The Efficient Set with Riskless Borrowing and Lending

The Efficient Set with Risk-less Borrowing and Lending 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 1

CML Slope • p* - portfolio in the tangency point of the CML and the portfolio frontier. • . f – slope CML , f is the maximized slope of a line that originates in rf point and touches the portfolio frontier. 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 2

Slope of CML • CML Slope • For optimal set: maximize the slope subject to the asset allocation constraint: wi - asset i weight – with N assets 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 3

The Case of Two Assets • Incorporating the constraint • Using the above the CML slope 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 4

Maximize the Slope • To maximized the slope take derivative with respect to the two weights and set to zero • Multiply both sides by (. )1/2 substitute for rp* and s 2 p* 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 5

Two Assets Efficient Set • Simplify (4) • Define constant • 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 6

Two assets Efficient Set • Define this definition allows • Rewrite (5) and rearrange 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 7

Two Assets Efficient Set • Rewrite in a Matrix form: • Solve • Use (6) to get the weights. 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 8

Example • Data: rf =8% and variance-covariance matrix Stock 1 11/26/2020 1 . 09 2 . 20 2 r 12% . 25 18% Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 9

Example • See Excel file “ 3 Asset Portfolio” 11/26/2020 Fin 355 -Efficient Set w/ Riskless Borrowing and Lending| Dr. Menahem Rosenberg 10