2 m 3 5 0 m 2 + 2 m 0 m 2 2 m 3 5 where = 0 m2 + is substituted. 3). 1 Let the number of portfolios in the set be K and let RKt be the (K 1) vector of time period t returns for the portfolios. Since the entire minimum variance boundary can be generated from the K portfolios, for any value of the constant y, there exists a combination of the portfolios with expected return y which is minimum variance with respect to the K portfolios plus the N assets. 11)) and denote this portfolio op.

12) holds for di erent values of op it must be the case that ( ; B ) = 0, that is the factor regression coe cients for each asset, including asset a, sum to one. 11) we have a = 0, that is the regression intercept will be zero for all assets including asset a. 4) 2 4 2 4 2 4 p 3 5 0 p2 p2 p2 0 p2 + 3 5 0 p2 2 p 2 p 3 5 where = 0 p2 + and = 0 h2 + I 2 . Given the N +1 assets, the maximum squared Sharpe ratio is 0 ;1 which is the squared Sharpe ratio of the tangency portfolio. 6) s2I = s2p + 12 a0 a + ( 2 h+(a 2 ) 0 ) : h 2 where sp is the squared Sharpe ratio of portfolio p.

4). 14), unless ! 1, so in general the assumed model will misprice options. For > 1 and/or n > 1, it will overstate the volatility of the future log bond price, hence overvaluing the option. This overvaluation increases with the expiration date of the option and/or the maturity of the underlying bond. This is true no matter what combination of the drift parameters we choose. Backus, Foresi and Zin (1996) use this result to caution against the popular practice among practitioners of augmenting standard arbitrage-free bond pricing models with time-dependent parameters to t exactly the yield curve.

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