Bvisa mafomula mune math latex markdown fomati kuti uverenge kudzoka kwehomwe uye kusiyanisa muModern Portfolio Theory https://en.wikipedia.org/wiki/Modern_portfolio_theory
Context:MPT inofungidzira kuti varimi vanodzivisa ngozi, zvichireva kuti kupiwa mapepa maviri anopa kudzoka kumwechete kunotarisirwa, varimi vachasarudza iyo ine njodzi shoma. Nokudaro, mutengesi achatora ngozi yakawedzerwa bedzi kana achiregererwa nemigumisiro yakakwirira inotarisirwa. Asiwo, mutengesi anoda mari yakawanda anofanira kubvuma kutarisana nengozi. Kuchinjanisa chaiko hakusati kuchizova kwakafanana nokuda kwavashambadzi vose. Vashambadziri vakasiyana-siyana vachaongorora kuwirirana kwakasiyana-siyana zvichienderana neunhu hwavo hwokusafarira ngozi. Mhedziso ndeyekuti muzvinashawo akangwara haazodyari muhomwe kana muhomwe yechipiri iripo ine njodzi-inotarisirwa kudzoka chimiro chakanakira, kureva kuti, kana pamwero iwoyo wehutarisiro pane imwe muhomwe ine zvirinani zvinotarisirwa kudzoka.  Pasi pomuenzaniso:  Kudzoka kwehomwe ndiko kusanganiswa kwakakosheswa kwezvibereko zvezvinhu zvinoumba. Kuchinja-chinja kwekudzoka kwehomwe \sigma _{p} ibasa rehukama ρij hwezvinhu zvinogadzira zvinhu, nokuda kwezvose zviviri zvezvinhu (i, j). Kusagadzikana kunopa nzwisiso mungozi inobatanidzwa nemari yacho. Kukwirira kwokushanduka-shanduka, ndiko kukwirira kwenjodzi.
- Kudzoka kunotarisirwa: $$ \mathrm{E}\left(R_p\right) =\sum_i w_i \mathrm{E}\left(R_i\right) $$ apo $R_p$ inodzoka pahomwe, $R_i$ inodzoka pazvinhu $i$ uye $w_i$ ndiyo kuyerwa kwezvinhu zvinoumba zvinhu $i$ (kureva, chikamu chezvinhu "i" muhomwe, kuitira kuti $\sum_i w_i=1$). - Kusiyana kwekudzoka kwehomwe: $$ "sigma_p^2=\sum_i w_i^2 \sigma_i^2+\sum_i \sum_{j \neq i} w_i w_j \sigma_i \sigma_j \rho_{i j} $$ apo $\sigma_i$ ndiyo (sampuli) mwero kutsauka kwokudzoka nguva nenguva pane chinhu $i$, uye $\rho_{i j}$ ndiyo kuwirirana coefficient pakati pokudzoka pazvinhu $i$ uye $j$. Zvimwewo, mashoko acho anogona kunyorwa seizvi: $$ \sigma_p^2=\sum_i \sum_j w_i w_i \sigma_i \sigma_j \rho_{i j} $$ apo $\rho_{i j}=1$ nokuda kwe $i=j$, kana kuti $$ \sigma_p^2=\sum_i \sum_j w_i w_j \sigma_{i j} $$ apo $\sigma_{i j}=\sigma_i \sigma_j \rho_{i j}$ ndiyo (sample) covariance yemari inodzoka nguva nenguva pazvinhu zviviri izvi, kana kuti inonzi $\sigma(i, j)$, $\operatorname{cov}{ }_{i j}$ kana $\operatorname{cov}(i, j)$. - Kuchinja-chinja kwemitero yehomwe (kutsauka kwakajairika): $$ \sigma_p=\sqrt{\sigma_p^2} $$ Nokuda kwechikwata chemidziyo miviri: - Mari inodzoka: $$\mathrm{E}\left(R_p\right) =w_A \mathrm{E}\left(R_A\right) +w_B \mathrm{E}\left(R_B\right) =w_A \mathrm{E}\left(R_A\right) +\left(1-w_A\right) \mathrm{E}\left(R_B\right) $$  - Kusiyana kwehomwe: $$\sigma_p^2=w_A^2 \sigma_A^2+w_B^2 \sigma_B^2+2 w_A w_B \sigma_A \sigma_B \rho_{A B}$$ Nokuda kwechikwata chemidziyo mitatu: - Mari inodzoka: $$\mathrm{E}\left(R_p\right) =w_A \mathrm{E}\left(R_A\right) +w_B \mathrm{E}\left(R_B\right) +w_C \mathrm{E}\left(R_C\right) $$ - Kusiyana kwehomwe: $$\sigma_p^2=w_A^2 \sigma_A^2+w_B^2 \sigma_B^2+w_C^2 \sigma_C^2+2 w_A w_B \sigma_A \sigma_B \rho_{A B}+2 w_A w_C \sigma_A \sigma_C \rho_{A C}+2 w_B w_C \sigma_B \sigma_C \rho_{B C}$$