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We find that funds hold stocks that outperform the market by 1.3 percent per year, but their net returns underperform by one percent. Source. Seems they don't work very well.

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Does savings always equal investment? If we are talking about national accounts then yes. Output of closed economy is given by: $$Y = C+I+G \tag{1}$$ Private saving $S_p$ is by definition is disposable that is not consumed (see further explanation in Blanchard et al. Macroeconomics an European Perspective pp 55) so: $$S_p= Y- T-C \tag{2}$$ Public saving $... 2$π = φ^{(σ−1)}*\frac{f_E+f_D}{φ^{(σ−1)}*(1+τ^{(1-σ)})}+φ^{(σ−1)}\frac{f_E+f_D}{φ^{(σ−1)}*(1+τ^{(1-σ)})}-f_E-2f_Dπ = 2\frac{f_E+f_D}{(1+τ^{(1-σ)})}-f_E-2f_D(1+τ^{(1-σ)})π = 2(f_E+f_D)-(1+τ^{(1-σ)})f_E-2(1+τ^{(1-σ)})f_D(1+τ^{(1-σ)})π = 2f_E+2f_D-f_E-τ^{(1-σ)}f_E-2f_D-2τ^{(1-σ)}f_D(1+τ^{(1-σ)})π = f_E-τ^{(1-σ)}f_E-2τ^{(1-σ)}f_D(1+τ^{(1-σ)})π = ...

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