Glass Shoes

Author

Affiliation

Hirotaka Fukui

Kobe University, Graduate School of Economics

Modified

April 16, 2026

지금부터 나를

가득 채운 너를 모두 모아서

그대가 날 찾을 수 있게

밝게 빛나 줄 거야

열두시가 넘어 나는 다시

돌아가야만 해요

내 마음을 남기고서 갈게 꼭 만나자

우리의 미래에서

Glass Shoes - fromis_9

Modern Portfolio Theory

Expected Return of a Single Asset

$J$ 個の実現可能な outcomes $j = 1, \dots J$ を持つ資産 $i$ を考えます. $R_{ij}$ をoutcome $j$ のリターン, $p_{ij}$ をその確率とします. リターンの期待値は

\[ \mathbb{E}[R_{i}] = \sum_{j=1}^{J} p_{ij} R_{ij} \tag{1}\]

と表されます. 資産 $i$ の分散は

\[ \sigma^{2}_{i} = \sum_{j=1}^{J} p_{ij} \left(R_{ij}-\mathbb{E}[R_{i}] \right)^{2} \]

と表されます.

Portfolio with two risky assets

2つのリスク資産からなるポートフォリオを考えます. リスク資産のインデックスを $i=A, B$ とします. $X_{i}$ をそれぞれのリスク資産のポートフォリオ内のウェイトとします ($X_{A}+X_{B}=1$). ポートフォリオ全体のリターンは

\[ R_{p, m} = X_{A} R_{A, m}+ (1-X_{A}) R_{B, m} \]

と表せます. ポートフォリオの期待リターンは

\[ \mathbb{E}[R_{p}] = X_{A} \mathbb{E}[R_{A}]+ (1-X_{A}) \mathbb{E}[R_{B}] \]

と書けます. ポートフォリオの分散は

\[ \begin{align} \sigma_{p}^{2} & = \left(X_{A} \sigma_{A}+ (1-X_{A}) \sigma_{B} \right)^{2} \\ & = X^{2}_{A} \sigma^{2}_{A} + (1-X_{A})^{2} \sigma^{2}_{B} + 2 X_{A} (1-X_{A}) \sigma_{AB} \end{align} \]

となります. $\sigma_{AB}$ はリスク資産 $A$ と $B$ の共分散です.

\[ \sigma_{AB} = \sum_{j=1}^{J} p_{j} \left(R_{A, m}-\mathbb{E}[R_{A}] \right)\left(R_{B, m}-\mathbb{E}[R_{B}] \right) \]

Portfolio with $N$ risky assets

Portfolio Possibility Curve

Minimum Risk Portfolio

Efficient Frontier

Portfolio with Riskless Lending and Borrowing

Single Index Model

Multi Index Model

Capital Asset Pricing Model (CAPM)

Arbitrage Pricing Theory (APT)

Empirical Tests of CAPM and APT

Consumption CAPM (C-CAPM)

Empirical Tests of C-CAPM

The Factor Zoo

Intermediary Asset Pricing

Production-based Asset Pricing

Demand System Approach to Asset Pricing

Important

All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me.

--- title: "Glass Shoes" engine: julia format: html: code-tools: true --- > 지금부터 나를 > > 가득 채운 너를 모두 모아서 > > 그대가 날 찾을 수 있게 > > 밝게 빛나 줄 거야 > > 열두시가 넘어 나는 다시 > > 돌아가야만 해요 > > 내 마음을 남기고서 갈게 꼭 만나자 > > 우리의 미래에서 > > [*Glass Shoes* - fromis_9](https://fromis9.fandom.com/wiki/Glass_Shoes) # Modern Portfolio Theory ## Expected Return of a Single Asset $J$ 個の実現可能な outcomes $j = 1, \dots J$ を持つ資産 $i$ を考えます. $R_{ij}$ をoutcome $j$ のリターン, $p_{ij}$ をその確率とします. リターンの期待値は $$ \mathbb{E}[R_{i}] = \sum_{j=1}^{J} p_{ij} R_{ij} $$ {#eq-ex-r-single} と表されます. 資産 $i$ の分散は $$ \sigma^{2}_{i} = \sum_{j=1}^{J} p_{ij} \left(R_{ij}-\mathbb{E}[R_{i}] \right)^{2} $$ と表されます. ## Portfolio with two risky assets 2つのリスク資産からなるポートフォリオを考えます. リスク資産のインデックスを $i=A, B$ とします. $X_{i}$ をそれぞれのリスク資産のポートフォリオ内のウェイトとします ($X_{A}+X_{B}=1$). ポートフォリオ全体のリターンは $$ R_{p, m} = X_{A} R_{A, m}+ (1-X_{A}) R_{B, m} $$ と表せます. ポートフォリオの期待リターンは $$ \mathbb{E}[R_{p}] = X_{A} \mathbb{E}[R_{A}]+ (1-X_{A}) \mathbb{E}[R_{B}] $$ と書けます. ポートフォリオの分散は $$ \begin{align} \sigma_{p}^{2} & = \left(X_{A} \sigma_{A}+ (1-X_{A}) \sigma_{B} \right)^{2} \\ & = X^{2}_{A} \sigma^{2}_{A} + (1-X_{A})^{2} \sigma^{2}_{B} + 2 X_{A} (1-X_{A}) \sigma_{AB} \end{align} $$ となります. $\sigma_{AB}$ はリスク資産 $A$ と $B$ の共分散です. $$ \sigma_{AB} = \sum_{j=1}^{J} p_{j} \left(R_{A, m}-\mathbb{E}[R_{A}] \right)\left(R_{B, m}-\mathbb{E}[R_{B}] \right) $$ ## Portfolio with $N$ risky assets ## Portfolio Possibility Curve ## Minimum Risk Portfolio ## Efficient Frontier ## Portfolio with Riskless Lending and Borrowing # Single Index Model # Multi Index Model # Capital Asset Pricing Model (CAPM) # Arbitrage Pricing Theory (APT) # Empirical Tests of CAPM and APT # Consumption CAPM (C-CAPM) # Empirical Tests of C-CAPM # The Factor Zoo # Intermediary Asset Pricing # Production-based Asset Pricing # Demand System Approach to Asset Pricing ::: {.callout-important} All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me. :::