Khipha amafomula ngefomethi ye-latex markdown yezibalo ukuze ubale ukubuya kwephothifoliyo nokuphambuka ku-Modern Portfolio Theory https://en.wikipedia.org/wiki/Modern_portfolio_theory
Context:I-MPT icabanga ukuthi abatshalizimali abafuni ukubeka engcupheni, okusho ukuthi uma kunama-portfolio amabili anikeza inzuzo efanayo elindelwe, abatshalizimali bazokhetha lowo onobungozi obuncane. Ngakho-ke, umtshali-zimali uzothatha ingozi eyengeziwe kuphela uma enxephezelwa yinzuzo ephakeme elindelwe. Ngakolunye uhlangothi, umtshali-zimali ofuna imbuyiselo ephakeme kulindeleke ukuba amukele ingozi eyengeziwe. Ukuhwebelana okuqondile ngeke kufane kubo bonke abatshalizimali. Abatshalizimali abahlukene bazohlola ukuhwebelana ngokuhlukile ngokusekelwe ezimfanelweni zomuntu ngamunye zokugwema ingozi. Lokhu kusho ukuthi umtshali-zimali onengqondo ngeke atshale imali kwiphothifoliyo uma kukhona iphothifoliyo yesibili enobungozi obuhle kakhulu - iphrofayili yokubuya okulindelekile - okungukuthi, uma kulelo zinga lobungozi kukhona iphothifoliyo ehlukile enenzuzo engcono elindelwe.  Ngaphansi komodeli:  Imbuyiselo yephothifoliyo iyinhlanganisela elinganiselwe yembuyiselo yezimpahla ezakha iphothifoliyo. Ukuguquguquka kwenzuzo yephothifoliyo \sigma _{p} umsebenzi wokusebenzelana ρij kwezimpahla zezakhi, kuwo wonke amabhangqa ezimpahla (i, j). Ukuguquguquka kunikeza ukuqonda ngengozi ehambisana nokutshalwa kwezimali. Lapho ukuguquguquka kukhulu, ingozi iba nkulu.
- Ukubuya okulindelekile: $$ \mathrm{E}\left(R_p\right) =\sum_i w_i \mathrm{E}\left(R_i\right) $$ lapho $R_p$ kungukubuya kwephothifoliyo, $R_i$ kungukubuya kwempahla $i$ futhi $w_i$ kungukugxila kwempahla yezakhi $i$ (okungukuthi, isilinganiso sempahla "i" kuphothifoliyo, ukuze $\sum_i w_i=1$). - Ukwehluka kokubuya kwephothifoliyo: $$ I-sigma_p^2=\sum_i w_i^2 \sigma_i^2+\sum_i \sum_{j \neq i} w_i w_j \sigma_i \sigma_j \rho_{i j} $$ lapho $\sigma_i$ kuyinto (isampula) ukuphambuka okujwayelekile kwembuyiselo yesikhathi sesikhathi sempahla $i$, futhi $\rho_{i j}$ kuyinto isilinganiso sokuhlobana phakathi kwembuyiselo yempahla $i$ no $j$. Okunye okungase kwenzeke ukubhala le nkulumo kanje: $$ \sigma_p^2=\sum_i \sum_j w_i w_i \sigma_i \sigma_j \rho_{i j} $$ lapho $\rho_{i j}=1$ for $i=j$, noma $$ \sigma_p^2=\sum_i \sum_j w_i w_i \sigma_{i j} $$ lapho $\sigma_{i j}=\sigma_i \sigma_j \rho_{i j}$ kuyinto (isampula) ye-covariance yembuyiselo ephindaphindwayo kuzimpahla ezimbili, noma enye indlela ekhonjiswe njenge $\sigma(i, j) $, $\operatorname{cov}{ }_{i j}$ noma $\operatorname{cov}(i, j) $. - Ukuguquguquka kwenzuzo yephothifoliyo (ukuphambuka okujwayelekile): $$ \sigma_p=\sqrt{\sigma_p^2} I-sigma_p=\sqrt{\sigma_p^2} I-sigma_p=\sqrt{\sigma_p^2} $$ Ukuze uthole iphothifoliyo enezimpahla ezimbili: - Inzuzo yephothifoliyo: $$\mathrm{E}\left(R_p\right) =w_A \mathrm{E}\left(R_A\right) +w_B \mathrm{E}\left(R_B\right) =w_A \mathrm{E}\left(R_A\right) +\left(1-w_A\right) \mathrm{E}\left(R_B\right) $$  - Ukwehluka kwesikhwama: $$\sigma_p^2=w_A^2 \sigma_A^2+w_B^2 \sigma_B^2+2 w_A w_B \sigma_A \sigma_B \rho_{A B}$$ Ukuze uthole iphothifoliyo enezimpahla ezintathu: - Inzuzo yephothifoliyo: $$\mathrm{E}\left(R_p\right) =w_A \mathrm{E}\left(R_A\right) +w_B \mathrm{E}\left(R_B\right) +w_C \mathrm{E}\left(R_C\right) $$ - Ukwehluka kwesikhwama: $$\sigma_p^2=w_A^2 \sigma_A^2+w_B^2 \sigma_B^2+w_C^2 \sigma_C^2+2 w_A w_B \sigma_A \sigma_B \rho_{A B}+2 w_A w_C \sigma_A \sigma_C \rho_{A C}+2 w_B w_C \sigma_B \sigma_C \rho_{B C}$$