The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 559 Friday, March 12, 2004 12:40 PM

Equity Portfolio Management 559

where

rp = return of the portfolio in excess of the constant risk-free rate

rm = return of the market index in excess of the constant risk-free rate

e = residual error term

β = beta of the portfolio

Subtracting market excess return (i.e., rm) from both sides, we get

ra = rp – rm = (β – 1)rm + e

where ra is the active return. Therefore,

σ^2 (rp – rm) = (β – 1)^2 σ^2 (rm) + σ^2 (e)

There would be no correlation between rm and the error term due to

the regression. The left hand side of the above equation is the portfolio

tracking error variance. So, we have

σ(rp – rm) = (β – 1)σ(rm)

As can be seen from the above equation, holding other things equal,

tracking error increases with market volatility.

To quantify the relationship between portfolio beta and tracking

error, look again at the formula for the tracking error from the market

model given above. Let w = weight of the portfolio invested in the

benchmark index; (1 – w) = weight of the portfolio invested in cash; rp =

portfolio return in excess of the risk-free return on cash, and; rb =

benchmark index return in excess of the risk-free return on cash.

Because the excess return on cash is zero, we know that

rp = wrb + (1 – w) 0 = wrb

If β is the portfolio beta versus the benchmark index, then letting

σ(.,.) denote the covariance,

β = σ(rp,rb)/σ^2 (rb) = wσ^2 (rb)/σ^2 (rb) = w

Next we know that, rp – rb = (w – 1)rb = (β – 1)rb

σ^2 (rp – rb) = (w – 1)^2 σ^2 (rb) = (β – 1)^2 σ^2 (rb)