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Drawbacks of the Markowitz Model
- Requires a huge # of estimates (cov)
- Does not provide guideline to forecast RP
- Errors in estimation of σij can lead to nonsensical results
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Index Models
- Decomposes a security's return into systematic and firm-specific component
- As valid as the assumption of normality of the rates of return
- To the extent that short-term 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
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Index Model Regression Equation
Ri(t) = αi + βiRM(t) + ei(t)
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The Single-Index-Model 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
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Single Index Model
- ri = E(ri) + βim + ei
- σi2 = βi2σm2 + σ2(ei)
- σP2 = βP2σm2 + ∑wiσ2(ei)
- Cov(ri, rj) = βiβjσm2
The index model is estimated by applying regression analysis to xs rates of return. β = slope, α = intercept
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Is the Index Model inferior to the Full-Covariance 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.
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8 steps to determine weights of a portfolio
- 1. Calculate initial weights based on α/σ2(ei)
- 2. Scale weights so that ∑wi = 1
- 3. Compute αP
- 4. Compute σ2(eA) (residual variance of PF)
- 5. wA° = [αA/σ2(eA)]/[E(RM)/σM2]
- 6. Calculate βA
- 7. Adjust weights to account for β
- wA* = wA°/[1 + (1 - βA)wA°]
- 8. Compute E(rOPF) and σOPF
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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)
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