单一指数模型

10.1 Guidelines
Ⅰ.单一指数证券市场 1. 证券的收益与方差 Ⅰ. A single-index security market
1. Security’s returns and its variance ① “均值-方差模型“的 ① The limitation of the “mean, variance 局限性 approach” ② 单一指数模型的主要 ② Main assumptions for single-index model 假设 2. Estimating the single index 2. 估计单一指数模型 model 3. Single index model and 3. 单一指数模型与多样化 diversification 4. 多样化组合对比单一资产 4. Well-diversified portfolio 3 versus single assets
1. Security’s Returns and its Variance
1) The limitation of the “mean, variance approach” “ The mean, variance approach” to portfolio analysis involves estimating and then selecting the portfolio that offers the best mean-variance combination. With “The mean, variance approach”, Consider what we need to do: (1) Mean and variance of each asset. (2) Correlation between each asset.
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2) Main Assumptions of Single-index model
(1) The returns on stocks tend to change in a similar fashion as an average return on the market, because the same economic factors affect almost all firms. This implies the excess return on stock i over risk-free rate can be decomposed into three components: ri－rf = αi + βi(rm－rf ) + ei (10.1)
(3) Mean and variance differing combinations of 4 assets.
This is fine when the portfolio consists of only two assets. What if there are 20 assets to construct? you need the information: n = 20 estimates of E(Ri) n = 20 estimates of variances n(n - 1)/2 = 190 estimates of covariances 230 estimates So for each combination we need 230 estimates.
Chapter 10
Single index model
单一指数模型
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The main goal of this chapter is to introduce Sharpe’s Single Index Model of Capital Asset Pricing.
According to the SingleHale Waihona Puke BaiduIndex Model, by using observable realized returns on a security to regress a relationship between realized returns on that security and the market index, we can examine how the returns on a particular asset or portfolio changes with respect to the returns of the market.
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Three components of the excess return on stock i over risk-free rate
(1)αi :A constant常数, which is different for each stock. (2)βi(rm-rf ) ：A component proportional to the excess return on market index, rm-rf. (3) ei :A random and unpredictable component due to unexpected events that are relevant only to this stock (firm specific).
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We further denote excess returns over the riskfree rate by R（因为股票市场的收益超出或低于 无风险资产收益的那部分的大小可以代表宏观经 济状况）. rewrite this equation as
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In light of the fact that a 20-security portfolio is relatively small, if n = 200 we need 20,300 estimates per combination. Single-index model enables us to dramatically reduce the number of parameters required to perform portfolio analysis.