Question

Unlevered mean and standard deviation: 10%, 20%.

If the borrowing rate is 6%, what is the mean and standard deviation of a portfolio where the investor has $10,000 equity and borrows $10,000 for a total investment of $20,000?

Using our approximation formula, what is the geometric mean for the levered portfolio?

R_L=_________________

?_L=_________________

Aprox.Geom.Mean=_____________

Answer 1

Borrowing is done at risk free rate, Rf = 6%

Market Rate is the unlevered mean of the portfolio, Rm = 10%

Volatility of Unlevered portfolio, ? = 20%

We have $200,000 to invest and we borrow $10,000 then ? would be (20,000 + 10,000)/20,000 = 1.5

Using Capital Allocation Line concept:

Expected return of the portfolio = E(R) = Rf + ?*(Rm - Rf)

                            = 6% + 1.5* (10% - 6%)

                                                      = 12%

Expected volatility of the portfolio = ?* ?

                                                                     = 1.5 * 20%

                                                                     = 30%

Using approximation formula, Geometric Mean = Arithmetic Mean – 0.5 * Variance of returns

                                                                                = 0.12 – 0.5* 0.3^2

                                                                                = 0.075

                                                                                = 7.5%