
Drawbacks of the Markowitz Model
 Requires a huge # of estimates (cov)
 Does not provide guideline to forecast RP
 Errors in estimation of σ_{ij} can lead to nonsensical results

Index Models
 Decomposes a security's return into systematic and firmspecific component
 As valid as the assumption of normality of the rates of return
 To the extent that shortterm returns are well approximated by normal distributions, index models can be used to select optimal portfolios nearly as accurately as the Markowitz algorithm
 Assumption: the rate of return of a broad index is a valid proxy for the common macroeconomic factor

Index Model Regression Equation
R_{i}(t) = α_{i} + β_{i}R_{M}(t) + e_{i}(t)

The SingleIndexModel Input List
 RP on the selected index (eg. S&P 500)
 σ of the selected index
 n sets of estimates of (a) β coefficients, (b) stock residual variances, and (c) α values

Single Index Model
 r_{i} = E(r_{i}) + β_{i}m + e_{i}
 σ_{i}^{2} = β_{i}^{2}σ_{m}^{2} + σ^{2}(e_{i})
 σ_{P}^{2} = β_{P}^{2}σ_{m}^{2} + ∑w_{i}σ^{2}(e_{i})
 Cov(r_{i}, r_{j}) = β_{i}β_{j}σ_{m}^{2}
 The index model is estimated by applying regression analysis to xs rates of return. β = slope, α = intercept^{}

Is the Index Model inferior to the FullCovariance Model?
 To add anouther index, we need both a forecast or the risk prem and estimates of β
 Using the full covariance matrix invokes estimation risk of thousands of terms
 Even if the full Markowitz is better in principle, it is very possible that cumulative effect of so many estimation error will results in an inferior PF.

8 steps to determine weights of a portfolio
 1. Calculate initial weights based on α/σ^{2}(e_{i})
 2. Scale weights so that ∑w_{i} = 1
 3. Compute α_{P}
 4. Compute σ^{2}(e_{A}) (residual variance of PF)
 5. w_{A}° = [α_{A}/σ^{2}(e_{A})]/[E(R_{M})/σ_{M}^{2}]
 6. Calculate β_{A}
 7. Adjust weights to account for β
 w_{A}* = w_{A}°/[1 + (1  β_{A})w_{A}°]
 8. Compute E(r_{OPF}) and σ_{OPF}

Evolution of β over time
 As time goes by, β → 1.0
 As firms grow, they diversify and therefore their β approaches the mkt β
 Merrill adjusted β = (2/3)E(β) + (1/3)(1.0)

